theory of seismic imaging -

Theory of Seismic Imaging

John A Scales

SamizdatPress

Theory of Seismic Imaging

John A Scales

Colorado School of [email protected]

Samizdat Press Golden White River Junction

Published by the Samizdat Press

Center for Wave PhenomenaDepartment of GeophysicsColorado School of MinesGolden Colorado

andNew England Research

Olcott DriveWhite River Junction Vermont

cSamizdat Press

Release January

Samizdat Press publications are available via FTPfrom hilbertminescoloradoedu or

Permission is given to freely copy these documents

Preface

Seismic imaging is the process through which seismograms recorded on the Earth s surface are mapped into representations of its interior properties Imaging methods arenowadays applied to a broad range of seismic observations from nearsurface environmental studies to oil and gas exploration even to longperiod earthquake seismology Thecharacteristic length scales of the features imaged by these techniques range over manyorders of magnitude Yet there is a common body of physical theory and mathematicaltechniques which underlies all these methods

The focus of this book is the imaging of reection seismic data from controlled sourcesAt the frequencies typical of such experiments the Earth is to a rst approximation avertically stratied medium These stratications have resulted from the slow constantdeposition of sediments sands ash and so on Due to compaction erosion change ofsea level and many other factors the geologic and hence elastic character of these layersvaries with depth and age One has only to look at an exposed sedimentary cross sectionto be impressed by the fact that these changes can occur over such short distances thatthe properties themselves are eectively discontinuous relative to the seismic wavelengthThese layers can vary in thickness from less than a meter to many hundreds of meters Asa result when the Earth s surface is excited with some source of seismic energy and theresponse recorded on seismometers we will see a complicated zoo of elastic wave typesreections from the discontinuities in material properties multiple reections within thelayers guided waves interface waves which propagate along the boundary between twodierent layers surface waves which are exponentially attenuated with depth waves whichare refracted by continuous changes in material properties and others The characterof these seismic waves allows seismologists to make inferences about the nature of thesubsurface geology

Because of tectonic and other dynamic forces at work in the Earth this rstorder view ofthe subsurface geology as a layer cake must often be modied to take into account bentand fractured strata Extreme deformations can occur in processes such as mountainbuilding Under the inuence of great heat and stress some rocks exhibit a taylikeconsistency and can be bent into exotic shapes without breaking while others becomeseverely fractured In marine environments less dense salt can be overlain by more densesediments as the salt rises under its own buoyancy it pushes the overburden out of the

i

ii

way severely deforming originally at layers Further even on the relatively localizedscale of exploration seismology there may be signicant lateral variations in materialproperties For example if we look at the sediments carried downstream by a river it isclear that lighter particles will be carried further while bigger ones will be deposited rstows near the center of the channel will be faster than the ow on the verge This givesrise to signicant variation is the density and porosity of a given sedimentary formationas a function of just how the sediments were deposited

Taking all these eects into account seismic waves propagating in the Earth will berefracted reected and diracted In order to be able to image the Earth to see throughthe complicated distorting lens that its heterogeneous subsurface presents to us in otherwords to be able to solve the inverse scattering problem we need to be able to undo all ofthese wave propagation eects In a nutshell that is the goal of imaging to transform asuite of seismograms recorded at the surface of the Earth into a depth section ie a spatialimage of some property of the Earth usually wavespeed or impedance There are twotypes of spatial variations of the Earth s properties There are the smooth changes smoothmeaning possessing spatial wavelengths which are long compared to seismic wavelengthsassociated with processes such as compaction These gradual variations cause ray pathsto be gently turned or refracted On the other hand there are the sharp changes shortspatial wavelength mostly in the vertical direction which we associate with changes inlithology and to a lesser extent fracturing These short wavelength features give rise tothe reections and diractions we see on seismic sections If the Earth were only smoothlyvarying with no discontinuities then we would not see any events at all in explorationseismology because the distances between the sources and receivers are not often largeenough for rays to turn upward and be recorded This means that to rst order reectionseismology is sensitive primarily to the short spatial wavelength features in the velocitymodel We usually assume that we know the smoothly varying part of the velocity modelsomehow and use an imaging algorithm to nd the discontinuities

The earliest forms of imaging involved moving literally migrating events around seismictime sections by manual or mechanical means Later these manual migration methodswere replaced by computeroriented methods which took into account to varying degreesthe physics of wave propagation and scattering It is now apparent that all accurateimaging methods can be viewed essentially as linearized inversions of the wave equationwhether in terms of Fourier integral operators or direct gradientbased optimization ofa waveform mist function The implicit caveat hanging on the word essentially inthe last sentence is this people in the exploration community who practice migrationare usually not able to obtain or preserve the true amplitudes of the data As a resultattempts to interpret subtle changes in reector strength as opposed to reector positionusually run afoul of one or more approximations made in the sequence of processing stepsthat make up a migration trace equalization gaining deconvolution etc On the otherhand if we had true amplitude data that is if the samples recorded on the seismogramreally were proportional to the velocity of the piece of Earth to which the geophone wereattached then we could make quantitative statements about how spatial variations in

iii

reector strength were related to changes in geological properties The distinction hereis the distinction between imaging reectors on the one hand and doing a true inverseproblem for the subsurface properties on the other

Until quite recently the exploration community has been exclusively concerned with theformer and today the word migration almost always refers to the imaging problem Themore sophisticated view of imaging as an inverse problem is gradually making its wayinto the production software of oil and gas exploration companies since careful treatmentof amplitudes is often crucial in making decisions on subtle lithologic plays amplitudeversus oset or AVO and in resolving the chaotic wave propagation eects of complexstructures

When studying migration methods the student is faced with a bewildering assortment ofalgorithms based upon diverse physical approximations What sort of velocity model canbe used constant wave speed v vx vx z vx y z Gentle dips Steep dips Shallwe attempt to use turning or refracted rays Take into account mode converted arrivalsD two dimensions D Prestack Poststack If poststack how does one eectoneway wave propagation given that stacking attenuates multiple reections Whatdomain shall we use Timespace Timewave number Frequencyspace Frequencywave number Do we want to image the entire dataset or just some part of it Are wejust trying to rene a crude velocity model or are we attempting to resolve an importantfeature with high resolution It is possible to imagine imaging algorithms that would workunder the most demanding of these assumptions but they would be highly inecient whenone of the simpler physical models pertains And since all of these situations arise at onetime or another it is necessary to look at a variety of migration algorithms in daily use

Given the hundreds of papers that have been published in the past years to do areasonably comprehensive job of presenting all the dierent imaging algorithms wouldrequire a book many times the length of this one This was not my goal in any caseI have tried to emphasize the fundamental physical and mathematical ideas of imagingrather than the details of particular applications I hope that rather than appearing as adisparate bag of tricks seismic imaging will be seen as a coherent body of knowledgemuch as optics is

Contents

Preface i

Introduction to Seismic Migration

The Reection Seismic Experiment

Huygens Principle

ZeroOset Data

Exploding Reector Model

Finite Oset

Stacking Velocity Analysis

Dipping Reectors

Resolution

Computer Exercise I

Harmonic Analysis and Delta Functions

Fourier Series

Fourier Transforms

Basic Properties of Delta Functions

The Sampling Theorem

Aliasing

Discrete Fourier Transforms

v

vi CONTENTS

Discrete Fourier Transform Examples

Trigonometric Interpolation

Computer Exercise II

Equations of Motion of the Earth

Computer Exercise III

Computer Exercise III An Example Solution

Elastic Wave Equations

Gravity Versus Elasticity

The Elastic Wave Equation

Scalar Wave Equations

Plane Waves

Spherical Waves

The Sommerfeld Radiation Condition

Harmonic Waves

Reection

Phase Velocity Group Velocity and Dispersion

Model A

Model B

Dispersion on a Lattice

Summary of Dispersion

A Quantum Mechanical Slant on Waves

Ray Theory

The Eikonal Equation

The Dierential Equations of Rays

CONTENTS vii

Fermat s Principle

Boundary Conditions and Snell s Laws

Fermat Redux

Kirchho Migration

The Wave Equation in Disguise

Application to Migration

Summary of Green Functions

Kirchho MigrationInversion

Case I Causal G S S S

Case II Causal G S S

Case III G Anticausal S S

The Method of Stationary Phase

Downward Continuation of the Seismic Wave eld

A Little Fourier Theory of Waves

WKBJ Approximation

PhaseShift Migration

OneWay Wave Equations

Paraxial Wave Equations

The Factored TwoWay Wave Equation

Downward Continuation of Sources and Receivers

Time Domain Methods

A Surfer s Guide to Continued Fractions

Fibonacci

Computer Exercise IV

viii CONTENTS

Plane Wave Decomposition of Seismograms

The Weyl and Sommerfeld Integrals

A Few Words on Bessel Functions

Reection of a Spherical Wave from a Plane Boundary

A Minor Digression on SlantStacks

Slantstacks and the Plane Wave Decomposition

Numerical Methods for Tracing Rays

Finite Dierence Methods for Tracing Rays

Initial Value Problems

Boundary Value Problems

Newton s Method

Variations on a Theme Relaxed Newton Methods

WellPosed Methods For Boundary Value Problems

Algebraic Methods for Tracing Rays

Distilled Wisdom

Eikonal Equation Methods

Computer Exercises V

Finite Dierence Methods for Wave Propagation and Migration

Parabolic Equations

Explicit TimeIntegration Schemes

Implicit TimeIntegration Schemes

Weighted Average Schemes

Separation of Variables for Dierence Equations

Hyperbolic Equations

The Full TwoWay Wave Equation

CONTENTS ix

Grid Dispersion

Absorbing Boundary Conditions

Finite Dierence Code for the Wave Equation

Computer Exercises VI

Computer Exercises VI Solution

References

Index

x CONTENTS

List of Figures

Relationship among the coordinates r s h and x Each indicates aseismic trace

Common midpoint ray paths under threefold coverage Sources are on theleft and receivers are on the right

In this example the shot spacing is half the group interval the distancebetween receivers on a given line Thus the CMP spacing is half thegroup interval too

Huygens Principle every point on a wavefront can be considered as a secondary source of spherical wavelets Here the plane wave moves a distancevt in the time increment t Its new position is found by drawing a linethrough the envelope of all of the spherical wavelets

Simple model of a syncline The velocity above and below the syncline isassumed to be constant

Synthetic zerooset data recorded over the syncline model shown in Figure Shown are traces sampled at seconds per sample spanning km on the Earth s surface

Constant velocity phase shift migration using the correct velocity

Hyperbolic travel time curve associated with a CMP gather over a horizontal constant velocity layer

The travel time curve for a constant velocity horizontal layer model is ahyperbola whose apex is at the oneway zerooset travel time t

Stacking along successive hyperbolae as we vary the velocity and zeroosettravel time gives a stack power Sv t as a function of v and t Findingthe best velocity model is then a matter of optimizing the stack powerfunction

xi

xii LIST OF FIGURES

common source gathers for a layer constant velocity model leftStacking velocity analysis for these gathers right

common source gathers for a layer model with a constant vz gradientleft Stacking velocity analysis for these gathers right

Unless the reector is horizontal the apparent dip on the time section isalways less than the true dip in the earth

In addition to increasing apparent dip migration also moves energy updip

The rst Fresnel zone is the horizontal distance spanned by one half awavelength

Fresnel zone construction

The kernel sinxx for and

A sinusoid sampled at less than the Nyquist frequency gives rise to spuriousperiodicities

The real left and imaginary right parts of three length time serieseach associated with a Kronecker delta frequency spectrum These timeseries are reconstructed from the spectra by inverse DFT At the top theinput spectrum is i in the middle i and at the bottom i

The real left and imaginary right parts of three time series of length each associated with a Kronecker delta frequency spectrum These timeseries are reconstructed from the spectra by inverse DFT At the top theinput spectrum is i in the middle i and at the bottom i

The top left plot shows an input time series consisting of a single sinusoidIn the top right we see the real part of its DFT Note well the wraparoundat negative frequencies In the middle we show the same input sinusoidcontaminated with some uniformly distributed pseudorandom noise andits DFT At the bottom left we show a Gaussian time series that we willuse to smooth the noisy time series by convolving it with the DFT of thenoisy signal When we inverse DFT to get back into the time domainwe get the smoothed signal shown in the lower right

Shot record and a cleaned up version

The raw data

CDP sorted data

LIST OF FIGURES xiii

Velocity analysis panels

Moveout corrected traces

The nal stack and zerooset migration

Plane wave propagation

A D pulse propagating towards the origin at speed c

A pulse incoming from the positive x direction and traveling with constantunit speed strikes a perfectly reecting barrier at the origin at t

Modulated sinusoid

A sample of the normal modes free oscillations of a homogeneous pointlattice with xed ends The lower frequency modes are purely sinusoidalthe higher frequency modes become modulated sinusoids as a result of thedispersive eects of this being a discrete system

Discrepancy between the dispersionfree string and the nite dimensionaldynamical system used to approximate it on a computer The lattice spacing was chosen to be unity

Synthetic Ricker lobes with noise added

Left A trace taken from the Conoco crosshole data set Right An enlarged view of the portion containing the rst arrival

Enlarged view of the squared trace Don t pay attention to the xaxislabeling The yaxis represents the normalized amplitude

Possible ray paths connecting the source and receiver

The minimum time path is by denition an extremum of the travel timeas a function of ray path

Stokes pillbox around an interface between two media Since the shapeis arbitrary we may choose the sides perpendicular to the interface to beinnitesimally small

The travel time along the ray path C connecting two points P and P isless than for any other path such as C

Geometry for Snell s law

xiv LIST OF FIGURES

Plane wave incident on a vertical boundary gives rise to a transmitted anda reected wave which must satisfy boundary conditions

Volume and surface dened in the specication of the wave equation

A Green function which vanishes on a planar surface can be calculated bythe method of images

A point scatterer in the Earth gives rise to a diraction hyperbola on thetime section Kirchho migration amounts to a weighted sum along thishyperbola with the result going into a particular location on the depthsection

The incident eld reects o a boundary at some depth The genericobservation point is r

Dene the surface S to be the union of the recording surface S and asurface joining smoothly with S but receding to innity in the z direction

Hypothetical phase function top real part of the integrand eigx for middle real part of the integrand for

A medium which is piecewise constant with depth

Claerbout s version of Gazdag s cz phaseshift algorithm

The continued fraction approximation to the square root dispersion formulagives an approximate dispersion formula whose accuracyreected in thesteepest angles at which waves can propagateincreases with the order ofthe approximation

An arbitrary number can be thought of as the slope of a line from the origin The continued fraction approximation for the number in this case thegolden mean consists of left and right polygons which approach an irrational number asymptotically and a rational number after a nite numberof steps

Spherical coordinates representation of the wave vector

Reection of a pulse from a planar surface

A curve can be specied either as the locus of all points such that t txor the envelope of all tangents with slope p and intercept

LIST OF FIGURES xv

Simple shooting involves solving the initial value problem repeatedly withdierent takeo angles iteratively correcting the initial conditions until asatisfactory solution of the boundary value problem is achieved

Newton s method for nding roots

Newton s method applied to a function with four roots and Thebottom gure shows the roots to which each of starting values uniformly distributed between and converged The behavior is complexin the vicinity of a critical point of the function since the tangent is nearlyhorizontal

Multipleshooting involves shooting a number of rays and joining themcontinuously

Finite dierence and analytic solution to the initialboundary value problemfor the heat equation described in the text The discretization factor r The errors are negligible except at the point x where the derivativeof the initial data is discontinuous

Finite dierence and analytic solution to the initialboundary value problemfor the heat equation described in the text The discretization factor r The errors are substantial and growing with time

Assuming r for the explicit method and r for the implicit hereis a plot showing the ratio of work between the two methods as a functionof the number of oating point operations per divide on a particular computer With a typical value of or oating point operations per dividewe can see that the work required for the two methods is very similar notwithstanding the fact that we need to use ten times as many time steps inthe explicit case

Domain of dependence of the rstorder hyperbolic equation

Hypothetical normalized source power spectrum The number of gridpoints per wavelength may be measured in terms of the wavelength atthe upper halfpower frequency

xvi LIST OF FIGURES

Waves on a discrete string The two columns of gures are snapshots intime of waves propagating in a D medium The only dierence betweenthe two is the initial conditions shown at the top of each column On theright we are trying to propagate a discrete delta function one grid pointin space and time On the left we are propagating a bandlimited versionof this The medium is homogeneous except for an isolated reecting layerin the middle The dispersion seen in the right side simulation is explainedby looking at the dispersion relation for discrete media Waves whosewavelengths are comparable to the grid spacing sense the granularity ofthe medium

The rst of common source gathers that comprise the nal computerexercise Figure out the model and produce a migrated image Drilling isoptional

The model that was used to produce the data shown in the previous gure

Wences Gouveia s migrated depth section

Tariq Alkhalifah s migrated depth section

Chapter

Introduction to Seismic Migration

The Reection Seismic Experiment

The essential features of an exploration seismic experiment are

Using controlled sources of seismic energy

Illumination of a subsurface target area with the downward propagating waves

Reection refraction and diraction of the seismic waves by subsurface heterogeneities

Detection of the backscattered seismic energy on seismometers spread out along alinear or areal array on the Earth s surface

On land the seismometers are called geophones Generally they work by measuring themotion of a magnet relative to a coil attached to the housing and implanted in the EarthThis motion produces a voltage which is proportional to the velocity of the Earth Mostgeophones in use measure just the vertical component of velocity However increasinguse is being made of multicomponent receivers in order to be able to take advantage ofthe complete elastic waveeld For marine work hydrophones are used which sense theinstantaneous pressure in the water due to the seismic wave

The receivers are deployed in clusters called groups the signal from each receiver in agroup is summed so as to a increase the signal to noise ratio and b attenuate horizontallypropagating waves On land horizontallypropagating Rayleigh waves are called groundroll and are regarded by explorationists as a kind of noise inasmuch as they can obscurethe soughtafter reection events The reason that receiver group summation attenuateshorizontally propagating signals and not the vertically propagating reected events is

CHAPTER INTRODUCTION TO SEISMIC MIGRATION

that the ray paths for the vertically propagating events strike the receivers at essentiallythe same time whereas if the receivers in a group are placed so that the horizontallypropagating wave is out of phase on the dierent receivers then summing these signals willresult in cancellation Sometimes the receivers are buried in the ground if the environmentis particularly noisyin fact burying the receivers is always a good idea but it s tooexpensive and timeconsuming in all but special cases The individual receiver groups areseparated from one another by distances of anywhere from a few dozen meters to perhaps meters The entire seismic line will be several kilometers or more long

Seismic sources come in all shapes and sizes On land these include for example dynamite weight drops large caliber guns and large resistive masses called vibrators whichare excited with a chirp or swept continuouswave signal For marine surveys vibratorsair guns electric sparkers and conned propaneoxygen explosions are the most commonsources The amount of dynamite used can range from a few grams for nearsurfacehighresolution studies to tens of kilograms or more The amount used depends on thetype of rock involved and the depth of the target Explosive charges require some sortof hole to be drilled to contain the blast These holes are often lled with heavy mudin order to help project the energy of the blast downward Weight drops involve a truckwith a crane from which the weight is dropped several meters This procedure can berepeated quickly Small dynamite blasts caps and large caliber guns pointed down are popular sources for very highresolution nearsurface studies By careful stacking andsignal processing the data recorded by a vibrating source can be made to look very muchlike that of an impulsive source Vibrators also come in all shapes and sizes from ton trucks which can generate both compressional and shear energy to highly portablehandheld devices generating roughly the power of a hi loudspeaker Sparkers whichrely on electric discharges are used for nearsurface surveys conducted before siting oshore oil platforms Airguns use compressed air from the ship to generate pulses similarto that of an explosion

The timeseries or seismogram recorded by each receiver group is called a trace Theset of traces recorded by all the receivers for a given source is called a common sourcegather There are three other standard ways of organizing traces into gathers We couldlook at all the traces recorded by a given receiver group for all the sources This is calleda common receiver gather We could look at all the traces whose sourcereceiverdistance is a xed value called the oset This called common oset gather Finallywe could look at all the traces whose sourcereceiver midpoint is a xed value This is acommon midpoint gather

Figure illustrates these coordinate systems for a twodimensional d marine surveyIt s called d since the goal is to image a vertical plane through the earth even though thedata were recorded along a line The sourcereceiver coordinates r and s are connected

For more details on sources and receivers see and

THE REFLECTION SEISMIC EXPERIMENT

o o o o o o o o o

o o o o o o o o o

o o o o o o o o o

o o o o o o o o o o o o o o o o o o

o o o o o o o o o




Receiver Coordinate r

Sh

ot

Co

ord

ina

te s

Half-Offset Coordinate h

Mid

poin

t Coo

rdin

ate

x

receivers shot

common midpoint gather

common offset gather

common receiver gather

common shot gather

* * * * * * *

o o o o o o o o o

Figure Relationship among the coordinates r s h and x Each indicates a seismictrace


Depth point

Common Midpoint

M

D

Figure Common midpoint ray paths under threefold coverage Sources are on theleft and receivers are on the right

S S R R R

Group Interval

R R R1 212 3 4 5 6

Figure In this example the shot spacing is half the group interval the distancebetween receivers on a given line Thus the CMP spacing is half the group interval too

to the osetmidpoint coordinates x and h via a rotation

x r s

h r s

For the most part in this course we will be discussing poststack or zerooset migrationmethods in which case the common midpoint domain CMP is generally used Thefold of a CMP gather is the number of source!receiver pairs for each midpoint This isillustrated in Figure which shows threefold coverage In this example the distancebetween sources is half the distance between receivers Figure Therefore the CMPspacing will be equal to the shot spacing and half the receiver spacing known as the groupinterval In Figure the source!receiver raypaths S R and S R are associatedwith the same CMP location at R

HUYGENS PRINCIPLE

Figure Huygens Principle every point on a wavefront can be considered as asecondary source of spherical wavelets Here the plane wave moves a distance vt in thetime increment t Its new position is found by drawing a line through the envelope of allof the spherical wavelets

Huygens Principle

Huygens principle that every point on a wavefront propagating in an isotropic mediumcan be considered a secondary source of spherical wavelets allows us to reduce problemsof wave propagation and depropagation migration to the consideration of point sourcesFigure shows Huygens construction for a plane wave propagating to the upper rightEach secondary spherical wavelet satises the equation

z z y y

x x vt t

where x y z is the origin of the spherical wavelet and t is its initiation time Ifthe medium is homogeneous we can use spherical wavelets of nite radius otherwise theconstruction must proceed in innitesimal steps

If we could view snapshots of the Earth s subsurface after a point source were initiatedwe would see an expanding spherical wavelet just as when a stone is thrown into a pondSo Equation denes a family of circles in xz space image space But that s nothow we actually see the seismic response of a point source We record data for a range oftimes at a xed depth position z usually z If z is xed Equation denes afamily of hyperbolae in x t space data space So just as the spherical wavelet is thefundamental unit out of which all other wave propagation phenomena can be built via


Huygens principle so the hyperbola is the fundamental feature of reection seismologyin terms of which all other reection!diraction events recorded at the surface can beinterpreted

To be precise what we really need is a generalization of Huygens principle due to Fresnelwho combined the wavelet construction with the Young s principle of interference Inother words the expanding spherical wavelets are allowed to interfere with one anotherFurther you will notice in Figure that the spherical wavelets have two envelopesone propagating in the same direction as the original plane wave and one propagating inthe opposite direction We only want the rst one of these Fresnel solved this problemby introducing an angledependent scaling factor called the obliquity factor We willsee later when we discuss integral methods of migration how this obliquity factor arisesautomatically

Fresnel was the rst person to solve problems of diraction which is any deviation fromrectilinear propagation other than that due to reection and refraction Fresnel s prediction that there should be a bright spot in the geometrical shadow of a disk was a majorblow to the corpuscular theory of light In fact this surprising prediction was regardedas a refutation of his memoir by French Academician Poisson Fortunately another member of the committee reviewing the work Arago actually performed the experiment andvindicated Fresnel "#

ZeroOset Data

So with the HuygensFresnel principle in hand we can think of any reection event as thesummation of the eects of a collection of point scatterers distributed along the reectingsurface Each one of these point scatterers is responsible for a diraction hyperbola Inorder to be able to make a picture of the reector which is the goal of migration weneed to be able to collapse these diraction hyperbolae back to their points of origin onthe reector There is one circumstance in which this procedure is especially easy tovisualize zerooset seismograms recorded in a constant velocity earth Imagine that wecan place a seismic source and receiver right next to one another so close in fact they areeectively at the same location With this arrangement we can record the seismic echosfrom the same point whence they originated To be precise if we start recording at thesame time that the source goes o we will record the direct arrival wave and no doubt

Geometrical optics is the limit of wave optics as the wavelength goes to zero In this limit there isno diraction there is no light in the shadow of a sharply dened object So in contrast to refractionwhich aects the shorter wavelengths more strongly diraction tends to deect the red end of thespectrum more than the blue end the deection being relative to the geometrical optics ray pathThe coronae around the sun and moon are diraction eects caused by the presence of water dropletsrandomly distributed in a layer of haze and are strongest when the droplets are of approximately uniformsize whereas the halos around the sun and moon are caused by refraction through ice crystals in thincirrus clouds See Chapter V

EXPLODING REFLECTOR MODEL

V

Earth’s Surface z=0

1

V2

Figure Simple model of a syncline The velocity above and below the syncline isassumed to be constant

shake up the receiver pretty thoroughly too Therefore in practice we must mute thedirect arrival either that or wait until just after things have settled down to turn on thereceiver Then we pick up our apparatus and move down the survey line a bit and repeatthe procedure This is called zero oset experiment In practice we can approximate thezerooset section by selecting the commonoset section having the minimum oset

Figure shows a model of a geologic syncline The velocity is assumed to have theconstant value v in the layer above the syncline and a dierent value v in the layerbelow This jump in the velocity gives rise to a reection coecient which causes thedowngoing pulse to be partially reected back to the surface where it is recorded on thegeophone

Figure is a computer generated zerooset section collected over this model Noticethat the travel time curve becomes multivalued as the source receiver combination movesover the syncline That s because there is more than one zerooset ray arriving at thereceiver in the time window we have recorded The goal of migration is to makeFigure look as much like Figure as possible

Exploding Reector Model

If T is the twoway time for a signal emanating at t to travel from the source tothe reector and back to the surface then it follows that T is the time it takes for asignal to travel from the source down to the reector or from the reector back up to thesource So apart from let s assume for the moment minor distortions the data recordedin the zerooset experiment are the same that would be recorded if we placed sources


0

1

2

3

Tim

e (s

ec)

1 2 3 4 5 6Distance (km)

Synthetic Data

Figure Synthetic zerooset data recorded over the syncline model shown in Figure Shown are traces sampled at seconds per sample spanning km on theEarth s surface

FINITE OFFSET

along the syncline and red them o at T with a strength proportional to the reectioncoecient at the syncline Equivalently we could re o the exploding reectors at t and halve the velocity

This means that if we somehow run the zerooset waveeld recorded at the surfacepx y z t backwards in time then by evaluating the results at t we would havea picture of the reection event as it occurred px y z t provided we use half thetrue velocity when we run the recorded eld backwards in time As we will see in detaillater on in the case of a constant velocity medium this time shifting of the wave eld canbe accomplished by a simple phase shift ie multiplying the Fourier transform of therecorded data by a complex exponential For now Figure must suce to show theresults of just such a procedure This is an example of phaseshift migration

You will notice that the vertical axis is time not depth This is referred to as migratedtime or vertical travel time In the simple case of constant velocity media there is a simplescaling relationship between depth and time t zv This extends to depthdependentor vz media via

t Z z

dz

vz

When we use this migrated time rather than depth for the output image the migrationis called time migration Evidently time migration will be a reasonable choice only if thevelocity shows no signicant lateral or vx variation Time migration has the advantagethat the migrated section is plotted in the same units as the input seismic section it hasthe disadvantage of being less accurate than depth migration The more the velocityvaries laterally the less accurate time migration will be

Finite Oset

The exploding reector analogy is a powerful one Unfortunately it only applies to zerooset data and typically exploration seismic surveys involve kilometers of oset betweeneach source and the farthest receivers Further there are even true zerooset situationswhere the exploding reector concept fails such as in the presence of strong lenses orwhen rays are multiply reected

Life gets more complicated when there is oset between the sources and receivers Insteadof there being a single twoway travel time for each source we have a travel time curveas a function of oset Figure shows a CMP gather associated with source!receiverpairs The dotted curve is the hyperbolic travel time versus oset curve for a constantvelocity layer

Invariably z is taken to be the vertical coordinate But in earth science z increases downward


0

1

2

3

Mig

rate

d T

ime

(sec

)

1 2 3 4 5 6Midpoint (km)

Phase Shift Migration

Figure Constant velocity phase shift migration using the correct velocity

FINITE OFFSET

1S S S S S S

R R R R RR M 321-1-2-3

-1 -3-223

Figure Hyperbolic travel time curve associated with a CMP gather over a horizontalconstant velocity layer

If the depth of the reecting layer is z and we let h denote half the distance betweensource and receiver then by the Pythagorean theorem it follows that the twoway traveltime as a function of halfoset h is given by

thv

h z

where t is now the twoway travel time and v is the constant velocity in the layer abovethe reector

Now the zerooset twoway travel time is simply

t z

v

So the dierence between the two is

th t

h

v

This is called the normal moveout correction and is the amount by which the real traceshave to be shifted in order to make them approximate the results of a zerooset experiment Normal moveout NMO refers generally to the hyperbolic moveout associated withhorizontal layers In the event that we have a stack of horizontal layers the travel timecurves are still approximately hyperbolic If we apply the NMO correction to traces sharing a common midpoint and sum them we get a plausible approximation to a zerooset


S

/2

M R

t/2

h

0t

Figure The travel time curve for a constant velocity horizontal layer model is ahyperbola whose apex is at the oneway zerooset travel time t

trace This means that we can apply zerooset migration to seismic data with nonzerooset Further stacking the traces in this way results in signicant improvement in signalto noise not only is random noise attenuated by stacking but nonhyperbolic events areattenuated as well Stacking is often the only way to see weak noisecontaminated eventsAnd the resulting poststack migration results are sometimes surprisingly good even incases where common midpoint CMP stacking should be a poor approximation to thezerooset trace

