jeff@sep.stanford.edu 1 riemannian wavefield migration: imaging non-conventional wavepaths and...

jeff@sep.stanford.edu1

Riemannian Wavefield Migration:Imaging non-conventional wavepaths and geometries

Jeff Shragge

Geophysics Department

University Oral Qualification Exam

jeff@sep.stanford.edu2

Agenda

• What is my problem? my solution?– Statement of Problem– Proposed Solution– Thesis Objectives– Potential Impact

Part I

Part II

• How am I going to solve it?– Riemannian Wavefield Extrapolation– Completed Work– Work to do

jeff@sep.stanford.edu3

3-D Seismic Imaging

Seismic targets increasingly

complex

Improved3-D Seismic acquisition

Improved3-D seismic

imaging

Improved3-D

interpretation

jeff@sep.stanford.edu4

3-D Seismic Imaging

Seismic targets increasingly

complex

Improved3-D Seismic acquisition

Improved3-D seismic

imaging

Improved3-D

interpretation

jeff@sep.stanford.edu5

Wave-Equation Migration – Review

Downward continuation

WavefieldU(t,s,g,z = 0)

WavefieldU(t,s,g,z = nΔz)

Apply recursive filter e(-ikz Δz)

n times

Imaging Condition

jeff@sep.stanford.edu6

3-D Imaging is Successful…

Distance – x axis

Dep

th

Dis

tan

ce –

y

axis

Distance – x axis

From: 3-DSI, Biondi (2004)

jeff@sep.stanford.edu7

Distance

Dep

th…so why are we needed?

From: 3-DSI, Biondi (2004)

Seismic imaging science is good, but not perfect– Irregular and sparse data– Illumination limitations – physical and imaging procedure– Topographic surface complexity– …

jeff@sep.stanford.edu8

Distance

Dep

th…so why are we needed?

From: 3-DSI, Biondi (2004)

Seismic imaging science is good, but not perfect– Irregular and sparse data– Illumination limitations – physical and imaging procedure– Topographic surface complexity– …

jeff@sep.stanford.edu9

Distance

Dep

thIllumination

Steep Dip reflector

Weak reflector

Subsalt Imaging

Complex Structure

From: 3-DSI, Biondi (2004)

• Poor illumination of subsurface because of :– poor physical illumination (Acquisition)– incomplete imaging procedure (Processing)

jeff@sep.stanford.edu10

Topographic surface complexity• How do we deal with topography directly in

wave-equation imaging?

jeff@sep.stanford.edu11

Why do we have Imaging Limitations?

Imaging illumination limitations

Topographic surface limitations

Coordinate system not conformal to propagation direction or acquisition surface

Migration physics decoupled from geometry

jeff@sep.stanford.edu12

Evidence – I

Problem • Non-conformal coordinate systems• Migration physics decoupled from geometry

Resulting Limitations • Inaccurate imaging of steep dips

– Downward continuation inaccurate at high angles– Overturning waves not used

• Extrapolation from acquisition surface topography– Hard to define extrapolation axis– Free-surface topography– Deviated well VSP geometry

jeff@sep.stanford.edu13

Steep Dip Imaging

Accuracy of wavefield extrapolation decreases as propagating waves tend to horizontal

Extrap

olatio

nD

irection

jeff@sep.stanford.edu14

Using Overturning Waves

Currently do not use potentially useful information provided by overturning waves

Extrap

olatio

nD

irection

jeff@sep.stanford.edu15

Proposed Solution

Use coordinate system conformal with wavefield propagation

Extrapolation

Direction

jeff@sep.stanford.edu16

Tilted Cartesian Example

ELF North-Sea Dataset

From:Shan and Biondi (2004)

0

1

2

3

4

5

2 4 6Distance [km]

Dep

th

[km]

