data-to-born transform for inversion and imaging with wavesmamonov/dtbdec2017.pdf · data-to-born...

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

A.V. Mamonov Data-to-Born transform 1 / 26

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


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)


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


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


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


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


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|

]


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


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


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


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


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

)


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


Numerical results: 1D true Born vs. DtB

Full suppression of multiple reflectionsBoth arrival times and amplitudes are matched exactly


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


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


Numerical results: 2D snapshotsσ c

Pq0V q0

Pq V q

Array with m = 50 sensors ×Snapshots plotted for a single source ◦


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


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


Numerical results: 2D Elasticity

Measured data DtB


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)


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]


data-to-born transform for inversion and imaging with wavesmamonov/dtbdec2017.pdf · data-to-born...

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