Scattered multi-static array imaging and eigenvalue problems - PDF document

Scattered multi-static array imaging and eigenvalue problems Wagner B. Muniz Department of Mathematics Federal University of Santa Catarina (UFSC) w.b.muniz@ufsc.br Brazilian Mathematics Colloquium 2013

Inverse scattering (acoustics, EM) u s u i D u i ( x ) = known incident wave u s ( x ) = measured scattered wave incident u i + scattered u s = total field u Time-harmonic assumption: ω = frequency { u ( x ) e − i ωt } acoustics: p ( x, t ) = ℜ e , { ( E , H )( x ) e − i ωt } EM: ( E , H )( x, t ) = ℜ e 1

Inverse scattering (acoustics, EM) u s i u D u i ( x ) = known incident wave u s ( x ) = measured scattered wave Direct problem: Given D (and its physical properties) describe the scattered field u s Inverse ill-posed problem : Determine the support (shape) of D from the knowledge of u s far away from the scatterer (far field region) 2

Outline 1 . Approaches for inverse scattering: − Traditional methods − Qualitative sampling methods 2 . Forward scattering − Radiating (outgoing) solutions − Rellich’s lemma 3 . The inverse scattering problem − Far field operator − Herglotz wave function 4 . Sampling formulation − Fundamental solution − Linear sampling method − Factorization method 5 . Failure (?) at eigenfrequencies − Modified Jones/Ursell far-field operator − Object classification at eigenfrequencies 6 . Applications 3

1. Approaches for inverse scattering Qualitative/sampling schemes Goal: try to • recover shape as opposed to physical properties • recover shape and possibly some extra info Fixed frequency of incidence ω : u s i u D Sampling: Collect the far field data u ∞ (or the near field data u s ) and solve an ill-posed linear integral equation for each sample point z 4

Inverse Scattering Methods Nonlinear optimization methods Kleinmann, Angell, Kress, Rundell, Hettlich, Dorn, Weng Chew, Ho- hage, Lesselier ... • need some a priori information − parametrization, # scatterers, etc • flexibility w.r.t. data • need forward solver (major concern) • full wave model • inverse crimes not uncommon! Asymptotic approximations (Born, iterated- Born, geometrical optics, time-reversal/mi- gration, ...) Bret Borden, Cheney, Papanicolaou, ... • need a priori information so linearizations be applicable (not for resonance region) • (mostly) linear inversion schemes • radar imaging with incorrect model? Qualitative methods (sampling, Factoriza- tion, Point-source, Ikehata’s, MUSIC?...) Colton, Monk, Kirsch, Hanke-Bourgeois, Cakoni, Pot- thast, Devaney, Hanke, Ikehata, Ammari, Haddar, ... • no forward solver • no a priori info on the scatterer • no linearization/asymptotic approx.: – full nonlinear multiple scattering model • need more data • do not determine EM properties ( σ, ϵ r ) 5

2. Forward wave propagation 101 Wave equation (pressure p = p ( x, t ), velocity c ) ∂ 2 ∂t 2 p − c 2 △ p = 0 Time-harmonic dependency: ω = frequency { u ( x ) e − i ωt } p ( x, t ) = ℜ e Helmholtz (reduced wave) equation: ( − i ω ) 2 u − c 2 △ u = 0 −△ u − k 2 u = 0 ⇒ where k = ω/c is the wavenumber. Plane wave incidence ’Plane wave’ in the direction d , | d | = 1, { e i kx · d e − i ωt } p ( x, t ) = cos { k ( x · d − c 0 t ) } = ℜ e Plane wave u i ( x ) = e i kx · d satisfies −△ u i − k 2 u i = 0 em R 3 , where k = ω/c 0 6

Forward scattering Incident field (say plane wave or point source) −△ u i − k 2 u i = f in R 3 , where k = ω/c 0 Helmholtz equation for the total field −△ u − k 2 u = 0 in R 3 \ D, B u = 0 on ∂D, Total field u = u i + u s , u s perturbation due to D Boundary condition (impenetrable) B u := ∂ ν u + i λu impedance (Neumann λ = 0) = u Dirichlet/PEC Analogous to Maxwell with ∇ × ∇ × E − k 2 E = F in R 3 \ D 7