Further if we knew that the reecting layer were horizontal then we could estimate thevelocity above it by measuring the slope of the asymptote to the hyperbola This isillustrated in Figure The slope of the hyperbolic asymptote is just one over thevelocity in the layer above the reector And the hyperbola s apex is at the onewayzerooset travel time

Stacking Velocity Analysis

In order to correct for normal moveout prior to stacking we must have a velocity modelv in Equation This is equivalent to stacking along the right hyperbola asshown in Figure If we choose too high or too low a velocity then stacking will notproduce a very high power in the output trace We could imagine doing a suite of stacksassociated with dierent velocities and zerooset travel times and plotting the resultingstack power At each zerooset travel time equivalent to depths in a constant v orvz medium we could scan across velocity and pick the v which gives the highest stack

DIPPING REFLECTORS

time t

Medium

Low

High

Offset x = 2h

Two-way

Figure Stacking along successive hyperbolae as we vary the velocity and zeroosettravel time gives a stack power Sv t as a function of v and t Finding the bestvelocity model is then a matter of optimizing the stack power function

power Proceeding down in depth or zerooset travel time in this way we could map outthe depth dependence of stacking velocity automatically

We can illustrate this stacking velocity analysis with a simple example Consider anEarth model which is composed of horizontal horizontal layers Figure shows common source gathers for such a model assuming a constant velocity of km!secondThe stacking velocity panel for this is shown on the right in Figure The sweepover velocity consisted of increments of m!s The stacking panel shows maximumcoherency at a constant or depthindependent velocity but the coherence maxima aresomewhat smeared out Next if we put in a depthdependent velocity dvdx thenthe common source gathers look like the left side of Figure The stacking velocitypanel is shown on the right It s clear that we would have no diculty picking out thelinear trend in velocity

Dipping Reectors

So far we have only considered the reection seismology of at ie horizontal layersBut because of tectonic and other forces such as buoyant material welling up under theinuence of gravity dipping layers are very commonin fact mapping apparent dip thatwhich we see on the zerooset seismic sections into true dip is one of the principle goalsof migration Unless the dip is actually zero true dip is always greater than apparentdip Consider the limiting case of reections from a vertical layer degree dip Since


0

0.5

1.0

1.5

Tim

e (s

ec)

5 10 15 20 25Distance (km)

Synthetic Data

0

0.1

0.2

0.3

0.4

0.5

0.6

1500 2000 2500 3000 3500

Velocity Analysis

Figure common source gathers for a layer constant velocity model leftStacking velocity analysis for these gathers right

0

0.5

1.0

1.5

Tim

e (s

ec)

5 10 15 20 25Distance (km)

Synthetic Data

0

0.1

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0.5

0.6

1500 2000 2500 3000 3500

Velocity Analysis

Figure common source gathers for a layer model with a constant vz gradientleft Stacking velocity analysis for these gathers right

DIPPING REFLECTORS

θ

θ

r

x

z

z = x tan( )

Figure Unless the reector is horizontal the apparent dip on the time section isalways less than the true dip in the earth

the reections must propagate along the surface the zerooset section will have a linearmoveout whose slope is inversely proportional to the velocity

More generally consider a reector dipping at an angle of in the true earth cf Figure The relation between depth and position is z x tan Now the normalincidence ie zerooset travel time for a ray propagating from x down to the reectorand back up again is t rv where r is the ray path length But r x sin So thatwe may compare apparent and true dip we need to convert this travel time to depth viaz vt Therefore in the unmigrated depth section we have z x sin The slopeof this event is by denition the tangent of the apparent dip angle Call this angle Therefore we have tan sin This is called the migrator s equation and shows thatapparent dip is always less than true dip

Let s carry this analysis one step further and consider a segment of dipping reector asin Figure The events associated with the two normal incidence rays drawn fromthe dipping reector to the two receivers will appear on the unmigrated section at theposition locations associated with the two receivers Therefore not only will migrationincrease the apparent dip of this segment of the reector but it will have to move theenergy horizontally updip as well

To recap what we have shown thus far in this introductory chapter migration will

Focus energy by collapsing diraction hyperbolae

Increase apparent dip

Move energy horizontally in the updip direction


R R

Apparent Dip True Dip

Figure In addition to increasing apparent dip migration also moves energy updip

Resolution

Just as the stacking velocity panels in the last section did not collapse to points so diraction hyperbolae do not collapse to points when migrated they focus The spread of thefocus tells us something about the resolution of our imaging methods Vertical resolutionis governed by the wavelength of the probing wave vf where f is the frequency Butwe need to remember to divide v by two for the exploding reector calculation we dividein two again to get the length of one half cycle So the eective vertical resolution of a Hz seismic wave propagating in a medium whose wavespeed is m!s is about m "#

Horizontal resolution is usually measured in terms of the Fresnel zone Imagine a sphericalwave striking the center of a hole punched in an innite plate How wide would the holehave to be to allow the sphere to penetrate to one half of its wavelength Answerby denition one Fresnel zone Alternatively we can measure the Fresnel zone acrossa diraction hyperbola as the distance spanning one zero crossing Figure Thisconcept goes back to the HuygensFresnel principle previously discussed Picture a pointsource in an innite homogeneous medium As we will see later the spatial part of thesolution of the wave equation in this case is eikrr where r is the distance from the originof the disturbance to the wavefront and k is the reciprocal wavelength Figure showsthe spherical wavelet at some point in time after its origination at the point P To get theeld at some observation point Q we consider each point on the spherical wavefront as asecondary source of radiation In other words the seismogram at point Q SQ must be

RESOLUTION

0

0.2

0.4

0.6

50 100 150 200

Fresnel Zone

Figure The rst Fresnel zone is the horizontal distance spanned by one half awavelength

proportional to the integral over the spherical wavefront of

eikr

r

Z

eiks

sKd

where refers to the surface of the spherical wavefront is the angle between theposition vector and the propagation direction and K is an angledependent fudge factorthat Fresnel had to add to make the wave have the proper directional dependence Therst exponential the one outside the integral is just the pointsource propagation from Pto the spherical wavefront The second exponential inside the integral is the propagationfrom the wavefront to the observation point Q

The key idea for this discussion is how Fresnel actually computed the integral As illustrated in Figure he broke up the sphere into zones whose radii measured fromthe observation point Q diered by onehalf wavelength The total radiation at theobservation point is then just the sum of the contributions from the individual zones Thssurprising conclusion is the total disturbance at Q is equal to half the disturbance dueto the rst zone alone And since the intensity is the square of the disturbance if we couldsomehow screen o all but the rst zone the intensity recorded at Q would be timesas large as if there were no screen there Claerbout "# page gives an analogy ofshouting at a hole in the Berlin wall and says that holes larger than one Fresnel zone causelittle attenuation This might be true for an irregularly shaped hole but the disturbancesdue to the even and odd numbered zones are of opposite signs In fact it can be shownthat the disturbances due to the rst two zones are nearly equal Therefore if all butthe rst two zones are somehow covered up the disturbance at the observation pointwill be nearly zero This is worth repeating if we go from having all but the rst zone

This obliquity factor will be explained rigorously later when we get to Kirchho migration


z

z

z

Fresnel Zones

3

2

1

P Q

sr

Figure Fresnel zone construction

screened o to having all but the rst two the intensity measured at Q will go from times the unobstructed intensity to zero For more details see "# Chapter VIII and "#Chapter V Especially note Section of "# which discusses diraction from sharpobstacles In this section Sommerfeld reproduces an astonishing photograph taken byE von Angerer using a circular sheetmetal disk

Exercises

What is the vertical travel time for a medium whose velocity increases linearly withdepth

What are the angles of incidence and reection for zerooset rays Does youranswer depend on the reector geometry

The travel time curve for a common source record in a model with a single horizontal constantvelocity layer is a hyperbola What is it if there are two horizontallayers instead What is the NMO correction in this case

A point scatterer in image space gives rise to a hyperbolic event in data spaceWhat reector geometry would give rise to a point in data space

What are the minimum and maximum vertical resolution in a layer model whosewavespeeds are and km!second

Look at Figure again Why are the hyperbolae associated with the deeperreectors atter looking than the ones associated with the shallower reectors

COMPUTER EXERCISE I

Figure shows the geometry for a reector with arbitrary dip The velocityabove the reecting layer is a constant v Consider the zerooset section recordedover such a reector Convert the curve of zerooset travel times versus osetto migrated times Let the dip of the resulting curve be called What is therelationship between and This is called the migrator s equation

Show that apparent dip is always less than migrated dip

Computer Exercise I

In most of the computer exercises in this course we will take advantage of the SU SeismicUnix processing system developed at the Center for Wave Phenomena Instructions onhow to obtain SU as well as a User s Guide are in Appendix A

To get a list of all the executables including graphics type suhelp Then simply typethe name of any executable with no command line arguments to get information on howto run the program

For example here is what happens if you type sulter at the Unix prompt

SUFILTER applies a zerophase sinesquared tapered filter

sufilter stdin stdout optional parameters

Required parameters

if dt is not set in header then dt is mandatory

Optional parameters

fff array of filter frequenciesHZ

ampsaa array of filter amplitudes

dt from header time sampling rate sec

Defaults fnyquist nyquist nyquist

nyquist nyquist calculated internally

amps trapezoidlike bandpass

filter

Examples of filters

Bandpass

sufilter data f

Bandreject

sufilter data f amps


Lowpass


Highpass


Notch

sufilter data f

amps

Here is an example shell script for generating some synthetic data and then migrating it

n

n

use susynlv to make synthetic data

susynlv ntn dt ft nxo

nxmn dxm fxm er ob

v dvdz dvdx smooth

ref

sushw keyd a junksusyn

supsimage junksusyn labelTime sec

labelDistance km

titleSynthetic Data syntheticdataps

apply gazdag

sugazmig junksusyn tmig vmig junkout

supsimage junkout labelMigrated Time sec

labelMidpoint km

titlePhase Shift Migration migrateddataps

In this example we ve used the postscript based plotting routine supsimage The lemigrated dataps for example can be displayed on an Xwindows device using ghostscriptgs or sent to a laser printer See the information in suhelp under plotting for more details

Exercises

Dip Use susynlv to make zerooset traces over a single at horizontal layerkm deep Make the model km across and use a constant velocity of km!secuse an oset or trace spacing of m and a sample interval of ms Then redothis experiment varying the dip of the layer from to degrees in degree stepsSo you should end up with zerooset sections Measure the apparent dip of eachsection

COMPUTER EXERCISE I

Oset For the same models as in the rst problem record one common sourcegather of traces with a maximum oset of km

Band limited data Take one of your zerooset sections from above and usesulter to band limit it to a reasonable explorationseismic range of hz

Migration Use susynlv to generate a zerooset section over a layer dipping at degrees Use a constant velocity of km!sec and record traces spaced m apart Migrate your zerooset section with sugazmig and verify that migrationincreases apparent dip

Chapter

Harmonic Analysis and DeltaFunctions

We collect here a few miscellaneous but essential results results about Fourier TransformsFourier Series the sampling theorem and delta functions

Fourier Series

Suppose f is a piecewise continuous function periodic on the interval " # Then theFourier coecients of f are dened to be

cn

Z

fteint dt

The Fourier series for f is then

ft X

ncne

int

One has to be a little careful about saying that a particular function is equal to itsFourier series since there exist piecewise continuous functions whose Fourier series divergeeverywhere However here are two basic results about the convergence of such series

Pointwise Convergence Theorem If f is piecewise continuous and has left and rightderivatives at a point c then the Fourier series for f converges converges to

fc fc

A right derivative would be limtfc t fct t Similarly for a left derivative

CHAPTER HARMONIC ANALYSIS AND DELTA FUNCTIONS

where the and denote the limits when approached from greater than or less than c

Another basic result is the Uniform Convergence Theorem If f is continuous withperiod and f is piecewise continuous then the Fourier series for f converges uniformlyto f For more details consult a book on analysis such as The Elements of Real Analysisby Bartle "# or Real Analysis by Haaser and Sullivan "#

Fourier Transforms

If a function is periodic on any interval that interval can be mapped into " # by alinear transformation Further if a function is not periodic but dened only on a niteinterval it can be replicated over and over again to simulate a periodic function Thisis called periodic extension In all of these cases the Fourier series applies But forfunctions which are not periodic and which are dened over an innite interval then wemust use a dierent argument In eect we must consider the case of a nonperiodicfunction on a nite interval say "r r# and take the limit as r If we do this theFourier series becomes an integral cf "# Chapter section The result is that afunction ft is related to its Fourier transform f via

ft

Z

f eit d

andf

Z

fteit dt

Here using time and frequency as variables we are thinking in terms of time series butwe could just as well use a distance coordinate such as x and a wavenumber k

fx

Z

fkeikx dk

with the inverse transformation being

fk Z

fxeikx dx

It doesn t matter how we split up the normalization For example in the interest ofsymmetry many people dene both the forward and inverse transform with a

p out

front It doesn t matter as long as we re consistent We could get rid of the normalizationaltogether if we stop using circular frequencies in favor of f measured in hertz or cyclesper second Then we have

gt Z

gfeift df

andgf

Z

gteift dt

FOURIER TRANSFORMS

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1

2

3

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10

20

30

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-50

50

100

150

200

250

300

10 100

1000

-1 -0.5 0.5 1

500

1000

1500

2000

2500

3000

10000

Figure The kernel sinxx for and

These transformations from time to frequency or space to wavenumber are invertible inthe sense that if we apply one after the other we recover the original function To see thisplug Equation into Equation

fx

Z

dkZ

fxeikx

x dx

If we dene the kernel function Kx x such that

Kx x

Z

eikx

x dk sinx x

x x

then we havefx

Z

fxKx xdx

where Kxx is the limit assuming that it exists of Kxx as In orderfor this to be true Kx x will have to turn out to be a Dirac delta function

We won t attempt to prove that the kernel function converges to a delta function and hencethat the Fourier transform is invertible you can look it up in most books on analysisBut Figure provides graphical evidence We show plots of this kernel function forx and four dierent values of and Clearly in the limit that the function K becomes a Dirac delta function

Were being intentionally fuzzy about the details to save time If youve never encountered a carefultreatment of generalized functions you should spend some time with a book like Volume I of Gelfandand Shilovs Generalized Functions Its quite readable and very complete


Basic Properties of Delta Functions

Another representation of the delta function is in terms of Gaussian functions

x lim

pe

x

You can verify for yourself that the area under any of the Gaussian curves associated withnite is one

The spectrum of a delta function is completely at sinceZ

eikxx dx

For delta functions in higher dimensions we need to add an extra normalization foreach dimension Thus

x y z

Z

Z

Z

eikxxkyykzz dkx dky dkz

The other main properties of delta functions are the following

x x

ax

jajx

xx

fxx a fax a Zx yy a dy x a Z

mfx dx mf m Z

x yy a dy x a

xx x

x

Z

eikx dk

x i

Z

keikx dk

The Sampling Theorem

Now returning to the Fourier transform suppose the spectrum of our time series ft iszero outside of some symmetric interval "fc fc# about the origin In other words

In fact the assumption that the interval is symmetric about the origin is made without loss of generality since we can always introduce a change of variables which maps an arbitrary interval into a

THE SAMPLING THEOREM

the signal does not contain any frequencies higher than fc hertz Such a function is saidto be band limited it contains frequencies only in the band "fc fc# Clearly a bandlimited function has a nite inverse Fourier transform

ft

Z fc

fcf eit d

Since we are now dealing with a function on a nite interval we can represent it as aFourier series

f X

nne

infc

where the Fourier coecients n are to be determined by

n

fc

Z fc

fcf einfc d

Comparing this result with our previous work we can see that

n fnfc

fc

where fnfc are the samples of the original continuous time series ft Putting allthis together one can show that the band limited function ft is completely specied byits values at the countable set of points spaced fc apart

ft

fc

Xn

fnfcZ fc

fceinfct d

X

nfnfc

sinfct n

fct n

The last equation is known as the Sampling Theorem It is worth repeating for emphasis any band limited function is completely determined by its samples chosen fcapart where fc is the maximum frequency contained in the signal This means that inparticular a time series of nite duration ie any real time series is completely speciedby a nite number of samples It also means that in a sense the information content ofa band limited signal is innitely smaller than that of a general continuous function

So if our bandlimited signal ft has a maximum frequency of fc hertz and the lengthof the signal is T then the total number of samples required to describe f is fcT

Aliasing

As we have seen if a timedependent function contains frequencies up to fc hertz thendiscrete samples taken at an interval of fc seconds completely determine the signal

symmetric one centered on


1 2 3 4

-1

-0.5

0.5

1

Figure A sinusoid sampled at less than the Nyquist frequency gives rise to spuriousperiodicities

Looked at from another point of view for any sampling interval $ there is a specialfrequency called the Nyquist frequency given by fc

The extrema peaks and

troughs of a sinusoid of frequency fc will lie exactly fc apart This is equivalent tosaying that critical sampling of a sine wave is samples per wavelength

We can sample at a ner interval without introducing any error the samples will beredundant of course However if we sample at a coarser interval a very serious kindof error is introduced called aliasing Figure shows a cosine function sampled at aninterval longer than fc this sampling produces an apparent frequency of ! the truefrequency This means that any frequency component in the signal lying outside theinterval fc fc will be spuriously shifted into this interval Aliasing is produced byundersampling the data once that happens there is little that can be done to correct theproblem The way to prevent aliasing is to know the true bandwidth of the signal orbandlimit the signal by analog ltering and then sample appropriately so as to give atleast samples per cycle at the highest frequency present

Discrete Fourier Transforms

In this section we will use the f cycles per second notation rather than the radiansper second because there are slightly fewer factors of oating around You shouldbe comfortable with both styles but mind those s Also up to now we have avoidedany special notation for the Fourier transform of a function simply observing whetherit was a function of spacetime or wavenumberfrequency Now that we are consideringdiscrete transforms and real data we need to make this distinction since we will generallyhave both the sampled data and its transform stored in arrays on the computer So forthis section we will follow the convention that if h ht then H Hf is its Fouriertransform

DISCRETE FOURIER TRANSFORMS

We suppose that our data are samples of a function and that the samples are taken atequal intervals so that we can write

hk htk tk k$ k N

where N is an even number In our case the underlying function ht is unknown allwe have are the digitally recorded seismograms But in either case we can estimate theFourier transform Hf at at most N discrete points chosen in the range fc to fc wherefc is the Nyquist frequency

fn n

$N n

N

N

The two extreme values of frequency fN and fN are not independent fN fN so there are actually only N independent frequencies specied above

A sensible numerical approximation for the Fourier transform integral is thus

Hfn Z

hteifnt dt

NXk

hkeifntk$

Therefore

Hfn $NXk

hkeiknN

Dening the Discrete Fourier Transform DFT by

Hn NXk

hkeiknN

we then haveHfn $Hn

where fn are given by Equation

Now the numbering convention implied by Equation has Nyquist at the extremeends of the range and zero frequency in the middle However it is clear that the DFT isperiodic with period N

Hn HNn

As a result it is standard practice to let the index n in Hn vary from to N with n and k varying over the same range In this convention frequency occurs atn positive frequencies from from n N negative frequencies run fromN n N Nyquist sits in the middle at n N The inverse transform is

hk

N

NXn

HneiknN


Mathematica on the other hand uses dierent conventions It uses the symmetric normalization

pN in front of both the forward and inverse transform and denes arrays

running from to N in Fortran fashion So in Mathematica the forward and inversetransforms are respectively

Hn pN

NXk

hkeiknN

and

hk pN

NXn

HneiknN

If you are using canned software make sure you know what conventions arebeing used

Discrete Fourier Transform Examples

Here we show a few examples of the use of the DFT What we will do is construct anunknown time series DFT by hand and inverse transform to see what the resulting timeseries looks like In all cases the time series hk is samples long Figures and show the real left and imaginary right parts of six time series that resulted frominverse DFT ing an array Hn which was zero except at a single point ie it s a Kroneckerdelta Hi ij if i j and zero otherwise here a dierent j is chosen for each plotStarting from the top and working down we choose j to be the following samples therst the second Nyquist Nyquist Nyquist the last We can see that the rstsample in frequency domain is associated with the zerofrequency or DC component of asignal and that the frequency increases until we reach Nyquist which is in the middle ofthe array

Next in Figure we show at the top an input time series consisting of a pure sinusoidleft and the real part of its DFT Next we add some random noise to this signal On theleft in the middle plot is the real part of the noisy signals DFT Finally at the bottomwe show a Gaussian which we convolve with the noisy signal in order to attenuate thefrequency components in the signal The real part of the inverse DFT of this convolvedsignal is shown in the lower right plot

Trigonometric Interpolation

There is yet is another interpretation of the DFT that turns out to be quite useful inapproximation theory Imagine that we have N samples spaced evenly along the x axis

xk fk k N


10 20 30 40 50 60

-0.4

-0.2

0.2

0.4

0.6

10 20 30 40 50 60

-0.4

-0.2

0.2

0.4

10 20 30 40 50 60

-0.1

-0.05

0.05

0.1

10 20 30 40 50 60

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-0.05

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10 20 30 40 50 60

-0.1

-0.05

0.05

0.1

10 20 30 40 50 60

-0.1

-0.05

0.05

0.1

Figure The real left and imaginary right parts of three length time series eachassociated with a Kronecker delta frequency spectrum These time series are reconstructed from the spectra by inverse DFT At the top the input spectrum is i in the middlei and at the bottom i

where the fk may be complex and xk kN Then there exists a unique polynomial

px eix e

ix NeiNx

such that pxk fk Moreover the j are given by

j

N

NXk

fkeijkN

Thus by performing a DFT on N data we get a trigonometric interpolating polynomialof order N

Exercises

Verify Equation

Find the Fourier transform pairs of the following for a generic function gt andwhere a b t are arbitrary scalars gat jbj gtb gt t gteit

Dene the convolution of two functions g and h as

g ht Z

g ht d


10 20 30 40 50 60

-0.1

-0.05

0.05

0.1

10 20 30 40 50 60

-0.4

-0.2

0.2

0.4

10 20 30 40 50 60

-0.1

-0.05

0.05

0.1

10 20 30 40 50 60

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-0.05

0.05

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10 20 30 40 50 60

-0.1

-0.05

0.05

0.1

10 20 30 40 50 60

-0.1

-0.05

0.05

0.1

Figure The real left and imaginary right parts of three time series of length each associated with a Kronecker delta frequency spectrum These time series arereconstructed from the spectra by inverse DFT At the top the input spectrum is iin the middle i and at the bottom i


10 20 30 40 50 60

-1

-0.5

0.5

1

10 20 30 40 50 60

0.25

0.5

0.75

1

1.25

1.5

10 20 30 40 50 60

-1

-0.5

0.5

1

1.5

10 20 30 40 50 60

1

2

3

4

10 20 30 40 50 60

0.2

0.4

0.6

0.8

10 20 30 40 50 60

-0.4

-0.2

0.2

0.4

Figure The top left plot shows an input time series consisting of a single sinusoidIn the top right we see the real part of its DFT Note well the wraparound at negativefrequencies In the middle we show the same input sinusoid contaminated with someuniformly distributed pseudorandom noise and its DFT At the bottom left we show aGaussian time series that we will use to smooth the noisy time series by convolving itwith the DFT of the noisy signal When we inverse DFT to get back into the timedomain we get the smoothed signal shown in the lower right


Show that the convolution of two functions is equal to the product of their Fouriertransforms

The correlation of two functions is just like the convolution except that the minussign becomes a plus Show that the correlation of two functions is equal to theproduct of the Fourier transform of the rst times the complex conjugate of theFourier transform of the second

Show that the autocorrelation of a function its correlation with itself is equal to theabsolute value squared of the Fourier transform of the function WienerKhinchinTheorem

Show that if a function is imaginary and even its Fourier transform is also imaginaryand even A function is said to be even if fx fx and odd if fx fx

What is the Fourier transform of pe

x

What is the Nyquist frequency for data sampled at ms ie seconds!sample

Computer Exercise II

In this exercise we re going to get a little experience playing with real data You willbe given instructions on how to ftp a shot record from the collection described in OzYilmaz s book Seismic Data Processing This book has data from all over the worldrecorded with many dierent types of sources both marine and land

The subject of this exercise is shot record number Here is its header le

inner offset zero trace added source at trace

phone spacing meters inner offset meters

surveyland areaAlberta sourceDynamite spreadsplit

n d f labelTime sec

n d f labelOffset km

n

titleOz

plottpow

The SU plotting programs will accept les of this form as command line arguments Thusonce you get your segy version of the data see below call it for example datasegyyou could do the following

At present these are available via ftp from hilbertminescoloradoedu in the directorypubdata

COMPUTER EXERCISE II

suximage datasegy parozH

or

supsimage datasegy parozH datasegyps

Exercises

The rst thing you need to do is put the data in segy format so that you can use allthe SU code on it This is easy just use suaddhead You only have to tell suaddheadto put in the number of samples nssomething and if the data are other than mil the sample interval Look closely at the example in the selfdoc of suaddheadhowever to see the way in which sample intervals are specied

Now that you ve got an segy data set you can look at it with various programs suchas supswigb supswigp supsimage and displayed using ghostscript in X windows oropen in NeXTStep or suximage in X windows However you ll immediately noticethat the traces are dominated by ground roll Rayleigh waves near the origin Infact that s about all you ll see So that you can see the other events too try usingthe perc option in the plotting For example

supswigb data perc

where data is the name of your segy data set

Ground roll is a relatively lowfrequency kind of noise so try using sulter onyour data to get rid of it Play around with the frequencies and see what pass bandworks best for eliminating these events

OK now that you ve eliminated the ground roll let s try eliminating the rst arrivals These are the diagonal linear moveout events associated with head wavespropagating along the base of the nearsurface weathering layers Here we canjust use a plain mute to kill these events The program sumute will do this for youIt expects an array of x locations and t values For example if we wanted to mutestraight across a section having traces at a time of seconds we might usesomething like

sumute data xmute tmute keytracl out

The use of the keyword tracl lets us input x locations in terms of trace numbersrather than osets If you d rather do things in terms of oset however that s upto you


0

1

2

3

20 40 60 800

1

2

3

20 40 60 80

Figure Shot record and a cleaned up version

Finally boost the amplitude of the deeper reections by running sugain over thedata In the gure below I ve used the tpow option You might have success withother options as well

Look around at the other SU programs and see if there are any things you d like to do tohelp clean up the data What you want to end up with is the section with the bestlookingreections ie hyperbolic events Be sure you keep track in which order you do thingsseismic processing operators are denitely NOT commutative

If you re using X windows suximage is nice You can enlarge an image by pushing theleft mouse button and dragging a box over the zone you want to enlarge To get backto unit magnication click the left mouse button You can view postscript images in Xwith gs which stands for ghostscript So typing gs leps will display it on your screenYou can send postscript les to be printed on the laser printer just by typing lpr lepsas you would for an ordinary le provided your default printer is equiped for postscriptIn Figure you will see my attempt at cleaning up these data The gure illustratessimple straightline mutes Perhaps you can do better These plots suer somewhat sinceI ve used supswigb instead of supswigp The latter does fancy polygon ll drawing butat the expense of a huge postscript le That s OK for you since once you print if o youcan delete the postscript But for these notes we must keep a copy of all the postscriptplots around so we ve chosen to use the more economical supswigb

Ghostscript is freely available over Internet from various archive locations if its not already on yoursystem Ghostview is a free X user interface for ghostscript You can view multipage documents withit


Sinc Function Interpolationvia

MathematicaThe sampling theorem says that if we multiplythe Nyquist-sampled values of a function by asuitably scaled sinc function, we get the whole

(band-limited) function

Here is the first example. 25 Hz data on the timeinterval [0,.5]. First, let’s look at two different sincfunctions.

fc = 25.;sinc[t_,n_] = Sin[Pi ( 2. fc t - n)]/

(Pi ( 2. fc t -n));plot1 = Plot[sinc[t,10],t,0.,.5,PlotRange->All];

0.1 0.2 0.3 0.4 0.5

-0.2

0.2

0.4

0.6

0.8

1


plot2 = Plot[sinc[t,11],t,0.,.5,PlotRange->All];

0.1 0.2 0.3 0.4 0.5

-0.2

0.2

0.4

0.6

0.8

1

The key point is that when we multiply each of theseby the function values and add them up, the wigglescancel out, leaving an interpolating approximation tothe function.

