Seismic inversion and imaging viamodel order reduction
Alexander V. Mamonov1,Liliana Borcea2, Vladimir Druskin3,
Andrew Thaler4 and Mikhail Zaslavsky3
1University of Houston,2University of Michigan Ann Arbor,
3Schlumberger-Doll Research Center,4The Mathworks, Inc.
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Motivation: seismic oil and gas exploration
Problems addressed:1 Inversion: quantitative
velocity estimation, FWI
2 Imaging: qualitative ontop of velocity model
3 Data preprocessing:multiple suppression
Common framework:Reduced Order Models(ROM)
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Forward model: acoustic wave equation
Acoustic wave equation in the time domain
utt = Au in Ω, t ∈ [0,T ]
with initial conditions
u|t=0 = B, ut |t=0 = 0,
sources are columns of B ∈ RN×m
The spatial operator A ∈ RN×N is a (symmetrized) fine griddiscretization of
A = c2∆
with appropriate boundary conditionsWavefields for all sources are columns of
u(t) = cos(t√−A)B ∈ RN×m
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Data model and problem formulations
For simplicity assume that sources and receivers are collocated,receiver matrix is also BThe data model is
D(t) = BT u(t) = BT cos(t√−A)B,
an m ×m matrix function of time
Problem formulations:1 Inversion: given D(t) estimate c2 Imaging: given D(t) and a smooth kinematic velocity model c0,
estimate “reflectors”, discontinuities of c3 Data preprocessing: given D(t) obtain F(t) with multiple
reflection events suppressed/removed
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Reduced order models
Data is always discretely sampled, say uniformly at tk = kτThe choice of τ is very important, optimally τ around Nyquist rateDiscrete data samples are
Dk = D(kτ) = BT cos(
kτ√−A)
B = BT Tk (P)B,
where Tk is Chebyshev polynomial and the propagator is
P = cos(τ√−A)∈ RN×N
A reduced order model (ROM) P, B should fit the data
Dk = BT Tk (P)B = BT Tk (P)B, k = 0,1, . . . ,2n − 1
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Projection ROMs
Projection ROMs are of the form
P = VT PV, B = VT B,
where V is an orthonormal basis for some subspaceWhat subspace to project on to fit the data?Consider a matrix of wavefield snapshots
U = [u0,u1, . . . ,un−1] ∈ RN×nm, uk = u(kτ) = Tk (P)B
We must project on Krylov subspace
Kn(P,B) = colspan[B,PB, . . . ,Pn−1B] = colspan U
The data only knows about what P does to wavefieldsnapshots uk
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ROM from measured data
Wavefields in the whole domain U are unknown, thus V isunknownHow to obtain ROM from just the data Dk?Data does not give us U, but it gives us inner products!Multiplicative property of Chebyshev polynomials
Ti(x)Tj(x) =12
(Ti+j(x) + T|i−j|(x))
Since uk = Tk (P)B and Dk = BT Tk (P)B we get
(UT U)i,j = uTi uj =
12
(Di+j + Di−j),
(UT PU)i,j = uTi Puj =
14
(Dj+i+1 + Dj−i+1 + Dj+i−1 + Dj−i−1)
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ROM from measured data
Suppose U is orthogonalized by a block QR (Gram-Schmidt)procedure
U = VLT , equivalently V = UL−T ,
where L is a block Cholesky factor of the Gramian UT U knownfrom the data
UT U = LLT
The projection is given by
P = VT PV = L−1(
UT PU)
L−T ,
where UT PU is also known from the dataCholesky factorization is essential, (block) lower triangularstructure is the linear algebraic equivalent of causality
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Problem 1: Inversion (FWI)
Conventional FWI (OLS)
minimizec
‖D? − D( · ; c)‖22
Replace the objective with a “nonlinearly preconditioned”functional
minimizec
‖P? − P(c)‖2F ,
where P? is computed from the data D? and P(c) is a (highly)nonlinear mapping
P : c → A(c)→ U→ V→ P
Similar approach to diffusive inversion (parabolic PDE, CSEM)converges in one Gauss-Newton iteration
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Conventional vs. ROM-preconditioned FWI in 1D
Conventional ROM-preconditioned
0 0.5 1 1.5 2 2.5 3
0.5
1
1.5
2
CG iteration 1, Er = 0.278869
CGtrue
0 0.5 1 1.5 2 2.5 3
0.5
1
1.5
2
CG iteration 1, Er = 0.272127
CGtrue
Automatic removal of multiple reflections.
