Inverse scattering for obstacles and cracks with impedance boundary - PowerPoint PPT Presentation

Introduction Obstacle scattering Inverse scattering for cracks List-to-do Inverse scattering for obstacles and cracks with impedance boundary conditions Fahmi Ben Hassen ENIT-LAMSIN, Tunis PICOF’12, Palaiseau April 2011

Introduction Obstacle scattering Inverse scattering for cracks List-to-do Introduction Obstacle scattering Forward and inverse scattering problems The Point Source Method Inverse scattering of multiple obstacles The Singular Sources Method Inverse scattering for cracks The Linear Sampling Method The Reciprocity Gap-Linear Sampling Method The Data Completion Algorithm and application to RG-LSM List-to-do

Introduction Obstacle scattering Inverse scattering for cracks List-to-do Introduction

Introduction Obstacle scattering Inverse scattering for cracks List-to-do Setting for reconstruction problem

Introduction Obstacle scattering Inverse scattering for cracks List-to-do Forward and inverse obstacle scattering problems

Introduction Obstacle scattering Inverse scattering for cracks List-to-do Acoustic scattering Given an incident field u i , a bounded obstacle D and a surface impedance λ , find a scattered field u s governed by • The time-harmonic Helmholtz equation ∆ u s + κ 2 u s = 0 in R 3 \ D . • A boundary condition, for example impedance BC ∂ u ∂ν + i λ u = 0 on ∂ D for the total field u = u i + u s • The Sommerfeld radiation condition (3d) r →∞ r ( ∂ u s ∂ r − i κ u s ) = 0 . lim Usually u i ( x , d ) = e i κ d · x is a plane wave with direction d ∈ S 2 or e i κ | x − z | u i ( x , z ) = Φ ( x , z ) := 1 | x − z | , x � = z in R 3 , a point source with source 4 π point z ∈ D .

Introduction Obstacle scattering Inverse scattering for cracks List-to-do Solution Techniques Assume that • D is a bounded open domain, R 3 \ D is connected and ∂ D is Lipschitz. • λ ∈ L ∞ ( ∂ D ) and Re ( λ ) > 0. • The wave number κ > 0. Under these assumptions • The exterior impedance obstacle problem is well posed. • Existence can be shown by variational methods, integral equation methods or several further special techniques. The scattered field has the asymptotic behaviour of an outgoing spherical wave „ 1 u s ( x ) = e i κ | x | „ «« u ∞ (ˆ x ) + O , | x | | x | x as | x | → ∞ uniformly in all directions of observation ˆ x = | x | . The function u ∞ is the far field pattern of u s .

Introduction Obstacle scattering Inverse scattering for cracks List-to-do The Inverse Problem Measured data are either the scattered field u s on some surface Γ or the far field pattern u ∞ on S 2 . Task: reconstruct the shape and properties of the unknown scatterer!!!!

Introduction Obstacle scattering Inverse scattering for cracks List-to-do Reconstruction Schemes • Iterative methods • Least Squares Method • Newton’s type Method • Level Set Methods • Evolutionary (Newton or Gradient) Methods • Successive Iteration Schemes (Kress-Rundell) • Decomposition methods • Series Expansion Methods: Spherical Waves, Bessel functions • Fourier - Plane Wave Expansion Methods • Kirsch-Kress Method or Potential Method • Point Source Method : Green representation and Point Source Approx. • Sampling and probe methods • Linear sampling method (Colton-Kirsch) • Factorization method (Kirsch) • Singular sources method (Potthast) - Probe method (Ikehata) • No response test (Luke-Potthast) • Orthogonality Sampling (Potthsat-Nakamura) • PSM uses measurements for one wave, but needs to know the boundary condition for shape reconstruction • SSM uses measurements for many incident waves, does not need to know the boundary condition

Introduction Obstacle scattering Inverse scattering for cracks List-to-do PSM : acoustics 3d, field reconstruction Point Source Method : Reconstruct an approximation to u s from u ∞ for one incident wave. Definition Consider a set of sampling domains G x parameterized by x ∈ R 3 , such that ∈ G x . We define the illuminated area x / E := { x ∈ R 3 : D ⊂ G x } . Theorem (FBH-Erhard-Potthast 2006) The back-projection operator Z x ) , x ∈ R 3 \ D , ( A ε u ∞ )( x ) := 4 π S 2 u ∞ ( − ˆ x ) g x ,ε (ˆ x ) ds (ˆ converges uniformly to u s on the illuminated area E | u s ( x ) − A ε u ∞ ( x ) | = 0 . ε → 0 sup lim x ∈E