Show[plot1,plot2]

0.1 0.2 0.3 0.4 0.5

-0.2

0.2

0.4

0.6

0.8

1


Suppose we are trying to approximate the constantfunction f(x) = 1. So we multiply the sinc functionsby 1.

myapprox[t_] = sinc[t,11] + sinc[t,10]

Sin[Pi (-11 + 50. t)] Sin[Pi (-10 + 50. t)]--------------------- + ----------------------------------------- -------------------- Pi (-11 + 50. t) Pi (-10 + 50. t)

Plot[myapprox[t],t,0.,.5,PlotRange->All]

0.1 0.2 0.3 0.4 0.5

0.2

0.4

0.6

0.8

1

1.2

Extend this. Sum a lot of these up.myapprox[t_] = Sum[sinc[t,n],n,1,25];Plot[myapprox[t],t,.01,.49,PlotRange->.1,.4,0,2]

0.15 0.2 0.25 0.3 0.35 0.40

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Chapter

Equations of Motion for the Earth

In the course of our study of migration methods we will encounter many dierent waveequations How can that be you say I know the wave equation it s

r

c

So what are you trying to pull

As you will see shortly this is only an approximate wave equation Implicit in it areassumptions about the frequency of the waves of interest the smoothness of the materialproperties and others Further migration involves many kinds of manipulations of thedata which aect the approximate wave equations that we use For example since stackingattenuates multiples if we re doing post stack migration we need a wave equation whichdoes not give rise to spurious multiples something which certainly cannot be said for theabove equation Therefore we thought it would be useful to give a careful derivation ofthe wave equation with all of the well almost all of them assumptions up front Ifderivations of the wave equation are old hat to you feel free to skip ahead to Section on page

So let us begin by deriving the equations governing the innitesimal elasticgravitationaldeformation of an Earth model which is spherically symmetric and nonrotating has anisotropic and perfectly elastic incremental constitutive relation and has an initial stresseld which is perfectly hydrostatic This is kind of a lot at one time and some of thesecomplications are not important in many seismological applications It is better in thelong run however to do the most complicated case initially and then simplify it later

We characterize the Earth s equilibrium nonmoving state by its equilibrium stress eldT its equilibrium density eld and its equilibrium gravitational potential eld

This chapter and the next two on elastic wave equations and rays were cowritten with MartinSmith of New England Research

CHAPTER EQUATIONS OF MOTION OF THE EARTH

These are related byr Tx xrx

and by Poisson s equationrx Gx

plus some boundary conditions We also suppose that T is purely hydrostatic Then

T pI p equilibrium pressure eld

This is reasonable if the Earth has no signicant strength The equilibrium equations arethen

rp r

r G

Suppose now that the Earth is disturbed slightly from its equilibrium conguration Wehave to be a little careful with our coordinates We express particle motion in the usualform

rx t x sx t

where x can be regarded as the particle s equilibrium position

We express the timedependent stress density and gravitational potential at a xedspatial point r as

r t r r t

r t r r t

Tr t Tr Tr t

prITr t

These equations dene the incremental quantities and T Note that the incremental quantities are dened as dierences from the equilibrium values at the observingpoint they are not in general the changes experienced by a material particle We alsoneed the particle acceleration

dtvr t t sx t to rst order in s

The equations of motion ie Fma are to rst order

t s r r r rp r T

using the fact that the incremental quantities are all of order s and keeping only termsthrough the rst order If we subtract the equilibrium condition we are left with

t s r r r T

Each term in this equation is evaluated at the spatial point r but the equation governsthe acceleration of the material packet which may be at r only for the single instant oftime t

We also must have

t r ts ts r

since the continuity equation for conservation of mass is

t r v

So to rst order

t r ts ts r

tf r sg

r s

since if s

Further for Newtonian gravity we have

r G

r G Gr s

The last bit we have to sort out is how to compute T the incremental stress at the pointr This is more complicated than it might appear because T is not simply the productof the local innitesimal strain tensor and the local elastic tensor The reason we havethis problem is the presence of an initial stress eld T which is not innitesimal but isof order zero in the displacement

Suppose for example we keep our attention on the point r while we rigidly translate theEarth a small amount After the translation we will be looking at the same spatial pointr but at a dierent piece of matter This new piece will in general have a dierent initialstress eld than our old piece If we translated the Earth an amount d we would perceivean incremental stress of

Tr t Tr dTr in general

However a rigid translation produces no strain as we well know and thus cannot causeany elastic stress Thus T above has nothing whatever to do with elasticity or strainbut is solely a result of the initial stress eld

To see how to handle this problem consider the stress tensor at a material particle whichwe express as

pxITm at r x sx t


We have expressed the total stress at the particle x which is now located at r as theinitial stress at the particle plus an increment Tm Note that we can use either Tmxor Tmr since Tm as we will see is rstorder in s We cannot use pr however ifwe want Tm to mean what we claim it to We now claim

Tm r sI rs sr

because the only way to alter the stress at a particle is by straining the continuum sris shorthand for rsT Remember that the stress tensor describes the forces betweenparts of the continuum across intervening surfaces These forces in turn are exactly aresult of the material being distorted out of its preferred state The expression for Tmin fact denes what we mean by an incremental constitutive relation

If pxITm is the stress tensor at a particle then the stress tensor at a point in spaceis simply

I s rpITm T T

which is just a Taylor series To rst order we must have

T Tm s rpI

So the full set of linearized equations is

t s r r r Tm r s rp

r s

r G

Tm r sI rs sr

rp r

Remember that we have assumed that the Earth

was not rotating

had a hydrostatic prestress eld

was selfgravitating

was spherically symmetric

had an isotropic perfectly elastic incremental constitutive relation

Remember too that in formulating these equations we encountered a situation in whichthe distinction between spatial and material concepts was crucial It is often said thatin linear innitesimal elasticity there is no dierence between Eulerian spatial andLagrangian material descriptions That is wrong There is no dierence in problemswhich do not have zeroeth order initial stress or other elds

COMPUTER EXERCISE III

Computer Exercise III

The purpose of this exercise is to make sure you are thoroughly familiar with segy headersand to prepare you for a later exercise in which you will need to process a blind data setYou will begin by making some synthetic data with susynlv Then you will add whateverheader information is necessary for you to complete the following processing ow

Do stacking velocity analysis using velan

NMO correct and stack the data to produce a zerooset section

Migrate the data with sugazmig

Repeat the migration and velocity analysis until you are satised with the image

These are just the main steps There will be other small steps such as guring out whichheaders are required sorting the data and so on

For creating the synthetic data with susynlv use the following description of the reectors

ref

ref

ref

ref

The sample interval is ms the shot spacing is the receiver spacing is You cando either a split spread or single ended survey For velocities use v dvdzdvdx Record enough samples so that you can see the deepest reector

A quick way of seeing what s going on might be to pick the near oset traces o of eachshot record and take these as a zerooset section

Susynlv will dump all the traces shot by shot into a single le You can examine all oftraces simultaneously with suximage or you can use suwind to pick o a smaller numberto look at Or you can use the zoom cababilities of suximage to zoom in on individualshot records

Computer Exercise III An Example Solution

Here is one solution to Exercise III contributed by Tagir Galikeev and Gabriel AlvarezAt the time this was done we had not begun using SUB in the class Try writing a SUBscript to do the preprocessing


Create Data

susynlv datasu nt dt nxs dxs

nxo dxo fxo ref

ref ref

ref v dvdz verbose

Reformat to SEGY

segyhdrs datasu

segywrite datasu tapedatasegy

Geometry Assignment CDP Sort

sushw datasu keysx a b c d j

sushw keyoffset a b j

sushw keygx a b c d j

suchw keycdp keygx keysx a b c

susort datageomcdpsu cdp offset

Perform Velocity Analysis

Analysis on CDP

suwind datageomcdpsu keycdp min max

suvelan semblancesu fv dv

Analysis on CDP



Analysis on CDP



Analysis on CDP



Apply NMO Correction Stack

sunmo datageomcdpsu cdp

vnmo

tnmo

vnmo

tnmo

vnmo

tnmo

vnmo

tnmo

sustack stacksu

COMPUTER EXERCISE III AN EXAMPLE SOLUTION

0

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e

20 40 60Channel

FFID 100

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e

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FFID 176

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e

20 40 60Channel

FFID 190

0

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e

20 40 60Channel

FFID 7

Figure The raw data

Gazdag Migration

sugazmig stacksu tmig vmig dx

supsimage stackmigps

titleMigrated Stack with True Velocities


0

0.5

1.0

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2.0

Tim

e

-1000 -500Offset

CDP 390 Gather

0

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e

-1000 -500Offset

CDP 290 Gather

0

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e

-1000 -500Offset

CDP 490 Gather

0

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e

-1000 -500Offset

CDP 590 Gather

Figure CDP sorted data


0

0.5

1.0

1.5

2.0

Tim

e

1000 2000 3000 4000 5000Velocity

Velocity Analysis at CDP 390

0

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e

1000 2000 3000 4000 5000Velocity


0

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e

1000 2000 3000 4000 5000Velocity


0

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e

1000 2000 3000 4000 5000Velocity


Figure Velocity analysis panels


0

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e

940 950 960

CDP 290

0

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e

3940 3950 3960

CDP 390

0

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e

6940 6950 6960

CDP 490

0

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e

9940 9950 9960

CDP 590

Figure Moveout corrected traces


0

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2.0

100 200 300 400

Migrated Stack with True Velocities

0

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1.0

1.5

2.0

100 200 300 400

Stack

Figure The nal stack and zerooset migration

Chapter

Elastic Wave Equations

Gravity Versus Elasticity

Our most recent version of the equations of motion

t s r r rs p r Tm

four more relations

is a little overwhelming and we might hope that not every geophysical application requiressolving the fullblown system

There is a very important class of problems requiring the simultaneous solution of theentire system namely the elasticgravitational normal mode problem On the other handas we will crudely demonstrate here at the higher range of frequencies of seismologicalinterest a large portion of the equation of motion can be neglected

To see this we shall have to make premature use of the notion of a propagating seismicwave Suppose that some sort of wave with angular frequency and wavevector k ispropagating through the Earth Then being deliberately vague

s goes like eikrt more or less

Let us consider how the various terms in the momentum equation scale with andk jkj We neglect the spatial variation of and and use S to denote the scale sizeof displacement

t s S

kS

r kgS where g gravity

rs rp kgS

r T kS or kS

CHAPTER ELASTIC WAVE EQUATIONS

We have skipped over for simplicity Generally speaking is less signicant than

The righthand side of the momentum equation is the sum of various elastic or gravitational terms Elastic terms scale like k gravitational terms scale like kg Thus theratio of gravitation to elastic forces goes like

gravity

elasticity kg

k

g

k

c the elastic velocity

g

k g

ck

I confess to reaching ahead somewhat We see that k results in an elasticallydominated system while k results in a gravitationallydominated system The twoinuences are crudely equal when

g

ck

or

k g

c

in the Earth

Let us further suppose that we are essentially in the elastic domain Then k c Thisis obviously somewhat contradictory but it does give us a useful idea of where gravitybecomes important Then we can argue

c

t

seconds

This asserts that at periods which are short compared to seconds the equationsof motion are dominated by elasticity In the highfrequency limit then we can make dowith

t s r Tm

Tm r sI rs sr

These are the equations of motion used in conventional shortperiod seismology secto sec At longer periods the inuence of gravity becomes nonnegligible and we mustnd ways to study the complete equations of motion A much better way to determinethe point at which gravity becomes signicant is to solve both versions of the equationsof motion and compare the solutions

THE ELASTIC WAVE EQUATION

The Elastic Wave Equation

We will rst boil down the equations of motion for high frequencies and therefore withoutgravity into a form called the elastic wave equation Then we will briey review whyequations of the form are called wave equations

For suciently high frequencies we have

t s r Tm

andTm r sI rs sr

where sr is shorthand for rsT Then

t s rr s r "rs sr#

t s r sr r rs sr

rr s r rs sr

The rst two terms result from spatial gradients of the elastic constants We are going todrop them for two reasons

As we could show by scale analysis they are relatively unimportant as long as thewavelength of the solution is short compared to the scale length of changes in theelastic properties For any continuous distribution of properties there will be somescale frequency above which we may ignore these terms

These terms can cause an incredible amount of grief

After surgery we have

t s rr s r rs sr

where and are suciently constant Since

r rs sr rr sr r s

t s rr s r r s

This is sometimes called the elastic wave equation although we know it to be only ahighfrequency pretender

Now express s ass r r r r r

where and are scalar elds The above expression is a representation for s bywhich we mean that we believe that for any s of interest there exist scalar elds


fullling the above Further we believe that recasting our problem in terms of thesescalars will do some good

You can verify if you wish that with this representation the elastic wave equation becomes

rt r r r

t r

r r rt r

If

t r

t r and

t r

then s as gotten from these scalars will be a solution of the elastic wave equation Howeveras a little thought shows we have not shown that every s satisfying the elastic waveequation can be found from scalar elds satisfying these three relationships Fortunatelyhowever it is so and we can replace if by if and only if

If terms involving the gradients of and appear all of this alas goes down the drainand we have a much more intractable system Qualitatively the gradient terms coupleshear and compressional motions together When this coupling is weak we can regardthose terms as small sources as is done in scattering theory If the coupling is strongenough the notions of distinct shear and compressional waves break down

The three scalar equations are called scalar wave equations This form of equation isdiscussed later Observe that three perfectly valid classes of solutions can be gotten bysetting two of the scalars to zero and requiring the third to be a nonzero solution of itsrespective wave equation Let fsg fsg fsg denote in the obvious way these threesets of solutions Because everything is linear any general solution to the elastic waveequation can be expressed as a linear combination of members of these three sets Notethat every s satises r s and every s s satises r s r s Also and satisfy exactly the same scalar wave equation

These solutions have certain characteristic properties which we will simply describe andnot prove here s is called a P wave or compressional wave It is a propagatingdilatation with phase velocity

VP q

s and s are called S waves or shear waves Particle motions for these waves are atright angles to the direction of propagation These waves have a phase velocity

VS q VP

There are two classes of shear waves simply because there are two possible orthogonalpolarizations Shear waves have no associated dilatation r s

SCALAR WAVE EQUATIONS

n z

x

r

y

Figure Plane wave propagation

Scalar Wave Equations

Plane Waves

We will now consider why equations of the form

t cr

are called scalar wave equations The factor c which we suppose to be nearly constantis called the wave speed

Let r be a position vector to any point in space let %n be a xed direction in space andlet fx be any twice dierentiable function of one variable If

r t f%n r ct

then

r %nf

r r %n %nf

f

and

t cf

t cf

The wave equation becomescf

cf

which is true Thus any of the above form satises the wave equation Such a isconstant on the planes dened by cf Figure

%n r ct A a constant

Also the plane associated with a xed A and therefore a particular value of moves inthe direction %n with speed c or to be pedantic velocity c%n Thus the pattern of in


space is propagating with the velocity c%n which is what we expect of a wave Similarlyf%n r ct represents a disturbance constant on the plane dened by %n r constantwhich is being propagated with velocity c in the negative %n direction

We generally think in terms of simple harmonic time dependence and use to denoteangular frequency If we dene the wave number

k

c

which is the number of wavelengths per some unit length and the wavenumber vector

k k%n

c%n

then

%n r ct c

k r t

a more conventional form

Now if the plane wave f%n r ct is a pressure disturbance propagating through anacoustic medium then it is related to the particle velocity v via Newton s second law

tv rp rf%n r ct

The kinetic and potential energy densities of the plane wave are equal to fc So thetotal acoustic energy density is just twice this

Spherical Waves

We begin with the scalar wave equation

t cr

where we assume c is constant We have already seen that if fx is any boundedtwicedierentiable function of a single variable then

r t f%n r ct

satises the wave equation for any direction %n

There are other simple wave solutions In spherical coordinates r if we assume is a function only of r and t ie spherical symmetry then

r r

rr

SCALAR WAVE EQUATIONS

Remember we are assuming that the wave is spherically symmetric about the origin ofthe coordinate system We might imagine a purely symmetric explosionlike source forsuch a wave The substitution

r t

rr t

produces

r

rr

and the wave equation reduces to the onedimensional wave equation

r

ct

So fr ct gr ct

and thus

rffr ct gr ctg

This solution represents two spherically symmetric wavesone traveling outward andone traveling inward The leading factor of

rtells us how the amplitudes of the waves

diminish as they progress

We can see that r

is a physically reasonable rate of decay Consider just the outwardtraveling wave

out

rfr ct

is virtually always proportional to a quantity such as displacement strain stresselectromagnetic eld strength etc and thus is usually a measure of energy densityThe energy in the wavefront crossing a sphere of radius r at time t is then

r fr ct

the size of which is independent of radius other than as a phase argument Thusenergy is conserved by the

rdependence

The Sommerfeld Radiation Condition

The other spherically symmetric solution

in

rgR ct

is a wave traveling inward toward the origin If the problem we were studying involved asource at the origin and if the medium was homogeneous and innite we might discardthe inwardbound solution on the grounds that energy must ow outward from the source


to innity and not the other way around This requirement is formally called the Sommerfeld radiation condition and is frequently applied in wave propagation problems

The Sommerfeld condition applies equally in nonspherically symmetric problems inthose cases the separation into incoming and outgoing radiation may be a little morecomplicated It is important to remember that the radiation condition only applies toradiation which is truly coming in from innity we cannot apply it inside of a regionwhere incoming energy might legitimately appear as a result of distant scattering unlessof course we wish to neglect that scattering

A rigorous statement of the Sommerfeld condition is this let p be a solution of theHelmholtz equation outside of some nite sphere Then it is required for a nite constantK that

jRpj K

R R ik p

uniformly with respect to direction as R where R is the radius of the sphere Inmany circumstances it is possible to show a priori that solutions of the Helmholtz equationautomatically satisfy this condition So it is not too risky to assume that it applies

Harmonic Waves

At a xed point in space r a wave disturbance is a function only of time

r t &t

Because they can be used to synthesize completely general disturbances an especiallyinteresting case is when & is periodic

&t P cos t

Here P is the amplitude and t is the phase The number of vibrations per second isthe frequency f The frequency is also the reciprocal of the period f T youcan readily verify that & remains unchanged when t t T

The most useful harmonic wave is the harmonic plane wave

r t P cos "k r t#

The set of harmonic plane waves for all frequencies is complete Thus any travelingwave can be built up out of a sum of plane waves For more details on this point see "#

If we allow the amplitudeP to be a function of space then the surfaces of constant amplitude no longernecessarily coincide with the surfaces of constant phase In that case the wave is called inhomogeneous

REFLECTION

Displacement X

speed = c

Figure A D pulse propagating towards the origin at speed c

Calculations with plane waves are simplied if we use complex notation Instead ofEquation we write

r t Re "P exp k r t#

As long as all the calculatons involved are linear we can drop the Re keeping in mindthat it s the real part of the nal expression that we want Nonlinear operations docome up however for example when computing the energy density we must square thedisplacement In such cases one must take the real part rst before performing thenonlinear operation

Reection

The imposition of boundary conditions at various surfaces in the medium generally givesrise to additional reected wave elds In elasticity boundaries also act in generalto convert one type of wave into another upon reection Reected energy plays anessential role in many important wave propagation phenomena

We will here examine only a simple case Suppose we have a semiinnite medium endingat x At the boundary surface x we impose the boundary condition Finally let

fx ct

be an incoming wave that is coming from the right as in Figure What is thesolution to the wave equation which satises the boundary conditions

We know that the general form of the solution of the D wave equation is

x t fx ct gx ct

where f is the known incoming wave and g is the unknown outgoing The boundarycondition has the form

t


0

1

2

3

4

5 2

4

6

8

-1

-0.5

0

0.5

1

0

1

2

3

4

5 2

4

6

8

-1

0.5

0

.5

1

Amplitude

Displacement

Time

Figure A pulse incoming from the positive x direction and traveling with constantunit speed strikes a perfectly reecting barrier at the origin at t

or

fct gct

gct fct

So

x t fx ct fct x

The rst term is just energy traveling to the left the second is energy traveling to theright Notice the reected wave has the same form as the incident wave but is of oppositesign obviously that is because the boundary condition we have applied requires the twowaves to exactly cancel at x The argument of fx is also reversed in sign in thereected wave this is because the spatial shape of the reected wave is the reverse of thatof the incident wave

We have illustrated these ideas with a simple numerical example in Figure whichshows an incoming Gaussian pulse bouncing o a perfectly reecting boundary at t

Phase Velocity Group Velocity and Dispersion

The next topic is somewhat outside of the subject of simple wave solutions to the scalarwave equation It is however one that we must deal with and now is as good a point asany

PHASE VELOCITY GROUP VELOCITY AND DISPERSION

Suppose that we have a system governed by a wave equation which diers from theobject of our current fascination in a rather strange way We suppose that this systemstill supports plane harmonic waves of the form

eitkr

where k is any vector but now we have

fk

where fk is some as yet unspecied function In our previous case

cjkj

We want to study the properties of wave motion in this system The dependence of onk is called dispersion for reasons that will emerge

This unusual behavior is not at all unusual and is more than just a fanciful assumptionMany types of wave propagation are noticeably dispersive including almost everything ofinterest to seismologists except perhaps body waves

Model A

We will rst construct a relatively uncomplicated model of the propagation of wave energyto see if we can infer some of the phenomena associated with dispersion Consider thesimultaneous propagation in the %x direction of two plane waves We have

x t eitkx eitkx

where fk fk

For simplicity we have assumed depends only upon jkj Now take

k k k k

where k

Then U U

where

fkU kf

So

eitkxeiUtx eiUtx

eitkx cosfx Utg


Figure Modulated sinusoid

The rst factor in is just the mean plane wave wiggling o to innity Because ofinterference between the two components however the amplitude of the wave is multipliedby a second factor which varies much more slowly in space since k Schematicallythis is shown in Figure which is a modulated version of the mean plane wave

The zerocrossings of the composite wave travel with the speed

C

k

whether the wave is modulated or not This quantity is called the phase velocity Themodulation peaks and troughs however travel with the speed

U

k

which is called the group velocity The modulation envelope controls the local energydensity of the wave where the bar denotes complex conjugation Hence we associategroup velocity with the speed of energy transport Notice that both C and U depend onlyupon k and not upon the wavenumber separation of the two interfering waves Thewavenumber separation does not control the modulation wavelength

Observe that when ck then

C U c

In this case both phase and group velocity are equal to the velocity appearing in thewave equation This condition is true for elastic waves propagating in an unboundedhomogeneous medium

This example serves more as a demonstration than a proof The result we derived indicates a sinusoidal modulation envelope which itself extends innitely far in both directionsand we might fairly wonder if our interpretation of the results is unfounded or at leastsuspect


Such suspicion while commendable would be in error The trouble with this demonstration lies in our use of only a nite number of interfering waves with a nite set two inthis case it is impossible to construct a spatially limited interference product

For a more precise statement we now present Model B If the following is too involvedfor you take heart in the fact that the answers are virtually the same as given aboveModel A is all that you really need

Model B

Construct a spatially bounded wave using a wavenumber integral over K the space ofwavenumber vectors

r t ZKkeitkrdvk

where fk All we are doing is adding up a continuum of plane waves since eachof these satises the wave equation by hypothesis so does the sum

In particular we take k to be a threedimensional Gaussian distribution centered aboutk

k a

eakk

Neakk for short

We have selected a Gaussian distribution because it is bandlimited in a practical senseand because its inverse transform is spatially limited the inverse is also Gaussian in thesame sense So

r t NZKea

kkeitkrdvk

Since k is eectively constrained to a region around k the size of which we controlby selecting a let us linearize around k

k k k U where

kU rk k

r t NZKea

ikriritiUtdv

r t NeitkrZKea

iriUtdv

r t NeitkrZKea

iPdv

where P Ut r


Call the integral HP Convert it to spherical polar coordinates by taking P asthe pole and let cos

HP Z

d

Z

ea

iPd

P jPj

Since Z

eiPd

PsinP

HP

P

Z

sinP ea

d

HP

P

Z

ieiP ea

d

Substitute

q a

iP

a

a

q iP

a

to get

HP

P

aeP

aZ

q iP

a

eq

dq

HP ia

peP

a

since Z

qeq

dq

So

r t iN

apeitkreUtra

This is the threedimensional version of our earlier result Note that U rkf is a vectorquantity and need not be parallel to k

Dispersion on a Lattice

This section may appear at rst to be a little o the beaten path but in fact it holdsthe key to a pervasive kind of error which will crop up when we consider nite dierencemethods of migration When we attempt to solve the wave equation numerically bynite dierences we immediately discover that any nite approximation to the continuumequations introduces an articial dispersion identical to the dispersion just discussed


This is a result of the fact that a nite dimensional system has only nitely many degreesof freedom and that any attempt to propagate energy of a wavelength smaller thanthe grid spacing will result in that energy being dispersed amongst the longer wavelengthcomponents of the solution Since it is important to recognize that this intrinsic drawbackto nite dierence approximation is indistinguishable from true physical dispersion in adynamical system we introduce the concept here in terms of wave propagation on a Dlattice which we use as a nite dimensional approximation of a continuous string iea D medium We will see that direct appeal to Newton s laws of motion gives riseto precisely the nite dierence equations that we will study in detail later and that thedispersion relation follows readily from these discrete equations of motion Further wewill show that as the wavelength of the disturbance propagating on the lattice gets largerand larger the disturbance no longer sees the granularity of the medium and propagatesnondispersively

The lattice will be populated with points of mass m uniformly distributed along the lineThe position of the ith particle will be xi whereas its equilibrium positions will be XiThe displacement from equilibrium is

ui xi Xi

The rst step is to consider small displacements from equilibrium If we denote theequilibrium energy of the system by V V X and expand V in a Taylor series aboutX retaining only terms to second order we have

V V

Xl

XlVllulul

where the sums are over all lattice elements and where

Vll

V

xlxl

X

We can reference the state of the system to its equilibrium by dening the interactionenergy U V V

U

Xl

XlVllulul

On the other hand the kinetic energy is

T

mXl

'ul

Since rotations are irrelevant in D the only conservation principles that we requireare invariance of the interaction energy under changes of particle numbering and rigidtranslations The particle numbering reects both a choice of origin and a direction of


increasing particle number that is l lp and l l Thus we require that the energysatisfy

Vll Vll Vll

This together with the requirement that rigid translations ul not change the energyimplies that

Xl

Xl

Vll

which in turn implies that

XlVll V

Xl l

Vll

Since we already have convenient expressions for the kinetic and potential energies theequations of motion follow directly from the EulerLagrange equations

d

dt

L

'ul L

ul

where L T U Substituting in the computed expressions for the kinetic and potentialenergies one readily veries that the equations of motion are

m(ul Xl

Vllul

We can put Equation in a more transparent form by rst putting Vll Vll inEquation Then using Equation which allows us to add and subtractP

l Vllul and changing the variable of summation to p we are left with

m(ul Xp

Vpulp ul ulp

By a similar argument one can show that the quadratic potential energy is

U

Xl

Xp

Vpulp ul

The interpretation of equation Equation is very simple since it represents forany p say p i the elastic force on the lth mass due to mass points symmetricallyplaced at l p and l p and with spring constants Vp It also gives us a very simple wayof quantifying the range of the molecular interactions Retaining only terms involving Vyields a model with nearest neighbor interactions Going to p gives nearestplusnextnearest neighbor interactions and so on


The plane wave dispersion relation for the perfect D lattice is obtained by substitutingthe trialsolution

ul Ueikalt

into Equation where a is the lattice spacing The calculation is straightforwardand Equation implies that

m ul ulXp

Vp" coskap#

This equation can be simplied somewhat by using the trigonometric identity cosA sinA Thus we arrive at the dispersion relation

m Xp

Vp"sinkap#

The simplest case to handle is that of nearest neighbor interactions Vp p where ijis the Kronecker delta function In which case

r

msinka

Owing to periodicity and symmetry it suces to consider the range of wavenumbers k

a the socalled rst Brillouin zone The group velocity is

vg d

dk

a

max coska

where max is simply q

m

Thus the maximum speed with which information can propag

ate on the lattice is

vmax max vg a max

a

r

m

Finally for a nite lattice we must impose boundary conditions at the endpoints Supposewe have N oscillators running from l to l N The simplest case is if we x theendpoints In which case the jth normal mode is

ujl Cl sin

jl

N

cos jtj N

And its eigenfrequency is

j

r

msin

j

N

It is easy to see that the total energy of each eigenmode is independent of time Aplot of some of these modes for a homogeneous lattice of mass points is given inFigure Notice especially that for the longer wavelengths the modes are pure sinusoidswhile for shorter wavelengths the modes become modulated sinusoids This is anothermanifestation of the dispersive eects of the discrete system


10 20 30 40 50

-0.2

-0.15

-0.1

-0.05

10 20 30 40 50

-0.2

-0.1

0.1

0.2

10 20 30 40 50

-0.2

-0.1

0.1

0.2

10 20 30 40 50

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-0.1

0.1

0.2

10 20 30 40 50

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-0.1

0.1

0.2

10 20 30 40 50

-0.2

-0.1

0.1

0.2

10 20 30 40 50

-0.2

-0.1

0.1

0.2

10 20 30 40 50

-0.2

-0.1

0.1

0.2

10 20 30 40 50

-0.2

-0.1

0.1

0.2

10 20 30 40 50

-0.2

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0.1

0.2

10 20 30 40 50

-0.2

-0.1

0.1

0.2

10 20 30 40 50

-0.2

-0.1

0.1

0.2

10 20 30 40 50

-0.2

-0.1

0.1

0.2

10 20 30 40 50

-0.2

-0.1

0.1

0.2

Figure A sample of the normal modes free oscillations of a homogeneous pointlattice with xed ends The lower frequency modes are purely sinusoidal the higherfrequency modes become modulated sinusoids as a result of the dispersive eects of thisbeing a discrete system


The Continuum Equations

It is always useful to have the contimuum limit of a given set of lattice equations Toarrive at the continuum equations for the nearest neighbor problem we simply replace ulby uxl in the equations of motion Equation but with Vp p and performTaylor series expansions of the terms about xl

ul uxl a

uxl u

xla

u

xla

Keeping only terms of second order or less the p term of Equation gives

mu

t a

u

xl

The last step is to approximate a density function by averaging the point mass overadjacent lattice sites m

a in which case Equation becomes

u

t

a

u

xl

from which we can identify Young s modulus as E a

The continuum dispersion relation is likewise computed by substituting a plane wave intoEquation with the result

k

sE

And the continuum group velocity is simply

d

dk

sE

a

r

m

which agrees with the lattice theoretic result of Equation at small wavenumbersThis is precisely what one would expect since at small wavenumbers or large wavelengthsa propagating wave does not see the granularity of the lattice The departure of the latticedispersion relation from the continuum is shown in Figure

Summary of Dispersion

Let us briey summarize the new eects associated with dispersive wave propagation


0.5 1 1.5 2 2.5 3

0.5

1

1.5

2

2.5

3

w

q

Figure Discrepancy between the dispersionfree string and the nite dimensionaldynamical system used to approximate it on a computer The lattice spacing was chosento be unity

If the medium is dispersive the phase velocity is dierent for each frequency component As a result dierent components of the wave move at dierent speeds andtherefore tend to change phase relative to one another

In a dispersive medium the speed with which energy ows may dier greatly fromthe phase velocity

Although we haven t talked about attenuation in a dissipative medium a pulse willbe attenuated as it travels with or without dispersion depending on whether thedissipative eects are sensitive to frequency