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Conventional vs. ROM-preconditioned FWI in 1D
Conventional ROM-preconditioned
0 0.5 1 1.5 2 2.5 3
0.5
1
1.5
2
CG iteration 5, Er = 0.265722
CGtrue
0 0.5 1 1.5 2 2.5 3
0.5
1
1.5
2
CG iteration 5, Er = 0.197026
CGtrue
Automatic removal of multiple reflections.
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Conventional vs. ROM-preconditioned FWI in 1D
Conventional ROM-preconditioned
0 0.5 1 1.5 2 2.5 3
0.5
1
1.5
2
CG iteration 10, Er = 0.273922
CGtrue
0 0.5 1 1.5 2 2.5 3
0.5
1
1.5
2
CG iteration 10, Er = 0.157774
CGtrue
Automatic removal of multiple reflections.
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Conventional vs. ROM-preconditioned FWI in 1D
Conventional ROM-preconditioned
0 0.5 1 1.5 2 2.5 3
0.5
1
1.5
2
CG iteration 15, Er = 0.268569
CGtrue
0 0.5 1 1.5 2 2.5 3
0.5
1
1.5
2
CG iteration 15, Er = 0.138945
CGtrue
Automatic removal of multiple reflections.
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Conventional vs. ROM-preconditioned FWI in 1D
Conventional ROM-preconditioned
0 0.5 1 1.5 2 2.5 30.2
0.4
0.6
0.8
1
1.2
1.4
1.6
CG iteration 1, Er = 0.173770
CGtrue
0 0.5 1 1.5 2 2.5 30.2
0.4
0.6
0.8
1
1.2
1.4
1.6
CG iteration 1, Er = 0.147049
CGtrue
Avoiding the cycle skipping.
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Conventional vs. ROM-preconditioned FWI in 1D
Conventional ROM-preconditioned
0 0.5 1 1.5 2 2.5 30.2
0.4
0.6
0.8
1
1.2
1.4
1.6
CG iteration 5, Er = 0.174695
CGtrue
0 0.5 1 1.5 2 2.5 30.2
0.4
0.6
0.8
1
1.2
1.4
1.6
CG iteration 5, Er = 0.105966
CGtrue
Avoiding the cycle skipping.
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Conventional vs. ROM-preconditioned FWI in 1D
Conventional ROM-preconditioned
0 0.5 1 1.5 2 2.5 30.2
0.4
0.6
0.8
1
1.2
1.4
1.6
CG iteration 10, Er = 0.174688
CGtrue
0 0.5 1 1.5 2 2.5 30.2
0.4
0.6
0.8
1
1.2
1.4
1.6
CG iteration 10, Er = 0.095547
CGtrue
Avoiding the cycle skipping.
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Conventional vs. ROM-preconditioned FWI in 1D
Conventional ROM-preconditioned
0 0.5 1 1.5 2 2.5 30.2
0.4
0.6
0.8
1
1.2
1.4
1.6
CG iteration 15, Er = 0.174689
CGtrue
0 0.5 1 1.5 2 2.5 30.2
0.4
0.6
0.8
1
1.2
1.4
1.6
CG iteration 15, Er = 0.086519
CGtrue
Avoiding the cycle skipping.