Introduction Obstacle scattering Inverse scattering for cracks List-to-do Acceleration of the PSM algorithm (1) Use fixed reference approximation domain G 0 and solve ( α I + H ∗ H ) g 0 ,α = H ∗ Φ ( · , 0 ) on G 0 Λ e i κ y · d g ( d ) ds ( d ) , R with Herglotz operator ( Hg )( y ) = y ∈ ∂ G 0 . (2) For G x = x + G 0 compute approximating density g x ,α ( d ) = e − i κ x · d g 0 ,α ( d ) . (3) Compute the reconstructed field A ε u ∞ , with density g x ,α . (4) Repeat (1)-(3) with a different parameter α 2 , calculate an approximation E c = { x ∈ R 3 : |I enl ( x ) | < c } of the illuminated area from I enl = u rec , 1 − u rec , 2 . « 1 „ R R E c x dx 3 E c dx 3 (5) Determine the ball B r ( x 0 ) given by x 0 = E c dx , r = q , 0 < q < 1 . R 4 π (6) Repeat the steps for different reference domains G ( j ) 0 ⇒ A ( j ) ε u ∞ and B r j ( x ( j ) 0 ) (7) A complete reconstruction of the field is given as j = 1 u s , ( j ) P n rec ( x ) χ B rj ( x ( j ) 0 ) ( x ) n B r j ( x ( j ) u s [ rec ( x ) = , x ∈ 0 ) . P n j = 1 χ B rj ( x ( j ) 0 ) ( x ) j = 1

Introduction Obstacle scattering Inverse scattering for cracks List-to-do PSM: acoustics 2d, field reconstruction Scattering by an obstacle ( λ = 1 + i , κ = 2, d = ( − 1 , 0 ) T )

Introduction Obstacle scattering Inverse scattering for cracks List-to-do PSM: acoustics 3d, field reconstruction Scattering by a sound-soft ball ( κ = 2)

Introduction Obstacle scattering Inverse scattering for cracks List-to-do PSM: acoustics 3d, shape reconstruction Reconstructions of a sound-soft torus ( κ = 2)

Introduction Obstacle scattering Inverse scattering for cracks List-to-do Reconstruction of a sound-hard or a complex obstacle by one incident wave : Kress R and Serranho P : A hybrid method for sound-hard obstacle reconstruction, J. Comp. Appl. Math. 2007 Serranho P : A hybrid method for inverse scattering for shape and impedance, IP 2006 1. Use the Potential Method to reconstruct the scattered wave from the far field pattern. 2. Use a regularized Newton method applied to a nonlinear operator equation with the operator that maps the unknown boundary onto the solution of the direct scattering problem. G : ( z , λ ) �→ z ′⊥ | z ′ | · ( ∇ u ◦ z ) + i ( λ u ) ◦ z in [ 0 , 2 π ] Find z such that G ( z ) = 0 where ∂ D = { z ( s ) : s ∈ [ 0 , 2 π ] } .

Introduction Obstacle scattering Inverse scattering for cracks List-to-do Limitation of the standard formulation of the PSM (a) Simulation of the scattered field (b) reconstruction of the the scattered field via PSM without taking into account the non-convexity of the scatterer. The non-convex part of the fields and domains cannot be reconstructed since it is outside of the illuminated area.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do Scattered Wave Splitting Assume that D = D 1 ∪ D 2 , D 1 ∩ D 2 = ∅ and D j is connected with C 2 -smooth boundary. Definition Consider two domains G 1 , G 2 with G 1 ∩ G 2 = ∅ and C 2 -smooth boundary are given s. t. ¯ D 1 ⊂ G 1 , ¯ D 2 ⊂ G 2 . Set G := G 1 ∪ G 2 . Theorem : Uniqueness of source splitting (FBH-Liu-Potthast) Assume that we are given a decomposition u s = u s 1 + u s 2 , s. t. : 1. u s j satisfies the radiation condition for j = 1 , 2; 2. u s j solves the Helmholtz equation in R n \ G j for j = 1 , 2; j ) + := u s 3. ( u s j ( x ) exists and is continuous on ∂ G . lim x �∈ G j , x → ∂ G j Then the splitting of u s is unique in R n \ G j , j = 1 , 2.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do Far field splitting via single-layer potentials Let us express the scattered wave by single-layer approach u s ( x ) := ( S ϕ )( x ) , x ∈ R n \ G The density ϕ lives on ∂ G = ∂ G 1 ∪ ∂ G 2 . We have S ϕ = S 1 ϕ 1 + S 2 ϕ 2 , where ϕ j ( y ) := ϕ ( y ) for y ∈ ∂ G j and Z x ∈ R n , ( S j ϕ j )( x ) := Φ( x , y ) ϕ j ( y ) ds ( y ) , ∂ G j The splitting of the far field of a scatterer D = D 1 ∪ D 2 is obtained from the following three steps.

Introduction Obstacle scattering Inverse scattering for cracks List-to-do Far field splitting via single-layer potentials 2 Algorithm (FBH-Liu-Potthast 2007) 1. Solve S ∞ ϕ = u ∞ to generate ϕ on ∂ G . 2. Define u s j ( x ) := ( S j ϕ j )( x ) , x �∈ G j , which can be considered as a scattered wave outside G j . 3. Compute the far field patterns of u s j defined by u ∞ := S ∞ j ϕ j , j = 1 , 2 . j • If G is chosen s. t. κ 2 is not the interior Dirichlet eigenvalue of − ∆ in G j then the far field equation is uniquely solvable. • The single-layer approach is a constructive method for the unique decomposition of the scattered field u s = u s 1 + u s in R n \ G 2

Introduction Obstacle scattering Inverse scattering for cracks List-to-do Far field splitting via single-layer potentials 3