Sommerfeld/Silver-M¨ uller conditions Exterior boundary value problem for u s Uniqueness: u s travels away from the obstacle −△ u s − k 2 u s = 0 in R 3 \ D, B u s = f := −B u i on ∂D, 2 � � ∫ ∂ ∂ru s − i ku s � � lim ds ( x ) = 0 � � � � R →∞ r := | x | = R (Sommerfeld radiation condition) Here x = | x | ˆ x = r ˆ x , ˆ x ∈ Ω Notation: Ω unit sphere Sommerfeld: ”... energy does not propagate from infinity into the domain ...” 8

Radiating solutions II Sommerfeld radiation condition on u s −△ u s − k 2 u s = 0 in R 3 \ D, B u s = f := −B u i on ∂D, 2 � � ∫ ∂ ∂ru s − i ku s � � lim ds ( x ) = 0 � � � � R →∞ r := | x | = R Asymptotic behavior of radiating solutions Def. u s is radiating if it satisifies – Helmholtz outside some ball and – Sommerfeld radiation condition If u s is radiating then Theor. ( ) e i k | x | 1 u s ( x ) = | x | u ∞ (ˆ x ) + O | x | 2 90 1.5 120 60 1 150 30 0.5 180 0 210 330 240 300 270 9

Rellich’s lemma [1943] Key tool in scattering theory: Identical far field patterns ⇓ Identical scattered fields (in the domain of definition) Rellich’s lemma (fixed wave number k > 0) If v 1 x ) = v 2 ∞ (ˆ ∞ (ˆ x ) for infinitely many ˆ x ∈ Ω then 2 ( x ), x ∈ R 3 \ D . v s 1 ( x ) = v s That is, if v 1 ∞ (ˆ x ) = 0 for ˆ x ∈ Ω then 1 ( x ) = 0, x ∈ R 3 \ D . v s Remark : R >> 1, ∫ ∫ | x | = R | v s ( x ) | 2 ds ( x ) ≈ x ) | 2 ds (ˆ Ω | v ∞ (ˆ x ) 10

3. The Inverse Scattering Problem Inverse problem: ill-posed and nonlinear Given several incident plane waves with dir. d u i ( x, d ) = e i kx · d , measure the corresponding far-field pattern u ∞ (ˆ x, d ) , ˆ x ∈ Ω and determine the support of D Re Im 350 350 300 300 250 250 200 200 150 150 100 100 50 50 100 200 300 100 200 300 11

Far field operator (data operator): F : L 2 (Ω) → L 2 (Ω) ∫ ( Fg )(ˆ x ) := Ω u ∞ (ˆ x, d ) g ( d ) ds ( d ) Remark 1: F is compact (smooth kernel u ∞ ) Remark 2: F is injective and has dense range whenever k 2 ̸ = interior eigenvalue Proof : Fg = 0 implies (Rellich) ∫ Ω u s ( x, d ) g ( d ) ds ( d ) = 0 , x ∈ R 3 \ D ∫ Ω u i ( x, d ) g ( d ) ds ( d ) = 0 , x ∈ ∂D −B that is, − B v g ( x ) = 0 , x ∈ ∂D where Herglotz wave function: ∫ Ω e ikx · d g ( d ) ds ( d ) , kernel g ∈ L 2 (Ω) v g ( x ) := so that v g satisfies the interior e-value problem −△ v g − k 2 v g = 0 in D, B v g = 0 on ∂D and v g = 0, g = 0, if k 2 ̸ = eigenvalue � 12

Far field operator (data operator): (SKP) F : L 2 (Ω) → L 2 (Ω) ∫ ( Fg )(ˆ x ) := Ω u ∞ (ˆ x, d ) g ( d ) ds ( d ) Obs.: F normal in the Dirichlet, Neumann and non-absorbing medium cases 13

