Pulse propagation in random waveguides with turning points and application to imaging

Transcription

1 Pulse propagation in random waveguides with turning points and application to imaging Liliana Borcea Mathematics, University of Michigan, Ann Arbor and Josselin Garnier Centre de Mathématiques Appliquées, Ecole Polytechnique. Support: Air Force Office of Scientific Research, under award FA

2 The overall goal Image in waveguides with unknown geometry. Small amplitude random perturbations of straight boundaries Alonso, B., Garnier 2011; B., Garnier 2014; Gomez Can we estimate larger variations? Cannot expect unique recovery. Bonnet Ben-Dhia, Chesnel, Nazarov, Pagneaux: show smooth and abrupt waveguide changes that are invisible from remote measurements. We can only hope to estimate a proxy waveguide that captures the kinematics of the recorded waves. 2

3 Imaging ideas With Cakoni and Meng: linear sampling imaging at single frequency in waveguide with sound hard boundary, for 50 propagating modes (i.e., waveguide width is 25 wavelengths). This method can handle mild variations of known geometry. With Garnier and Meng: Tomographic estimation of waveguide from travel times of the recorded propagating modes. Question: Can we do this in waveguides with random boundary? 3

4 Setup: Slowly changing waveguide with random boundary Array of sensors probes with pulses the left (narrowing) part of the waveguide, which is possibly terminating. It records the reflected (turning) waves. Ω + r D x 0 Ω z What is the effect of random boundary on the pulses? Need weak scattering (negligible mode coupling) to hope to be able to estimate travel times. Random boundary can still affect turning waves! 4

5 Setup Pulse f(t) emitted from x generates wave, ( 1 ) c 2 2 t p(t, x) = f(t)δ(x x ), x Ω, t R, with p(t, x) 0 for t 0. Coordinates x (r, z) with z along axis, r in normal direction. D Ω + Ω 0 x r z At boundary: p(t, x) = 0 Ω r + (z) = D(z/L) 2 r (z) = D(z/L). 2 [ 1 + σν ( )] z D is increasing and smooth, ν is mean zero, stationary, with integrable autocorrelation R ν, smooth and bounded a.s. Bounded, differentiable curvature κ(z/l) of axis, L = scale of propagation, l = correlation length of fluctuations of strength σ. l 5

6 Scaling regime We use stochastic asymptotic analysis in = l/l 1. Pulse f(t) = cos(ω o t)f ( Bt) with envelope F of O(1) support. - Modulation at central frequency ω o and f(ω) supported at frequencies ω ω o B/2 for B ω o. - Small bandwidth B same number of propagating components (modes) of wave. Pulse support is O( ) smaller than travel time L/c. D(z)/λ o 1 determines number of propagating modes. λ o l efficient interaction with random boundary. Amplitude of random fluctuations is σ = σ. Mode coupling at σ 1 but turning waves sees an effect even at σ 1. 6

7 Random change of variables To satisfy the boundary conditions at all change variables r = ρ + (2ρ + D(z/L)) σν 4 ( ) z l, ρ D(z/L) 2, where in our scaling z L = O(1), z l = O ( 1 ), ρ D l, σ = σ. The homogeneous Dirichlet (sound soft) boundary conditions are at ρ = ±D/2, independent of. This change of variables maps the random fluctuations to the wave operator, which will have an asymptotic expansion in. 7

8 The perturbed wave equation We obtain, frequency by frequency, for k(ω) = ω/c, z 2 + k2 (ω) + ρ σ L 1 3/2 + L p (ω, ρ, z) = f (ω) δ(ρ ρ )δ(z) L 1 is differential operator with coefficients proportional to ν. L 2 is differential operator that accounts for curvature and has coefficients proportional to ν 2. At boundary p ( ω, ±D(z) 2, z ) = 0 and radiation conditions. Effect of random boundary seen after removing leading part. 8

9 Waveguide modes Mode decomposition: Expansion in basis {y j (ρ, z)} j 1 of eigenfunctions of k 2 (ω) + 2 ρ : p(t, x) = dω 2π p(ω, x)e iωt, p (ω, ρ, z) = j=1 û j (ω, z)y j(ρ, z), - Modes û j (ω, z)e iωt are time harmonic 1-D waves: - Wavenumbers k 2 (ω) ( ) πj 2 D(z) are real for j N(z) = D(z)k(ω) π. - For j > N(z) modes are evanescent. They contribute only in strong scattering regime (mode coupling). To analyze turning wave let first N(z) = 1 near source (z 0). 9

10 Propagating mode equation and turning point Propagating mode û (ω, z)e iωt satisfies [ 2z ( k 2 (ω) π2 D 2 (z) ) + π2 σ 3/2 D 2 (z) ν ( ) z +... ] û (ω, z) = 0 with source conditions at z = 0 and radiation conditions. Turning point z T (ω) where k(ω) = π/d(z T (ω)). For z > z T (ω) propagating wave and for z < z T (ω) evanescent. z T z 10

11 Global turning wave decomposition Decomposition uses flow M (ω, z) that cancels leading term ( û (ω, z) i z û (ω, z) ) = ( M (ω, z) M ) ( (ω, z) a (ω, z) i z M (ω, z) i z M (ω, z) b (ω, z) ) Assuming simple turning point i.e., D (z T ) > 0, M (ω, z) = π η ω (z) 1 4 β 1/2 (ω, z) e iπ 4 iφ ω(0) [ Ai ( η ω (z)) ib i ( η ω (z))] where β(ω, z) = k 2 (ω) π 2 /D 2 (z), A i and B i are Airy functions, z φ ω (z) = ds β(ω, s), η ω (z) = sign(z z T ) z T (ω) [ 3 φω (z) 2 ]

