Aspects of seismic inversion

Paul Childs*Schlumberger Cambridge Research

With acknowledgements to: Colin Thomson*, ZhongMin Song †, Phil Kitchenside † Henk Keers*† Schlumberger WesternGeco, Gatwick

HOP, Newton Institute, June 19th 2007

2 PC3/20/07

Survey configuration

Marine seismic

3 PC3/20/07

Spectral interference notches from receiver-side free-surface ghost;

O/U receivers allow for up/down separation, hence ghost removal;

flatter, broader spectrum shows Earth structure better in seismic sections (=> better “attribute analysis”).

4 PC3/20/07

GPS

Streamer Steering

•Streamer control - with IRMA

(Intrinsic Ranging by Modulated

Acoustics) and Q-fin:GPS

IRMArange data

Birdcontroller Closing the loop

IRMAcontroller

TRINAV

TRIACQ

5 PC3/20/07

velocity

Seismic record

Seismic“Image”

time

time

6 PC3/20/07

Complexity• Acoustic approximation is often made

Ray methods: James Hobro, Chris Chapman, Henrik Bernth

7 PC3/20/07

(Acoustic) Problem definition

• Recover inhomogeneous

subsurface velocity

(density, impedance, …)

field from surface

measurements

• Born/Fréchet Kernel

• Green function from

– Full wave equation

– One-way wave equations

– Asymptotic ray theory

– Maslov

– Gaussian beam…

.x

(x,y)s r

z s: sourcer: receiverx: scatterer/reflector

)()(

22

2

sxuuxc

dxxsGc

crxGu ),;(),;(

32

3

),(

Rx

xu

8 PC3/20/07

Fréchet kernels for multi-scale waveform inversion

• Full wave equation inversion

• Acoustic wave equation

– Frequency domain

• Helmholtz equation

– Multigrid solver

• Multiscale approach

• Ray modeling

– Turning waves

– Maslov asymptotic approx.

• Sensitivity, resolution & influence

2

22

2

22

2

11.

ˆ1)(

v

fp

fp

t

p

fufu

t

u

9 PC3/20/07

Frequency domain formulation• Frequency domain adjoint formulation

(Pratt):

• Forward model:

PriorumuumumE obs

d

obs C *1))(),(())(),((2

1)(

)))(),((Re( *1 obsd

Tm umuCJE

muJ

u

EAu

m

AE H

T

m

um

AAJu

m

A

m

uAfmumA

10)(),(),(

Back-propagationForward

propagation

10 PC3/20/07

Multi-scale approach*• Low frequency => large wavelength

=> large basin of convergence

• Multiscale continuation

– Solve for low frequency ~3 Hz

– Increase frequencies incrementally

• Use last [subsurface velocity] as initial guess for new frequency

f0 f1 f2 …. fn …..

J

*Sirgue, L and Pratt, R.G. (2004) “Efficient waveform inversion and imaging: A strategy for selecting temporal frequencies”, Geophysics 69(1), pp.231-248*Pratt, R.G. et al (1998, 1999)*Ghattas et al. & Tromp et al

11 PC3/20/07

Dept

h

Depth

Vp

Test model• Exact velocity model • Traveltime tomography starting

model

• Sensitivities

Surf

ace

Surface

SurfaceSurface

Surface Surface

Surface

Vp

Depth

12 PC3/20/07

Inversion results f < 20 Hz

Vp@1.5Hz

Vp@5Hz

Vp@16Hz

Vp@Truth

13 PC3/20/07

Inversion results Vp vs Depth

1/4 1/2 3/4

Wavelength

0

500

1000

1500

2000

2500

0 2 4 6 8 10 12 14 16 18

Frequency (Hz)

Wav

elen

gth

(m

)

max wavelength

min wavelength

ave wavelength

Dep

th

Offset

14 PC3/20/07

Optimization• Quasi-Newton, LBFGS

– Solve for vp, ρ, source wavelet,…

• Project constraints

• L2, H2 + TV regularization

• Gauss-Newton + line search

• Constrained by modelling cost

• Multiple right hand sides

– Direct solver (SuperLU/MUMPS)

– Multigrid solvers

Tma EIHm .)( 1

vAuA H 1

)()( uTVβEE LS

15 PC3/20/07

Multigrid Preconditioner (Erlangga et

al, 2006)*

• H: Helmholtz– Indefinite– Not coercive– Non-local

• C: Complex shifted Laplace, improved spectral properties

• Preconditioner for H is C solved by Multigrid

),(),()1(),( 22 xfxikx

rki 221 )(

02 01

bCxHC )( 1

*Y.A. Erlangga and C.W. Oosterlee and C. Vuik (2006).