A Quantum Mechanical Slant on Waves

What do we mean when we say that a pulse arrives at a certain time Following "#let us consider a probabilistic approach to this question inspired by quantum mechanicsThe eikonal equation

r)r sr

describes the high frequency propagation of seismic waves here ) is the wavefront and sis the slowness In seismic applications we almost always treat the depth z as a preferredcoordinate since our media are vertically stratied for the most part So let s think of zas being more of a parameter of motion and rewrite the eikonal equation as

)

z Hx y px py z

where px )x and py )y and H is given by

Hx y px py z hsx y z px py

i

A QUANTUM MECHANICAL SLANT ON WAVES

H is called the Hamiltonian and Equation is an example of what is known inphysics as a HamiltonJacobi equation A good reference for this sort of thing is Chapter of Goldstein s book Classical Mechanics "#

Since the Hamiltonian has been obtained from the eikonal equation it is clear that it givesa representation in terms of rays The classical mechanical analog of rays is particlesNow when we make the transition from the classical mechanics of particles to the quantummechanics of wave packets we interpret the generalized momenta px and py in the Hamiltonian as operators

px i h

x

py i h

y

H i h

z

where h is Planck s constant except that in the mechanics problem we would be thinkingin terms of time as the parameter not depth These operators are interpreted strictlyin terms of their action of some wave function & so that for example when we write acommutation relation such as

"px x# ih

this is really shorthand for

"px x#& fpxx& xpx&g i

h

f& x& x&g i

h

&

In our case we can identify Planck s constant with one over the frequency as the frequency goes to innity we must recover classical ie raytheoretic physics So we have

px i

x

py i

y

H i

z

If we plug these operators into the denition of the Hamiltonian we arrive at none otherthan the Helmholtz equation

r& k& k s

But now we re in business because we can interpret the solution of the wave equationprobabilistically as in quantum mechanics For example at a xed point in space we can


normalize the wave function & as

Z

&& dt

Then the function j&j can be interpreted as the probability density of the arrival of awavefront at a time t This allows us to make a rigorous wavetheoretic concept of arrivaltime since now we can say that the expectation value of the arrival time is

hti Z

&t& dt

So naturally we want to identify the width of a pulse with its standard deviation

Z

&t hti& dt

Let us therefore dene the arrival time of a pulse as

ta hti

Well it looks plausible on paper but does it work In Figure we show some syntheticwavelets These are just the rst lobe of a Ricker wavelet perturbed by uniformlydistributed noise On the right of each plot is the average time and arrival time asdened above We will leave the plausibility of these numbers to your judgement Ofcourse there is nothing magical about one standard deviation We could use a dierentmeasure say three standard deviations on the grounds that this would contain almost allof the energy of the pulse The main point is that Equation is a quantitative andeasily computed measure

As usual real data are somewhat more problematic In Figure we show a trace takenfrom the Conoco crosshole data We looked around for one that looked especially cleanOn the right of the gure you see a zoomedin view around the rst break Hard to tellfrom this view so we just squared the trace amplitude Figure shows a further zoomof the squared trace Here the rise in amplitude associated with the arrival is relativelyeasy to pick out The quantum mechanical calculation gives an arrival time of around in this particular picture For more details of this quantum mechanical approach see"#

Exercises

What is the physical signicance of the fact that shear waves are dilationfree rs

Do shear waves ever propagate faster than compressional waves If not why not

Derive the wave equation in spherical coordinates r Then specialize to thecase of a spherically symmetric medium

A QUANTUM MECHANICAL SLANT ON WAVES

10 20 30 40

0.002

0.004

0.006

0.008

0.01

0.012

0.014

<t> = 32 arrival time = 28

10 20 30 40

0.002

0.004

0.006

0.008

0.01

0.012


10 20 30 40

0.002

0.004

0.006

0.008

0.01

0.012

0.014


Figure Synthetic Ricker lobes with noise added

0 100 200 300 400 500

-0.002

0

0.002

milliseconds

ampl

itude

80 90 100 110 120 130 140 150

milliseconds

-0.002

0

0.002

milliseconds

ampl

itude

Figure Left A trace taken from the Conoco crosshole data set Right An enlargedview of the portion containing the rst arrival


2.5 5 7.5 10 12.5 15

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Figure Enlarged view of the squared trace Don t pay attention to the xaxislabeling The yaxis represents the normalized amplitude

In Chapter you computed the Fourier transform of a Gaussian Imagine that thisGaussian is the initial state of a pulse traveling down the xaxis ux Comparethe width of jux j call this $x with the width of its Fourier transform squaredjAkj call this $k What conclusions can you draw about how $x and $k arerelated

Show that ifr t f%n r ct

then

r %nf

r r %n %nf

f

and

t cf t cf

Show that the peaks and troughs of the modulated sinusoid propagate with the speedU k

Chapter

Ray Theory

The Eikonal Equation

We will concentrate on the canonical scalar wave equation

t cr

where is some potential We Fouriertransform from time t to angular frequency so that

t i

and we have

cr

Planewave solutions for transformed are of the general form

Pei

) k r

jkj

c

for the case where c the wave speed is everywhere constant

Now suppose that c varies with position but slowly enough that we can use the waveequation

crr

This is called the Helmholtz equation Basically such a supposition amounts to neglectingthe coupling terms involving things like rc or r which most exact wave equationspossess In general as frequency increases we expect this approximation to become

CHAPTER RAY THEORY

increasingly valid Dene the index of refraction with respect to some arbitraryreference speed c by

r ccr

Now we will investigate the possibility that may have possibly approximate solutionsof the form

P reir

where P r is a real slowlyvarying wave amplitude and )r is a real generalizedphase Remember that we are in the frequency domain we regard as xed of course

We plug our assumed form of into the wave equation and see what comes out

r rP ei ir)Pei

r r r

rP ei irP r)ei

ir)Pei r) r)Pei

so

Pei crP r) r)P ei

icrP r) Pr)ei

If P ) satisfy this equation then as specied above satises the wave equation tobe more precise the approximate wave equation when c is a slowlyvarying function ofposition Equating real and imaginary parts individually we get

P crP cPr) r)

or

r)

crP

P

for the real part andr) r log P r)

for the imaginary

We now suppose that

r) rP

P

to such an extent that the second term in Equation is negligible When c is aconstant in space the P is also a constant and rP is exactly zero while r) is exactlyequal to c When c is no longer constant but still reasonably smooth neglectingrPP is of course an approximation We justify this by claiming that at sucientlyhigh frequencies the wavenumber of the solution is much much greater than the rate atwhich wave amplitude or energy varies in space There are two important features ofthis approximation

THE EIKONAL EQUATION

The approximation in general improves as frequency increases In the limit of unbounded frequency it becomes arbitrarily good Because of this feature we sometimes call this approach asymptotic ray theory

At any nite frequency there are always cases in which ray theory fails We mustalways be careful in exploiting this technique

The equation

r)

c

is called the eikonal equation for the Greek word * meaning image The solution) is sometimes called the eikonal The solutions ) to this nonlinear partial dierential equation are the phase surfaces associated with highfrequency wave propagationthrough a medium with at worst a slowlyvarying wave speed

Note that r) is essentially the local wavenumber To see this we neglect possiblevariation in P and expand

r P expif)r g

P expif)r r)rg

Basically the eikonal equation is telling us how the local solution wavenumber is relatedto the local and only the local material properties

Equation is called the transport equation since once we solve the eikonal equationfor the phase we can compute the slowly varying change in amplitude which is due togeometrical spreading

It s worth pointing out that we could have proceeded in an entirely equivalent derivationof the eikonal equation by dening the phase in Equation relative to the frequencyie use use eik

r instead Plugging this ansatz into the Helmholtz equation we wouldhave seen several terms with a k in front of them Then in the limit that the frequencygoes to innity only these terms would have been nonnegligible

In fact Equation is the rst in an innite series of transport equations obtained by makingthe more general ansatz

eirp pk pk

It is not obvious that this expression is legitimate either as a convergent or an asymptotic series Nevertheless experience is overwhelmingly in its favor For more details see

CHAPTER RAY THEORY

The Dierential Equations of Rays

Consider the surfaces associated with constant values of the eikonal The normal %n tothese surfaces are given by

%n r)

jr)j r)

kwhere k

c

We dene a trajectory by the locus of points r where is arclength such that

dr %nr

We call this trajectory a ray We will see later why we attach so much signicance tothe ray Right now we wish to nd a way to construct the ray without having to directlysolve the eikonal equation We do this as follows

dr %n r)

k

dr

kr)

ddr

kdr)

Since

d %n r

ddr

k%n rr)

kr) rr)

k

rr) r)

why

krk

soddr r

ord%n r

This equation describes the precise manner in which a ray as we have dened it twistsand turns in response to the spatial variation of the wave speed c Imagine starting atsome initial point r and in some initial direction %n Then in the presence of someparticular r this equation tells us how to take little local steps along the unique rayspecied by r %n

FERMATS PRINCIPLE

From Equation it is obvious that if the index of refraction or velocity or slownessis constant then the equation of a ray is just dr which is the equation of a straightline Notice that once the rays have been found the eikonal can be evaluated since itsvalues at two points on a ray dier by

Rd Finally we note that the curvature of a

ray is dened to be dr But this must point in a direction normal to the ray and in thedirection of increasing curvature If we dene a new unit normal vector along the raypath % then the radius of curvature is dened by the equation

dr

%

Using the ray equation Equation and the eikonal equation it can be shown that

% r log

which shows that rays bend towards the region of higher slowness or index of refractionin optics Also notice that Equation is precisely the equation of a particle ina gravitational potential So in a deep sense geometrical optics is equivalent to theclassical mechanics of particlesFermat s principle of least time being the exact analog ofHamilton s principle of least action about which more in the next section

Fermats Principle

We begin with Huygen s Principle which asserts that as a wave disturbance spreadsthrough a medium we may at each instant regard the points disturbed by the ray atthat moment as a new set of radiating sources To take a very simple example consider apulse from some source which at time t has spread outward to the surface V Huygen sPrinciple as we have already seen says that at any later time the pulse may be found bycomputing the signal from a set of sources appropriately distributed over the disturbedarea contained by V In this form Huygen s Principle is simply a statement of theobvious fact that we can regard the calculation for times later than some moment t asbeing the solution of an initialvalue problem starting at t having initial displacementsand velocities equal to those they actually had at t

Huygen s Principle suggests a rather incoherent jumbled progress of wave energy outwardfrom the source It implies that wave energy traverses every possible path available to itWe on the other hand hope to nd some more orderly scheme of wave propagation

Harmony among these divergent notions comes as follows Consider a medium in whichthe wave speed strictly increases with depth but slowly of course and in which wehave a source and a receiver both at some common depth In Figure we have drawnin three trial paths by which radiated energy might reach the receiver one of them issomewhat fanciful Any continuous piecewise dierentiable curve is obviously a possible

CHAPTER RAY THEORY

Γ

S R

Figure Possible ray paths connecting the source and receiver

path Because of the multitude of unrelated paths we might expect and it is so that thesignals arriving from the vast majority of them will interfere destructively to produce nonet observable eect at the receiver

Now contemplate the path with the distinguishing property that it is the path from sourceto receiver having the shortest timeofight traveltime In our particular case this pathwill not be a straight line from source to receiver but because the wave speed increasesdownward the minimumtime path will dip down somewhat into the higher velocityregion The path is sketched qualitatively as the lowermost path on the gure

Let + symbolically denote this path and let be a small pathvariation which vanishes ateither end We are playing a little fast and loose with algebra here but it will work outall right Symbolically the set of paths

+

for small is a group of paths from source to receiver near to and containing theminimumtime path + Since + is a minimumtime path none of the nearby paths canbe any faster than + and most must be slower For any particular traveltime as afunction of must look like Figure because the bottom of this curve is at as isalways the case for a minimum thus there must be a cluster of paths all having the sametraveltime and which will therefore interfere constructively to produce a signal at thereceiver One of the most important properties of a minimumtime path then is that itdelivers a signicant amount of coherent energy at the far end

Notice that this argument only really required that the minimumtime path be a localminimum Between any pair of points in a given medium there may be a number ofminimumtime in the local sense paths and each should contribute an arrival

Fermat s Principle asserts that a ray is a minimumtime path through the medium Whenwe have shown that this is so we shall have found out why ray theory is useful

We begin by proving Lagrange s integral invariant Since

%n

kr)

FERMATS PRINCIPLE

α

t

Figure The minimum time path is by denition an extremum of the travel time asa function of ray path

d

2

lda

Ωδ Ω

1

Figure Stokes pillbox around an interface between two media Since the shape isarbitrary we may choose the sides perpendicular to the interface to be innitesimallysmall

where k is constant we must have

r %n

and therefore by Stokes theorem using the notation dened in Figure

Zr %n da

Z

%n dl

if r) is continuous and where the integral is around any closed path This result is calledLagrange s integral invariant

We can immediately use this to prove that the ray path connecting two points P and

CHAPTER RAY THEORY

Q

P

P Q

Q

Q

1

Q’

2

1

1 2

2

2 _

_ C C

_ . . . . .

Figure The travel time along the ray path C connecting two points P and P is lessthan for any other path such as C

P is the minimum time path Figure shows a bundle of rays one of which connectsthe two points P and P These rays are intersected at right angles by two surfaces ofconstant phase To get the general result that the length of C is inevitably greater thanor equal to the length of C and hence that the travel time T C is greater than or equalto T C it suces to show that

dl QQ dlQQ

Using Lagrange s invariant on the innitesimal triangle dened by QQQ we haveZ

QQQ

%n dl %n dlQQ %n dlQQ

dlQQ

The last term doesn t involve any dot product because along the segment QQ the line

element dl and the tangent to the ray the normal of the isophasal surface %n are parallelSimilarly along the path QQ

the line element is perpendicular to the ray Therefore

%n dlQQ

Thus we have

%n dlQQ dlQQ

Now for any path%n dlQQ

dlQQ

ThereforedlQQ

dlQQ

dl QQ

The last equality holds because the phase dierence between two isophasal surfaces alongany two raypaths must be the same Thus we have shown thatZ

Cdl

ZCdl

This discussion comes straight out of Born and Wolf Nota bene however the typo in theirEquation of Section

FERMATS PRINCIPLE

and assuming constant speed T C T C Fermat s principle

Also note that although the equality strictly holds only when both paths are the same theamount of inequality when C is not the same as C depends upon the amount by which

%n dlQQdiers from jdljQQ

As you can easily show when %n and dl dier in directions by only a small angle thatis when C diers only slightly in direction from the local ray then

%n dlQQ jdljQQ

so the inequality is only aected to second order in That is why the set of paths closeto a ray path all have the same traveltime to rst order in the deviation from the raypath

Boundary Conditions and Snells Laws

When we derived Lagrange s invariant the region , was completely arbitrary Howeverlet s consider the case in which this region surrounds a boundary separating two dierentmedia If we let the sides of the Stokes pillbox perpendicular to the interface go tozero then only the parts of the line integral tangential to the interface path contributeAnd since they must sum to zero that means that the tangential components must becontinuous

%n %ntangential

where the subscripts and refer to a particular side of the boundary This is equivalentto saying that the vector

%n %n

must be normal to the boundary

Now imagine a ray piercing the boundary and going through our Stokes pillbox as shownin Figure If and are the angles of incidence and transmission measured fromthe normal through the boundary then continuity of the tangential components meansthat sin sin which is called Snell s law of refraction Similarly one can showthat in the case of a reected ray the angle of incidence must equal the angle of reection

Strictly speaking the above argument is not legal If there are true discontinuities in themedium then the conditions under which the ray approximation are valid slow variationin material properties relative to the wavelength of the ray are violated To be precise weneed to consider the boundary conditions of the full wave equation It turns out that wewill get the same result so we can wa-e a little bit and say something to the eect thatwhat we are really talking about are plane waves refracting!reecting at the boundary

CHAPTER RAY THEORY

d l2

δ Ω

^

ray path

n

θ

θ1

Figure Geometry for Snell s law

FERMATS PRINCIPLE

r

θt

θi

θ

Figure Plane wave incident on a vertical boundary gives rise to a transmitted and areected wave which must satisfy boundary conditions

To see that this is true consider a plane pressure wave of unit amplitude i in D incidentupon a vertical boundary at x at some angle of incidence i

i eikcos ixsiniz

Similarly for the reected and transmitted plane waves we have cf Figure for thegeometry

r Ceikcos rxsinr z

andt Ce

ikcos txsintz

Convince yourself that i and r must have the same wavenumber and that all three planewaves must have the same frequency Now in order for the pressure to be continuousacross the boundary we must have

"i r t#x

See Section for discussion of this issue The other relevant boundary condition is continuity ofdisplacement for a welded contact or merely continuity of normal displacement for the contact betweenan inviscid uid and a solid

CHAPTER RAY THEORY

The only way that this can be true for arbitrary z is if k sin i k sin r k sin t Inother words i r and sin t cc sin i

Fermat Redux

We end this section by showing that Fermat s principle of least time leads back preciselyto the ray equations which we derived before by making a highfrequency asymptoticapproximation This discussion is patterned on the one in "# Let us begin by writingthe travel time between any two points A and B along the ray as

t

c

Zray path

r d

where is a dimensionless parameter that increases monotonically along the ray Usingdots to denote dierentiation with respect to path length we have

d q

'x 'y 'z d j 'rj d

In terms of the travel time can be written

t

c

Z B

Ar 'r j 'rj d

c

Z B

Afr 'r d

where f j'rj Fermat s principle that the ray is locally the least time path impliesthat variations in t must be zero

t

c

Z B

A"rrf rr rf 'r# d

Integrating this by parts gives

t

c

Z B

A

rrf d

dr rf

r d

In order that this integral be zero for arbitrary variations r it is necessary that the partof the integral within square brackets be identically zero

rrf d

dr rf

This is the EulerLagrange equation of variational calculus "#

FERMAT REDUX

Using the denition of the function f as j'rj it follows that

rrf j 'rjr

r rf 'r

j 'rj

Finally we observe that d j 'rj d so that the stationarity of t implies

d

d

dr

d

r

which is none other than Equation Thus we have shown that asymptotic ray theoryand Fermat s principle lead to the same concept of rays

Exercises

Using the ray equationddr r

show that in a homogeneous medium rays have the form of straight lines

Show that in a medium in which depends only on depth z the quantity

%z dr

d

is constant along rays This shows that rays are conned to vertical planes and thatthe ray parameter p z sin iz is constant where iz is the angle between theray and the z axis

Suppose that a medium has a spherically symmetric index of refraction or slowness Show that all rays are plane curves situated in a plane through the origin andthat along each ray r sin constant where is the angle between the positionvector r and the tangent to the ray at the point r You may recognize this result asthe analog of conservation of angular momentum of a particle moving in a centralforce

What two independent equations must the amplitude P and the phase ) satisfy inorder that the Helmholtz equation hold

Next show that if rPP

can be neglected in comparison to r) then one of the

two equations you just derived reduces to r)

c the eikonal equation

Use Lagrange s invariant to prove Snell s law of refraction

CHAPTER RAY THEORY

Chapter

Kirchho Migration

The Wave Equation in Disguise

In this rst pass through the Kirchho migration algorithm we will restrict attention tothe constant velocity case and follow more or less Schneider s original derivation "#In later chapters we will discuss generalization to variable velocity media The derivationitself begins with the forward problem for the wave equation Although this discussionwill focus on the acoustic wave equation it applies equally well to the migration of elasticor even electromagnetic scattering data

Suppose that we wish to solve a boundary value problem for the scalar wave equationon the interior of some closed volume , in R with boundary conditions specied on asmooth boundary , We re looking for a function such that

r

ct r ,

The geometry is illustrated in Figure

A complete specication of the problem requires that we specify the initial values of andits normal derivative n in , and either or n on , Specifying the functionon the boundary is called the Dirichlet problem specifying the normal derivatives iscalled the Neumann problem Some dierential equations allow one to specify both typesof conditions simultaneously Cauchy problem but the wave equation does not "# Forstarters we will imagine our sources lying on the free surface of the Earth which is goingto be part of , so that we can use the homogeneous sourcefree form of the waveequation

Now when we use Huygen s principle we are synthesizing a general solution of the waveequation from the cumulative eects of an innite number of point sources This canbe made mathematically systematic by dening an impulse response function or Green

CHAPTER KIRCHHOFF MIGRATION

W

dW

Figure Volume and surface dened in the specication of the wave equation

function which is the solution of the wave equation with a point source term Irrespectiveof the boundary conditions involved we dene a Green function depending on both sourceand observer coordinates +r t r t as the solution of

r+

ct + r rt t

where is the Dirac delta function and r t are the source coordinates

If we multiply this equation by and the wave equation by + and subtract one equationfrom the other we get

r rt t +r r+

c

+t t+

or

r rt t r "+r r+#

ct "+t t+#

To get rid of the delta function we d like to integrate this equality over some spacetimevolume , is an obvious choice for the space part and we ll integrate over all time

rr t Z

Zr "+r r+# dv dt

We have decided to swim upsteam and abandon the awkward and singular Greens function in favorof Green function

THE WAVE EQUATION IN DISGUISE

c

Z

Zt "+t t+# dv dt r R

The lefthand side comes from integrating the delta function The value of r is calculated by a limiting argument and turns out to be if r is in the interior of , if r ison the boundary and otherwise

The rst integral on the right of Equation can be converted to a surface integralover , via the divergence theorem The second integral can be integrated by parts withrespect to time giving a term of the form

Z

+t t+dv

Clearly we can assume that and t are zero until sometime after the source is redo say at t so the lower limit is zero And provided the Green function is causalor assuming a Sommerfeld radiation condition on the upper limit must be zero too

The result of all this is the Kirchho Integral Theorem

rr t Z

Z

"+r r+# n da dt r R

where n is the outward pointing unit normal of ,

The Kirchho Integral Theorem gives the solution of the wave equation everywhere inspace once the values of and its normal derivative are known on the boundary Butsince we cannot in general specify both of these consistently Cauchy conditions we needto solve for one in terms of the other This we do by taking the limit of Equation as the observation point r approaches the surface Using the notation

j f

nj g

we then have for the Neumann problem

fr t

Z

Z

"+g fn+# da dt r ,

So to recap since we can t specify both the function and its normal derivative we mustspecify one and solve for the other If we specify for example the normal derivativeie g then Equation is an integral equation for f If we then solve for f wecan use the Kirchho Integral to compute the solution to the wave equation There isa comparable equation such as Equation for the Dirichlet problem where wespecify f and solve for g

This will come back to haunt us later


So far we haven t said anything about what boundary conditions the Green function mustsatisfy But if you look back over the derivations thus far we haven t assumed anythingat all in this regard The expressions derived so far are true for any Green functionwhatsoever ie any function satisfying Equation Now in special cases it may bepossible to nd a Green function which vanishes on part or all of the boundary in questionor whose normal derivative does If the Green function vanishes on , then we can solvethe Dirichlet problem for the wave eld directly from Kirchho s Integral since the termin that integral involving the unknown normal derivative of the wave eld is cancelledIn other words if the Green function vanishes on the boundary then Equation reduces to

r t

Z

Z

fr tn+r t r t da dt

which is not an integral equation at all since by assumption we know f Similarly if thenormal derivative of the Green function vanishes on the boundary then we can immediately solve the Neumann problem for the wave eld since the unknown term involving itsboundary values is cancelled In either event having a Green function satisfying appropriate boundary conditions eliminates the need to solve an integral equation If we don thave such a special Green function we re stuck with having to solve the integral equationsomehow either numerically or by using an approximate Green function as we will dolater when we attempt to generalize these results to nonconstant wavespeed media

Application to Migration

Consider now the problem of data recorded on the surface z To begin we willconsider the case of media with constant wavespeed so that there are no boundaries inthe problem Of course this is really a contradiction we ll see how to get around thisdiculty later Then we can take , to consist of the plane z smoothly joined toan arbitrarily large hemisphere extending into the lower halfspace If we let the radiusof the hemisphere go to innity then provided the contribution to the integral fromthis boundary goes to zero suciently rapidly the total integral over , reduces to anintegral over the plane z Showing that the integral over the lower boundary canreally be neglected is actually quite tricky since even if we use a Green function whichsatises Sommerfeld the eld itself may not We will discuss this in detail later in PaulDocherty s talk on Kirchho inversion

In the special case of a constant velocity Earth two Green functions which vanish on thesurface z can be calculated by the method of images They are

+rr t r t

t t Rc

R t t Rc

R

and

+ar t r t

t t Rc

R t t Rc

R

APPLICATION TO MIGRATION

Figure A Green function which vanishes on a planar surface can be calculated bythe method of images

whereR

qx x y y z z

andR

qx x y y z z

We won t derive these Green functions here that will be left for an exercise Howeverit is obvious from our discussion of spherical waves that these are spherically symmetricsolutions of the wave equation The subscripts a and r refer respectively to advancedand retarded The retarded Green function is causal propagating outward from its originas t increases The advanced Green function is anticausal It represents a convergingspherical wave propagating backwards in time

Now all we have to do is plug one of these Green functions into Equation Andsince migration involves propagating diraction hyperbolae back to their origin it is clearthat we should be using +a Notice too that Rz Rz so that we can combinethe normal ie z derivatives of the two delta functions with the result for r in thesubsurface

r t

Z

Zz

fr tzt t Rc

Rda dt

And since z appears only in the combination z z

r t

z

Z

Zz

fr tt t Rc

Rda dt


Finally using the properties of the delta function to collapse the time integration we have

r t

z

Zz

fr t Rc

Rda

This is Schneider s Kirchho migration formula "#

This has been a rather involved derivation to arrive at so simple looking a formulaso let s recap what we ve done We started by converting the wave equation into anintegral equation by introducing an arbitrary Green function We observed that if wewere clever or lucky enough to nd a particular Green function which vanished orwhose normal derivative did on the surface that we were interested in then the integralequation collapsed to an integral relation which simply mapped the presumed knownboundary values of the wave eld into a solution valid at all points in space For theproblem of interest to us data recorded at z it turned out that we were able to writedown two such special Green functions at least assuming a constant velocity Earth Thequestion now is what boundary values to we use

We have already observed that CMP stacking produces a function sx y z t whosevalue at each midpoint location approximates the result of a single independent zerooset seismic experiment So the boundary data for the migration procedure do notcorrespond to the observations of a real physical experiment and the solution to theinitialboundary value problem that we have worked so hard on is not an observable waveeld From an abstract point of view we may simply regard the Kirchho migrationformula Equation as a mapping from the space of possible boundary values intosolutions of the wave equation

For poststack migration we identify f in Equation with the CMP stacked datas Then we set c c and t for the exploding reector model And nally weperform the integration over the recording surface x y

depth section r

z

Zz

sr RcR

da

The normal derivative outside of the integral is responsible for what we called the obliquity factor in our discussion of the HuygensFresnel principle Remember R is thedistance between the output point on the depth section and the particular receiver ortrace location on the surface z So as we integrate along the surface z we arereally summing along diraction hyperbolae in the zerooset data This is illustrated inFigure

Equation can also be viewed as a spatial convolution which downward propagatesthe recorded data from one z level to the next Mathematically there is nothing specialabout z it just happens to be where we recorded the data So if we were facedwith a layered medium one in which the velocity were constant within each layer wecould use this integral to downward propagate the recorded data one layer at a time

APPLICATION TO MIGRATION

(x,z)

R

x

z t

x

x x

z t

.

