S. Marmorat (POEMS - INRIA Rocquencourt) F. Collino (CERFACS) J.-D. Benamou (POEMS - INRIA Rocquencourt)

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1 Numerical MicroLocal Analysis (NMLA) S. Marmorat (POEMS - INRIA Rocquencourt) F. Collino (CERFACS) J.-D. Benamou (POEMS - INRIA Rocquencourt) Given frequency domain wave data, the proposed algorithm gives a pointwise estimate of the the number of rays, their slowness vectors and corresponding wavefront curvature. With time domain wave data and assuming the source wavelet is given, the method also estimates the traveltime

2 Geometric optics equations / Far field models asymptotic (large ω) solutions of ω 2 ω) + û(x; ω) = 0 c2 (x)û(x; û replaced by ansatz yields û û ray (x; ω) = A(x)e iωϕ(x) ϕ(x) = 1 c(x) 2 ϕ(x) A(x) + A(x) ϕ(x) = 0 : Find local For constant medium and far field data, linear (plane wave) phase approximation is a popular choice (beamforming, DOA) : û û ray (x; ω) = B(x)e i k x, k = ω c 2

3 Rq. : Plane wave approximation around a point x 0 ϕ(x) ϕ(x 0 ) + (x x 0 ) ϕ(x 0 ) (x x 0) T Hϕ(x 0 ) (x x 0 ) +... yields û(x; ω) B(x 0 )e iω(x x 0) ϕ(x 0 ) where B(x 0 ) = A(x 0 )e iωϕ(x 0). The first general N-rays ansatz we consider is : û(x; ω) N n=1 B n (x 0 )e iω(x x 0) ϕ n (x 0 ) x near x 0 3

4 NMLA observable in the frequency domain The observable data ( s { s = 1}) U α ( s) = c(x 0) û iω r (x 0 + r s; ω) + û(x 0 + r s; ω), r = αc(x 0) should fit the ansatz form ω. U α ( s) N n=1 ( s d n + 1) B n e iα s d n, dn = c(x 0 ) ϕ n (x 0 ) 4

5 Relaxation towards a linear system Set d = (cos θ d, sin θ d ). U α ( s) 2 π 0 ( s d + 1) β( d) e iα s d dθ d U = K α β. (Discretization) K α,m,n = ( d m d n + 1) e iα d m d n, U m = U α ( d m ), B m = β(θ m ). K α is compact but a regularized inverse is easy to compute (Filter + truncate Fourier modes of β) and its norm is bounded independantly of α (Stability). 5

6 NMLA filter β := with where 1 2 L(α) + 1 F 1 ({ˆβ l }), ˆβ l = H l F({U}) l H l = D 1 l = 0 else. if l < L(α) D l (α) = 2πi l (J l (α) i J l (α)) and L(α) = min{α, α + α 1/3 2.5} Chosen such that K 1 α < 3 (Stability Theorem) 6

7 For an exact Plane wave : If wavefield is a perfect plane wave of direction d with amplitude A then and U plwa ( s) = ( d s + 1) A e iωϕ(x 0) e iα d s β plwa ( s) = A e iωϕ(x 0) S α (θ d θ s ) S α (θ) = sin((l(α) )θ) (2L(α) + 1) sin( θ 2 ) where L(α) is the (explicitely given) Number of Fourier Modes used to filter β. Fourier modes are also given analytically ˆβ l = A e iωϕ(x 0) e ilθ d 7

8 Test 2 sources, homogeneous medium NMLA stability. β(θŝ) White noise (20%-40%) Correlated noise (20%-40%) Red lines : exact ray angles. 8

9 Varying α = ωr c(x 0 ) L(α) : 10, 20, 50 9

10 NMLA 2 nd order / near field application? ωr α = L(α) bounds the number of Fourier modes, while c(x 0 ) we hope to recover dirac masses... Cannot increase α because of the plane wave approximation. ϕ(x) ϕ(x 0 )+(x x 0 ) ϕ(x 0 )+ 1 2 (x x 0) T Hϕ(x 0 ) (x x 0 )+... Recall x x 0 = α c(x 0) ω, r = x x 0. Need to estimate 2 nd order terms. 10

11 The simplest 2 nd order approximation Assume only one ray in the solution. Constant curvature HF asymptotics corresponds to the point source solution H 1 0 (ω c x x s ). 11

12 Approximate NMLA data with H 1 0 U α ( s) i 4 H(1) 0 A 0 (x 0 ) ( ωc d d + r s ) e iω(ϕ(x 0) d) i 4 H(1) 0 ( ω c d d ) + r s d and d yet to be found θ d and 1 d are the local ray direction and mean curvature. 12

13 Use FMM type asymptotic expansions γ = c(x ω d is the large 0 ) parameter. This new ansatz yields a curvature correction to the NMLA Fourier modes ˆβ l Ae iωϕ(x0) e i(lθ d + (l2 2γ 1 4 ) ) i log( ˆβ l ˆβ 0 ) versus l Recall : Plane wave approximation Fourier modes were ˆβ l Ae iωϕ(x 0) e ilθ d 13

14 Test 1 source, homogeneous medium Black : NMLA - Blue : NMLA 2nd order - Red : exact direction.. zoom 14

15 Synthetic data numerical simulation (snapshots) Generated using standard FDTD + ABCs 15

16 Source Point localization in Heteogeneous medium Synthetic data - backward ray tracing using NMLA output (red) versus Radon (green) and PWD (yellow). Ray backward propagation and velocity model Center of red circle is the exact source localization. 16