SVD for elastic inversion of seismic data

Ilya Silvestrov and Vladimir A. Tcheverda

SWLIM VIIJune 21-26, 2010

Motivation: Cross-well seismic data inversion example

Acquisition system

λ µ ρ

True model

Result of inversion

λ µ ρ

Strong false footprints in Lame parameters occur after solution of

inverse problem

sour

ces

rece

iver

s

B nonlinear forward modeling operator(e.g. 2D isotropic elastodynamic equations)

,obsumB rr=

mr elastic properties of the Earth’s interior (e.g. Lame parameters and Density)

obsur observed seismograms (e.g. X and Z component data)

Seismic inverse problem

Solution of seismic inverse problem

Linear inversion (Least-squares migration):

,umJ rrδδ =

where J is a Freshet derivative (Jacobian in finite-dimensional case).

)()( 1 kobsT

kkkkT

k mBuJmmJJ rrrr−=−+

min)(21 2

→− mBu obs rrNon-linear inversion (Full-waveform inversion):

kkkk mmH ∇=−+ )(~1

rrGauss-Newton method:

In any case we have to invert linearized forward modeling operator J

For least-squares we have:

Linearization using Born’s approximationδρρρδµµµδλλλ +=+=+= 000 , ,

,0 uuu obs rrrδ+=

mJu rrδδ =

=

=

δρδµδλ

δδδ

δ muu

u rr ,2

1

=

232221

131211

JJJJJJ

J

,)(),,,(),,( xdxmxxxKmJxxu jX

rsijjijrsirrrrrrr

δωδωδ ∫==

Assuming that small perturbations in the model:

causes small perturbations in the wavefield:

the linearized forward modeling operator has the form:

where kernels of the integral operators of first kind are determined by Green’s function in the reference model.

ijK

The inversion problem is ill-posed

uu

Am

m errorerror

r

r

r

r

δ

δµ

δ

δ)(≤

nssA 1)( =µ - condition number,

0...21 ≥≥≥≥ nsss - singular values

Truncation of singular value decomposition (SVD)

∑=

=r

iii

r mm1

][ v)v,( rrrrδδ

Stability of solution

umJ rrδδ =

Because of ill-posedness, the matrix approximation of J will be ill-conditioned

- right singular vectorsivr

- stable component of solution

Model parameters:Background media: Vp = 3000m/s, Vs = 1700m/s, density = 2300kg/m^3Target area grid size: 10m x 10mFrequencies interval: [ 0.1Hz; 70Hz ]Frequencies sample rate: 0.2HzSource wavelet: Ricker, central frequency 30HzNumber of sources: 1Number of receivers: 51

Inversion of offset VSP data for look-ahead scenario

•Homogeneous background medium

•P-wave incidence

(PP and PS scatterings are considered)

In this case matrix approximation of operator J may be constructed explicitly

The following parameterizations will be considered:

);;(1 ρδρ

µδµ

λδλ

=M

=

ρδρδδ ;;2 Vs

VsVpVpM

=

ρδρδδ ;;2 IS

ISIPIPM

Elastic medium parameterization

Isotropic elastic medium may be parameterized by set of three parameters.

Singular values of Jacobian

Level of round-off error

Parameterization using Lame parameters210=cond

Strong coupling of parameters is observed

01 >mδ 02 =mδ 03 =mδ 0>δλ 0=δµ 0=δρ

0=δλ 0>δµ 0=δρ 0=δλ 0=δµ 0>δρ

01 >mδ 02 =mδ 03 =mδ 0>Vpδ 0=Vsδ 0=δρ

210=cond

Parameterization using velocities

Strong coupling of parameters is observed

0=Vpδ 0>Vsδ 0=δρ 0=Vpδ 0=Vsδ 0>δρ

210=condParameterization using impedances

There is no coupling of parameters. Density is not recovered.

01 >mδ 02 =mδ 03 =mδ 0>IPδ 0=ISδ 0=δρ

0=IPδ 0>ISδ 0=δρ 0=IPδ 0=ISδ 0>δρ

Profile of recovered perturbation and trend/reflectivity decomposition

Profiles of true and recovered perturbationsof P impedance along vertical line X = 150m

IPIPδ

No trend component

Real velocity profile

Macro-velocity (trend)

Reflectivity

Trend/reflectivity decomposition

Inversion of offset VSP data for look-ahead scenario

Singular values

Right singular vectors of Jacobian

Density component is zero for

high-order singular vector

Frequency content of singular vectors

There are no low-frequencies in high-order singular vectors

Inversion of cross-well data

Model parameters:

Vp = 3100m/s, Vs = 1700m/s, density = 2000kg/m^3Target area grid size: 10m x 10mFrequencies interval: [ 0.1Hz; 80Hz ]Frequencies sample rate: 0.5HzSource wavelet: Ricker, central frequency 40HzNumber of sources: 30Number of receivers: 30

Wenyi Hu, Aria Abubakar, and Tarek M. Habashy, 2009. Simultaneous multifrequency inversion of full-waveform seismic data. Geophysics, 74, R1– R14

Singular values of Jacobian

Singular values for cross-well problem Singular values for VSP problem

Inversion of cross-well data is much favorable task

Parameterization using impedances110=cond

01 >mδ 02 =mδ 03 =mδ 0>IPδ 0=ISδ 0=δρ

0=IPδ 0>ISδ 0=δρ 0=IPδ 0=ISδ 0>δρ

Coupling of P impedance and density is observed

Parameterization using velocities

01 >mδ 02 =mδ 03 =mδ 0>Vpδ 0=Vsδ 0=δρ

0=Vpδ 0>Vsδ 0=δρ 0=Vpδ 0=Vsδ 0>δρ

110=cond

There is no coupling of parameters. Density is not recovered.Result for shear velocity is not good because pressure sources were used.

Summary

SVD analysis of linearized forward modeling operator allows us to explain following features of seismic inverse problem:

• Pressure and shear impedances are appropriate parameters for inversion using reflected waves. Parameterization using velocities of Lameparameters may give unreliable results because of parameters coupling

• In case of OVSP data inversion for look-ahead: – only impedances discontinuities can be inverted– Low-frequency component of solution can not be inverted– Density can not be inverted

• Velocities are appropriate parameters for inversion using transmitted waves

Conclusions

• A reliable solution of seismic inverse problem requires careful prior study in each particular case

• Major drawbacks of seismic inverse problem occurred even in the linear statement

• SVD analysis may be (should be?) used as a powerful tool for analyzing the new inversion algorithms

Acknowledgments

The research was done in cooperation with Schlumberger Moscow Research and was partly supported by Russian Fund of Basic Researches