.

point scatterer

zero-offset section migrated image

t=2R/c

diffraction hyperbola

Figure A point scatterer in the Earth gives rise to a diraction hyperbola on the timesection Kirchho migration amounts to a weighted sum along this hyperbola with theresult going into a particular location on the depth section

Since the Fourier transform of a convolution is just the product of the Fourier transformsit makes sense to recast this downward propagation operation in the spatial Fourier orwavenumber domain In the appendix to his paper "# Schneider shows that the Fouriertransform of Equation can be written

kx ky z $z kx ky z Hkx ky$z

where

H eiz

qc

kxky

This means that for constant velocity media downward propagation is purely a phaseshift operation

Another commonly seen form of the Kirchho migration procedure is obtained by performing the z dierentiation Using the chain rule we nd that

r

Zz

cos

Rc

ts

c

Rs

tRc

da

where is the angle between the z axis and the line joining the ouput point on the depthsection x y z and the receiver location x y Because of the R in the denominatoryou will often see the second term inside the brackets ignored

All of these formulae for Kirchho migration can be interpreted as summing along diffraction curves We look upon every point in the depth section say r as a possible point


diractor Since we presumably know the velocity above this point we can nd the apexof its hypothetical diraction hyperbola Then we sum along this curve according to anintegral such as Equation and deposit the value so obtained at the point r ButKirchho migration is signicantly more accurate than simply summing along diractionhyperbolae even in constant velocity media because of the presence of the obliquityfactor cos and the derivative of the data

When the wavespeed c is not constant the Kirchho formulae must be generalized Essentially this amounts to replacing the straight ray travel time Rc with a travel timeaccurately computed in the particular medium

t T x y zx y z

or what amounts to the same thing using a Green function valid for a variable velocitymedium We will discuss this later in the section on Kirchhotype inversion formulae

Summary of Green Functions

The freespace Green functions for the Helmholtz equation

rGkr r kGkr r

r r

are

Gkr r

eikjrrj

jr rj

iH k jr rj

i

keikjxx

j

in respectively and dimensions where H is the zeroth order Hankel function

of the rst kind

The freespace Green functions for the wave equaton

rGr t r t

ctGr t r t r rt t

are

Gr t r t

jr rj "jr rj c t t#

cq

ct t jr rj.

t t jr rj

c

c.

t t jx xj

c

SUMMARY OF GREEN FUNCTIONS

in respectively and dimensions where . is the step function .x for x and .x for x For more details see Chapter of Morse and Feshbach "#

Exercises

In this extended exercise you will derive following "# the freespace Green function for both the Helmholtz equation and the wave equation First we deneGr r k to be a solution to the Helmholtz equation with a point source

r kGr r k r r

Now show that if there are no boundaries involved G must depend only on r r

and must in fact be spherically symmetric Dene R r r and R jRj Nextshow that the Helmholtz equation reduces to

R

d

dRRG kG R

Show that everywhere but R the solution of this equation is

RGR AeikR BeikR

The delta function only has inuence around R But as R the Helmholtzequation reduces to Poisson s equation which describes the potential due to pointcharges or masses From this you should deduce the constraint A B Theparticular choice of A and B depend on the time boundary conditions that specifya particular problem In any event you have now shown that the freespace Greenfunction for the Helmholtz equation is

GR AeikR

R B

eikR

R

with AB It is convenient to consider the two terms separately so we dene

GR eikR

R

Now show that the timedomain Green function obtained from the Fourier transform

GR t t GR

Z

eikR

Rei d

where t t is the relative time between the source and observation point Nowwhat is the nal form in the time domain of the Green functions Gr t r t Gr r t t Which one of these is causal and which one is anticausal andwhy

For more details the interested reader is urged to consult the chapter on Greenfunction methods in Volume I of Morse and Feshbach "#


Go through the same derivation that leads up to Schneider s constant velocity migration formula but use the D Green function given above You should end upwith a migrated depth section that looks like

x z

z

Zdx

ZRc

sx tqt Rc

dt

This result is often written in the following fareld form

x z Z cos p

Rctsx Rc dx

where t is a fractional derivative The derivation of this is simplest in the fre

quency domain where we can use the far eld approximation to the Hankel function

H kr

s

/krdikr

When we Fourier transform this back to the time domain thep that results from

applying this asymptotic expression ca be interpreted as a a halfderivative operator

Fractional derivatives and integrals are dened in a purely formal fashion usingCauchy s theorem for analytic functions Let f be analytic everywhere within andon a simple closed contour C taken in a positive sense If s is any point interior toC then

fs

i

ZC

fz

z sdz

By dierentiating this expression n times we get

f ns n

i

ZC

fz

z sndz

Now the expression on the right is welldened for nonintegral values of n providedwe replace the factorial by the Gamma function Therefore we identify again purelyformally the expression on the left as the fractional derivative or fractional integralwhen n is negative of the indicated order

The Gamma function z is dened via the integral

z

Z

tzet dt

It can be shown that and z zz The Gamma function is analytic everywhere exceptat the points z where it has simple poles For more details see

SUMMARY OF GREEN FUNCTIONS

A pointsource in D is equivalent to a linesource in D Therefore in an innitehomogeneous medium the timedomain D Green function can be obtained byintegrating the D Green function along a line as follows

gp tp t Z

"Rc t t#

Rdz

where p and p are radius vectors in the x y plane and

R q

x x y y z z

You should expect that g is a function only of P jp pj and t t

Show that gP cpc P if P c and gP otherwise Hint

introduce the change of variables z z Then

R P ddR R

These will allow you to do the integration over the R coordinate so that the deltafunction can be evaluated

In cylindrical coordinates z the Laplacian is

rf

f

f

f

z

Assuming an innite ie no boundaries constantvelocity medium Show thataway from the source and at suciently low frequency the frequencydomain ieHelmholtz equation Green function is proportional to log

Schneider s Kirchho migration formula is

depth section

z

Zz

sr t Rc

Rda

where R q

x x y y z z s is the stacked data and the integral

is over the recording surface Show that in the far eld ie when R is large thedepth section is proportional to

Zz

cos

R"ts#tRc da

Chapter

Kirchho MigrationInversion

We will begin by looking at Claerbout s heuristic imaging condition and show that this isequivalent to the formulae derived in the theory of Kirchho inversion Inversion meansthat we provide a geometrical image of the reectors as well as quantitative estimates ofthe reection coecients A source is excited at some point xs Let uI denote the incidenteld and uS the scattered eld Suppose we know the scattered eld at some depth pointr as illustrated in Figure Claerbout s imaging condition is that the migrated imageat r is given by

mr

Zd F

uSrxS

uIrxS

where F is a lter emphasising the bandlimited nature of the scattering eld

This looks as if we are deconvolving the scattered eld by the incident eld If the mediumis slowly varying in the WKBJ sense then we can approximate

uI AIrxseiIrxs

This chapter is based on talks that Paul Docherty gave before the class in and which wereadapted from his paper

S

r

x s

D D D D D D D D

Figure The incident eld reects o a boundary at some depth The genericobservation point is r

CHAPTER KIRCHHOFF MIGRATIONINVERSION

where I is the travel time from the source to the observation point r In this case wehave

mr

AI

Zd F uSrxS eiIrxs

This has the form of a Fourier transform and amounts to evaluating the scattered eldat a time I If we continue along these lines and use the WKBJ approximation for thescattered eld then we have

mr AS

AI

Zd F eiIrxsSrxs

But this integral is just a delta function bandlimited by the presence of F Denoting thisby B and identifying the ratio AS

AIas the reection coecient we have the fundamental

resultmr RrBI S

The interpretation of this expression is clear it is nonzero only where the reectioncoecient is nonzero and then only when I S The last is because of the bandlimited nature of the deltafunction The peak value certainly occurs when I S andis given by

mpeakr

Rr

Zd F

or

mpeakr

Rr Area of lter F

Now we can pretend we know AI and I but how do we get the scattered eld at depthThe answer is using Green functions We will proceed in much the same way as in thepreceeding chapter except that now we take explicit account of the fact that the velocityis not constant We will denote the true wavespeed by vx it will be assumed that vis known above the reector We will introduce a background wavespeed cx whichis also nonconstant but assumed to be known and equal to v above the reector Bydention the total eld u satises the wave equation with v

r

v

uxxs x xs

whereas the incident eld satises the wave equation with the background wavespeed

r

c

uIxxs x xs

For this hypothetical experiment we can make the source whatever we want so we ll makeit a deltafunction This is without loss of generality since we can synthesize an arbitrarysource by superposition

The basic decomposition of the total eld is

uxxs uIxxs uSxxs

When we talked about the planewave boundary conditions in the chapter on rays wedecomposed the eld into an incident and scattered or reected eld on one side of theboundary and a transmitted eld on the other side of the boundary Here we are usinga dierent decomposition neither more nor less validjust dierent Below the reectorthe scattered eld is dened to be the total eld minus the incident eld We can imaginethe incident eld being dened below the reector just by continuing the backgroundmodel c below the reector This is a mathematically welldened concept it just doesn tcorrespond to the physical experiment As we will see though it let s us get to the heartof the migration problem

Introduce a quantity which represents in some sense the perturbation between thebackground model and the true model

cx

vx

so that

c

v

c

Since c v above the reector is zero in this region Whereas is nonzero below thereector In terms of the equation for the scattered eld can be written

r

c

uSxxS

cuxxS

Now let us introduce a Green function for the background modelr

c

Gx r x r

As usual we convert this into an integral relation via Green s identity

uSrxS ZVGx r

cuxxS dV

ZS

uSxxS

Gx r

nGx r

uSxxS

n

dS

At this point we have complete exibility as to how we choose the volume!surface ofintegration We also have yet to decide which Green function to use For example itcould be causal or anticausal We have a great variety of legitimate choices all givingrise to dierent formulae Some of these formulae will be useful for migration and somewill not We have to see essentially by trial and error which ones are which


S

r

x

S

S

0

0 0

s

D D D D D D D D

Figure Dene the surface S to be the union of the recording surface S and a surfacejoining smoothly with S but receding to innity in the z direction

Case I Causal G S S S

Let S be the union of S the data recording surface z and a surface S which issmoothly joined to S but which recedes to innity in the z direction Figure

First we observe that the surface integral over S must be zero To see this applyGreen s identity to the volume enclosed by S S S the corresponding surfacegoing to innity in the upper half plane Then in the volume enclosed by S is zeroand the scattered eld is zero too since the Green function has no in support this domainTherefore the total surface integral

ZSS

uSxxS

Gx r

nGx r

uSxxS

n

dS

is equal to zero

But as S goes to innity the elds G and u must go to zero by virtue of the Sommerfeldradiation condition since in this case both G and u are causal Therefore the surfaceintegral over S must be zero itself So now if we use S S S since the surfaceintegral over S must be zero by the same Sommerfeld argument we have that

uSrxS ZVGx r

cuxxS dV

This expression requires that we know the total eld uxxS and everywhere withinthe volume V If we approximate the total eld by the incident eld the socalled Bornapproximation then we have

uSrxS ZVGx r

cuIxxS dV

CASE II CAUSAL G S S

which is a just a linear function of the unknown perturbation In other words we haveus L where L is a linear function So by choosing a causal Green function and asurface extending to innity in the z direction we end up with a Born inversion formulafor the unknown wavespeed perturbation

Case II Causal G S S

If we now take the volume V to be the area bounded by recording surface and the reectingsurface then the volume integral goes away since is zero in this region This gives

uSrxS

ZS

uSrxS

Gx r

nGx r

uSxxS

n

dS

Now the integral over S is zero by the same argument we used in Case I So in fact

uSrxS

Z

uSxxS

Gx r

nGx r

uSxxS

n

dS

But to make use of it we need the scattered eld at depth So we plug in the WKBexpressions for uSxxS ASxxSeiSxxS and G AGrxse

iGrxs thencomputing the normal derivatives is straightforward

nG i %n rGAGeiG

neglecting the gradient of the amplitude WKBJ again and

nuS i %n rSASeiS

Plugging these into the surface integral we arrive at

uSrxS Zi ASAG "%n rG %n rS# eiGS

This gives us a formula for doing WKBJ modeling we take AS RAI where R isthe reection coecient and we would have to compute AI G and S I on viaraytracing


Case III G Anticausal S S

In this case the volume V is still the region between the recording surface S and thereecting surface so the volume integral goes to zero What we would really like to beable to do is show that the integral over is negligible so that we would have a surfaceintegral just over the recording surface This would be perfect for migration!inversionformulae This issue is treated somewhat cavalierly in the literature is sent o toinnity and some argument about Sommerfeld is given But for migration!inversion wewant an anticausal Green function so as to propagate the recorded energy backwards intime And as we will now show only the causal Green function satises the Sommerfeldcondition as rAs the observation point goes to innity the causal Green function has the form

G

ReikR

Also we can approximate GR

by Gn

So as R we have

G

n

R

ReikR ikeikR

ReikR

ik

R

and henceG

n i

cG K

R

as R goes to innity K is just some constant factor By the same argument

uSn

i

cus K

R

with a dierent constant factor K

Now the integral whose value at innity we need to know is

ZS

uSxxS

Gx r

nGx r

uSxxS

n

dS

If we add and subtract the term i cGus we haven t changed anything but we end upwith Z

SuS

G

R i

cG

G

uSR

i

cus

dS

The elds themselves G and us decay as one over R for large R Therefore the completeintegrand of this surface integral must decay as one over R The surface area of Sobviously increases as R So we conclude that the surface integral itself must decay asone over R as R goes to innity In other words if we use the causal Green function wecan neglect surface integrals of this form as the surface of integration goes to innity

CASE III G ANTICAUSAL S S

But for migration we want an anticausal Green function And for the anticausal Greenfunction G goes as eikRR we have

G

R i

cG K

R

where the sign is especially important Because of it GR i

cG must decay as one over

R not R The point is that we are stuck with using a causal expression for the scatteredeld since that is the result of a real physical experiment But for the Green function wehave a choice Choosing the causal Green function results in an integrand which decaysas one over R making the surface integral go to zero as R goes to innity But choosingthe anticausal Green function results in an integrand which decays only as one over R

In Schneider s original paper "# he rst did the argument with the causal Green fucntiontook the surface o to innity and argued quite properly that this contribution from thesurface at innity was zero But then when it came time to do migration he slippedin an anticausal Green function As we have just seen this argument is false So thereis more to the question than this Somehow we must show that the contribution to thescattered eld from the integration over is negligible compared to the integration overthe recording surface S The details of the argument are given in "# but the upshot isthat by using a stationary phase argument one can indeed show that the integral over is negligible

Chapter

The Method of Stationary Phase

We have various occasions in this course to consider approximating integrals of the form

I Z b

afxeigx dx

where is a large parameter Such integrals arise for example in the Green functionintegral representation of solutions of the wave equation in slowly varying media Slowlyvarying material properties imply large relative wavenumbers And in the chapter onphaseshift methods we saw that highfrequency solutions of the cz wave equation canbe written in the form

pkzeRkz dz

There is a wellknown method for treating integrals of this form known as the methodof stationary phase The basic idea is that as the parameter gets large the integrandbecomes highly oscillatory eectively summing to zero except where the phase functionhas a stationary point We can illustrate this with a simple example shown in Figure The top gure shows a Gaussian which we take to be the phase function gx Clearlythis has a stationary point at the origin In the middle gure we show the real part ofthe hypothetical integrand at a relatively small value of In the bottom gure hasbeen increased to You can see that only in the neighborhood of the origin is theintegrand not highly oscillatory

Stationary phase methods have been around since the th century The name derivesfrom the fact that the most signicant contribution to I comes from those points wheregx ie the phase is stationary We want an approximation that gets better as theparameter gets larger If we integrate I by parts we get

I fxeigx

igx

b

a

i

Z b

ax

fx

gx

eigx dx

This discussion follows that of Jones

CHAPTER THE METHOD OF STATIONARY PHASE

-2 -1 1 2

0.2

0.4

0.6

0.8

1

-1 -0.5 0.5 1

-1

-0.5

0.5

1

-1 -0.5 0.5 1

-1

-0.5

0.5

1

Phase function

Integrand low frequency

Integrand high frequency

Figure Hypothetical phase function top real part of the integrand eigx for middle real part of the integrand for

The rst term is clearly O The second term will be O once we integrate byparts againprovided g doesn t vanish in the interval "a b#

If g is nonzero in "a b# we re done since as we have the estimate

I eigx

igx

b

a

But suppose gx at some point x in "a b# We can break op the integral I intothree parts

Z x

afxeigx dx

Z x

xfxeigx dx

Z b

xfxeigx dx

where is a small number In the rst and last of these integrals we can use the previousresult since by assumption g is nonstationary in these intervals The only question thatremains is how the integral over the interval "x x # compares to these othercontributions Let s try approximating g by a Taylor series in the neighborhood of x

gx gx gxx x

gxx x

The second term in the Taylor series is zero by assumption If we introduce the newvariable t x x then we haveZ x

xfxeigx dx fx

Z

ei gxg

xt dt

We can pull the fx out of the integral since can be assumed to be as small as welike this works provided f is not singular in the neighborhood of x If it is then wemust resort to stronger remedies Now if the functions f and g are wellbehaved thenwe can make the terms due to the integration by parts as small as we like since theyare all at least O There are many complications lurking in the shadows and wholebooks are written on the subject for example "# but we will assume that we onlyhave to worry about the contribution to I from the integral around the stationary pointEquation In other words we are making the approximation that we can write Ias

I fxZ

ei gxg

xt dt

But this integral can be done analytically since

H Z

ex

dx

H Z

ex

dx Z

ey

dy

Z

Z

ex

y dx dy

Therefore

H Z

Z

er

r dr d

Z

Z

e d d

CHAPTER THE METHOD OF STATIONARY PHASE

So H p and Z

eiat

dt

r

aei

So we re left with

I s

gxfxe

igx

If g turns out to be negative we can just replace g in this equation with jgj ei

If we re unlucky and g turns out to be zero at or very near to x then we need to takemore terms in the Taylor series All these more sophistacated cases are treated in detailin Asymptotic Expansions of Integrals by Bleistein and Handelsman "# But this cursoryexcursion into asymptotics will suce for our discussion of Kircho and WKBJ migrationresults

Exercises

Consider the integral

I Z b

afxeigx dx

Assuming that the phase gx is nowhere stationary in the interval "a b# then I isOp for large Compute p

If x is a stationary point of g in the interval "a b# then

I s

gxfxe

igx

Show that this term dominates the endpoint contribution computed above forlarge

Use the method of stationary phase to approximate the integral

Z

eigx dx

where gx et

Chapter

Downward Continuation of the SeismicWaveeld

A Little Fourier Theory of Waves

Just to repeat our sign conventions on Fourier transforms we write

r t Z Z

eikrtk dk d

The dierential volume in wavenumber space dk dkx dky dkz is used for convenienceWe re not going to use any special notation to distinguish functions from their Fouriertransforms It should be obvious from the context by simply looking at the arguments ofthe functions

Now if the wavespeed in a medium is constant then as with any constant coecientdierential equation we assume a solution of exponential form Plugging the plane waveeikrt into the wave equation r ct we see that our trial form does indeedsatisfy this equation provided the wavenumber and frequency are connected by the dispersion relation k k k c Or if you prefer to think of it in these terms wehave Fourier transformed the wave equation by replacing r with ik

We can always simplify the analysis of the wave equation by Fourier transforming awayone or more of the coordinates namely any coordinates upon which the coecients inthis case just c do not depend In exploration geophysics it s safe to assume the timeindependence of c so we can always make the replacement t i When it comes tothe spatial variation of c we have to be more careful After a constant velocity mediumthe next simplest case to treat is a velocity which depends only on the depth z coordinateIf velocity and hence the wavenumber depends only on z we can Fourier transform the

CHAPTER DOWNWARD CONTINUATION OF THE SEISMIC WAVEFIELD

x

z

V

V

V

1

2

3

Figure A medium which is piecewise constant with depth

x y dependence of the Helmholtz equation r k to get

d

dzhk kx ky

i

Here we nd that the trial solution

eikzz

works provided that k kz kx ky as it must and dkzdz So how can we usethis result in a cz medium There are two possibilities The rst and most common inpractice assumption is that the velocity is actually piecewise constant ie the mediumis made up of a stack of horizontal layers within each of which c is constant as shownin Figure Or put another way if we let denote the value of the waveeld atany depth z then provided the velocity is constant in the strip z z $z the depthextrapolated waveeld at z $z is

eikzjzj

Choice of the plus sign means that the plane wave

eikzzt

will be a downgoing wave because the phase stays constant if z is increased as t increasesTo nd the vertical wavenumber kz we must Fourier transform the data and extract thehorizontal wavenumbers kx and ky kz is then given by kz

qk kx ky

WKBJ APPROXIMATION

The other possibility for using the plane wavelike solutions in a cz medium is withinthe WKBJ approximation

WKBJ Approximation

In fact we ve already encountered the WKBJ approximation in another guise in the derivation of the eikonal equation where we argued that the solution of the wave equationin a slowly varying medium could be represented in terms of slowly varying plane wavesNow we apply this idea to the cz wave equation The zeroth order approximation wouldbe for kz to be constant in which case the solution to

d

dz kz

is just eikzz

Higher orders of approximation are achieved by making the slowly varying plane waveassumption ez It doesn t matter whether we put an i in the exponent or not itall works out in the end Then in order for to be a solution of Equation it isnecessary that

kz

where primes denote dierentiation with respect to z The rst order of approximationis to neglect the term Then we can integrate the ODE exactly to get

ikz

and hence ei

Rkzz dz

The second order approximation is a little harder to get Using ikz we can write ikz So the nonlinear ODE for is now

ikz kz

and therefore

ikz

s i

kzkz

If we make the slowlyvaryingmedia approximation again then the second term in thesquare root is small compared to the rst and we can simplify the square root using arst order Taylor approximation

iZkz

kzkz

dz

iZkz dz

ln kz

The letters stand for G Wentzel H Kramers L Brillouin and H Jereys


So the second order WKBJ solution is

pkzeiRkz dz

The singularity at kz is just a reection of the breakdown of the approximation thatwe made to achieve the result Putting this all together we have the plane wave likeWKBJ solution in the spacetime domain

pkzeikrt

where k %xkx %yky %zRkz dz As usual we choose the sign for downgoing waves

and the sign for upgoing waves The fact that these two waves propagate throughthe medium independently as if there were no reections is the crux of the WKBJ orgeometrical optics approximation Physically the WKBJ wave corresponds to the primaryarrival and is not the result of any internal reections within the model In explorationgeophysics amplitude corrections are usually made on other grounds than WKBJ theoryso you will often see the square root term neglected The WKBJ solution is then just apure phaseshift

PhaseShift Migration

We have already seen that for piecewise constant cz media downward continuation ofthe waveeld is achieved by applying a series of phaseshifts to the Fourier transformeddata This is the essence of Gazdag s phaseshift method "# perhaps the most popularcz method But let s formally introduce the method in its most common guise D timemigration Once again we introduce the migrated or vertical travel time zv Insteadof downward extrapolating the waveeld a depth increment $z we use a vertical timeincrement $ $zc Then the downward extrapolation operator we derived in the lastsection becomes

exp

i $

vuutckx

To nd the migrated image at a nite n$ we apply this operator n times insuccession But unlike Kirchho migration where we applied the imaging condition onceand for all when we downward extrapolate the waveeld we must image at every depthFortunately selecting the exploding reector imaging condition t can be done bysumming over all frequencies

For time migration the migration depth step $ is usually taken to be the same as thesampling interval So it will usually be the case that the complex exponential which does

If ft Reitfd then putting t gives f

Rf d

ONEWAY WAVE EQUATIONS

PhaseShift Migration a la Claerbout

skx Fourier Transformsx tfor max $ f

for all kx finitialize Imagekx for all f

C exp"i $

r

ckx

#

Imagekx Imagekx C skx gImagex FourierTransformImagekx

gg

Figure Claerbout s version of Gazdag s cz phaseshift algorithm

the downward extrapolation need not be recomputed for each $ Pseudocode for thephaseshift algorithm is given in Figure

The main drawback to the phaseshift migration would appear to be the improper wayin which amplitudes are treated at boundaries Rather than account for the partitioningof energy into reected and transmitted waves at each boundary as must surely beimportant if we have a layered model we simply pass all the energy as if the transmissioncoecient were unity Presumably if the velocity is indeed slowly varying we would bebetter using a migration method based more explicitly on WKBJ approximations Theother possibility is to keep both the upgoing and downgoing solutions to the cz waveequation say with coecients A and A and then apply the proper boundary conditionsso that new A and A are determined at each layer to conserve total energy Classicphaseshift would then be the limiting case of setting the upgoing A coecient to zero

OneWay Wave Equations

When we can Fourier transform the wave equation it is easy to select only downgoingwave propagationwe choose the positive square root in the dispersion relation Unfortunately the Fourier transform approach is limited to homogeneous media A technique tosort of reverseengineer oneway wave propagation suggested by Claerbout and describedin "# section is to take an approximate dispersion equation containing only downgoing wavenumbers and see what sort of wave equation it gives rise to upon makingthe substitution ikx x iky y and so on For example start with the dispersion


relation kz kx ky

c Then solve for kz and select downgoing wavenumbers by

choosing the positive square root

kz

c

s ckx

and making the replacement ikz z we get the following downward continuation operator

z i

c

s ckx

valid for depthdependent media

The problem with extending this to laterally varying media is the presence of the squareroot One could imagine making the replacement ikx x but how do we take the squareroot of this operator It is possible to take the square root of certain kinds of operatorsbut the approach most often taken in exploration geophysics is to approximate the squareroot operator either by a truncated Taylor series or by a continued fraction expansionThe latter goes as follows

Write the dispersion relation

kz

c

s ckx

askz

c

p X

cR

The nth order continued fraction approximation will be written Rn and is given by

Rn X

Rn

You can readily verify that this is the correct form of the approximation by noting that ifit converges it must converge to a number say R which satises

R X

R

which means that R X

The rst few approximate dispersion relations using continued fraction approximationsto the square root are given by

kz

cn

kz

c ckx

n

kz

c kx

c ckx

n

For slightly more on continued fractions see the section at the end of this chapter

ONEWAY WAVE EQUATIONS

-1 -0.5 0.5 1

0.2

0.4

0.6

0.8

1

exact

15 degree

45 degree

60 degree

Figure The continued fraction approximation to the square root dispersion formulagives an approximate dispersion formula whose accuracyreected in the steepest anglesat which waves can propagateincreases with the order of the approximation

It is clear from the n term that these approximations rely to a greater small n orlesser larger n extent on the approximation that waves are propagating close to verticallyIn fact it can be shown that the n term is accurate for propagation directions of within degrees of vertical The n term is accurate out to degrees and the n termis accurate to degrees For more details on this aspect see Claerbout "# Chapter The dierent accuries of these dispersion relations are shown in Figure whichwas computed with the following Mathematica code

rx rnx x !" rnx

where I have taken k for convenience For example

r" x# x

x

x

To make use of these various approximations to the square root for cz x media weproceed in two steps First since we are always going to be extrapolating downward indepth so that we can assume that the velocity is constant in a thin strip $z we replacethe vertical wavenumber kz with iz This gives us the following downward extrapolatorsaccording to Equations

iz

c n

iz

c ckx

n

iz

c kx

c ckx

n

Next to account for the cx variation we replace kx with ix For example using then or degree approximation we have

z i

c

c

x


Paraxial Wave Equations

Yet another approach to achieving oneway propagation in laterally varying media isclosely related to the WKBJ methods In this approach we make the by now standardnearplane wave guess but allow a laterally varying amplitude

z xeitzc

If we plug this form into the wave equation what we discover is that in order for tosatisfy the wave equation it is necessary that satisfy

r i

cz

So all we ve done is rewrite the wave equation in terms of the new variable The nearplane wave approximation comes in dropping the term z in comparison to x Thisresults in a parabolic equation for

z ic

x

This is called a paraxial approximation since we are assuming that wave propagationoccurs approximately along some axis in this case the z axis The paraxial approximationis very popular in ray tracing since by making it we can generate a whole family of raysin the neighborhood of a given ray for almost no extra cost In other words we begin bytracing a ray accurately Then we generate a family of paraxial rays in a neighborhoodof this ray How big this family can be made depends on the extent to which we can dropthe transverse Laplacian in comparison with the axial Laplacian

The Factored TwoWay Wave Equation

Another trick wellknown from quantum mechanics is to factor the cz wave equationinto an upgoing and a downgoing part Clearly we can write

d

dzhk kx ky

i

d

dz ik

d

dz ik

where q

kx kyk

If we denote the rst dierential operator in square brackets as L and the second as Lthen the cz wave equation Equation which is LL can be written as acoupled rst order system

downgoing L

upgoing L

DOWNWARD CONTINUATION OF SOURCES AND RECEIVERS

As usual for migration we neglect the coupling of upgoing and downgoing waves and justconsider the rst order system describing downgoing waves

If you re having trouble keeping straight which is the upgoing and which is the downgoingequation just remember what the two plane wave solutions of the wave equation look likein the spacetime domain eikzt and eikzt These both satisfy the wave equationprovided k c In the rst of these z must increase as t increases in order to keepthe phase constant In the second z must decrease as t increases in order to keep thephase constant Therefore eikzt must be a downgoing wave and eikzt must be anupgoing wave

Downward Continuation of Sources and Receivers

In this section we consider the nonzerooset generalization of phaseshift migrationThis is discussed in Section of Claerbout s book "# we follow the simpler notationof "# however

Generally for D problems we have written the recorded data as x z t Forslightly greater generality we can write this as r zr t where zr is the depth coordinateof the receiver Implicitly the data also depend on the source coordinates so we shouldreally write s zs r zr where zs is the depth coordinate of the source For a xedsource the wave equation is just

t chr zr

i

The principle of reciprocity states that we should be able to interchange source and receivercoordinates and leave the observed waveeld unchanged This is intuitively obvious onray theoretic grounds in practice it is very dicult to verify with eld data since sourcesand receivers have dierent antenna characteristics For example it would seem that aweight drop should be reciprocal to a vertical geophone a marine airgun reciprocal to ahydrophone Claerbout says that reciprocity is veried in the lab to within the precision ofthe measurements and that in the eld small errors in positioning the sources and receiverscan create discrepancies larger than the apparent discrepancies due to nonreciprocity Inany event we will assume that reciprocity holds That being the case we can write

t chs zs

i

Assuming that the velocity depends only on depth so that we can Fourier transform awaythe lateral coordinates we end up with the two equations

zr

c

krc


and

zs

c

ksc

where kr and ks are the horizontal wavenumbers in receiver and source coordinates

Now these are the same cz wave equations that we have dealt with many times beforejust expressed in new coordinates So we already know that both of these equationshave upgoing and downgoing solutions With zerooset data the source and receivercoordinates are the same so when we downward continued in the exploding reectormodel we only needed a single wave equation But now we want to allow for a greaterdegree of generality in order that we might be able to apply the methods we ve alreadydiscussed to faroset data So let us imagine downward continuing both the sources andreceivers simultaneously At the reector location the source and receiver coordinatesmust coincide Coincident sources and receivers imply that t This means that wecan still use the exploding reector imaging condition All we have to do is come upwith oneway versions of the two equations above Clearly then we can use the onewaywave equations that we ve already derived The choice of which one to use is ours Itis common however to use the factored version of the wave equation Equation Then we end up with

zr i

c

h r

i

and

zs i

c

h s

i

where r ckr and s cks

When the source and receiver coordinates are coincident at the reector then zr zs zOr put another way we can imagine lowering the sources and receivers simultaneouslyIn either case

z zr zs

The way to think about this equation is that at a xed receiver location a small change inthe vertical location of a source or receiver is equivalent to a small change in the observedtravel times in the data Therefore

z ztt

and so the total change in travel time associated with a perturbation dz of both sourcesand receivers is given by

dt "zr t zst# dz

This allows us to write

z i

c

h s

ih r

i

DOWNWARD CONTINUATION OF SOURCES AND RECEIVERS

which eects the simultaneous downward continuation of the sources and receivers referredto by Claerbout as survey sinking Equation is called the doublesquarerootequation

The doublesquareroot equation is easily solved for cz mediathis is just our old friendthe phase shift Thus we can do nonzerooset phaseshift migration since just as for thezerooset case we can say that the data at any depth z $z can be obtained from thedata at z by

z $z zeic

h s

r

iz

We re almost nished now All we have to do is apply the imaging condition Part of thisis just the usual exploding reector t which we accomplish in the frequency domainby reintroducing and summing The other part of the imaging condition is that r sThis is conveniently expressed in terms of the half oset via h r s So thefully migrated section is

x h t z Xkx

Xkh

X

kx kh zeikxx

where x is the midpoint r s r kx khc and s kx khc Thesummations over kh and are the imaging conditions The summation over kx inverseFourier transforms the waveeld back into midpoint space