Herglotz wave function Superposition with kernel g ∫ ∫ ∫ e ikx · d g ( d ) ds ( d ) u s ( x, d ) g ( d ) ds ( d ) u ∞ (ˆ x, d ) g ( d ) ds ( d ) ❀ ❀ Ω Ω Ω ∥ ∥ ∥ v s ( x ) v g ( x ) ( Fg )(ˆ x ) ❀ ❀ By superposition the incident Herglotz func- tion v g ( x ) induces the far field pattern ( Fg )(ˆ x ) ( R 3 ): The fundamental solution e i k | x − z | Φ( x, z ) := 4 π | x − z | , x ̸ = z, is radiating in R 3 \ { z } . Fixing the source z ∈ R 3 as a parameter, then Φ( · , z ) has far field pattern x, z ) = 1 4 πe − i k ˆ x · z Φ ∞ (ˆ 14

4. Linear Sampling Method (LSM) Let z ∈ R 3 . Consider Far field equation Fg z (ˆ x ) = Φ ∞ (ˆ x, z ) It is solvable only in special cases, if z = z 0 and D is a ball centered at z 0 . In general a solution doesn’t exist. Ex. 2D Neumann obstacle: ( k = 3 . 4, k = 4) k =3.4 k =4 −3 −3 60 60 −2 −2 50 50 −1 −1 40 40 0 0 30 30 1 1 20 20 2 2 10 10 3 3 −2 0 2 −2 0 2 z inside D , || g z || remains bounded z outside D , || g z || becomes unbounded Nevertheless the regularized algorithm is nu- merically robust and the following approxima- tion theorem holds 15

LSM theorem (SKP) If − k 2 ̸ = Dirichlet eigenvalue for Theorem the Laplacian in D then (1) For any ϵ > 0 and z ∈ D , there exists a g z ∈ L 2 (Ω) such that - ∥ Fg z − Φ ∞ ( · , z ) ∥ L 2 (Ω) < ϵ, and - lim z → ∂D ∥ g z ∥ L 2 (Ω) = ∞ , lim z → ∂D ∥ v g z ∥ H 1 ( D ) = ∞ . (2) For any ϵ > 0, δ > 0 and z ∈ R 3 \ D , there exists a g z ∈ L 2 (Ω) such that - ∥ Fg z − Φ ∞ ( · , z ) ∥ L 2 (Ω) < ϵ + δ and - lim δ → 0 ∥ g z ∥ L 2 (Ω) = ∞ , lim δ → 0 ∥ v g z ∥ H 1 ( D ) = ∞ where v g z is the Herglotz function with kernel g z . 16

LSM motivation (Dirichlet) • Assume u ∞ (ˆ x, d ) known for ˆ x, d ∈ Ω corresponding to u i ( x, d ) = e ikx · d • Let z ∈ D and g = g z ∈ L 2 (Ω) solve Fg = Φ ∞ ( · , z ): ∫ Ω u ∞ (ˆ x, d ) g ( d ) ds ( d ) = Φ ∞ (ˆ x, z ) • Rellich’s lemma: ∫ x ∈ R 3 \ D Ω u s ( x, d ) g ( d ) ds ( d ) = Φ( x, z ) , • Boundary condition u s ( x, d ) = − e i kx · d on ∂D implies: ∫ Ω e ikx · d g ( d ) ds ( d ) = Φ( x, z ) , − x ∈ ∂D, z ∈ D. If z ∈ D and z → x ∈ ∂D then || g || L 2 (Ω) → ∞ since | Φ( x, z ) | → ∞ Same analogy: Neumann, impedance, inho- mogeneous medium 17

Factorization method (Dirichlet) f ∈ H 1 / 2 ( ∂D ) Generalized scattering problem: ∆ v + k 2 v = 0 in R 3 \ D, v = f on ∂D, v radiating Data to far-field operator: takes f into v ∞ G : H 1 / 2 ( ∂D ) → L 2 (Ω) , f ❀ Gf := v ∞ Theorem z ∈ D iff Φ ∞ ( · , z ) ∈ Range ( G ) Proof: Rellich + singularity of Φ( · , z ) at z . 18