“A Novel Multigrid Based Preconditioner For Heterogeneous Helmholtz Problems”,

SIAM J. Sci. Comput.,27, pp. 1471-1492, 2006

16 PC3/20/07

Multigrid Helmholtz solver - subsalt

Convergence (Cost)

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

0 1000 2000 3000 4000

Iterations

-lo

g(R

/R0

)

NONE

MG5

WavefieldSigsbee salt velocity model

Convergence independence on number of grids

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

1.00E+00

1.00E+01

0 50 100 150 200 250 300 350 400

iterations

log

(R/R

0) MG5

MG4

MG3

MG2

MG1

Multiple grids

Depth

Offset

Vp

17 PC3/20/07

Plane wave synthesis

source

rece

iver

xs xr X,p,T

19 PC3/20/07

Results• Low frequency starting

model

– caustics & pseudo-causticsSurface

Initial velocity model

20 PC3/20/07

Densified rays show stabilityeven in such a complicated model;waveformsshow back-scattering

CJT, 1999

time

time

time

21 PC3/20/07

Maslov* waveforms• Integral over plane waves

• Sensitivity

• Asymptotic theory

dqemqAemxu mqxii

rr

),,()sgn(

42

),(~

),,(2

1

)),((),(),(),,( mqxxmqpmqTmqx rr

),(),(

),(

)(),()(),,(

),()(),,(

mxTimxTi

mxTi

em

Ae

m

TmxAi

m

mxu

emxAmxu

m

T

m

x

m

p

,,

mq

p

mq

x

m

A22

,,

*C.H.Chapman & R.Drummond, “Body-wave seismograms in inhomogeneous media using Maslov asymptotic theory”, Bull. Seism. Soc. America, vol 72, no. 6, pp.S277-S317, 1982.

22 PC3/20/07

Ray sensitivity equations (1)

• Hamiltonian system

• Dynamic ray tracing

• Paraxial sensitivity

x

p

H

H

p

x

dt

d

qp

qx

qp

qx

HH

HH

qp

qx

dt

d

xpxx

pppx

D

22

2

1 )(xcpH

mxH

mpH

mp

mx

mxH

mpH

mp

mx

mp

mx

dt

d2

2

2

2

DD

23 PC3/20/07

Ray tracing for gradients

• Calculating the kernel

• Solve ODEs

• Propagator solves for

• Sparse automatic differentiation evaluates

m

mqxpmqxx

m

p

m

mqT

m

mqxr

r

),()),((

),(),,(

mp

mx

1det(P) I,P DP,P ),( 0 TTdt

d

m

T

m

p

m

x

,,

24 PC3/20/07

Fréchet derivatives

• Basis functions

• Regularize over wavepaths

• Regularized gradient (5Hz)

26 PC3/20/07

Measures of resolution

• Resolution matrix

– Posterior covariance

– Lanczos solver

– Hessian vector products only

uqm

AAJqu

m

AAJJqJHq TTT

11 ,

*)Re( JJH Ta

uqm

AAAu

m

A TT

1

Gauss Newton

Diag(R)

Velocity model

Depth

Offset

Depth

Vp

Offset

27 PC3/20/07

Resolution of inverse operatorSingular Values

0

500

1000

1500

2000

2500

3000

0100200300400500600

)Re( *uJHm

EHm T

GNGNGN

)Re()Re( * JJHJJHR HGN

TGN

28 PC3/20/07

Closure• Frequency domain finite difference (FD) full

waveform inversion– regularization– multi-scale optimization

• Full wave equation FD(FE, SEM…) may be too detailed– reduced physics for forward models

• Which approximations inform the inverse solution ?• Are Maslov waveforms effective for turning ray

waveform inversion ?• Uncertainty estimates

End

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