This downward continuation of sources and receivers together with the zerooset imagingcondition gives a prestack migration algorithm Unfortunately it is of little utility as itstands because it requires that the velocity be laterally invariant there s not much pointin doing prestack migration if the velocity is really cz One possible generalizationwould be to use a dierent velocity function for sources and receivers in Equation and Equation We can do this by downward continuing sources and receivers inalternating steps with the dierent velocity functions The algorithm we end up with is

Sort data into commonsource gathers r s t z and extrapolate downward adistance $z using Equation

Resort data into commonreceiver gathers r s t z and extrapolate downwarda distance $z using Equation

Compute the imaged CMP section by mx z r x s x t z

Clearly this method puts a tremendous burden on computer I!O if we re going to be continually transposing data back and forth between commonsource and commonreceiverdomains It might make more sense to simply migrate the individual commonsourcerecords We can certainly downward continue a common source record using any of theoneway way equations that we ve developed already The only problem is what imaging


condition do we use The eld scattered from any subsurface point r z will imagecoherently at a time t which equals the travel time from that point to the source pointts r s where s is a xed source location Here is the complete algorithm

Calculate the travel time ts r s from a given source location to all points inthe subsurface

Downward continue using Equation for example the shot record r s t z This results in the scattered eld at each point in the subsurface r s t z

Apply the imaging condition mr z s r s ts r s z

If we used the correct velocity model to downward continue each shot record then weshould be able to sum all of the migrated commonsource gathers and produce a coherentimage of the subsurface So we might write the nal migrated image as

mr z Xs

mr z s

The main theoretical drawback of this approach to prestack shot migration is that itpresupposes the existence of a singlevalued travel time function If however there ismore than one ray between a source and any point in the subsurface then there is nolonger a welldened imaging time In this case we must resort to Claerbout s imagingcondition described by Paul Docherty in his lecture on Kirchho migration!inversionformulae

Time Domain Methods

In the discussion of Kirchho migration methods we found that provided we had a Greenfunction satisfying certain boundary conditions on the recording boundary z wecould generate a solution of a boundary value problem for the wave equation anywherein the subsurface and at all times The boundary values that we use are the stackedseismic records Therefore if we use an anticausal Green function the solution of the waveequation that we get corresponds to the propagation of the zerooset data backwards intime Choosing t and c c yields an image of the subsurface reector geometryFurther the Green function solution is computed by integrating over the recording surfaceNow since the time and the spatial coordinates are coupled via the travel time equationthis integration amounts to summing along diraction hyperbolae Nevertheless the formof the integration is that of a spatial convolution We then saw that in the Fourier domainthis spatial convolution could be shown to be a phaseshift operation And under theassumptions usually associated with time migration we could think of this convolutionas being a downward propagation of the recorded data

TIME DOMAIN METHODS

Clearly there is a fundamental equivalence between propagating the recorded data backward in time and propagating it downward in depth If the velocity model is constantor only depth dependent then this equivalence is easily demonstrated by replacing timeincrements with migrated time or depth increments suitably scaled by the velocity Inlaterally heterogeneous media the connection between the two is more complicated inessence it s the wave equation itself which connects the two approaches

When we discussed phaseshift migration methods everything was done in the frequencydomain where it s dicult to see clearly how backpropagation in time is related Downward propagation at least for vz media is still a phaseshifting operation howeverSo as we downward propagate the stacked seismic data from depth to depth we have toinverse Fourier transform the results at each depth in order to apply the imaging condition Fortunately plucking o just the t value of the inverse transformed data iseasy just sum over frequencies The point here is that we downward propagate the entirewaveeld from depth to depth at any given depth applying the imaging condition resultsin a coherent image of that part of the waveeld which was reected at that depth

This all begs the question of why we can t simply use the time domain wave equation tobackward propagate the stacked seismic data to t performing the imaging once andfor all We can This sort of migration is called reversetime migration But rememberin a fundamental sense all migration methods are reverse time Certainly Kirchho isa reversetime method and as we have seen by properly tracing the rays from sourceto receiver we can apply Kirchho methods to complicated velocity modelscertainlymodels where time migration is inappropriate So the distinctions that are made in theexploration seismic literature about the various kinds of migration methods have littleto do with the fundamental concepts involved and are primarily concerned with how themethods are implemented

It appears that the rst published account of what we now call reversetime migrationwas given by Hemon "# His idea was to use the full wave equation with zero initialvalues for and t specied at some time T The recorded data sx t z are thentake as boundary values for a boundary value problem for the full wave equation but inreverse time This should cause no diculty since the wave equation is invariant under ashift and reversal of the time coordinate t T t

The early papers on reverse time migration approximated the wave equation by an explicit nite dierence scheme that was accurate to second order in both space and timeWe will discuss such approximations in detail later for now we simply exhibit such anapproximation

xk zj ti Axk zj ti xk zj ti

A "xk zj ti xkzj ti

xk zj ti xk zj ti#

where A cxk zj$th where h is the grid spacing in the x and z directions This sort


of approximation is called explicit because it gives the eld at each gridpoint at a giventimestep in terms of the eld at nearby gridpoints at the previous timestep at eachtimestep ti everything to the right of the equal sign is known Explicit nite dierencemethods are very easy to code and the above scheme can be used for both modeling andmigration In the latter case we simply run time backwards

Initially at the greatest time tmax recorded in the data the eld and its time derivativeare set to zero in the entire subsurface Then data at tmax provide values for the eldat z and data at three times ending at tmax provide second order estimates of thetime derivative of the eld The nite dierence equation propagates these values intothe subsurface for a single backward timestep $t as data at tmax $t are introducedat z We continue this procedure until t This has the eect of propagating therecorded data into the subsurface

There are two extremely serious drawbacks with the nite dierence approximation thatwe ve introduced The rst is that low order schemes such as this require around nitedierence gridpoints per the shortest wavelength present in the data in order to avoidnumerical dispersion We can get around this problem by using a higher order schemeA scheme which is accurate to fourth order in space and time requires only about gridpoints per wavelength The second problem is that using the full wave equation gives riseto spurious reections at internal boundaries These internal reections are consideredspurious because even if the model boundaries have been correctly located the stackeddata is presumed to have had multiples attenuated through the stacking procedure Thesolution to this problem proposed by Baysal et al "# was to use a wave equation inwhich the density was fudged in order to make the normalincidence impedance constant across boundaries The idea being that since density is not something we normallymeasure in reection seismology we can make it whatever we want without aecting thevelocities

A Surfers Guide to Continued Fractions

We illustrate here a few properties of continued fractions adapted from the fascinatingbook History of Continued Fraction and Pade Approximants by Claude Brezinski "#

Continued fractions represent optimal in a sense to be described shortly rational approximations to irrational numbers Of course we can also use this method to approximaterational numbers but in that case the continued fraction expansion terminates after anite number of terms To get a feel for how these approximations work let s start withone of the oldest continued fractions one which we can sum analytically The rst fewterms are

S

A SURFERS GUIDE TO CONTINUED FRACTIONS

S

x

S

x x

S

x x

x

S

x x

x x

The modern notation for this continued fraction expansion is

S jjx

jjx

jjx

Each successive term Sn in this development is called a convergent Clearly each convergent can by appropriate simplication be written as a rational number For example

S x

x

and

S x

x x

It is easy to see that the following recursion holds

S

x S S

x S S

x S

If this sequence converges then it must be that

S

x S

which implies that

S xS

It is important to understand that we haven t actually proved that this continued fractionconverges we ve simply shown that if it converges it must converge to the quadratic inEquation More generally we can write the solution of the quadratic form

S bS a

as

S ajjb

ajjb

ajjb


Some examples Let a and b in The rst few convergents are

S S S

S

S

This sequence converges to the positive root of the quadratic namelyp

whichwas known as the golden number to the Greeks In fact the even and odd convergentsprovide respectively upper and lower bounds to this irrational number

S S S p

S S S

Similarly for a and b the even and odd convergents of provide upper andlower bounds for the positive root of namely

p to which they converge

This turns out to be a general result in fact the evenodd convergents of the continuedfraction expansion of a number always provide upperlower bounds to Also one canshow that the rational approximations to generated by the convergents are optimal inthe sense that the only rational numbers closer to than a given convergent must havelarger denominators

Since continued fractions are rational approximations we can give a nice geometrical illustration of what s going on in the approximation Figure shows the positive quadrantwith a large dot at each integer Each convergent being a rational number can be thoughtof as a line segment joining the dots The asymptote of these successive line segments willbe a line from the origin whose slope is the number we re approximating If that numberhappens to be rational the continued fraction series terminates otherwise not There areinnitely more irrational numbers than rational ones since the rationals are denumerablecan be put into onetoone correspondence with the integers while the irrationals arenot Therefore if we choose a number or slope at random it is very unlikely that it willintersect one of the dots since that would imply that its slope were rational This mayappear counterintuitive however since it looks like there are an awful lot of dots

In this particular example the polygonal segments represent the convergents of the continued fraction for the Golden mean The left and right vertices are given by the evenand odd convergents

left p q p q p q

right p q p q p q

There is a simple way to construct the continued fraction representation of any number Begin by separating out the integral and decimal part of the number

n r where r

Mathematica uses the golden number or golden mean as the default aspect ratio for all plots Tothe Greeks this number represented the most pleasing aspect or proportion for buildings and paintingsThis view held sway well into the Renaissance

A SURFERS GUIDE TO CONTINUED FRACTIONS

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

. . . . . . . . . . . (p1,q1)

(p2,q2)

(p0,q0)

Figure An arbitrary number can be thought of as the slope of a line from the originThe continued fraction approximation for the number in this case the golden meanconsists of left and right polygons which approach an irrational number asymptoticallyand a rational number after a nite number of steps

If the remainder r is nonzero we repeat this procedure for r

r n r where r

We repeat this procedure until either it terminates with a zero remainder or we get tiredThe result is

n

n n

n

As an example for we get

Fibonacci

The numbers appearing in the continued fraction expansion of the golden ratio might lookfamiliar They are also known as the Fibonacci numbers and they satisfy the recursion

F F F j F j F j


The Fibonacci numbers are named after Leonardo of Pisa also known as Fibonacci acontraction of lius Bonacci son of Bonacci Leonardo was born and died in PisaItaly ourishing in the rst part of the th century He was a merchant who traveledwidely in the Near East and became well acquainted with Arabic mathematical work TheFibonacci numbers were introduced in his study of the breeding habits of rabbits "# HisLiber abaci was written in but not published until

As well as Fibonacci numbers there are Fibonacci sequences For example

S AS B Sj Sj Sj

So instead of adding the previous two iterates we simply concatenate the previous twosequences Fibonacci layered media are widely used to study disordered media sincealthough deterministic they are very complex as the order of the sequence becomes large

Exercises

Make the substitution kx ix and derive the degree downward extrapolationequation Hint multiply both sides of the equation by the denominator of thecontinued fraction approximation so none of the derivatives are in the denominator

Write your downward extrapolator as

z i

c another term

In optics these separate parts are known as respectively the thin lens term andthe diraction term In practice the downward extrapolation for cz media mightproceed in two phases for each $z step rst we extrapolate using the thin lensterm then we apply the diraction term

Derive the timedomain version of your degree equation by making the substitution it

Starting with the wave equation derive Gazdag s phaseshift migration method forcz media

Give a pseudocode algorithm for phaseshift depth migration What changes arenecessary to make this a time migration algorithm Under what circumstanceswould such a time migration algorithm be accurate

The degree approximate dispersion relation is

kz

c kx

c ckx

What is the frequencydomain wave equation associated with this dispersion relation

COMPUTER EXERCISE IV

If r s t z is the downward continuation of the shotrecord r s t what isthe migrated image associated with this shot record

Computer Exercise IV

Below is reproduced a complete migration code from the SU library written by JohnStockwell and Dave Hale Study the layout of this code and start thinking about how youwill implement a straightray version of Kirchho migration You should try to copy theoutlines of this code or other SU migration codes such as sumigtkc and sustoltc insofaras the way parameters are read in memory allocated trace headers processed and thenal migrated section output Do not reinvent these wheels both because it would be awaste of time and also because the person who writes the best code will have the pleasureof seeing it added to the SU library Try to be as modular in your writing as possible sothat in a few weeks after we ve talked about numerical methods for tracing rays you willbe able to replace your straight ray calculation of travel times with a more sophisticatedone For this exercise stick within the zerooset approximation but keep in mind howyou might implement Claerbout s prestack imaging condition

As a rst step write a Kirchho migration code that will use a completely general cx zvelocity model but don t worry about tracing rays to compute travel times For thisexercise you may either compute travel times along straight rays or simply use the traveltime associated with the rms velocity between the source and receiver It would probablybe best to assume the velocity model is dened on a grid That should smoothly pavethe way for the next exercise which will involve accurate travel time calculations inmoderately complex structures

! SUGAZMIG Revision Date !! !

!

Copyright c Colorado School of Mines

All rights reserved

This code is part of SU SU stands for Seismic Unix a

processing line developed at the Colorado School of

Mines partially based on Stanford Exploration Project

SEP software Inquiries should be addressed to

Jack K Cohen Center for Wave Phenomena

Colorado School of Mines

Golden CO

jkc#dixminescoloradoedu


!

include suh

include segyh

include headerh

! self documentation !

char sdoc $

SUGAZMIG SU version of GAZDAGs phaseshift migration

for zerooffset data

sugazmig infile outfile vfile optional parameters

Optional Parameters

dtfrom headerdt or time sampling interval

dxfrom headerd or midpoint sampling interval

ft first time sample

ntauntfrom data number of migrated time samples

dtaudtfrom header migrated time sampling interval

ftauft first migrated time sample

tmig times corresponding to interval velocities

in vmig

vmig interval velocities corresponding to times

in tmig

vfile name of file containing velocities

Note ray bending effects not accounted for in this

version

The tmig and vmig arrays specify an interval velocity

function of time Linear interpolation and constant

extrapolation is used to determine interval velocities

at times not specified Values specified in tmig must

increase monotonically

Alternatively interval velocities may be stored in

a binary file containing one velocity for every time

sample in the data that is to be migrated If vfile

is specified then the tmig and vmig arrays are

ignored


NULL%

! end self doc !

!

Credits CWP John Stockwell Oct

Based on a constant v version by Dave Hale

!

segy tr

! prototypes for functions defined and used below !

void gazdagvt float k

int nt float dt float ft

int ntau float dtau float ftau

float vt complex p complex q

! the main program !

main int argc char argv

$

int nt ! number of time samples !

int ntau ! number of migrated time samples !

int nx ! number of midpoints !

int ikixititauitmig! loop counters !

int nxfft ! fft size !

int nk ! number of wave numbers !

int np ! number of data values !

int ntmignvmig

float dt ! time sampling interval !

float ft ! first time sample !

float dtau ! migrated time sampling interval !

float ftau ! first migrated time value !

float dk ! wave number sampling interval !

float fk ! first wave number !

float tk ! timewave number !

float tmig vmig ! arrays of time int velocities !

float dx ! spatial sampling interval !

float vt ! velocity vt !

float pq ! input output data !

complex cpcq ! complex inputoutput !


char vfile ! name of file containing velocities !

FILE vfp ! velocity file pointer !

FILE tracefp ! fp for trace storage !

FILE headerfp ! fp for header storage file !

! hook up getpar to handle the parameters !

initargsargcargv

requestdoc

! get info from first trace !

if &gettrtr errcant get first trace

nt trns

! let user give dt and!or dx from command line !

if &getparfloatdt dt $

if trdt $ ! is dt field set' !

dt float trdt !

% else $ ! dt not set assume ms !

dt

warntrdt not set assuming dt

%

%

if &getparfloatdxdx $

if trd $ ! is d field set' !

dx trd

% else $

dx

warntrd not set assuming d

%

%

! get optional parameters !

if &getparfloatftft ft

if &getparintntauntau ntau nt

if &getparfloatdtaudtau dtau dt

if &getparfloatftauftau ftau ft

! store traces and headers in tempfiles while

getting a count !


tracefp etmpfile

headerfp etmpfile

nx

do $

""nx

efwritetrHDRBYTESheaderfp

efwritetrdata FSIZE nt tracefp

% while gettrtr

erewindtracefp

erewindheaderfp

! determine wavenumber sampling for real

to complex FFT !

nxfft npfarnx

nk nxfft!"

dk PI!nxfftdx

fk

! allocate space !

p allocfloatntnxfft

q allocfloatntaunxfft

cp alloccomplexntnk

cq alloccomplexntaunk

! load traces into the zerooffset array

and close tmpfile !

efreadp FSIZE ntnx tracefp

efclosetracefp

! determine velocity function vt !

vt eallocfloatntau

if &getparstringvfilevfile $

ntmig countparvaltmig

if ntmig ntmig

tmig eallocfloatntmig

if &getparfloattmigtmig tmig

nvmig countparvalvmig

if nvmig nvmig

if nvmig&ntmig errnumber tmig and vmig must be equal

vmig eallocfloatnvmig

if &getparfloatvmigvmig vmig

for itmig itmigntmig ""itmig

if tmigitmigtmigitmig


errtmig must increase monotonically

for itt itntau ""itt"dt

intlinntmigtmigvmigvmigvmigntmig

tvtit

% else $

if freadvtsizeoffloat ntfopenvfiler &nt

errcannot read (d velocities from file (sntvfile

%

! pad with zeros and Fourier transform x to k !

for ixnx ixnxfft ix""

for it itnt it""

pixit

pfarcntnxfftpcp

! migrate each wavenumber !

for ikkfk iknk ik""k"dk

gazdagvtkntdtftntaudtauftauvtcpikcqik

! Fourier transform k to x including FFT scaling !

pfacrntaunxfftcqq

for ix ixnx ix""

for itau itauntau itau""

qixitau ! nxfft

! restore header fields and write output !

for ix ixnx ""ix $

efreadtrHDRBYTESheaderfp

trns ntau

trdt dtau

trdelrt ftau

memcpytrdataqixntauFSIZE

puttrtr

%

%

void gazdagvt float k

int nt float dt float ft

int ntau float dtau float ftau

float vt complex p complex q

!


Gazdags phaseshift zerooffset migration for one

wavenumber adapted to vtau velocity profile

Input

k wavenumber

nt number of time samples

dt time sampling interval

ft first time sample

ntau number of migrated time samples

dtau migrated time sampling interval

ftau first migrated time sample

vt velocity vtau

p arraynt containing data to be migrated

Output

q arrayntau containing migrated data

$

int ntfftnwititauiw

float dwfwtmaxwtauphasecoss

complex cshiftpp

! determine frequency sampling !

ntfft npfant

nw ntfft

dw PI!ntfftdt

fw PI!dt

! determine maximum time !

tmax ft"nt dt

! allocate workspace !

pp alloccomplexnw

! pad with zeros and Fourier transform t to w

with w centered !

for it itnt it""

ppit it( ' cnegpit pit

for itnt itntfft it""

ppit cmplx

pfaccntfftpp

! account for nonzero ft and nonzero ftau !


for itau itau ftau itau"" $

for iwwfw iwnw iw""w"dw $

if w w e!dt

coss pow vtitau k!w

if cosspowftau!tmax $

phase wftftausqrtcoss

cshift cmplxcosphase sinphase

ppiw cmulppiwcshift

% else $

ppiw cmplx

%

%

%

! loop over migrated times tau !

for itautauftau itauntau itau""tau"dtau $

! initialize migrated sample !

qitau cmplx

! loop over frequencies w !

for iwwfw iwnw iw""w"dw $

! accumulate image summed over frequency !

qitau caddqitauppiw

! compute cosine squared of propagation angle !

if w w e!dt

coss pow vtitau k!w

! if wave could have been recorded in time !

if cosspowtau!tmax $

! extrapolate down one migrated time step !

phase wdtausqrtcoss

cshift cmplxcosphase sinphase

ppiw cmulppiwcshift

! else if wave couldnt have been recorded in time !

% else $

! zero the wave !

ppiw cmplx


%

%

! scale accumulated image just as we would

for an FFT !

qitau crmulqitau!nw

%

! free workspace !

freecomplexpp

%

Chapter

Plane Wave Decomposition ofSeismograms

As we have seen throughout this course wave equation calculations are greatly simpliedif we can assume planewave solutions This is especially true in exploration seismology since to rst order the Earth is vertically stratied Horizontal boundaries makeCartesian coordinates the natural choice for specifying boundary conditions And if weapply separation of variables to the wave equation in Cartesian coordinates we naturallyget solutions which are represented as sums or integrals of plane waves

One problem we face however is that plane waves are not physically realizable exceptasymptotically If we are far enough away from a point source to be able to neglect thecurvature of the wavefront then the wavefront is approximately planar The sources ofenergy used in exploration seismology are essentially point sources but it is not clearwhen we can be considered to be in the far eld

On the other hand plane waves spherical waves cylindrical waves all represent basisfunctions in the space of solutions of the wave equation This means that we can representany solution of the wave equation as a summation or integral of these elementary wavetypes In particular we can represent a spherical wave as a summation over plane wavesThis means that we can take the results of a seismic experiment and numerically synthesizea plane wave response We will begin this chapter by deriving the Weyl and Sommerfeldrepresentations for an outgoing spherical wave These are respectively the plane waveand cylindrical wave representations Then following the paper by Treitel Gutowskiand Wagner "# we will examine the practical consequences of these integral relationsIn particular we will examine the connection between plane wave decomposition and theslantstack or p transformation of seismology

Migrating plane wave sections is no more dicult than migrating commonsource recordsThe imaging condition is essentially the same but expressed in terms of the plane wave

CHAPTER PLANE WAVE DECOMPOSITION OF SEISMOGRAMS

propagation angle The comparison between shot migration and plane wave migrationare nicely described in the paper by Temme "# Once we have the plane wave decomposition we can migrate individual plane wave components One reason this turns out to beimportant is that it seems that we can achieve high quality prestack migrations by migrating a relatively small number of plane wave components In other words plane wavemigration appears to be more ecient than shot record migration since we can achievegood results with relatively few plane wave components Commonsource records on theother hand have a very limited spatial aperture

The Weyl and Sommerfeld Integrals

The starting point for the plane wave decomposition is the Fourier transform of thespherical wave emanating from a point source at the origin

eikr

rZ

Z

Z

Akx ky kze

ikxxkyykzzdkxdkydkz

This equation looks like a plane wave decomposition already It would appear that if wecan invert the transform for the coecients A in other words solve the integral

Akx ky kz Z

Z

Z

eikr

reikxxkyykzzdx dy dz

then we will be done But look again The integral for A involves all possible wavenumbers kx ky and kz not just those satisfying the plane wave dispersion relation Soin fact Equation is not a plane wave decomposition We need to impose thedispersion relation as a constraint by eliminating one of the integrals over wave numberThis is the approach taken in Chapter of Aki and Richards "# who evaluate the z

integral using residue calculus and make the identication kz qk kx ky If you are

comfortable with residue calculus then you should have a look at this derivation Here wewill follow a slightly simpler approach from "# We will rst evaluate Equation in the xy plane then appeal to the uniqueness of solutions to the Helmholtz equation toextend the result to nonzero values of z Many excellent textbooks exist which cover theuse of complex variables in the evaluation of integrals for example Chapter of Morseand Feshbach "#

Let s begin by considering the special case z Then the problem is to evaluate

Akx ky

Z

Z

eikr

reikxxkyydxdy

We can evaluate this integral in polar coordinates We need one set of angles for thewavevector k and another for the position vector r kx q cos ky q sin where

THE WEYL AND SOMMERFELD INTEGRALS

kx ky q and x r cos y r sin The element of area dxdy rdrd So wehave

kxx kyy rq"cos cos sin sin # rq cos

Therefore

Akx ky

Z

Z

eirkq cosdrd

The radial integral can be done by inspection

Z

eirkq cosdr

Z

eirBdr

irB

iB

provided we can treat the contribution at innity Many books treat this sort of integralin an ad hoc way by supposing that there is a small amount of attenuation This makesthe wave number complex and guarantees the exponential decay of the antiderivative atinnity k ik with k If you want to look at it this way ne A more satisfyingsolution is to use the theory of analytic continuation The integral in Equation isthe real part of the integral Z

eirB

dr

where B is a complex variable B i The integral clearly converges to iB

provided And this result is analytic everywhere in the complex plane except B Thismeans that we can extend by analytic continuation the integral in Equation tothe real B axis In fact we can extend it to the entire complex plane except B Wecan even extend it to zero values of B by observing that this integral is really the Fouriertransform of a step function The result is that at the origin this integral is just pi timesa delta function The proof of this will be left as an exercise

So we have

Akx ky iZ

d"k q cos #

i

k

Z

d

qk

cos

where The integral is now in a standard form

Z

dx

a cosx

p a

if a Putting this all together we end up with

Akx ky i

qk kx ky

and henceeikr

r

i

Z

Z

eikxxkyyqk kx ky

dkxdky


q

f

a

2

2

z

x

y

k

Figure Spherical coordinates representation of the wave vector

This is the plane wave decomposition for z propagation We can extend it to nonzerovalues of z in the following way By inspection the general expression for this integralmust have the form

eikr

r

i

Z

Z

Z

eikxxkyykz jzj

kzdkxdkydkz

where we have made the identication kz qk kx ky

The spherical wave eikrr satises the Helmholtz equation for all z the Sommerfeld radiation condition and has certain boundary values at z given by Equation Youcan readily verify that the integral representation Equation that we speculate isthe continued version of Equation also satises Helmholtz Sommerfeld and hasthe same boundary values on z It turns out that solutions of the Helmholtz equationwhich satisfy the Sommerfeld condition are unique This means that Equation must be the valid representation of eikrr for nonzero values of z

Equation is know as the Weyl integral We would now like to recast it as anintegration over angles to arrive at an angular spectrum of plane waves In sphericalcoordinates one has cf Figure

kx k cos sin ky k sin sin kz k cos

We have no problem letting run from to But a moment s reection will convinceyou that we can t restrict to real angles The reason for this is that as kx or ky go

THE WEYL AND SOMMERFELD INTEGRALS

to innity as they surely must since we are integrating over the whole kx ky plane kzmust become complex But

cos kzk

Thus we see that complex kz values imply complex plane wave angles And if theangle of propagation of a plane wave is complex it must decay or grow exponentiallywith distance These sorts of waves are called evanescent or inhomogeneous waves Thisis an extremely important point The physical explanation given in Treitel et al "#is that these evanescent waves are source generated and that provided we are well awayfrom the source we can ignore them Thus in "# the evanescent waves are ignored andthe resulting integration over the angle is restricted to the real axis But rememberplane waves are nonphysicalthey do not exist in nature They are purely mathematicalabstractions Plane waves can be approximated in nature by various procedures Forexample we could place a large number of point sources along a line and set them otogether This would generate a downgoing plane wave Or we could apply a slight timedelay between each source to simulate a plane wave incident at some angle Anotherpossibility is to be so far removed from a point source that the wavefront curvature isnegligible

A point source on the other hand is easily realizable drop a pebble into a pond if youwant to see one So it should come as no surprise that we will have diculty interpretingthe results of a decomposition of a physical wave into nonphysical waves We havealready seen that essentially any transient pulse can be synthesized from elementaryFourier components but these don t exist in nature either Any real signal must have abeginning Remember also that evanescent plane waves are not really waves at all Theydo not suer any phase shift upon translation and therefore have no travel time delayassociated with them This causes no practical problem so long as the origin of theseterms is far enough away from any boundary that the exponential decay will have killedo any energy associated with them

As to the direction of decay of these evanescent plane waves that is completely arbitraryIn order to apply the dispersion relation as a constraint we needed to integrate out one ofthe wavevector components We chose kz because we like to think in terms of verticallylayered media where z is the naturally favored coordinate The result of this was exponential decay of the inhomogeneous components in the z direction But clearly we couldhave chosen any direction in k space and ended up with an analogous result

The net result of all of this is that when kx or ky go to innity kz i which impliesthat cos i Now using the fact that

cos

ix sinix i sinhx

i

ex ex

it follows that when cos i i So that in spherical coordinates

eikr

r

ik

Z i

Z

eikxxkyykz jzj sindd


This rst integral above is now taken to be a contour integral in the complex plane Weintegrate along the Re axis from to then we integrate to innity in the directionof negative Im

Finally there is one other very useful form for the Weyl integral It is achieved byintroducing the radial wave number kr k cos where is the angle of emergenceshown in Figure Then using an integral representation of the cylindrical Besselfunction cf "# the Weyl integral becomes

eikr

r

Z

Z

eirkr cosez

pkrk krdkrdq

kr k

Z

Jkrrezp

krkkrdkrqkr k

In this form the integral is known as the Sommerfeld integral and represents a sum overcylindrical waves The cylindrical Bessel functions arise from the separation of variablesof the wave equation in either cylindrical or spherical coordinates

A Few Words on Bessel Functions

In any of the standard books on special functions or what used to be called the functionsof mathematical physics you will see the following solutions of Bessel s equation

Jx x

Xj

j

j +j

x

j

Jx x

Xj

j

j +j

x

j

The limiting forms of these functions are

x Jx

+

x

x Jx s

xcosx

The transition from the small argument behavior to the large argument behavior occursin the region of x Also notice that the largeargument asymptotic form demonstratesthe existence of an innite number of roots of the Bessel function These are approximatedasymptotically by the formula

xn n

REFLECTION OF A SPHERICAL WAVE FROM A PLANE BOUNDARY

where xn is the nth root of J You will nd more than you ever wanted to know aboutBessel Functions in Watson s A Treatise on the Theory of Bessel Functions "# Any ofthe classic textbooks on analysis from the early part of this century will have chapters onthe various special functions Morse and Feshbach "# is another standard reference

Reection of a Spherical Wave from a Plane

Boundary

The reason for going to all this trouble is so that we can treat the problem of the reectionof a pulse from a planar boundary Clearly this is the crucial problem in reectionseismology We can write the total potential as the incident plus reected eld

eikr

r re

This is illustrated in Figure A plane wave reecting from a boundary producesa reected plane wave Therefore must be representable as a superposition of planewaves since the incident pulse is All we need do is to take into account the phase changeassociated with the travel time from the source location and the boundary and to scalethe amplitude of the reected plane waves by the angledependent reection coecientDenoting the angle of incidence by we have from the Sommerfeld integral

re Z

Jkrrehz

pkrkRkrdkrq

kr k

Z

JkrreihzkzR

ikzkrdkr

where R is the reection coecient and h is the distance of the source above the reectinglayer

The plane wave incidence angle is a function of both kr and In fact

kr

ccos

So we can write R ) kr Further so far we ve neglected the frequency content ofour point source so if we want to interpret Equation as for example the verticalpwave displacement potential or pressure of a source plus reected compressional wavesay r z then we need to scale the integral by the Fourier transform of the sourceexcitation function F This leads us to the rst equation of Treitel et al "# and is thereal jumping o point for the practical implementation of the plane wave decompositionof seismograms

r z F Z

) kr

Jkrreihzkz

ikzkrdkr


a a

source

receiver

R

h

z

(r, -z)

(0,-h) . .