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Problem 2: Imaging
ROM is a projection, we can use backprojection
If span(U) is suffiently rich, then columns of VVT should be goodapproximations of δ-functions, hence
P ≈ VVT PVVT = VPVT
Problem: U and V are unknown
We have a rough idea of kinematics, i.e. the travel times
Equivalent to knowing a smooth kinematic velocity model c0
For known c0 we can compute
U0, V0, P0
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Backprojection imaging functional
Take backprojection P ≈ VPVT and make another approximation:replace unknown V with V0
P ≈ V0PVT0
For the kinematic model we know V0 exactly
P0 ≈ V0P0VT0
Take the diagonals of backprojections to extract approximateGreen’s functions
G( · , · , τ)−G0( · , · , τ) = diag(P−P0) ≈ diag(
V0(P− P0)VT0
)= I
Approximation quality depends only on how well columns ofVVT
0 and V0VT0 approximate δ-functions
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Simple example: layered modelTrue sound speed c Backprojection: c0 + αI
RTM imageA simple layered model, p = 32sources/receivers (black ×)Constant velocity kinematicmodel c0 = 1500 m/sMultiple reflections from wavesbouncing between layers andsurfaceEach multiple creates an RTMartifact below actual layersA.V. Mamonov ROMs for inversion and imaging 20 / 31
Snapshot orthogonalizationSnapshots U Orthogonalized snapshots V
t = 10τ
t = 15τ
t = 20τ
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Snapshot orthogonalizationSnapshots U Orthogonalized snapshots V
t = 25τ
t = 30τ
t = 35τ
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Approximation of δ-functionsColumns of V0VT
0 Columns of VVT0
y = 345 m
y = 510 m
y = 675 m
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Approximation of δ-functionsColumns of V0VT
0 Columns of VVT0
y = 840 m
y = 1020 m
y = 1185 m
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High contrast example: hydraulic fracturesTrue c Backprojection image I
Important application: hydraulic fracturing
Three fractures 10 cm wide each
Very high contrasts: c = 4500 m/s in the surrounding rock,c = 1500 m/s in the fluid inside fractures
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High contrast example: hydraulic fracturesTrue c RTM image
Important application: hydraulic fracturing
Three fractures 10 cm wide each
Very high contrasts: c = 4500 m/s in the surrounding rock,c = 1500 m/s in the fluid inside fractures
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Backprojection imaging: Marmousi model
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Problem 3: Data preprocessing
Use multiple-suppression properties of ROM to preprocess dataCompute P from D and P0 from D0 corresponding to c0
Propagator perturbation
Pε = P0 + ε(P− P0)
Propagate the perturbation
Dε,k = BT Tk (Pε)B
Generate filtered data
Fk = D0,k +dDε,k
dε
∣∣∣∣ε=0
Can show that Fk corresponds to data that a Born forwardmodel will generate
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Example: seismogram comparison
Three direct arrivals +three multiplesDirect arrival from smallscatterer masked by thefirst multiple
Dk − D0,k Fk − D0,k
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Conclusions and future work
ROMs for inversion, imaging, data preprocessingTime domain formulation is essential, linear algebraic analoguesof causality: Gram-Schmidt, CholeskyImplicit orthogonalization of wavefield snapshots: suppressionof multiples in backprojection imaging and data preprocessingAccelerated convergence, alleviated cycle-skipping inROM-preconditioned FWI
Future work:Non-symmetric ROM for non-collocated sources/receiversNoise effects and stabilityROM-preconditioned FWI in 2D/3D
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References
1 Nonlinear seismic imaging via reduced order modelbackprojection, A.V. Mamonov, V. Druskin, M. Zaslavsky, SEGTechnical Program Expanded Abstracts 2015: pp. 4375–4379.
2 Direct, nonlinear inversion algorithm for hyperbolic problems viaprojection-based model reduction, V. Druskin, A. Mamonov, A.E.Thaler and M. Zaslavsky, SIAM Journal on Imaging Sciences9(2):684–747, 2016.
3 A nonlinear method for imaging with acoustic waves via reducedorder model backprojection, V. Druskin, A.V. Mamonov,M. Zaslavsky, 2017, arXiv:1704.06974 [math.NA]
4 Untangling nonlinearity in inverse scattering with data-drivenreduced order models, L. Borcea, V. Druskin, A.V. Mamonov,M. Zaslavsky, 2017, arXiv:1704.08375 [math.NA]
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