12 Properties of propagator M is invertible, with det M (ω, z) = 2, for all z. It ensures energy conservation z [ a (ω, z) 2 b (ω, z) 2] = 0. For z > z T (ω) + O(1) it gives û (ω, z) a (ω, z) β 1/2 (ω, z) e z i 0 dz β(ω, z ) b (ω, z) β 1/2 (ω, z) e i z 0 dz β(ω, z ) M grows at z < z T (ω). To counter growth set radiation cond: a (ω, z b ) = ie 2iφ ω(0) 0 b (ω, z b ), φ ω (0) = ds β(ω, s) z T (ω) for some z b < z T (ω). Result independent of particular z b! 12

13 The reflection coefficient Radiation condition and conservation of energy give R (ω, z) = a (ω, z) b (ω, z) = ie2iφω(0) +iψ ω (z) Random phase ψ ω (z) satisfies for z > z b, z ψ ω (z) = 2π3 η [ ω (z) 1 2 A 2 i ( η ω (z)) + B2 i ( η ω (z))] D 2 (z) β(ω, z) cos 2 { ψ ω (z) 2 σ ν ( ) z arg [ A i ( η ω (z)) + ib i( η ω (z))]} with initial condition ψ ω (z b) = 0. Source gives b (ω, 0). Reflected wave: a (ω, 0) = R (ω, 0)b (ω, 0). 13

14 The reflection coefficient When z T z O( 2/3 ), arguments of Airy functions are η ω(z) = 3 2 zt z ds π2 D 2 (s) k2 (ω) Using asymptotic expansions of A i and B i we obtain z ψ ω(z) = O ( exp [ 4 3 η ω(z) 3/2]) starting from z b, ψ ω (z) 0 until z approaches z T from the left. In the remaining domain the problem is similar to a diffusion (central limit type) limit. 14

15 The results Theorem 1: ψ (0) is asymptotically Gaussian distributed in the ω limit 0, with mean zero and variance υ 2 = k4 (ω o ) γ ω R ν (0) lim 0 σ 2 ln ( 1 0 z T (ω) dz β(ω, z) where R ν = power spectral density of ν and γ ω = z π 2 ), D 2 (z) z=zt (ω). Theorem 2: Let ψ j = ψ ωo+ w j (0) since bandwidth is B. Ψ = ( ψ 1,..., ψ m) converges in distribution as 0 to Gaussian vector with mean zero and covariance matrix C = υ2 3 (I m + 2J m ) where I m = identity, J m = matrix with all entries equal to one. Use diffusion limit theorem due to J. H. Kim 1996.

16 The reflected wave The array at z = 0 receives the reflected wave p ref (t, ρ) sin ( πρ D(0) [ 2i 0 exp ) z T (ωo+ ω) dω [ ( ) ] 2πB e i(ω o+ ω)t ω F + c.c. B dz β(ω o + ] ω, z) + iψ ωo+ (0) ω Previous theorems give distribution of ψ ωo+ ω as 0. Expand phase in powers of and recall β(ω, z) = k(ω) π2 D 2 (z) Observe wave at t = T + t near travel time T = 2 c 0 z T (ω o ) dz 1 [ π D(z)k(ω o ) ] 2. 16

17 The pulse stabilization result Theorem 3 The reflected wave at t is, up to multiplicative constant independent of, ( ) [ i p πρ ref (t, ρ) sin Re e D(0) [ 2 0 z T (ωo) dz β(ω o,z) ωot ] F ref (t) ]. It oscillates at frequency ω o / as emitted pulse, with envelope F ref (t) = dw 2πB ( ) w F e iw2 θ+iψ ωo+ w (0) iwt. B As 0, Fref (t) converges in distribution, in space of continuous functions on compact sets in R, to F ref (t) = e iϕ υ2 6 dw 2πB ( ) w F e iw2 θ ω o iwt, ϕ N ( 0, 2υ 2 /3 ). B 17

18 Sketch of proof t 1,..., t m calculate finite-order moments E m j=1 Fref dw1 (t j) = 2πB E ( ) w1 F... B [ e i m j=1 ψj ] dwm 2πB ( wm F )e i ( m j=1 w 2 j θ ωo w j t j B ) By Theorem 2 [ lim E e i m j=1 ψ ] [ m j = e m(2m+1)υ2 6 = E e υ2 6 +iϕ] 0 j=1 We obtain lim E 0 m j=1 Fref (t j) = E m j=1 F ref (t j ) Moreover sup t t t of {Fref (t)} for t in compact set in R. F ref (t ) F ref (t) C t tightness of laws 18

19 Setup for travel time tomography in multimode waveguide Source emits N = π D(0)k(ω o ) travel times of turning modes propagating modes. Measure T j = 2 c 0 z j (ω o ) dz [ 1 ( jπ D(z)k(ω o ) ) 2 ] 1/2, j = n,..., N. Non-turning modes j n are reflected if waveguide terminates. D Ω + x r 0 Ω z Mode speed depends on waveguide geometry. Can invert explicitly with Abel transform in special situations. 19

20 Conclusions Pulse stabilization result in waveguide displays strong interaction of the turning wave with the random boundary. If we did not have the turning point, the waves would see no effect at L = l/, for random fluctuations of amplitude σ = σ, with σ 1/ ln() 1/2. At σ the modes are coupled by the random fluctuations. We analyzed this case (with Garnier and Wood) for time harmonic waves. Again, the turning wave picks up a random phase in vicinity of z T, but there is no pulse stabilization in this case. Non-monotone D(z) trapped modes. This is difficult. Lots to do for the inverse problem. 20