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Data-to-Born transformfor inversion and imaging with waves
Alexander V. Mamonov1,Liliana Borcea2, Vladimir Druskin3, and Mikhail Zaslavsky3
1University of Houston,2University of Michigan Ann Arbor,
3Schlumberger-Doll Research Center
Support: NSF DMS-1619821, ONR N00014-17-1-2057
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Introduction
Inversion with waves: determine acoustic properties of amedium in the bulk from response measured at or near the surface
Highly nonlinear problem due to, in part, multiple scattering
Given the full waveform response, can we deduce what responsewill the same medium have if waves propagated under the singlescattering approximation, i.e. in Born regime?
Turns out we can!
A highly nonlinear transform takes full waveform data to singlescattering data: Data-to-Born (DtB) transform
Acts as a data preprocessing algorithm, can be integrated intoexisting workflows
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Forward model: 1D
Acoustic wave equation for pressure
(∂tt + A)p(t , x) = 0, x ∈ (0, `), t > 0,
with operator
A = −σ(x)c(x)∂x
[c(x)σ(x)
∂x
],
and initial conditions
p(0, x) =√σ(x)b(x), pt(0, x) = 0
Wave speed: c(x), impedance: σ(x)Solution: pressure wavefield
p(t , x) = cos(t√
A)√σ(x)b(x)
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Liouville transform and travel time coordinates
Using Liouville transform and travel time coordinatesT (x) =
∫ x0
dsc(s) , can write everything in first order form
∂t
(PU
)=
(0 −Lq
LTq 0
)(PU
),
where
Lq = −∂T +12∂T q,
LTq = ∂T +
12∂T q,
withq(T ) = lnσ
(x(T )
)Why use Liouville transform? Lq is affine in q!We consider Born approximation w.r.t. q around some q0
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Data sampling and propagator
Data model for collocated sources/receivers:
D(t) =⟨
b, cos(
t√
LqLTq
)b⟩
Data is sampled discretely at tk = kτ :
Dk = D(tk ) =⟨b, cos
(k arccos
(cos(τ√
LqLTq)))
b⟩=⟨b, Tk (P)b
⟩,
where the propagator (Green’s function) is
P = cos(τ√
LqLTq
)Works best with τ approximately around Nyquist samplingfor source b
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Wavefield snapshots and time stepping
Data sampling induces wavefield snapshots
Pk (T ) = P(tk ,T ) = Tk (P)b(T )
Snapshots satisfy exactly a time-stepping scheme
1τ2
[Pk+1(T )− 2Pk (T ) + Pk−1(T )
]= −ξ(P)Pk (T ),
withξ(P) =
2τ2
(I −P
)� 0
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Reduced order model
Factorizeξ(P) = LqLT
q
Simple Taylor expansion
Lq = Lq + O(τ2)
Why work with ξ(P)?Can estimate P just from the sampled data Dk !
Reduced order model (ROM): matrix P̃PP ∈ Rn×n and vectorb̃ ∈ Rn satisfying data interpolation conditions
Dk =⟨b, Tk (P)b
⟩= b̃TTk (P̃PP)b̃, k = 0,1, . . . ,2n − 1
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Projection ROM
ROM interpolating the data is a projection
P̃PP i,j =⟨Vi ,PVj
⟩where Vk form an orthonormal basis for
Kn(P,b) = span{b,Pb, . . . ,Pn−1b} = span{P0,P1, . . . ,Pn−1}
Problem: snapshots Pk are unavailable, thus orthogonalizedsnapshots Vk are unknownSolution: data gives us inner products of snapshots, entries ofmass and stiffness matrices
Mi,j =⟨Pi ,Pj
⟩, Si,j =
⟨Pi ,PPj
⟩A.V. Mamonov Data-to-Born transform 8 / 26
Mass and stiffness matrices from data
Snapshots and data in terms of Chebyshev polynomials
Pk = Tk (P)b, Dk =⟨b,Pk
⟩=⟨b, Tk (P)b
⟩Chebyshev polynomials obey a multiplication property
Ti(P)Tj(P) =12
[Ti+j(P) + T|i−j|(P)
]Can express mass and stiffness matrix entries in terms of data
Mi,j =⟨Pi ,Pj
⟩=
12
[Di+j + D|i−j|
]Si,j =
⟨Pi ,PPj
⟩=
14
[Di+j+1 + D|i+j−1| + D|i−j+1| + D|i−j−1|
]
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Implicit snapshot orthogonalization
Consider snapshots “matrices”
P = [P0,P1, . . . ,Pn−1], V = [V0,V1, . . . ,Vn−1],
formallyM = PT P, VT V = I (1)
If snapshots were known, we could use Gram-Schmidtorthogonalization (QR factorization)
P = VR, V = PR−1 (2)
with upper trangular R ∈ Rn×n
Combine (1)–(2) to getM = RT R,
a Cholesky factorization of the mass matrix known from data
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ROM from data
ROM is a projection, formally
P̃PP = VT PV (3)
We have Cholesky factor R of the mass matrix M = RT RSubstitute V = PR−1 into (3):
P̃PP = VT PV = R−T (PT PP)R−1 = R−T SR−1,
where S = PT PP is known from dataROM sensor vector
b̃ = Re1 = (D0)1/2e1
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Factorization of ξ(P̃PP)
Once P̃PP is computed, form
ξ(P̃PP) =2τ2
(I− P̃PP
)� 0
and take another Cholesky factorization
ξ(P̃PP) = L̃qL̃Tq
Note: propagator ROM P̃PP is tridiagonal, thus its Cholesky factorL̃q ∈ Rn×n is lower bidiagonalSince
