Seismic wave propagation in heterogeneous media:

Modeling, signal processing, and inversion

Christian PoppeliersAngelo State University Physics Dept.

21 April, 2006

Seismic wave propagation

• Why?– Listening to waves tells us about the Earth

• 1) seismic wave velocity tomography• 2) seismic imaging discrete structure

– All involve scattering of mechanical waves

Which types of waves?

• Acoustic (sound waves)– Travel through all materials

• Elastic (mechanical waves)– Travel only through solids

--> Most wave modeling in exploration seismology is acoustic

--> Wave propagation in most solids is actually elastic

Some trivia:

1) Main exploration tool of the petroleum industry2) Typical industry scale survey contains 1000’s shots3) 100’s geophones per shot4) datasets can get up to a terrabyte in size!

petroleum industry is the single biggest user of electronic data storage medium

5) $4+ billion a year industry

Assumptions in reflection seismology

• 1) layered structure– Deterministic (i.e. large-scale structures that can be

“mapped”, or delineated)

• 2) “single” scattering• Linear scattering

– i.e. no internal reverberations, etc.

• 3) Acoustic waves ONLY• 4) seismic velocity is smoothly varying

Reality1) Lot’s of internal reflections, and mode conversions

com

pres

siona

l wav

e

shea

r wav

e

Reality, continued2) The Earth might not be perfectly, or vaguely, layered!!

most of the Earth’s crust is crystalline!!!

The Point:Seismic scattering in the earth can be

• the result of STOCHASTIC (“random”) impedance• very STRONG• lots of mode conversions

So what ?…quantifying scattering in stochastic medium can help us “map”this portion of the Earth

This is called statistical seismic imaging

In complex geologic material this can be used as in interpretation tool.

a different paradigm in seismic data analysis:assumptions of horizontal layering, and deterministic structure are removed

- crystalline crust - near-surface, unconsolidated sediments/gravels)- mantle

why is this useful?• a useful interpretational tool...

Inversion:

• Given the data, can we estimate the stochastic fabric?

• First step: develop forward model

• 1) Linear scattering:– reflections are from small velocity perturbations:

– assume impedance is proportional to velocity:

– which means:

Current inversion techniques for statistical seismic imaging assume:

v(x) : Earth’s 3D velocity fieldvo(x) : Earth’s background velocity field

δv(x) : perturbations of velocity about the background

)()()( xxx vvv o

xpxv

xpxpxp 0

linear scattering assumptions, con’t

• Stochastic field is defined as:

1

x

x

x

x

p

p

v

v

reflections occur where δp(x) ≠0

Linear scattering, in 1-D:

dtetwts )()()(

dtetwts )()()(

dtptwts )()()(

)()()( tptwts convolution

In the frequency domain

)()()( fpfwfs

if δp(x)/p(x) <<1, then scattering is “weak”, or approximately linear

Problem: mode conversions and multiple scattering aren’t accounted forSolution: scattering is weak/linear, so they aren’t a problem

“single” or “weak” scattering Born Approximation

s(t) : recorded seismogramw(t) : input seismic pulseδp(t) : acoustic impedance contrasts of the Earth

For now, we simply avoid P-to-S mode conversions:

• P to S mode conversions are avoided if offset is small:

shot

receiver

Ref

lect

ed P

-wav

e

Goal: θ < 200

…and…2) The earth is semi-fractal

• Earth texture of δp is fractal– Model = Von Kármán

2

1

2

122222 )1)((2

)2()(

ak

akp

k = vector wavenumber Γ = Gamma functiona = characteristic lengthυ = Hurst number

)0(

))(()(

2

G

rGHChh

]1,0[,0,)()(

rrKrrG

power spectrum autocorrelation

τ = lagH = rms impedance variationG = Bessel functionr = radius

What does this mean?

