Data-to-Born transform for inversion and imaging with waves

Transcription

1 Data-to-Born transform for inversion and imaging with waves Alexander V. Mamonov 1, Liliana Borcea 2, Vladimir Druskin 3, and Mikhail Zaslavsky 3 1 University of Houston, 2 University of Michigan Ann Arbor, 3 Schlumberger-Doll Research Center Support: NSF DMS , ONR N A.V. Mamonov Data-to-Born transform 1 / 26

2 Introduction Inversion with waves: determine acoustic properties of a medium 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 response will the same medium have if waves propagated under the single scattering approximation, i.e. in Born regime? Turns out we can! A highly nonlinear transform takes full waveform data to single scattering data: Data-to-Born (DtB) transform Acts as a data preprocessing algorithm, can be integrated into existing workflows A.V. Mamonov Data-to-Born transform 2 / 26

3 Forward model: 1D Acoustic wave equation for pressure with operator and initial conditions ( tt + A)p(t, x) = 0, x (0, l), t > 0, [ c(x) ] A = σ(x)c(x) x σ(x) x, p(0, x) = σ(x)b(x), p t (0, x) = 0 Wave speed: c(x), impedance: σ(x) Solution: pressure wavefield p(t, x) = cos(t A) σ(x)b(x) A.V. Mamonov Data-to-Born transform 3 / 26

4 Liouville transform and travel time coordinates Using Liouville transform and travel time coordinates T (x) = x ds 0 c(s), can write everything in first order form ( ) ( ) ( ) P 0 Lq P t = U L T, q 0 U where with L q = T T q, L T q = T T q, q(t ) = ln σ ( x(t ) ) Why use Liouville transform? L q is affine in q! We consider Born approximation w.r.t. q around some q 0 A.V. Mamonov Data-to-Born transform 4 / 26

5 Data sampling and propagator Data model for collocated sources/receivers: ( ) D(t) = b, cos t L q L T q b Data is sampled discretely at t k = kτ: D k = D(t k ) = b, cos ( k arccos ( cos ( ))) τ L q L T q b = b, Tk (P)b, where the propagator (Green s function) is ( ) P = cos τ L q L T q Works best with τ approximately around Nyquist sampling for source b A.V. Mamonov Data-to-Born transform 5 / 26

6 Wavefield snapshots and time stepping Data sampling induces wavefield snapshots P k (T ) = P(t k, T ) = T k (P)b(T ) Snapshots satisfy exactly a time-stepping scheme with 1 [ ] τ 2 P k+1 (T ) 2P k (T ) + P k 1 (T ) = ξ(p)p k (T ), ξ(p) = 2 τ 2 ( I P ) 0 A.V. Mamonov Data-to-Born transform 6 / 26

7 Reduced order model Factorize ξ(p) = L q L T q Simple Taylor expansion Why work with ξ(p)? L q = L q + O(τ 2 ) Can estimate P just from the sampled data D k! Reduced order model (ROM): matrix P R n n and vector b R n satisfying data interpolation conditions D k = b, T k (P)b = b T T k ( P) b, k = 0, 1,..., 2n 1 A.V. Mamonov Data-to-Born transform 7 / 26

8 Projection ROM ROM interpolating the data is a projection P i,j = V i, PV j where V k form an orthonormal basis for K n (P, b) = span{b, Pb,..., P n 1 b} = span{p 0, P 1,..., P n 1 } Problem: snapshots P k are unavailable, thus orthogonalized snapshots V k are unknown Solution: data gives us inner products of snapshots, entries of mass and stiffness matrices M i,j = P i, P j, Si,j = P i, PP j A.V. Mamonov Data-to-Born transform 8 / 26

9 Mass and stiffness matrices from data Snapshots and data in terms of Chebyshev polynomials P k = T k (P)b, D k = b, P k = b, Tk (P)b Chebyshev polynomials obey a multiplication property T i (P)T j (P) = 1 2 [ ] T i+j (P) + T i j (P) Can express mass and stiffness matrix entries in terms of data M i,j = 1 ] P i, P j = [D i+j + D 2 i j S i,j = 1 ] P i, PP j = [D i+j+1 + D 4 i+j 1 + D i j+1 + D i j 1 A.V. Mamonov Data-to-Born transform 9 / 26

10 Implicit snapshot orthogonalization Consider snapshots matrices P = [P 0, P 1,..., P n 1 ], V = [V 0, V 1,..., V n 1 ], formally M = P T P, V T V = I (1) If snapshots were known, we could use Gram-Schmidt orthogonalization (QR factorization) with upper trangular R R n n Combine (1) (2) to get P = VR, V = PR 1 (2) M = R T R, a Cholesky factorization of the mass matrix known from data A.V. Mamonov Data-to-Born transform 10 / 26

11 ROM from data ROM is a projection, formally P = V T PV (3) We have Cholesky factor R of the mass matrix M = R T R Substitute V = PR 1 into (3): P = V T PV = R T (P T PP)R 1 = R T SR 1, where S = P T PP is known from data ROM sensor vector b = Re 1 = (D 0 ) 1/2 e 1 A.V. Mamonov Data-to-Born transform 11 / 26