08

jeff@sep.stanford.edu18

Tilted Cartesian Example

Downward Continuation Reverse-time

Plane-wave w/ dipping Cartesian coordinates

From:Shan and Biondi (2004)

jeff@sep.stanford.edu19

Evidence – II

Problem• Non-conformal coordinate systems• Migration physics decoupled from geometry Resulting Limitations • Inaccurate imaging of steep dips

– Downward continuation inaccurate at high angles– Overturning waves not used

• Extrapolation from acquisition surface topography– Hard to define extrapolation axis– Free-surface topography– Deviated well VSP geometry

jeff@sep.stanford.edu20

Free Surface Topography

• How to define extrapolation surface orthogonal to free surface?

jeff@sep.stanford.edu21

VSP Deviated Well Topography

• Receiver wavefield acquired in well deviated in 3-D• How to define wavefield extrapolation from borehole surface?

jeff@sep.stanford.edu22

Proposed Solution

Use coordinate system conformal with borehole geometry

jeff@sep.stanford.edu23

Summary of Problem

Imaging Illumination Limitations

Topographic Surface Limitations

Coordinate system not conformal to propagation direction or acquisition surface

Migration Physics decoupled from Geometry

DifficultSteep DipImaging

No use of Overturning

waves

Extrapolationfrom complex free-surface

Extrapolationfrom deviated

boreholes

jeff@sep.stanford.edu24

Summary of Solution

Reduce Imaging Illumination Limitations

Enable W.E. imaging directly from Topographic Surfaces

Perform Migration on Coordinate systems conformal to propagation direction/acquisition surface

Couple Migration Physics with Geometry

ImproveSteep DipImaging

Use Overturning

waves

Eliminate need for free-surface

datuming

W.E. Imaging for massive 3-D VSP data

jeff@sep.stanford.edu25

What am I going to do?

Handle multipathing/ triplication

Applicable to 3-D field

data

Handle arbitrary geometry

Handle large data volumes

Improvesteep dipImaging

Eliminate need for free-surface datuming

W.E. imaging for massive 3-D VSP data

Req

uirem

ent

s

AVA studies Physical property

analysis

Imp

act

3-D curvilinear coordinate wave-equation migration

method

jeff@sep.stanford.edu26

End of Part I

Handle multipathing/ triplication

Applicable to 3-D field

data

Handle arbitrary geometry

Handle large data volumes

Improvesteep dipimaging

Eliminate need for free-surface datuming

W.E. imaging for massive 3-D VSP data

Req

uirem

ent

s

AVA studies Physical property

analysis

Imp

act

3-D curvilinear coordinate wave-equation migration

method

jeff@sep.stanford.edu28

Part II – Agenda

• Riemannian Wavefield Extrapolation (RWE)

• Completed Work– Generating Coordinate systems– Extrapolation examples– Deal with or avoid Triplication

• Work to do– Tackling the Geophysical Imaging problem– Technical issues

jeff@sep.stanford.edu29

Part II – Agenda

• Riemannian Wavefield Extrapolation (RWE)

• Completed Work– Generating Coordinate systems– Extrapolation examples– Deal with or avoid Triplication

• Work to do– Tackling the Geophysical Imaging problem– Technical issues

jeff@sep.stanford.edu30

RWE in 2-D ray-coordinates

RiemannianCartesian

x

z Extrapolation Direction

Orthogonal Direction

z

x

jeff@sep.stanford.edu31

RWE: Helmholtz equation

i j j

ij

i

UgU

g

g

1

(associated) metric tensor

UsU 22

Laplacian

)( kii x

Coordinate system

Sava and Fomel (2004)

jeff@sep.stanford.edu32

RWE: (Semi)orthogonal coordinates

i j j

ij

i

UgU

g

g

1

200

0

0

GF

FE

gij

2

2

0

0

J

gij

jeff@sep.stanford.edu33

1st order 2nd order2nd order 1st order

RWE: Helmholtz equation

UsU

J

U

JJ

UJ

J

U 222

2

22

2

2

1111

UsU

cU

cU

cU

c 222

2

2

2

Ray-coordinate Interpretationα = velocity function J = geometric spreading or Jacobian