• 1) Earth texture is self-similar on all scales– up to the “characteristic length”

• 2) Parameterized by – power spectrum “rollover” size parameter, a– “roughness” --> fractal dimension => D=E+1-v

log wavenumber

log

ampl

itude

1a

“slope” is a functionof fractal dimension

lag

lag, τ

power spectrum auto correlation

steepness controlled by fractal dimension

half-width controlled by a

example: numerical Earth:

a=100mD=2.9

a=100mD=2.5

color corresponds to seismic velocity

4.9 m/s

5.1 m/s

a=500mD=2.5

Is Earth continuously varying?

not really

most crystalline materials contain an n-modal distribution of velocities (n=2,3)

adjust our fractal model

• Goff et. al., 1994; transform continuous fields into binary fields

effects on statistical parameters?

continuous binary

1/a = kc 1/a ≈ 1.15kd

D=E+1-vc D ≈ E+1-(vc/2)

velo

city

, km

/s

P-w

ave velocity, km

/sec

The point, forward model:

Source

Receiver

explosive source

“fine” model “coarse” model

geophones basic observation: the size-scale of the heterogeneities have an affect on the wavefield

Linear inversion:

Color corresponds to estimated “textural size”

Given seismic data that has propagated through a stochastic earth, can we estimate the equivalent fractal “textural parameters”?

Work on this so far

• 1) inversion assuming “white” reflectivity

• 2) iterative inversion with non-white reflectivity

• 3) non-linear inversion

First Attempt; binary model, simple scattering

• recall convolution model:

)()()( tptwts

what this really means:

)(*)()( trtwts

)(2

1)( tp

dt

d

vtr

where

s(t) = seismic reflection dataw(t) = seismic pulseδp(t) = stochastic impedancer(t) = reflectivity

thus, in 1-D:

stochastic velocity:

with corresponding reflectivity

convolved with an illuminating wavelet

will yield this data:

*

But we record data• have to go backwards (inversion)

1) Given data, estimate reflectivity

)()()( krkwks

)(

)()(

kw

kskr known as DEconvolution

since )(2

1)( tp

dt

d

vtr

T

dttrvtp )(2)(ˆ

from δp(t), we can estimate equivalent stochastic model

note that w(k)-1 is known as an “inverse filter”.Its really an inverse operator

The recipe:given data:

deconvolve:

numerically integrate

“clean up” and normalize via a thresholding operator

R

RR

R

T

Tzr

TzrT

Tzr

zr

)(ˆ,1

)(ˆ,0

)(ˆ,1

)(ˆ

from estimate of δp, obtain equivalent fractal model:

grid search through autocorrelations of fractal models

N

zhhvzi aCCN

axE

1

2),:(),(1

),:(

x

where Ei is a minimum, record D, υ

autocorr. of theoretical model

autocorr. of estimated δp,

least-squares

Does it work? • kind of...

known wavelet

UNknown wavelet

Poppeliers and Levander, 2004

Limitations:1) decon requires the wavelet2) wavelet is usually unknown3) deconvolution assumes white reflectivity4) fractal reflectivity is not white5) very sensitive to uncorrelated noise6) not good at estimating fractal dimension

major problem: deconvolution

• reflectivity is NOT white– in fact, reflectivity in the Earth is fractal

• “solution”? – modify the inverse operator in deconvolution

by the non-white, fractal model

The recipe:assume that data is given by

)()()( kWkRkS

with a power spectrum of222

)()()( kRkWkS

initially assume that reflectivity is white, so |R(k)|2 = 1:

2

0

2)()( kWkS

))(()(ˆ 10

11 kWFzw

form inverse filter:

inverse filter

F-1(.) = inverse Fourier transform

deconvolve data to obtain initial estimate of reflectivity:

)(*)(ˆ)(ˆ 10 zszwzr

from r0, threshold, integrate and find a and D as before

From a and D, form reflectivity power spectrum model:

),,(4

)(20

222

1 Dakpv

kkR

and use to modify the inverse operator:

2

1

02

1)(

)()(

kR

kWkW

and form new inverse filter:

)()( 11

11 kWFzw

von Karman model

itera

te

does it work better?

autocorr oforiginal model

autocorr fromspiking decon

autocorr fromiterative decon

Yes! ...1-D simulation observations:A) spiking decon method underestimates char. length, cannot obtain fractal dimension

B) iterative decon method comes closer to original char. length, can estimate fractal dimension

2-D simulationspiking decon

iterative decon

iterative method more accurate and more precise

a (m)

a Da

Synthetic model:a=300mD=2.7

field data: 1986 PASSCAL Basin and Range seismic experiment

Results agree with drill cores

Finally, what if velocities are not discrete?