Figure Reection of a pulse from a planar surface

Treitel et al interpret ) kr as the Fourier transform of the velocity potential of theplane wave component with angular frequency and horizontal wave number kr For asingle layer we know that in fact the reection coecient is independent of frequency Fora stack of layers it turns out that the eective reection coecient does indeed dependon the frequency And in that case we end up with a sum of phase shift terms associatedwith transmission of the plane wave through the layers above the reecting layers This isthe basis of the wellknown reectivity method for synthetic seismogram generation "#Let s not worry about the source spectrum F for now let us assume that the sourcehas a broadband spectrum so that we can assume F Then Equation hasthe form of a FourierBessel transform pair "# p For any function gr we candene a FourierBessel transform Gkr via the invertible transformation pair

gr Z

GkrJkrrkrdkr

Gkr Z

grJkrrrdr

This works just like a Fourier transform and with it we can invert the integral in Equation analytically to achieve

) kreihzkz

ikzZ

r zJ

r

csinr dr

For the moment let s not worry about the interpretation of ) If the medium is dened bya stack of horizontal layers then we can calculate ) straightforwardly using the methods

REFLECTION OF A SPHERICAL WAVE FROM A PLANE BOUNDARY

Figure A curve can be specied either as the locus of all points such that t txor the envelope of all tangents with slope p and intercept

in "# That s not the main goal of "# their main goal is to illuminate the connectionbetween slantstacks and plane wave decomposition It will turn out that the slantstacking procedure is essentially equivalent to plane wave decomposition That meansthat instead of attempting to evaluate the Sommerfeld integral numerically we can applythe slantstacking procedure which is widely used already in the exploration seismologycommunity

A Minor Digression on SlantStacks

Suppose we have some curve tx For now tx is a purely geometrical object we ll worryabout the seismic implications later Now tx represents an object the curve There isanother completely equivalent representation of this object namely as the envelope ofall tangents to the curve Figure This equivalent geometry of slopes and interceptsis known as Pluecker line geometry The conventional representation is known as pointgeometry Line geometry forms the basis of the Legendre transformation which is widelyused in physics to switch between the Lagrangian and Hamiltonian pictures and fromintensive to extensive variables An excellent discussion of the Legendre transformationcan be found in Callen s book Thermodynamics "#

If we know the analytic form of a curve it is straightforward to calculate its Legendretransformation into slopeintercept space Let s consider a few examples Since a straightline has only one slope and one intercept the Legendre transformation of a line is a pointIn other words the line t px gets mapped into the point p Next considerthe quadratic t x The family of tangents to this curve must satisfy t px wherep x In order that the lines t x intersect the curve t x it is necessary that x Eliminating x between the expressions for and p we see that the quadratict x gets mapped into the curve p Finally consider the family of hyperbolae

The reason for using p to represent the slope is that in seismology the slope is just the ray parameteror horizontal phase slowness


given by t x The family of tangents to these curves are given by t px wherep x

px In order for these lines to intersect the hyperbolae it is necessary and

sucient that px Eliminating x between the expressions for and p we

obtain the result that the Legendre transformation of a t x hyperbola is a p ellipse

p

In the case of seismic data we don t have an explicit relationship between the travel timet and the horizontal coordinate x or r Of course to the extent that we believe thatdiraction signatures are hyperbolic we should expect these to be mapped into pellipses But we really need a numerical way to compute the p transformation ofseismic data This is accomplished with the slantstack we sum along straight lineseach parameterized with a slope p and an intercept and deposit the resulting value inthe p section The reason this works is that unless the line we are summing alongis approximately tangent to the curve the summation results in a zero contribution Inother words the intersection of the line of integration and the curve we wish to slantstackis a set of measure zero and therefore contributes nothing to the slantstack On the otherhand a tangent has a higher order contact with a curve

If we were simply dealing with plane curves which could be parameterized by tx thenwe would have a complete picture slantstack equals Legendre transformation Withseismic data the curve we are slantstacking has some amplitude associated with it Soit sounds as if we should make the slantstack a summation of amplitudes along lines Inthe limiting case of an innite frequency wavelet we just recover the kinematical slantstack already discussed In the more general case we have what is known as a dynamicalslantstack

Slantstacks and the Plane Wave Decomposition

To show the connection between slantstacks and the plane wave decomposition followingTreitel et al "# we begin with an example of a slantstack operator

st cx Z

Z

ux y t xcxdx dy

where cx is the horizontal phase velocity and u is our seismic data it could be pressure orone component of displacement We are clearly summing along lines of constant horizontalphase velocity and parameterizing the result by that phase velocity Introducing polar

For the theory behind this check out a book on dierential geometry such as Struiks where theresult we need is on page

SLANTSTACKS AND THE PLANE WAVE DECOMPOSITION

coordinates

x r cos

y r sin

dx dy r dr d

we havest

Z

Z

ur t r cos sincrr dr d

or since we are summing along a line we can equivalently write this as a convolutionwith a delta function

st Z

Z

ur t t t cos r dr d

where t r sinc

We can do the angular integral easily Dene the intermediate function ht by

ht Z

t t cos d

Z

t t cos d

Letting w t cos we have

ht Z t

tt wtw dw

The result is that

ht

t t if t t otherwise

Thus the slantstack reduces to

st Z

"ut r ht r # r dr

Believe it or not the function we re convolving the data with ht r is a knownFourier transform pair of the th order cylindrical Bessel function One hesitates to saywellknown but in Oberhettinger s table of integrals edition p we nd that

ht r Fourier Transform

r sin

c

t

J

r

csin

We can now write the slantstack in a form which is almost identical to the version of theplane wave decomposition we had in Equation

s Z

u rJ

r

csin

r dr


For comparison we had for the plane wave decomposition of a pulse reected from ahorizontal boundary

) kreihzkz

ikzZ

r zJ

r

csin

r dr

where r z is the vertical pwave displacement potential or pressure ie eikrr and) was the angledependent reection coecient The dierences between the right sidesof these two equations are a that in the slantstack we only know the eld at the surfacez whereas for the plane wave decomposition the potential or pressure is assumedknown everywhere is space and b if we interpret as a potential say the vertical pwavedisplacement potential then we have to dierentiate it with respect to z in order to getthe observed data u Dierentiation with respect to z just cancels the factor of ikz in thedenominator leaving

) kreihzkz

Z

u r zJ

r

csin

r dr

We can interpret the phaseshift on the left side of this equation as an upward continuationoperator Once again following the notation of "# we suppose that the point source isburied at r z h and redene the vertical distance between the point source andthe receiver to be z Then we have

) kreikzz ) eizc cos

Z

u r zJ

r

csin

r dr

We ll simply dene this to be ) z Then the plane wave component at z ) is given exactly by the slantstack of the data

) Z

u r J

r

csin

r dr

Alternatively starting with Equation

) kreihzkz

ikzZ

r zJ

r

csinr dr

we can dierentiate with respect to z to get

) kreihzkz

Z

u r zJ

r

csinr dr

From this we infer that at the surface z we have

) kreihkz

Z

u r J

r

csinr dr

SLANTSTACKS AND THE PLANE WAVE DECOMPOSITION

Or we could start directly with

r z Z

Jkrr

ikzeihzkzRkr dkr

dierentiate this with respect to z to get rid of the ikz and convert the potential into adisplacement Then inverting the FourierBessel transform we end up with

ReihkzZ

u r Jkrrr dr

We ve only just scratched the surface of planewave decompositions reectivity methodsand slantstacks In addition to the paper by Fuchs and M(uller already cited "# ChrisChapman has written a whole series of really outstanding papers on these subjects Ofspecial note are his papers on Generalized Radon transforms and slant stacks "# and Anew method for computing synthetic seismograms "#

Chapter

Numerical Methods for Tracing Rays

On page we showed that the eikonal equation which is a nonlinear partial dierential equation could be reduced to an ordinary dierential equation by introducing newcoordinates dened in terms of the surfaces of constant phase The normals to theseisophasal surfaces were called rays Later we saw that the same ray equations followedautomatically from Fermat s principle Now we need to address the issue of how to solvethese ray equations for realistic media

There are essentially two classes of numerical methods for the ray equations In the rsta heterogeneous medium slowness as a function of space is approximated by many pieceswithin each of which the ray equations are exactly solvable For example in a piecewiseconstant slowness medium the exact solution of the ray equation must consist of a piecewise linear ray with Snell s law pertaining at the boundary between constantslownesssegments One can show that in a medium in which the slowness has constant gradientsthe ray equations have arcs of circles as solutions So if we approximate our medium byone in which the slowness has constant gradients then the raypath must be make up oflots of arcs of circles joined via Snell s law For lack of a better term let s call thesemethods algebraic methods On the other hand if we apply nite dierence methods tothe ray equations we can simply regard the slowness as being specied on a grid andnumerically determine the raypaths Let s call these methods of solving the ray equationsnite dierence methods

In practice both algebraic and nite dierence methods are applied to tomography whereraypaths must be known and to migration where we only need to know travel timesHowever it is probably true that algebraic methods are more popular for tomographyand nite dierence methods are more popular for migration One reason for this is thatnite dierence methods are easier to write and since Fermat s principle guarantees thatperturbations with respect to raypath are of higher order than perturbations with respectto the slowness the inaccuracies in raypaths due to not applying Snell s law exactly atboundaries have a relatively minor eect on the quality of a migration On the other hand

CHAPTER NUMERICAL METHODS FOR TRACING RAYS

in travel time tomography having accurate raypaths is sometimes vital for resolution

Finite Dierence Methods for Tracing Rays

We will now describe methods for solving the ray equations Equation when theslowness s is an arbitrary dierentiable function of x and z The generalization to raytracing in three dimensions is completely straightforward The methods to be discussedare classic and are described in much greater detail in the works of Stoer and Bulirsch "#and Gear "# We will consider rays dened parametrically as a function of arclengthor time or any other parameter which increases monotonically along the ray In otherwords a ray will be the locus of points x z as ranges over some interval If wewere to write the rays as simple functions zx where x is the horizontal coordinate thennumerical diculties would inevitably arise when the rays became very steep or turnedback on themselves if the ray becomes very steep then nite dierences $z$x maybecome unstable forcing us to switch to a xz representation and if the ray turns backon itself there will not be a singlevalued representation of the ray at all Arclengthparameterization avoids all this diculty allowing one for example to decide on a stepsize depending only on the intrinsic properties of the medium and not on the directionthe ray happens to be propagating

The rst step is to reduce the second order Equation

d

d

sr

d

dr

rs

to a coupled system of rst order equations The position vector r in Cartesian coordinatesis simply x%x z%z Denoting dierentiation with respect to the arclength parameter bya dot one has

dr

d 'x%x 'z%z

Carrying out the dierentiations indicated in Equation and equating the x and zcomponents to zero we get

s(x sx 'x sz 'z 'x sx

On the other hand a zx or xz representation is certainly a little simpler and results in a lowerorder system of equations to solve To get the ray equations in this form replace d withp

dx dz p xz dz or

p zx dx

in the travel time integral

t

Zray

sx zd

and proceed with the derivation of the EulerLagrange equations as we did at the end of the chapter onray theory

FINITE DIFFERENCE METHODS FOR TRACING RAYS

and

s(z sz 'z sx 'z 'x sz

Now to reduce a second order equation say y f to two rst order equations it isnecessary to introduce an auxiliary function say z Letting y z the equation y fis completely equivalent to the two coupled equations

y z

z f

Carrying out this procedure with Equation we arrive at the four coupled rstorder ordinary dierential equations ODE

'z z

'z sxz szzs

z sxs

'z z

'z szz sxzs

z szs

where z x 'z 'x z and so on

These are the coupled nonlinear ray equations in arclength parameterization We cansimplify the notation by considering the four functions zi i as being components of a vector z and the right hand side of Equation as being the componentsof a vector f Then Equation reduces to the single rst order vector ODE

dz

d fz s sx sz

Thus we can dispense with vector notation and without loss of generality restrict attention to the general rst order ODE

dz

dx fx z

where we will now use x as the independent variable to conform with standard notationand use primes to denote dierentiation The remainder of this chapter will be devotedto numerically solving initial value problems IVP and boundary value problems BVPfor Equation


Initial Value Problems

The initial value problem for Equation consists in nding a function zx whichsatises Equation and also satises

zx z

It can be shown that Equations and have a unique solution provided thata f satises the Lipschitz condition on the closed nite interval "a b# b zx iscontinuously dierentiable for all x in "a b# c zx fx zx for all x in "a b# andd zx z It follows from the mean value theorem of calculus that a sucientcondition for f to be Lipschitz is that all the partial derivatives of f with respect to zexist and be bounded and continuous

In addition to being much easier to solve than boundary value problems initial valueproblems have the important property that the solution always depends continuously onthe initial value In other words initial value problems are well posed For a completediscussion see "# The potential for illposedness will turn out to be a very seriousproblem for twopoint boundary value problems

The simplest solution method for an IVP is to approximate the derivative with a rstorderforward dierence so that Equation becomes

zx h zx

h fx zx

where h is the step size This gives zx h zx hfx zx which in turn givesEuler s method

z

i i hfxi i i

xi xi h

Thus the approximate solution consists of polygonal or piecewise linear approximationto the curve zx We start with an initial value of x and integrate using the recursionformula above for until we reach the desired ending value of x Euler s method is a

Remember x and z no longer refer to the Cartesian coordinates They are simply the independentand dependent variables of an abstract ODE

f is said to satisfy the Lipschitz condition if there exists a number N such that for all x jfx zfx zj N jz zj

INITIAL VALUE PROBLEMS

special case of the general onestep method

z

i i hxi i f h i

xi xi h

is the iteration function in other words is simply a rule which governs how theapproximation at any given step depends on the approximations at previous steps It canbe determined in various ways via quadrature rules truncation of Taylor series etc Forthe Euler method is simply the righthandside of the ODE The global discretizationerror of the method is errx h x h zx The method is convergent if the limitof errx hn where hn x xn is zero as n goes to innity If errx hn is hpnp the method is said to be of order p

Perhaps the most widely used onestep methods are of the RungeKutta type An exampleof a fourth order RungeKutta scheme is obtained by using the iteration function

x z h "k k k k#

where

k fx z

k fx h z kh

k fx h z kh

k fx h z kh

One way to organize the myriad of one step methods is as numerical integration formulaeFor example the solution of the IVP

zx fx zx z

has the solutionzx zx

Z x

xftdt

The RungeKutta method in Equation is simply Simpson s rule applied to Equation

Thus far we have given examples of singlestep integration methods the approximatesolution at any given step depends only on the approximate solution at the previous stepThe natural generalization of this is to allow the solution at the say kth step to dependon the solution at the previous r steps where r To get the method started one


must somehow compute r starting values r We will not consider multistepmethods in any further detail the interested reader is referred to "# and "# and thereferences cited therein

We have now developed sucient machinery to attack the initial value problem of raytracing By applying one of the singlestep solvers such as the RungeKutta method Equation to the ray equations Equation we can compute the raypath for anarbitrarily complex medium We can think of shooting a ray o from a given point inthe medium at a given takeo angle with one of the axes and tracking its progress withthe ODE integrator This turns out to be all we really need to do migration We don treally care where a ray goes we just want to know what travel time curves look like sowe can do migration On the other hand if we are doing travel time tomography then weneed to know not where a given ray goes but which ray or rays connects a xed sourceand receiver Thus we need to be able to solve the boundary value problem BVP forEquation

Boundary Value Problems

The numerical solution of twopoint BVP s for ordinary dierential equations continuesto be an active area of research The classic reference in the eld is by Keller "# Thedevelopment given here will only scratch the surface it is intended merely to get theinterested reader started in tracing rays

We now consider boundary value problems for the ray equations The problem is tocompute a ray zx such that

dz

dx fx z subject to za za and zb zb

The simplest solution to the BVP for rays is to shoot a ray and see where it ends upie evaluate zb If it is not close enough to the desired receiver location correct thetakeo angle of the ray and shoot again One may continue in this way until the ray endsup suciently close to the receiver Not surprisingly this is called the shooting methodThis idea is illustrated in Figure which shows the geometry for a reection seismicexperiment

The method of correcting the takeo angle of subsequent rays can be made more rigorousas follows By shooting we replace the BVP Equation with an IVP

dz

dx fx z za za and za

and adjust Thus we can view a solution of this IVP as depending implicitly on z zx The BVP then is reduced to the problem of nding roots of the functionzb zb

BOUNDARY VALUE PROBLEMS

s r

Figure Simple shooting involves solving the initial value problem repeatedly withdierent takeo angles iteratively correcting the initial conditions until a satisfactorysolution of the boundary value problem is achieved

Having now reduced the problem to one of nding roots we can apply the many powerfultechniques available for this purpose among which Newton s method is certainly one ofthe best Newton s method is so important underlying as it does virtually all methodsfor nonlinear inversion that it merits a brief digression

Newtons Method

Suppose you want to nd roots of the function fx ie those points at which f is zeroIf f is the derivative of some other function say g then the roots of f correspond tominima of g so root nding and minimization are equivalent

Let denote the root of f to be computed Expanding the function f about in a Taylorseries

f fx f x x

we see that neglecting higher order terms

x fx

f x

To see intuitively what this expression means consider the parabolic function shown inFigure The tangent to the curve at x fx satises the equation

y f xx *y

where *y is the intercept with the yaxis Eliminating *y from the following two simultaneousequations

fx f xx *y

f xx *y


x x

1 0

Figure Newton s method for nding roots

gives

x x fx

f x

So we can interpret Newton s method as sliding down to the xaxis along a tangent tothe curve associated with the initial point x Thus will be closer to one of the tworoots for any starting point we choose save one At the midway point between the tworoots the slope is zero and our approximation fails Having computed call it wecan insert it back into the above expression to arrive at the next approximation for theroot

f

f

Continuing in this way we have the general rst order Newton s method for nding theroots of an arbitrary dierentiable function

n n fn

f n

Newton s method can be shown to be quadratically convergent for suitably wellbehavedfunctions f provided the initial guess is close enough to the root "# p

Higher order Newtontype methods can be obtained by retaining more terms in the truncated Taylor series for f For example retaining only the linear terms amounts to approximating the function f with a straight line Retaining the second order term amountsto approximating f with a quadratic and so on It is easy to see that the second orderNewton method is just

n n f n "f n fnf n#

f n


Newton s method has a very natural generalization to functions of more than one variablebut for this we need the denition of the derivative in Rn ndimensional Euclidean spaceWriting the function fx which maps from Rn into Rm in terms of components as

fx

fx x xn

fx x xn

fmx x xn

the derivative of f is dened as the linear mapping D which satises

f fx Dfx x

in the limit as x Dened in this way D is none other than the Jacobian of f

Dfx

fx

fxn

fmx

fmxn

So instead of the iteration

n n fn

f n

we have

n n Dfn

fn

provided of course that the Jacobian is invertible

Example

Let s consider an example of Newton s method applied to a D function

qx x x

We haveqx x x

and thereforeqx

qx

x x

x x

Newton s iteration is then

xi xi xi xixi xi


The function q has four roots at and p If we choose a starting point equal to say we converge to

p to an accuracty of one part in a million within iterations Now

the point Newton s method converges to if it converges at all is not always obvious Toillustrate this we show in Figure a plot of the function q above a plot of the points towhich each starting value uniformly distributed along the abscissa converged In otherwords in the bottom plot there tickmarks representing starting values distributeduniformly between and along the xaxis The ordinate is the root to which each ofthese starting values converged after iterations of Newton s method Every startingpoint clearly converged to one of the four roots zero is carefully excluded since q is zerothere But the behavior is rather complicated It s all easy enough to understand bylooking at the tangent to the quadratic at each of these points the important thing tokeep in mind is that near critical points of the function the tangent will be very nearlyhorizontal and therefore a single Newton step can result in quite a large change in theapproximation to the root One solution to this dilemma is to use steplength relaxationIn other words do not use a full Newton step for the early iterations See "# for details

Variations on a Theme Relaxed Newton Methods

The archetypal problem is to nd the roots of the equation

gx

The most general case we will consider is a function g mapping from Rm to Rn Such afunction has n components gix x xm

Newton s method can be written in the general case as the iteration

xk xk Hkgxk

where Hk is the inverse of the Jacobian matrix Jxk Now let us relax the strict denitionof H and require that

limk

kI HkJxk

Such an algorithm will have similar convergence properties as the full Newton s methodThe details of this theory can be found in books on optimization such as "# We give afew important examples here

Hk Jx An example of this would be step length relaxation An exampleof step length relaxation on the function shown in Figure is given in "# HereHk akJ

x where ak


100 200 300 400 500 600

-3

-2

-1

1

2

3

-3 -2 -1 1 2 3

5

10

15

Figure Newton s method applied to a function with four roots and Thebottom gure shows the roots to which each of starting values uniformly distributedbetween and converged The behavior is complex in the vicinity of a critical pointof the function since the tangent is nearly horizontal


Use divided dierences to approximate the derivatives For example with n wehave

xk xk xk xkgxk

gxk gxk

Adjoint approximation Take Hk akJT xx This approximation is extremely

important in seismic inversion where it corresponds essentially to performing aprestack migration given that g is a waveform mist function See "# for moredetails

WellPosed Methods For Boundary Value Problems

Conceptually the simplest method for solving twopoint boundary value problems for theray equations is shooting we solve a sequence of initial value problems which come closerand closer to the solution of the boundary value problem Unfortunately shooting canbe illposed in the sense that for some models minute changes in the takeo angles ofrays will cause large changes in the endpoints of those rays This is especially true nearshadow zones In fact one is tempted to say that in practice whenever a shooting coderuns into trouble it s likely to be due to the breakdown in geometrical optics in shadowzones Put another way the rays which cause shooting codes to break down almost alwaysseem to be associated with very weak events

On the other hand there are many techniques for solving boundary value problems forrays which are not illposed and which will generate stable solutions even for very weakevents in shadow zones We will now briey mention several of them First there ismultipleshooting This is an extension of ordinary shooting but we choose n pointsalong the xaxis and shoot a ray at each onejoining the individual rays continuously atthe interior shooting points as illustrated in Figure

Bending is an iterative approach to solving the ray equations based on starting with someinitial approximation to the twopoint raypathperhaps a straight linethen perturbingthe interior segments of the path according to some prescription This prescription mightbe a minimization principle such as Fermat or it might employ the ray equation directlyOne especially attractive idea along these lines is to apply a nite dierence scheme tothe ray equations This will give a coupled set of nonlinear algebraic equations in place ofthe ODE The boundary conditions can then be used to eliminate two of the unknownsThen we can apply Newton s method to this nonlinear set of equations the initial guessto the solution being a straight line joining the source and receiver For more details see"#

ALGEBRAIC METHODS FOR TRACING RAYS

x x x x x x 1 2 3 4 5 6

Figure Multipleshooting involves shooting a number of rays and joining them continuously

Algebraic Methods for Tracing Rays

If the slowness or velocity assume certain functional forms then we can write downanalytic solutions to the ray equation Cerveny "# gives analytic solutions for variousmedia including those with a constant gradient of velocity c constant gradient of cn andln c and constant gradient of slowness squared s For example if the squared slownesshas constant gradient

s a ax az ay

then the raypaths are quadratic polynomials in arclength This is probably the simplestanalytic solution for an inhomogeneous medium Similarly if the velocity has a constantgradient then the raypath is an arc of a circle However the coecients of this circular representation are much more cumbersome than in the constant gradient of squaredslowness model

We can readily extend this sort of approach to media that satisfy these special propertiesin a piecewise fashion For example if the model is composed of blocks or cells withineach of which the velocity has a constant gradient and across which continuity of velocityis maintained then the analytic raypath will consist of smoothly joined arcs of circlesFor more details of this approach see the paper by Langan Lerche and Cutler "# whichinuenced many people doing seismic tomography

Distilled Wisdom

We have presented all of the theoretical tools necessary to write an ecient and accurateraytracer for heterogeneous media However there are many practical considerationswhich arise only from years of experience in attempting to apply these methods There


is probably no better source of this distilled wisdom than the short paper by Sam Grayentitled Ecient traveltime calculations for Kirchho migration "# Gray shows howinterpolation of traveltime curves and downward continuation of rays can signicantlyreduce the computational costs of traveltime based migration methods

Eikonal Equation Methods

During the past or years there has been an explosion of interest in computing traveltimes by solving the eikonal equation directly The main reason for this is that nite difference methods for the eikonal equation are well suited for both vector oriented machinessuch as the Cray !Cray XMP and massively parallel machines such as the CM!CM In this brief discussion we will content ourselves to look at just one aspect of thisproblem as described in the work of Vidale and Houston "#

Suppose we want to solve the eikonal equation krtk s on a square D grid of size hConsider a single unit of this grid bounded by four points whose travel times are goingcounter clockwise starting at the lower left t t t t So points and are diagonallyopposite Approximating the travel times at grid points and as rstorder Taylorseries about the point we have

t t ht

y

t t ht

x

and

t t h

t

x

t

y

Putting these together we have

ht

y t t t t

and

ht

x t t t t

which gives

h

t

x

t

y

t t t t

t t t t

This is our nite dierence approximation to the right hand side of the eikonal equationIf you carry though the algebra you will see that most of the cross terms cancel so that

COMPUTER EXERCISES V

we are left with just t t t t Now the squared slowness we approximateby its average value at the grid points

s s

s s s s

The result is Vidale s approximation for the eikonal equation

t t t t

sh

ort t

qsh t t

For recent work on the use of the eikonal equation in Kirchho migration see "# and"# Also see "# for a massively parallel implementation

Computer Exercises V

Now extend your Kirchho code from Exercise IV by accurately calculating travel timesin a general cx z medium Whether you do this by tracing rays or by solving the eikonalequation is up to you At this point you should not be concerned about eciency Tryto get something that works then worry about making it fast

Chapter

Finite Dierence Methods for WavePropagation and Migration

The most widely used technique for modeling and migration in completely heterogeneousmedia is the method of nite dierences We will consider nite dierences applied totwo generic equations the parabolic equation

u

t K

u

x

which arises in this context from Claerbout s paraxial approximation to the wave equationand the hyperbolic wave equation

u

t c

u

x

where K and c are functions of space but not of time or u Our treatment of nitedierences will be extremely terse For more details the reader is urged to consult thestandard references including Smith s Numerical solution of partial dierential equations nite dierence methods "# from which much of the rst part of this chapter isdirectly drawn Mitchell s Computational methods in partial dierential equations "#and Richtmyer and Morton s Dierence methods for initial value problems There arecountless ne references to the theory of partial dierential equations but an especiallyuseful one by virtue of its brevity readability and rigor is John s Partial DierentialEquations "# In particular we have relied on John s discussion of characteristics andthe domain of dependence of PDE s and their associated nite dierence representations

First a little notation and nomenclature We use U to denote the exact solution of somePDE u to denote the exact solution of the resulting dierence equation and N to denotethe numerical solution of the dierence equation Therefore the total error at a grid pointi j is

Uij Nij Uij uij uij Nij

CHAPTER FINITE DIFFERENCE METHODS FOR WAVE PROPAGATION AND MIGRATION

which is just the sum of the discretization error Uij uij and the global rounding erroruij Nij

Let Fiju denote the dierence equation at the i j mesh point The local truncationerror is then just FijU If the local truncation error goes to zero as the mesh spacinggoes to zero the dierence equation is said to be consistent

Probably the most important result in numerical PDE s is called Lax s Theorem If alinear nite dierence equation is consistent with a wellposed linear initial value problemthen stability guarantees convergence A problem is said to be wellposed if it admits aunique solution which depends continuously on the data For more details see "#

Ultimately all nite dierence methods are based on Taylor series approximations of thesolutions to the PDE For example let U be an arbitrarily dierentiable function of onereal variable then including terms up to order three we have

Ux h Ux hU x hU x hU x

and

Ux h Ux hU x hU x hU x

First adding then subtracting these two equations we get respectively

Ux h Ux h Ux hU x

and

Ux h Ux h hU x

Thus we have the following approximations for the rst and second derivative of thefunction U

U x Ux h Ux Ux h

h

and

U x Ux h Ux h

h

Both of these approximations are secondorder accurate in the discretization ie OhThis means that the error will be reduced by onefourth if the discretization is reduced byonehalf Similarly if in Equations we ignore terms which are Oh andhigher we end up with the following rstorder forward and backward approximations toU x

U x Ux h Ux

h

and

U x Ux Ux h

h

PARABOLIC EQUATIONS

Parabolic Equations

In the rst part of this chapter we will consider functions of one space and one timevariable This is purely a notational convenience Where appropriate we will give themore general formulae and stability conditions The discretization of the space and timeaxes is x ix ih t jt jk for integers i and j Then we have Uih jk UijThe simplest nite dierence approximation to the nondimensional constant coecientheat equation

u

t

u

x

is thusuij uij

k

uij uij uijh

Explicit TimeIntegration Schemes

Now suppose that we wish to solve an initialboundary value problem for the heat equation In other words we will specify Ux t for x at the endpoints of an interval say " #and Ux for all x Physically this corresponds to specifying an initial heat prolealong a rod and requiring the endpoints of the rod have xed temperature as if theywere connected to a heat bath at constant temperature

We can rewrite Equation so that everything involving the current time step j ison the right side of the equation and everything involving the new time step j is onthe left

uij uij r"uij uij uij#

where r kh

tx

As an illustration of this nite dierence scheme consider the following initialboundaryvalue problem for the heat equation

U

t

U

x

Ux t Ux t

Ux

x if x x if x

Figure shows the solution computed for time steps using an x discretization x and a t discretization t thus giving r The errors are negligible except atthe point x where the derivative of the initial data are discontinuous On the otherhand if we increase t to so that r the errors become substantial and actually


FD r=.1

Analytic

RMS difference

24

6

8

102

4

6

8

10

0

0.25

0.5

0.75

1

24

6

8

102

4

6

8

10

0

25

.5

75

1

00.2

0.40.6

0.8

1 0

0.002

0.004

0.006

0.008

0.01

00.25

0.5

0.75

1

00.2

0.40.6

0.8

1 0

0.002

0.004

0.006

0.00

0.