jeff@sep.stanford.edu34

RWE: Dispersion relationR

iem

anni

anC

arte

sian

2222 skckickickc

2222 skk xz 1

0

cc

cc

jeff@sep.stanford.edu35

RWE: Dispersion relationR

iem

anni

anC

arte

sian

sk

cc

cc

o

1

0

22

2

k

c

ck

c

cik

c

cik o

222xz ksk

jeff@sep.stanford.edu36

RWE: Wavefield extrapolationR

iem

anni

anC

arte

sian

sk

cc

cc

o

1

0

zikxx

ze ω)z,,U(k=ω)z,z,U(k ΔΔ

τΔγγ

τττΔτ ike ω),,U(k=ω),,U(k

jeff@sep.stanford.edu37

Part II – Agenda

• Riemannian Wavefield Extrapolation (RWE)

• Completed Work– Generating coordinate systems– Extrapolation examples– Deal with or avoid Triplication

• Work to do– Tackling the Geophysical Imaging problem– Technical issues

jeff@sep.stanford.edu38

Generating Coordinate Systems

Single Arrival

Multiple Arrival

Monochromaticray tracing

To what degree can wave propagation be modeled in a coordinate system?

Broad-band ray tracing

Cartesian Coordinates

(SEP-114)

RayCoordinates

(SEP-115)

jeff@sep.stanford.edu39

Monochromatic ray tracing

• Use local velocity and WAVEFIELD PHASE information to calculate coordinate system

• Create a coordinate system conformal with wavefield propagation direction

jeff@sep.stanford.edu40

Monochromatic ray tracing

0

Distance

Dep

th

Distance

Dep

th

NOTE: Gradient of monochromatic wavefield phase shows orientation of propagation direction

jeff@sep.stanford.edu41

What is a phase-ray?

Distance

Dep

th

Phase-ray

eAU )()( x,x, xi

jeff@sep.stanford.edu42

Calculating phase-rays

Cartesian

ray equations

z

Uy

Ux

U

U

1Im

z)dτy,v(x,

dz

dy

dx

φ

• Decoupled system of 1st order ODEs

• Ray solution: – Specify an initial point– Numerically integrate dx and dz – Output x and z coordinates of ray

• Rayfield explicitly dependent on frequency

jeff@sep.stanford.edu43

Phase-ray example

Distance

Dep

th

jeff@sep.stanford.edu44

Part II – Agenda

• Riemannian Wavefield Extrapolation (RWE)

• Completed Work– Generating Coordinate systems– Extrapolation examples– Deal with or avoid Triplication

• Work to do– Tackling the Geophysical Imaging problem– Technical issues

jeff@sep.stanford.edu45

Adaptive phase-ray extrapolation

Calculate 1Phase-ray

step

Calculate1 wavefield

step

Bootstrapping procedure

jeff@sep.stanford.edu46

Example – Point SourceDistance

Dep

th

jeff@sep.stanford.edu47

Example – Plane wave

Dep

th

Distance

jeff@sep.stanford.edu48

Example – Salt

Dep

th

Distance

jeff@sep.stanford.edu49

Part II – Agenda

• Riemannian Wavefield Extrapolation (RWE)

• Completed Work– Generating Coordinate systems– Extrapolation examples– Deal with or avoid Triplication

• Work to do– Tackling the Geophysical Imaging problem– Technical issues

jeff@sep.stanford.edu50

Coordinate System Triplication

UsU

J

UJ

J221

ωγ

α

γτατα

Dep

th

Distance

jeff@sep.stanford.edu51

Coordinate System Triplication

SmoothVelocity

Ray-trace on single arrival

Eikonal solvers

Coordinate System Triplication:Avoid or Handle?