ξ(P) = LqLTq ≈ LqLT
q ,
we conclude that L̃q approximates
Lq = −∂T +12∂T q,
affine in q!A.V. Mamonov Data-to-Born transform 12 / 26
Data-to-Born transform
Assume known c (travel-time coordinates) and choose areference impedance q0, consider the perturbation
L̃ε = L̃q0 + ε(L̃q − L̃q0
)Solve
ξ(P̃PP) =2τ2
(I− P̃PP
)= L̃qL̃T
q
for P̃PP and perturb the propagator correspondingly
P̃PPε= I− τ2
2L̃εL̃εT
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Data-to-Born transform
Recall data interpolation for unknown q:
Dk = b̃TTk (P̃PP)b̃,
similarly Dq0
k for reference q0
The Data-to-Born transform (DtB) is given by
Bk = Dq0
k + b̃T[ d
dεTk
(P̃PP
ε)∣∣∣∣ε=0
]b̃
Chain rule does not apply to matrix functions, use Chebyshevpolynomial three-term recurrence to differentiate
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Numerical results: 1D Born approximation
Compare DtB to data generated using true Born approximationin q = lnσ with known c and reference non-reflective q0:
∂t
(PBorn
UBorn
)=
(0 −Lq0
LTq0 0
)(PBorn
UBorn
)+
12∂T
((q0 − q)Uq0
(q − q0)Pq0
),
where as before
Lq0 = −∂T +12∂T q0, LT
q0 = ∂T +12∂T q0,
and
∂t
(Pq0
Uq0
)=
(0 −Lq0
LTq0 0
)(Pq0
Uq0
)
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Numerical results: 1D snapshots
σ P20 P80
V20 V80Referencec = σ ≡ 1, q0 ≡ 0
Source/receiver atx = 0 (T = 0)
Spatial axis in k ,integer units of τ :Tk = kτ
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Numerical results: 1D true Born vs. DtB
Full suppression of multiple reflectionsBoth arrival times and amplitudes are matched exactly
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Higher dimensions
Approach generalizes naturally to higher dimensionsData recorded at an array of m sensorsData is an m ×m matrix function of time
Di,j(t) =⟨
bi , cos(
t√
LqLTq
)bj
⟩, i , j = 1, . . . ,m,
where
Lq = −c(x)∇ · +12
c(x)∇q(x)·,
LTq = c(x)∇+
12
c(x)∇q(x)
No travel time coordinate transformation in higher dimensions
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Higher dimensions: ROM
If sampled data is Dk = D(kτ) ∈ Rm×m, ROM satisfies matrixinterpolation conditions
Dk = b̃TTk (P̃PP)b̃, k = 0, . . . ,2n − 1,
with block matrices P̃PP ∈ Rmn×mn, b̃ ∈ Rmn×m
ROM and DtB transform are computed with block versions of thesame algorithms as in 1D
All linear algebraic procedures (Gram-Schmidt, Cholesky) arereplaced with their block counterparts
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Numerical results: 2D snapshotsσ c
Pq0V q0
Pq V q
Array with m = 50 sensors ×Snapshots plotted for a single source ◦
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Numerical results: 2D true Born vs. DtB
Single row of data matrixcorresponding to source ◦Vertical: time (in units of τ )Horizontal: receiver index(out of m = 50)
Measured data Born data DtB
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Numerical results: 2D DtB + RTM
Reverse time migration (RTM)image computed from bothmeasured full waveform dataand DtB transformed data
RTM from measured data RTM from DtB
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Numerical results: 2D Elasticity
Measured data DtB
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Numerical results: 2D Elasticity
Measured data DtB
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Conclusions and future work
Data-to-Born: transform acoustic full waveform data to singlescattering (Born) data for the same mediumBased on techniques of model order reductionData-driven approach relying on classical linear algebraalgorithms (Cholesky, QR), no computations in the continuumEasy to integrate into existing workflows as a preprocessing stepEnables the use of linearized inversion algorithms
Future work:Reverse direction, Born-to-Data: use a cheap single scatteringsolver to generate full waveform dataTest performance of linearized inversion algorithms (e.g. LS-RTM)on DtB dataExtend to frequency domain wave equation (Helmholtz)
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References
Untangling the nonlinearity in inverse scattering with data-drivenreduced order models, L. Borcea, V. Druskin, A.V. Mamonov,M. Zaslavsky, 2017, arXiv:1704.08375 [math.NA]
Related work:1 Direct, nonlinear inversion algorithm for hyperbolic problems via
projection-based model reduction, V. Druskin, A. Mamonov, A.E.Thaler and M. Zaslavsky, SIAM Journal on Imaging Sciences9(2):684–747, 2016.
2 Nonlinear seismic imaging via reduced order modelbackprojection, A.V. Mamonov, V. Druskin, M. Zaslavsky, SEGTechnical Program Expanded Abstracts 2015: pp. 4375–4379.
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]
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