025

.5

5

1

2

4

6

8

10

2

4

6

8

10

0

0.02

0.04

2

4

6

8

10

2

4

6

8

10

0

02

4

Figure Finite dierence and analytic solution to the initialboundary value problemfor the heat equation described in the text The discretization factor r The errorsare negligible except at the point x where the derivative of the initial data isdiscontinuous

PARABOLIC EQUATIONS

Analytic

RMS difference

FD r=.5

2

4

6

8

10

2

4

6

8

10

00.050.1

0.150.2

0.25

2

4

6

8

10

2

4

6

8

10

005.11525

00.2

0.40.6

0.8

1 0

0.002

0.004

0.006

0.008

0.01

00.25

0.5

0.75

1

00.2

0.40.6

0.8

1 0

0.002

0.004

0.006

0.00

0.

025

.5

5

1

24

6

8

102

4

6

8

10

0

0.25

0.5

0.75

1

24

6

8

102

4

6

8

10

0

25

.5

75

1

Figure Finite dierence and analytic solution to the initialboundary value problemfor the heat equation described in the text The discretization factor r The errorsare substantial and growing with time


grow with time Figure To see just what is happening let us write this explicittime integration as a linear operator Equation can be expressed as

ujujujuNj

r r r r r r r r

r r

ujujujuNj

or as

uj A ujwhere we use a boldface u to denote a vector associated with the spatial grid The singlesubscript now denotes the time discretization We have by induction

uj Aj u

To investigate the potential growth of errors let s suppose that instead of specifying uexactly we specify some v which diers slightly from the exact initial data The erroris then e u v Or at the jth time step

ej Aj e

A nite dierence scheme will remain stable if the error ej remains bounded as j This in turn requires that the eigenvalues of A be less than since ej results from therepeated application of the matrix to an arbitrary vector

Let us rewrite the tridiagonal matrixA as the identity matrix plus r times the tridiagonal

Suppose xs is the sth eigenvector of A and s is the sth eigenvalue Any vector in particular theinitial error vector can be written as a summation of eigenvectors assuming the matrix is full rank withcoecients cs

e X

csxs

Therefore

ej X

csjsxs

PARABOLIC EQUATIONS

matrix Tn where

Tn

The matrix Tn has been wellstudied and its eigenvalues and eigenvectors are known

exactly "# p In particular its eigenvalues are sinsN

The more general

result is that the eigenvalues of the N N tridiagonal

a b c a b c a b

c a

are

s a pbc cos

s

N

This means that s the eigenvalues of A are given by

s r sin

s

N

So the question is for what values of r is this expression less than or equal to one Itis not dicult to show that we must have r sin

sN

and hence if r then

s

This feature of the explicit timeintegration scheme that the stepsize must be strictlycontrolled in order to maintain stability is common of all explicit schemes whether forthe parabolic heat equation or the hyperbolic wave equation It is possible however toderive nite dierence schemes which are unconditionally stable Ie stable for all nitevalues of r It is important to keep in mind however that just because a method might bestable for a large value of r does not mean that it will be accurate But as we will nowshow it is possible to derive nite dierence methods where the constraints on r dependonly on the desired accuracy of the scheme and not its stability

Implicit TimeIntegration Schemes

The method of Crank and Nicolson involves replacing the second derivative term in theheat equation with its nite dierence representation at the jth and j st time stepsIn other words instead of Equation we have

uij uijk


uij uij uij

huij uij uij

h

This is an example of an implicit timeintegration scheme It is said to be implicit becausethe nite dierence equations dene the solution at any given stage implicitly in terms ofthe solution of a linear system of equations such as Equation

Using the same notation as for the explicit case we can write the CrankNicolson timeintegration as

I rTnuj I rTnuj

Formally the solution of this set of equations is given by

uj I rTn I rTnuj

We don t need to calculate the inverse of the indicated matrix explicitly For purposes ofthe stability analysis it suces to observe that the eigenvalues of the matrix

A I rTn I rTn

will have modulus less than one if and only if

r sinsN

r sin

sN

And this is clearly true for any value of r whatsoever

The fact that these eigenvalues have modulus less than for all values of r implies unconditional stability of the CrankNicolson scheme However do not confuse this stabilitywith accuracy We may very well be able to choose a large value of r and maintain stability but that does not mean the nite dierence solution that we compute will accuratelyrepresent the solution of the PDE Now clearly x and t must be controlled by thesmallest wavelengths present in the solution So it stands to reason that if the solutionof the PDE varies rapidly with space or time then we may need to choose so small avalue of the grid size that we might as well use an explicit method The tradeo here isthat with CrankNicolson we must solve a tridiagonal system of equations to advance thesolution each time step

But solving a tridiagonal especially a symmetric one can be done very eciently Algorithms for the ecient application of Gaussian elimination can be found in NumericalRecipes or the Linpack Users Guide The latter implementation is highly ecient Howcan we judge this tradeo If we count the oating point operations op implied by theexplicit timeintegration scheme Equation we arrive at N oating point operations per time step where N is the number of x gridpoints Now x and hence N canbe taken to be the same for both the explicit and the implicit methods The dierence

We count a multiply as one op and an add as one op So the expression ax y implies ops

PARABOLIC EQUATIONS

Floating point operations per divide

Explicit/Implicit ratio of work

12 14 16 18 20

1.4

1.6

1.8

Figure Assuming r for the explicit method and r for the implicit here isa plot showing the ratio of work between the two methods as a function of the number ofoating point operations per divide on a particular computer With a typical value of or oating point operations per divide we can see that the work required for the twomethods is very similar notwithstanding the fact that we need to use ten times as manytime steps in the explicit case

will come in t The number of oating point operations required to perform Gaussianelimination on a symmetric tridiagonal is approximately F N

where F is the

number of oating point operations required to do a divide This will vary from machineto machine but is likely to be in the range of Now N will be large so we can ignorethe dierence between N and N Therefore the ratio of work between the explicit andthe implicit methods will be approximately

N

F N

F

per time step But the whole point is that we may be able to use a much larger timestep for the implicit method Suppose for example that we need to use an r of for theexplicit methods but that we can use an r of for the implicit approach That meansthat we ll need ten times as many time steps for the former as the latter Therefore theratio of work between the two methods to integrate out to a xed value of time is

F

or more generally

F

where is the ratio of the number of explicit time steps to the number of implicit timesteps


Weighted Average Schemes

A class of nite dierence approximations that includes both the fully explicit and theCrankNicolson methods as special cases is based upon a weighted average of the spatialderivative terms

uij uijk

uij uij uij

h

uij uij uij

h

If we are left with the usual explicit scheme while for we have CrankNicolson For we have a socalled fully implicit method For the method isunconditionally stable while for stability requires that

r t

x

Separation of Variables for Dierence Equations

It is possible to give an analytic solution to the explicit nite dierence approximationof the heat equation in the case that the solution to the PDE is periodic on the unitinterval Just as in the case of classical separation of variables we assume a solution ofthe dierence equation

uij ruij ruij ruij

of the formuij figj

so thatgjgj

rfi rfi rfi

fi

As in the standard approach to separation of variables we observe at this point thatthe left side of this equation depends only on j while the right side depends only on iTherefore both terms must actually be constant

gjgj

rfi rfi rfi

fi C

where C is the separation constant The solution to

gj Cgj

isgj ACj

HYPERBOLIC EQUATIONS

The other equation

fi r C

rfi fi

presumably must have a periodic solution since the original PDE is periodic in spaceTherefore a reasonable guess as to the solution of this equation would be

fi B cosi D sini

We won t go through all the details but it can be shown using the known eigenvalues ofthe tridiagonal that

C r sins

N

and hence a particular solution of the dierence equation is

uij E

r sins

N

jsin

si

N

We can generalize this to a particular solution for given initial data by taking the coecients Es to be the Fourier coecients of the initial data Then we have

uij Xs

Es

r sin

s

N

jsin

si

N

Hyperbolic Equations

Now let us consider the rstorder hyperbolic equation

Ut cUx

where for the time being c is a constant Later we will generalize the discussion to allowfor arbitrarily heterogeneous wave speed

Along the line dened by x ct constant we have

dU

dt

d

dt"Uct t# cUx Ut

Therefore U is constant along such lines Dierent values of give dierent such linesAs a result the general solution of Equation can be written

Ux t fx ct f

where the function f denes the initial data since Ux fx


x

x x - ct =

(x,t)

t

x

Figure Domain of dependence of the rstorder hyperbolic equation

The lines x ct dene what are known as characteristics The value of the solution U atan arbitrary point x t depends only on the value of f at the single argument In otherwords the domain of dependence of U on its initial values is given by the single point This is illustrated in Figure

Following John "# we now show that a simple rstorder forwarddierence scheme ofthe type employed for the heat equation leads to an unstable algorithm when appliedto a hyperbolic equation This leads to the notion that the domain of dependence of adierence equation must be the same as that of the underlying PDE this is the CourantFriedrichsLewy condition

Proceeding in the same fashion as for the heat equation let us consider the followingrstorder nitedierence approximation of Equation

uij uij rc"uij uij#

where r kh tx Dening the shift operator E via Euij uij we have

uij rc rcE uij

and hence by inductionuiN rc rcEN ui

orux t uxNk rc rcEN fx

Using the binomial theorem we have

ux t NX

m

Nm

rcmrcENmfx

HYPERBOLIC EQUATIONS

and hence

ux t NX

m

Nm

rcmrcNmfx N mh

The rst indication that something is wrong with this scheme comes from the observationthat the domain of dependence of the dierence equation appears to be

x x h x h x Nh

whereas the domain of dependence of the PDE is xct which in this case is xcNkSince this is not even in the set x ih it is clear that in the limit as k the dierencescheme cannot converge to the solution of the PDE Further the errors associated withsome perturbed value of the initial data grow exponentially according to

NX

m

Nm

rcmrcNm rcN

On the other hand if we use a backward dierence approximation in space then we have

ux t uxNk rc rcEN fx

and

ux t NX

m

Nm

rcmrcNmfx N mh

Now the domain of dependence of the dierence equation is

x x h x h xNh x tr

This set has the interval "x tr x# as its limit as the mesh size goes to zero providedthe ratio r is xed Therefore the point x ct will be in this interval if

rc

This is referred to variously as the CourantFriedrichsLewy condition the CFL condition or simply the Courant condition

Using the backward dierence scheme Equation we see that the error growthis governed by

NX

m

Nm

rcmrcNm rc rcN

which gives us condence in the stability of the method provided the CFL criterion issatised

This is a worstcase scenario in which the error has been cleverly chosen with alternating sign tocancel out the alternating sign in the binomial result


The Full TwoWay Wave Equation

By this point it should be clear how we will solve the full wave equation by nite differences First we show a nite dierence formula for the wave equation in two spacedimensions which is secondorder accurate in both space and time Then we show ascheme which is fourthorder accurate in space and second in time In both cases we takethe grid spacing to be the same in the x and z directions x z

uijk Cijuijk

Cij "uijk uijk uijk uijk#

uijk

whereCij cijdtdx

cij is the spatially varying wavespeed and where the rst two indices in u refer to spaceand the third refers to time The CFL condition for this method is

t x

cmaxp

where cmax is the maximum wavespeed in the model

A scheme which is fourthorder accurate in space and secondorder accurate in time is

uijk Cijuijk

Cij f "uijk uijk uijk uijk#

"uijk uijk uijk uijk#g uijk

The CFL condition for this method is

t x

cmax

q

For more details see "# A straightforward extension of these explicit nite dierenceschemes to elastic wave propagation is contained in the paper by Kelly et al "# Thispaper exerted a profound inuence on the exploration geophysics community appearing

THE FULL TWOWAY WAVE EQUATION

1

.5

0 0 5 10 15 20 25 30 35 40

Frequency (hz)

Figure Hypothetical normalized source power spectrum The number of grid pointsper wavelength may be measured in terms of the wavelength at the upper halfpowerfrequency

as it did at a time when a new generation of digital computers was making the systematicuse of sophisticated nite dierence modeling feasible

Strictly speaking the boxed nite dierence equations above are derived assuming ahomogeneous wavespeed Nevertheless they work well for heterogeneous models in somecases especially if the heterogeneities are aligned along the computational grid Moregeneral heterogeneous formulations require that we start with a wave equation which stillhas the gradients of the material properties Although this generalization is conceptuallystraightforward the resulting nite dierence equations are a lot more complicated Fordetails see "# In particular this paper gives a nite dierence scheme valid for PSVwave propagation in arbitrarily heterogeneous D media It requires more than a fullpage just to write it down

Grid Dispersion

The question remains as to how ne the spatial grid should be in order to avoid numericaldispersion of the sort we rst studied in the section on the D lattice page This measure of the grid size for dispersion purposes is given in terms of the numberof gridpoints per wavelength contained in the source wavelet But which wavelengththat associated with Nyquist with the peak frequency or some other Alford et alintroduced the convention of dening the number of grid points per wavelength in termsof the wavelength at the upper halfpower frequency This is illustrated in Figure fora hypothetical source wavelet Measured in these terms it is found that for best results atleast grid points per wavelength are required for the secondorder accurate schemeshown in the rst box above Whereas using the fourthorder scheme this could berelaxed to around grid points per wavelength This means that using the fourthorder


scheme reduces the number of grid points by a factor of whereas the number of oatingpoint operations per time step goes up by only 0

Absorbing Boundary Conditions

We won t go into much detail on the problem of absorbing boundary conditions Theidea is quite simple Whereas the real Earth is essentially unbounded except at the freesurface our computational grid needs to be as small as possible to save storage If wedidn t do anything special at the sides and bottom of the model the wave propagatingthrough this grid would think it was seeing a free surface when it reached one of thesethree sides Clayton and Engquist "# proposed replacing the full wave equation with itsparaxial approximation near the nonfreesurface edges of the model Using a paraxialequation ensures oneway propagation at the edges of the model Since the paraxialapproximations they use are rst order in the spatial extrapolation direction only thenearest interior row of grid points is needed for each absorbing boundary

Making paraxial approximations denes a preferred direction of propagation typicallyalong one of the spatial axes This means that waves incident normally upon one of theabsorbing sides of the model can be attenuated quite nicely But wave incident at anangle will only be approximately attenuated If one knew the direction of propagation ofthe waves which were to be attenuated it would be possible to rotate coordinates in sucha way that waves propagating in that direction were preferentially attenuated SimilarlyClayton and Engquist show how a rotation can be used to take special account of thecorners of the model The detailed formulae for the ClaytonEngquist scheme are givenin "# and implemented in an example in the next section The key idea however is thatwe use a standard explicit time integration scheme for all the interior points except forthose immediately next to one of the sides we wish to be absorbing

Finite Dierence Code for the Wave Equation

In this appendix we give an extremely simple but nonetheless functioning code for solvingthe acoustic wave equation via nite dierences This code is for D and is secondorderaccurate in both space and time It is written in Fortran for use on the ConnectionMachine It runs at around Gigaops on a full CM and was written by JacekMyzskowski of Thinking Machines Inc

c Fortran implementation of a second order

c spacetime explicit finite difference scheme

FINITE DIFFERENCE CODE FOR THE WAVE EQUATION

Figure Waves on a discrete string The two columns of gures are snapshots in timeof waves propagating in a D medium The only dierence between the two is the initialconditions shown at the top of each column On the right we are trying to propagate adiscrete delta function one grid point in space and time On the left we are propagating abandlimited version of this The medium is homogeneous except for an isolated reectinglayer in the middle The dispersion seen in the right side simulation is explained by lookingat the dispersion relation for discrete media Waves whose wavelengths are comparableto the grid spacing sense the granularity of the medium


c In this version the boundary conditions are

c ignored The ony reason for using Fortran

c in this case is that it runs fast on a

c Connection Machine

parameter f

parameter nxnyntnratensnap

real pnxny pnxny pnxny cnxny pmax

logical masknxny

pi atan

printmake a simple layered model

cnxny!

cnxny!"ny!

cnxny!"ny!

cnxny!"ny

printfinish model generation

cmin minvalc

cmax maxvalc

printcmincmaxcmincmax

c

c grid points per wavelength max frequency f

c

dx cmin!f

dt dx!cmaxsqrt

dtbydxdt!dx

c cdtbydx

c cc

c put a point source in the middle of the grid

ix nx!

iy ny!

c Start time iterations

c

c Make the ricker wavlet

c

t !f

w pif

c

mask false

maskixiy true


c begin time integration

do itnt

c

c Calculate the source amplitude

c

t it dt t

arg minwt!

arg minwt " pi!w !

arg minwt pi!w !

amp arg exparg

amp amp arg exparg

amp amp arg exparg

c

c Loop over space

c

p c p

" ccshiftp "cshiftp

" cshiftp "cshiftp

p

wheremask p p " amp

pp

pp

continue

stop

end

In a more sophisticated version we dene auxilliary arrays of weights in order to eectClaytonEngquist absorbing boundary conditions on three of the four sides of the computational grid So rst we dene the weights then we put them into the nite dierencestencil

c

c Precompute this to save run time


c Inside weights

c

pcc

c p

c p

c p

c p

c p

c

p

ifitypeeq then

c Top side weights

mask false

maskny true

wheremask

c

c

c

c

c

c

end where

c Bottom side weights

mask cshiftmask

wheremask

c c

c

c

c c

c

c

end where

c Left side weights

mask false

masknx true

wheremask

c c

c

c c


c

c

c

end where

c Right side weights

mask cshiftmask

wheremask

c c

c c

c

c

c

c

end where

c Right lower corner weights

mask false

masknxny true

masknxny true

masknxny true

wheremask

c c

c c

c

c c

c

c

end where

c Right upper corner weights

masknxny false

mask cshiftmask

maskny true

wheremask

c c

c c

c

c

c c

c

end where


c Left upper corner weights

maskny false

mask cshiftmask

mask true

wheremask

c c

c

c c

c

c c

c

end where

c Left lower corner weights

mask false

mask cshiftmask

masknx true

wheremask

c c

c

c c

c c

c

c

end where

end if

stuff deleted

c

c Loop over space

c

p cp

"ccshiftp "ccshiftp

"ccshiftp "ccshiftp

"cp

wheremask p p " amp

pp

pp

COMPUTER EXERCISES VI

0

1

2

3

4

20 40 60 80 100Trace number

Tim

e (s

econ

ds)

Figure The rst of common source gathers that comprise the nal computerexercise Figure out the model and produce a migrated image Drilling is optional

continue

etc

Computer Exercises VI

Figure shows the rst common source gather from a synthetic data set containing such gathers Each gather has traces whose receiver locations are xed throughoutthe experiment The source moves across the model from left to right The receiverspacing is The source spacing is The traces were recorded at ms sampling samples per trace Your job is to discover the model and produce a migrated imageusing your nowfancy Kirchho migration codes

These data are available via ftp from hilbertminescoloradoedu in the directory pubdata


Computer Exercises VI Solution

Figure shows the model that was used to produce the data in Figure In thenext two gures we see the results obtained by Wences Gouveia and Tariq AlkhalifahThe recording aperature was limited strictly to the km shown so no reections fromthe outside of the structure were recorded Because of the ray bending and the dip ofthe syncline it is very dicult to get a good image of the sides of the syncline and thedipping reectors below the syncline These are both extremely good images under thecircumstances Further no information was given on the velocity model all this had tobe deduced from the data

Here is the processing sequence that Gouveia applied to achieve the result shown inFigure

NMO correction with a constant velocity the velocity of the rst layerestimated by a sequence of Stolt migrations as a preprocessing forDMO

Constant velocity DMO in the frequency domain to remove the eect ofdip on the stacking velocities

Inverse NMO with the velocity used in step

Velocity analysis via the standard semblance methods

Conversion of the stacking velocities to interval velocities using the Dixequation as a rst approximation

Post stack depth migration using modied Kirchho for handling lateralvelocity variations with the velocities estimated in the previous itemRenements on this velocity prole was accomplished by repeating thepost stack migration for slightly dierent velocity models

And here is Alkhalifah s account of his eorts to achieve the results shown in Figure

Because the synthetic data containing only three layers seemed at rst to bereasonably easy I went through a normal sequence This sequence is basedon applying constant velocity NMO constant velocity DMO this step is independent of velocity inverse NMO velocity analysis NMO with velocitiesobtained from velocity analysis stack and nally Kirchho poststack migration Caustics were apparent in the data and only with a multiarrivaltraveltime computed migration such as a Kirchho migration based on raytracing that we can handle the caustics The model also had large lateralvelocity variations in which the velocity analysis and NMO parts of the abovementioned sequence can not handle Such limitations caused clear amplitude

COMPUTER EXERCISES VI SOLUTION

0

0.2

0.4

0.6

0.8

1.0x10 4

Dep

th (

ft)

0 0.2 0.4 0.6 0.8 1.0x10 4Range (ft)

Primary RaysFigure The model that was used to produce the data shown in the previous gure

degradations on the data especially for the deeper events these events aremostly eected by the lateral velocity variations The velocities used in themigration were obtained from velocity analysis and a lot of guessing Alsorunning the migration with dierent velocities helped in guring out whichvelocities gave the best nal image After all is said and done the model didnot seem easy at all It might have been straightforward when one uses aprestack migration with a migration velocity analysis however with the toolsI used it was basically a nightmare


Figure Wences Gouveia s migrated depth section

COMPUTER EXERCISES VI SOLUTION

Figure Tariq Alkhalifah s migrated depth section

Bibliography

"# K Aki and P G Richards Quantitative seismology theory and practice Freeman

"# R M Alford K R Kelly and D M Boore Accuracy of nitedierence modelingof the acoustic wave equation Geophysics

"# R G Bartle The elements of real analysis Wiley

"# E Baysal D D Koslo and J W Sherwood Reverse time migration Geophysics

"# J Berryman Lecture notes on nonlinear inversion and tomography

"# N Bleistein and R A Handelsman Asymptotic expansions of integrals Dover

"# M B1ath Mathematical aspects of of seismology Elsevier

"# M Born and E Wolf Principles of optics Pergamon

"# C Brezinski History of continued fraction and Pade approximants SpringerVerlag

"# H B Callen Thermodynamics Wiley

"# V Cerveny Ray tracing algorithms in threedimensional laterally varying layeredstructures In G Nolet editor Seismic tomography Reidel

"# C Chapman A new method for computing synthetic seismograms GeophysicalJournal R Astr Soc

"# C Chapman Generalized Radon transforms and slant stacks Geophysical JournalR Astr Soc

"# J Claerbout Imaging the Earths interior Blackwell

BIBLIOGRAPHY

"# R Clayton and B Engquist Absorbing boundary conditions for the acoustic andelastic wave equations Bulletin of the Seismological Society of America

"# A M Correig On the measurement of body wave dispersion J Geoph Res

"# M B Dobrin and C H Savit Introduction to geophysical propsecting McGrawHill

"# P C Docherty Kirchho migration and inversion formulas Geophysics

"# M Ewing W S Jardetzky and F Press Elastic waves in layered media McGrawHill

"# K Fuchs and G M(uller Computation of synthetic seismograms with the reectivitymethod and comparison with observations Geophysical J Royal Astr Soc

"# J Gazdag Wave equation migration with the phaseshift method Geophysics

"# J Gazdag and P Sguazzero Migration of seismic data Proceedings of the IEEE

"# C W Gear Numerical initial value problems in ordinary dierential equationsPrenticeHall

"# I M Gel fand and G E Shilov Generalized functions Volume I Academic Press

"# H Goldstein Classical Mechanics AddisonWesley

"# S H Gray Ecient traveltime calculations for kirchho migration Geophysics

"# S H Gray and W P May Kircho migration using eikonal equation traveltimesGeophysics

"# N B Haaser and J A Sullivan Real analysis Van Nostrand Reinhold

"# C Hemon Equations d onde et modeles Geophysical Prospecting

"# M Hestenes Conjugate direction methods in optimization SpringerVerlag Berlin

"# J D Jackson Classical Electrodynamics Wiley

BIBLIOGRAPHY

"# F John Partial dierential equations SpringerVerlag

"# D S Jones Acoustic and electromagnetic waves Oxford University Press

"# H Keller Numerical methods for twopoint boundary value problems Blaisdell

"# K R Kelly R W Ward S Treitel and R M Alford Synthetic seismograms anitedierence approach Geophysics

"# R Langan I Lerche and R Cutler Tracing rays through heterogeneous media anaccurate and ecient procedure Geophysics

"# A R Mitchell Computational methods in partial dierential equations Wiley

"# P M Morse and H Feshbach Methods of theoretical physics McGraw Hill

"# R D Richtmyer and K W Morton Dierence methods for initial value problemsInterscience

"# E Robinson T S Durrani and L G Peardon Geophysical signal processingPrentice Hall

"# J A Scales and M L Smith Introductory geophysical inverse theory SamizdatPress

"# W A Schneider Integral formulation of migration in two and three dimensionsGeophysics

"# G D Smith Numerical solution of partial dierential equations nite dierencemethods Oxford

"# A Sommerfeld Optics Academic Press

"# J Stoer and R Bulirsch Introduction to numerical analysis SpringerVerlag

"# D Struik Lectures on classical dierential geometry AddisonWesley

"# A Tarantola Inverse Problem Theory Elsevier New York

"# P Temme A comparison of commonmidpoint singleshot and planewave depthmigration Geophysics

"# S Treitel P Gutowski and D Wagner Planewave decomposition of seismogramsGeophysics

"# J van Trier A massively parallel implementation of prestack kirchho depth migration In rd Mtg Eur Assoc Expl Geophys Abstracts pages Eur AssocExpl Geophys

BIBLIOGRAPHY

"# J van Trier and W W Symes Upwind nitedierence calculation of traveltimesGeophysics

"# J Vidale and H Houston Rapid calculation of seismic amplitudes Geophysics

"# G N Watson A treatise on the theory of Bessel functions Cambridge UniversityPress

Index

absorbing boundary conditions aliasing angular spectrum asymptotic ray theory

Bessel functions Bessel functions asymptotic approxima

tion Bessel functions zeros boundary value problems wave equation

Brillouin zone

Cauchy problem characteristics of hyperbolic PDE s CMP CMP spacing common midpoint gather common oset gather common receiver gather common source gather consistency nite dierence equations continued fractions continuum limit of lattice CourantFriedrichsLewy condition CrankNicolson curvature of a ray cylindrical wave decomposition

depth migration diraction summation dilatation Dirac delta function Dirichlet problem discretization error dispersion relation dispersion relation approximate

dispersion relation lattice dispersion continuum dispersion on a lattice doublesquareroot equation downward continuation

eigenfrequency lattice eigenfunctions lattice eikonal equation equations of motion high frequency equations of motion linearized error discretization error global rounding error local truncation error total Eulerian coordinates evanescent waves explicit nite dierence methods exploding reector model

Fermat s principle nite dierence methods nite dierence methods elastic waves

nite dierence schemes D fourthorder

nite dierence schemes D secondorder

fold Fourier series Fourier series convergence Fourier Transforms FourierBessel transform fractional calculus Fredholm integral equation Fresnel Zone fullwaveform inversion ii

INDEX

fully implicit method

Gamma function generalized momenta geophones global rounding error golden mean gravitational potential equilibrium gravity versus elasticity Green function Green function causal anticausal Green function constant velocity Green functions Helmholtz equation Green functions wave equation group velocity

HamiltonJacobi equations harmonic waves Helmholtz equation Helmholtz equation cz Huygen s principle Huygens principle hydrophones

implicit nite dierence methods incremental constitutive relation incremental elds inhomogeneous waves

Kirchho Integral Theorem Kirchho migration Kirchho migration Schneider s formula

Lagrange s integral invariant Lagrangian coordinates Lax s Theorem least action principle least time principle Legendre transformation local truncation error

method of images migrated time vertical travel time migration

migration common source migration commonsource migration Kirchho migration phaseshift migration phaseshift algorithm migration plane wave migration prestack phaseshift migration reverse time migration time domain migration phase shift migration travel time migration phase shift

Neumann problem Newton s method Newton s second law normal moveout Nyquist frequency

P waves paraxial approximation phase velocity phase velocity P wave phase velocity shear wave phaseshift migration prestack plane wave decomposition plane wave pressure plane waves plane waves angular spectrum Pluecker line geometry Poisson s equation

quantum mechanics

radiation condition Sommerfeld Radon transformation ray equation boundary value problems

ray equation rst order system ray equation initial value problems ray equations ray equations arclength parameteriza

tion ray parameter ray theory

INDEX

raytracing numerical methods reection at a boundary reverse time migration

Sampling Theorem scalar wave equation Schneider s migration formula seismic sources separation of variables nite dierences

shear waves shooting methods shot record migration slantstack Snell s law Sommerfeld integral Sommerfeld radiation condition spatial convolution spherical waves stationary phase method stress tensor at a material point stress tensor equilibrium stress tensor incremental survey sinking

taup method tridiagonal matrix eigenvalues

uniqueness of solution to IVP

vibrators

wave equation cz wave equation elastic wave equation integral equation form wave equations dip dependent wave equations factored wave equations oneway wave equations paraxial weighted average nite dierence meth

ods wellposedness of initial value problem for

rays Weyl integral WKBJ approximation

zerooset data

theory of seismic imaging -

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