Avoid Triplication

Avoid Zero Division:

Add ε(Sava &

Fomel,2003)

Iterative coord.

updatingSEP-115

Handle Triplication

Numerically isolate triplication branches

SEP-115

Regularization through Inversion

jeff@sep.stanford.edu52

Regularization through Inversion

0

Tim

e [s]

X-Coordinate

6000

4000

0 50-50

0

Tim

e [s]

3000

1000

0 50-50

Z-Coordinate0 4000

0

2000

4000

80000

0

00

0

1

2

0

1

2

Shooting Angle [deg] Shooting Angle [deg]

jeff@sep.stanford.edu53

Regularization through Inversion

0

1

2

Tim

e [s]

X-Coordinate

6000

4000

0 50-50 00

1

2

Shooting Angle [deg]

0

Tim

e [s]

3000

1000

0 50-50

Z-Coordinate

0

0

1

2

Data: Xd and Zd coordinates from ray-tracing

Model parameters to be fit:Xm and Zm

Xm and Zm are triplication-free if they satisfy Laplace equation

ΔXm = 0 = ΔZm

jeff@sep.stanford.edu54

Regularization through Inversion

0

1

2

Tim

e [s]

X-Coordinate

6000

4000

0 50-50 00

1

2

Shooting Angle [deg]

0

Tim

e [s]

3000

1000

0 50-50

Z-Coordinate

0

0

1

2

ΔXm = 0 = ΔZm

Geometric Regularization

Minimize Curvature of Xm and Zm

jeff@sep.stanford.edu55

Fitting Goals

Equation Type X-Coordinate Z-Coordinate

Physical Data Fitting

W(J)(Xm-Xd) 0 W(J)(Zm-Zd) 0

Geometric Regularization

ΔXm 0 ΔZm 0

W(J) dependent on Jacobian of coordinate system, e.g.,n

0JJ

J)J(W

J0 = initial spreadingn = curve adjusting parameter

jeff@sep.stanford.edu56

Part II – Agenda

• Riemannian Wavefield Extrapolation (RWE)

• Completed Work– Generating Coordinate systems– Extrapolation examples– Deal with or avoid Triplication

• Work to do– Tackling the geophysical imaging problem– Technical issues

jeff@sep.stanford.edu57

Imaging – Overturning waves• 3-D surveys consist of 1000s of shot + receiver locations

– Shot-profile migration often prohibitively expensive

• Plane-wave migration (Liu et al., 2002)– Phase encoding scheme (Romero et al., 2000)

• independent of coordinate-system

jeff@sep.stanford.edu58

Imaging – Overturning waves

• Source wavefield

• Receiver wavefield

• Imaging Condition

)xx(piN

1jjxall

0xe)0z,x,(S)p,0z,x,(S

ωωω

)xx(piN

1jjxall

0xe)0z,x,(R)p,0z,x,(R

ωωω

)p,,z,x(R)p,,z,x(S)p,z,x(I xallx*allxp ωω

ω

jeff@sep.stanford.edu59

Imaging – Overturning waves

jeff@sep.stanford.edu60

Imaging – Overturning waves

• Multiple plane-wave imaging

• 3-D Extension – use 2-D plane-wave filter

– Cylindrical wave migration (Duquet et al., 2001)

maxx

minx

p

pxxx p)p,z,x(I)p(f)z,x(I Δ

)]yy(p)xx(p[iyx

yxe)p,p,(g 00 ωω

jeff@sep.stanford.edu61

Imaging – Massive 3-D VSPExperiment Setup

– 2-D areal surface source pattern

– 3-C downhole receivers

– Record forward- and backscattering wavefield

jeff@sep.stanford.edu62

VSP Imaging – Reciprocity • Use reciprocity to create common receiver profiles

– Large computational savings

• Source wavefield – plane wave coordinates• Receiver wavefield – quasi-spherical coordinates

