Aspects of seismic inversion
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Transcript of Aspects of seismic inversion
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
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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