12 Factorization of ξ( P) Once P is computed, form ξ( P) = 2 τ 2 ( I P ) 0 and take another Cholesky factorization ξ( P) = L q LT q Note: propagator ROM P is tridiagonal, thus its Cholesky factor L q R n n is lower bidiagonal Since ξ(p) = L q L T q L q L T q, we conclude that L q approximates affine in q! L q = T T q, A.V. Mamonov Data-to-Born transform 12 / 26

13 Data-to-Born transform Assume known c (travel-time coordinates) and choose a reference impedance q 0, consider the perturbation L ε = L q 0 + ε ( Lq L ) q 0 Solve ξ( P) = 2 τ 2 ( I P ) = L q LT q for P and perturb the propagator correspondingly P ε = I τ 2 2 L ε LεT A.V. Mamonov Data-to-Born transform 13 / 26

14 Data-to-Born transform Recall data interpolation for unknown q: D k = b T T k ( P) b, similarly D q0 k for reference q 0 The Data-to-Born transform (DtB) is given by B k = D q0 k + b [ T d dε T k ( P ε) ε=0 ] b Chain rule does not apply to matrix functions, use Chebyshev polynomial three-term recurrence to differentiate A.V. Mamonov Data-to-Born transform 14 / 26

15 Numerical results: 1D Born approximation Compare DtB to data generated using true Born approximation in q = ln σ with known c and reference non-reflective q 0 : ( ) ( ) P Born (P ) ( ) 0 Lq Born 0 t = U Born L T q U 0 Born 2 (q 0 q)u q0 T (q q 0, )P q0 where as before L q 0 = T T q 0, L T q 0 = T T q 0, and t ( P q0 U q0 ) = ( 0 Lq 0 L T q 0 0 ) ( ) P q0 U q0 A.V. Mamonov Data-to-Born transform 15 / 26

16 Numerical results: 1D snapshots σ P 20 P 80 Reference c = σ 1, q 0 0 Source/receiver at x = 0 (T = 0) Spatial axis in k, integer units of τ: T k = kτ V 20 V 80 A.V. Mamonov Data-to-Born transform 16 / 26

17 Numerical results: 1D true Born vs. DtB Full suppression of multiple reflections Both arrival times and amplitudes are matched exactly A.V. Mamonov Data-to-Born transform 17 / 26

18 Higher dimensions Approach generalizes naturally to higher dimensions Data recorded at an array of m sensors Data is an m m matrix function of time ( D i,j (t) = b i, cos t L q L T q )b j, i, j = 1,..., m, where L q = c(x) c(x) q(x), L T q = c(x) c(x) q(x) No travel time coordinate transformation in higher dimensions A.V. Mamonov Data-to-Born transform 18 / 26

19 Higher dimensions: ROM If sampled data is D k = D(kτ) R m m, ROM satisfies matrix interpolation conditions D k = b T T k ( P) b, k = 0,..., 2n 1, with block matrices P R mn mn, b R mn m ROM and DtB transform are computed with block versions of the same algorithms as in 1D All linear algebraic procedures (Gram-Schmidt, Cholesky) are replaced with their block counterparts A.V. Mamonov Data-to-Born transform 19 / 26

20 Numerical results: 2D snapshots σ c P q0 V q0 P q V q Array with m = 50 sensors Snapshots plotted for a single source A.V. Mamonov Data-to-Born transform 20 / 26

21 Numerical results: 2D true Born vs. DtB Single row of data matrix corresponding to source Vertical: time (in units of τ) Horizontal: receiver index (out of m = 50) Measured data Born data DtB A.V. Mamonov Data-to-Born transform 21 / 26

22 Numerical results: 2D DtB + RTM Reverse time migration (RTM) image computed from both measured full waveform data and DtB transformed data RTM from measured data RTM from DtB A.V. Mamonov Data-to-Born transform 22 / 26

23 Numerical results: 2D Elasticity Measured data DtB A.V. Mamonov Data-to-Born transform 23 / 26

24 Numerical results: 2D Elasticity Measured data DtB A.V. Mamonov Data-to-Born transform 24 / 26

25 Conclusions and future work Data-to-Born: transform acoustic full waveform data to single scattering (Born) data for the same medium Based on techniques of model order reduction Data-driven approach relying on classical linear algebra algorithms (Cholesky, QR), no computations in the continuum Easy to integrate into existing workflows as a preprocessing step Enables the use of linearized inversion algorithms Future work: Reverse direction, Born-to-Data: use a cheap single scattering solver to generate full waveform data Test performance of linearized inversion algorithms (e.g. LS-RTM) on DtB data Extend to frequency domain wave equation (Helmholtz) A.V. Mamonov Data-to-Born transform 25 / 26

26 References Untangling the nonlinearity in inverse scattering with data-driven reduced order models, L. Borcea, V. Druskin, A.V. Mamonov, M. Zaslavsky, 2017, arxiv: [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 Sciences 9(2): , Nonlinear seismic imaging via reduced order model backprojection, A.V. Mamonov, V. Druskin, M. Zaslavsky, SEG Technical Program Expanded Abstracts 2015: pp A nonlinear method for imaging with acoustic waves via reduced order model backprojection, V. Druskin, A.V. Mamonov, M. Zaslavsky, 2017, arxiv: [math.na] A.V. Mamonov Data-to-Born transform 26 / 26