Source wavefield Receiver wavefield

jeff@sep.stanford.edu63

VSP Imaging – Phase Encoding • Use algorithm similar to plane-wave

– Random time shifts

Source wavefield Receiver wavefield

)z,x(tie))z,x(t,(g ΔωΔω

jeff@sep.stanford.edu64

VSP Imaging – Passive Seismic Mode• Back-propagate wavefield from borehole to source• Record P- and S-waves – multimode imaging

– Permutations of S + R velocity models, causal/acausal propagation– Examine 6 candidate scattering modes (Shragge and Artman, 2003)

Source wavefield Receiver wavefield

jeff@sep.stanford.edu65

Part II – Agenda

• Riemannian Wavefield Extrapolation (RWE)

• Completed Work– Generating Coordinate systems– Extrapolation examples– Deal with or avoid Triplication

• Work to do– Tackling the Geophysical Imaging problem– Technical issues

jeff@sep.stanford.edu66

Technical Issues - I• 3-D ray-coordinate system non-orthogonality

– May be problematic because have cross partial differential terms– Solution – Use a Nth order polynomial to approximate the coordinates– Analytical derivatives

0

1

2

Tim

e [s]

X-Coordinate

6000

4000

0 50-50 00

1

2

Shooting Angle [deg]

0

1

2

Tim

e [s]

3000

1000

0 50-50

Z-Coordinate

0

0

1

2

nx

N

1nn),(X εαγτ

nz

N

1nn),(Z εβγτ

jeff@sep.stanford.edu67

Technical Issues - II• Code and test different extrapolation operators

– Ray-coordinate 15 degree equation used– Examine more accurate operators, e.g.,

• 45 degree, • split-step Fourier• Fourier Finite Differencing

– 2-D methods that approximate 3-D solutions, e.g.,• Splitting

– Examine stability of operators in 2-D and 3-D

jeff@sep.stanford.edu68

Field Data Verification• Overturning Waves

– 3-D ELF North Sea synthetic + field data sets (@ SEP)– 3-D Exxon/Mobil data set (@ SEP)

• Massive 3-D VSP– Paulsson Geophysical Inc. has agreed to make available a data set

acquired in Long Beach, CA• 5 wells – straight to deviated in 3-D• 240 receivers, 1000s of shot • Adequate wavefield sampling ~ 50 feet/receiver

jeff@sep.stanford.edu69

Timeline for Proposed Work

Activity 2004

Test 2-D code on synthetic/field data

Develop other extrapolation operators

Develop 2-D migration code

Test methods for handling triplications

Test non-orthogonality of 3-D coords.

3-D field tests – Overturning/VSP

Develop 3-D migration code

Graduate

3-D synthetic tests – Overturning/VSP

Write Thesis

2005 2006 2007

Internship

jeff@sep.stanford.edu70

Questions?

jeff@sep.stanford.edu72

Summary of ProblemProblem• Coordinate system not conformal to wavefield propagation• Migration physics decoupled from geometry

Resulting Limitations• Inaccurate imaging of steeply dipping structure• Difficult to extrapolate from topographic surfaces

– free-surface topography, deviated well VSP

Extrapolation

Direction

jeff@sep.stanford.edu73

Summary of SolutionProposed Solution• Define Migration on generalized coordinate systems

• Couple physics of migration to geometry– Curvilinear coordinate system more conformal with:

• orientation of propagating wavefield• topography of acquisition surface

Extrapolation

Direction

jeff@sep.stanford.edu74

Potential Imaging Improvements• Use more wavepaths to improve steep dip imaging

– More information to contribute to imaging– Increase migration aperture

• Extrapolate direction from surfaces with topography

– Directly apply wave-equation migration– Eliminate need for datuming

Extrapolation

Direction

jeff@sep.stanford.edu75

Potential Impact

Improved steep dip imaging• Better interpretation geologic structure• Assist downstream processing tasks

– AVA studies– Physical property analysis

Massive 3-D VSP imaging• W.E. imaging directly applicable even in deviated boreholes• Provide additional tools

– Angle-domain CIGs, etc.