• Previous methods limited to bi-modal velocity distribution

• what about tri-modal, n-modal, or continuous velocity distributions?

new approach:• Like before, assume that data is from

convolution (linear scattering):

x =

time/space domain

power spectal domain

* =

222

,,ˆ,, kDSkDRkW

“make up” pulse and fractal model who’s product is equal to the data!

given

222,,,,ˆ kaRkWkaS

details:

1) data, wavelet, and reflectivity are discretely sampled in the wavenumber domain2) wavenumber sample interval for the wavelet is much greater than than that of the data3) i.e. more data samples than wavelet samples4) the problem is grossly over-determined5) no information about wavelet’s phase (must assume)

forward model:

222

,,,,ˆsws kaRkWkaS

2

0

2222 4

)(,,ˆV

kkPkWkaS s

sws

data S is a product of the wavelet W and reflectivity R

The reflectivity is a function of the impedance contrasts, δP(k)

2

1

2

122222 )1)((2

)2()(

ak

akp

von

Karm

an

impe

danc

e m

odel

von Karman reflectivity

inverse method:

KkkSkaSE Nn ,0,,ˆ

K

NNRnNn dkkSkaPkWSkSkaSO0

22)(),,()(ˆ;,,ˆ

ig

ii dJm

TnkWkWkWam ,,,,, 21

TniiiT

Ni kaDkaDkaDkDkDkDd ,,ˆ,,,,ˆ,,,ˆ,,, 2121

nNkaSm

J Nn

,,,ˆ2,1

nnn

nnN m

SJ

,...4,3...4,3,

ˆ

inversion must minimize difference between model and data:

K=nyquist wavenumber

objective function:

Newton’s Method:

partial derivatives of von Karman reflectivity

partial derivatives of interpolated pulse power spectrum, or individual spline functions

inversion estimates:

• 1) characteristic length

• 2) fractal dimension

• 3) power spectrum of illuminating wavelet!– but not the phase...

Preliminary tests/results:

a=30m

a=120msimulated shot data:

single shot, at surface fc=10Hz, zero phase illum. wavelet

Poppeliers, 2006

test true parameters recovered az recovered Daz (m), D

noise free 30, 2.7 43.0 ± 5.7 2.61 ± 0.05noise free 120, 2.7 142.1 ± 1.8 2.60 ± 0.15

15% colored noise 30, 2.7 55.7 ± 7.2 2.60 ± .08 15% colored noise 120, 2.7 151.9 ± 12.2 2.59 ± .30

Poppeliers, 2006

Po

pp

elie

rs,

20

06

Poppeliers, 2006

Future work: Question

• Why can’t we use scattering principles from optics?

• Because, of mode conversion issues!– analysis indicates this approach is inappropriate for

large offsets

• What about strongly scattered seismic data?– For large offsets, S converted modes will dominate

the data

If the seismic wavefield that propagates though a stochastic earth is 1) NOT acoustic and 2) NOT singly scattered.

• Then,

• More likely

)()()( krkwks

)()(exp],,,[:)( kwkretcxakFks nPath length

The linear model

x =

but strong scattering looks like

=Some function (with parameters)

x F(f:[a,v,x,etc])expn x

The point:

• 1) Current model for inversions are only good for weakly scattered wavefields– Scattering is not linear

• 2) Forward modeling of seismic waves in a stochastic medium is necessary– First step towards developing inversion– We need a model (based on lots of statistics)

of the function F(f[a,v,x,ect])expn

approach?

Build a family of curves, based on the results of numerical tests, a “find” a function that describes these curves in a general sense.

Results of multiple tests

“test” function

Why?

• Finding the correct function F(f[a,v,x,ect])expn is key to the inverse problem– Need a “forward” model before we can invert

data.• Invert data: given the data, can we determine the

stochastic parameters?

T=30sec

T=30sec

T=100sec

T=100sec

Lightly scattering dimple

Heavily scattering dimple

Notice the difference in the wavefield here. Can we say something about the texture of the dimple?

concluding remarks:

• statistical seismic imaging is a new paradigm of seismic data analysis

• rich in mathematical ideas

• will form a useful tool in interpreting existent data (cheap!)