View
6
Download
0
Category
Preview:
Citation preview
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
Recommended