Multiscale Analysis of Wave Propagation and Imaging in Random Media - PowerPoint PPT Presentation

Multiscale Analysis of Wave Propagation and Imaging in Random Media Josselin Garnier (Ecole Polytechnique, France) • In random media, correlation-based imaging aims at exploiting the information carried by incoherent wave fluctuations when the coherent wave is vanishing. • Fourth-order moment analysis makes it possible to understand when correlation-based imaging is effective. HKUST May 2019

Wave propagation in random media • Wave equation: ∂ 2 u 1 x = ( x , z ) ∈ R 2 × R ∂t 2 ( t, � x ) − ∆ � x u ( t, � x ) = F ( t, � x ) , � c 2 ( � x ) x ) = δ ( z ) f ( x ) e − iωt . • Time-harmonic source in the plane z = 0: F ( t, � x ) e − iωt and ˆ → u ( t, � u ( � ֒ x ) = ˆ u satisfies ω 2 x = ( x , z ) ∈ R 2 × R x ) ˆ u ( � x ) + ∆ � x ˆ u ( � x ) = − δ ( z ) f ( x ) , � c 2 ( � • Random medium model: � � 1 x ) = 1 1 + µ ( � x ) c 2 ( � c 2 o c o is a reference speed, µ ( � x ) is a zero-mean random process. • The statistical properties of the wave field ˆ u can be studied by multiscale analysis [1,2]. [1] G. Papanicolaou, SIAM J. Appl. Math. 21 (1971) 13. [2] J.-P. Fouque et al., Springer, 2007.

Wave propagation in the random paraxial regime • Consider the time-harmonic form of the scalar wave equation (with � x = ( x , z )): � � u + ω 2 ( ∂ 2 z + ∆ ⊥ )ˆ 1 + µ ( x , z ) u = − δ ( z ) f ( x ) . ˆ c 2 o • The function ˆ φ (slowly-varying envelope of a plane wave) defined by � � u ( x , z ) = ic o 2 ω e i ωz co ˆ ˆ φ x , z satisfies � � φ + ω 2 2 i ω = 2 i ω z ˆ ∂ z ˆ φ + ∆ ⊥ ˆ µ ( x , z )ˆ ∂ 2 φ + φ δ ( z ) f ( x ) . c 2 c o c o o • If µ is mixing and smooth , if � x � � x � ω → ω ε 2 , z µ ( x , z ) → ε 3 µ ε 4 , , f ( x ) → f , ε 2 ε 2 φ ε converges in distribution in C 0 ( R + , L 2 ( R 2 )) (or C 0 ( R + , H k ( R 2 ))) as ε → 0 to then ˆ the solution of: φ = ic o φdz + iω d ˆ 2 ω ∆ ⊥ ˆ ˆ φ ◦ dB ( x , z ) 2 c o with B ( x , z ) Brownian field E [ B ( x , z ) B ( x ′ , z ′ )] = γ ( x − x ′ ) min( z, z ′ ), � ∞ −∞ E [ µ ( 0 , 0) µ ( x , z )] dz , and ˆ γ ( x ) = φ ( z = 0 , x ) = f ( x ) [1]. [1] J. Garnier et al., Ann. Appl. Probab. 19 (2009) 318.

First-order moments in the paraxial regime Consider φ = ic o φdz + iω d ˆ 2 ω ∆ ⊥ ˆ ˆ φ ◦ dB ( x , z ) 2 c o starting from ˆ φ ( x , z = 0) = f ( x ). • By Itˆ o’s formula, φ ] − ω 2 γ ( 0 ) d φ ] = ic o dz E [ˆ 2 ω ∆ ⊥ E [ˆ E [ˆ φ ] 8 c 2 o and therefore � � � ˆ � − γ ( 0 ) ω 2 z = ˆ φ ( x , z ) φ hom ( x , z ) exp , E 8 c 2 o � ∞ −∞ E [ µ ( 0 , 0) µ ( x , z )] dz and ˆ where γ ( x ) = φ hom is the solution in the homogeneous medium. • Strong damping of the coherent wave. 8 c 2 = ⇒ Identification of the scattering mean free path Z sca = γ ( 0 ) ω 2 [1]. o = ⇒ Coherent imaging methods (such as Kirchhoff migration or Reverse-Time migration) fail. [1] A. Ishimaru, Academic Press, 1978.

Second-order moments in the paraxial regime • The mean Wigner transform defined by � � �� � � � � ˆ � r + q r − q ˆ W ( r , ξ , z ) = R 2 exp − i ξ · q φ 2 , z φ 2 , z d q , E is the angularly-resolved mean wave energy density. By Itˆ o’s formula, it solves a radiative transport equation � � � ω 2 ∂ W ∂z + c o ω ξ · ∇ r W = R 2 ˆ γ ( κ ) W ( ξ − κ ) − W ( ξ ) d κ , 4(2 π ) 2 c 2 o starting from W ( r , ξ , z = 0) = W 0 ( r , ξ ), the Wigner transform of f . ω 2 = ⇒ Identification of the scattering cross section o ˆ γ ( κ ) [1]. 4 c 2 • The fields at nearby points are correlated and their correlations contain information about the medium. = ⇒ One should use cross correlations for imaging in random media (CINT [2], ...). [1] A. Ishimaru, Academic Press, 1978. [2] L. Borcea et al., Inverse Problems 21 (2005) 1419.

Remarks on fourth-order moments • Physical conjecture: the wave field has Gaussian statistics; therefore we know everything when the first two moments are characterized. → the conjecture may be wrong. • The statistical second-order moments and the mean Wigner transform are not observed directly, only empirical quantities are observed. • Calculations of fourth-order moments are useful to: • test the Gaussian conjecture. • quantify the statistical stability of empirical second-order moments, Wigner transforms, and correlation-based imaging methods. • implement intensity-correlation-based imaging methods (when only intensities can be measured, as in optics). HKUST May 2019

Fourthd-order moments in the random paraxial regime • Consider φ = ic o φdz + iω d ˆ 2 ω ∆ ⊥ ˆ ˆ φ ◦ dB ( x , z ) 2 c o starting from ˆ φ ( x , z = 0) = f ( x ). • Let us consider the fourth-order moment: � � r 1 + r 2 + q 1 + q 2 � ˆ � r 1 − r 2 + q 1 − q 2 � ˆ M 4 ( r 1 , r 2 , q 1 , q 2 , z ) = φ , z φ , z E 2 2 �� � r 1 + r 2 − q 1 − q 2 � ˆ � r 1 − r 2 − q 1 + q 2 × ˆ φ , z φ , z 2 2 By Itˆ o’s formula, � � M 4 + ω 2 ∂M 4 = ic o ∇ r 1 · ∇ q 1 + ∇ r 2 · ∇ q 2 U 4 ( q 1 , q 2 , r 1 , r 2 ) M 4 , 4 c 2 ∂z ω o with the generalized potential U 4 ( q 1 , q 2 , r 1 , r 2 ) = γ ( q 2 + q 1 ) + γ ( q 2 − q 1 ) + γ ( r 2 + q 1 ) + γ ( r 2 − q 1 ) − γ ( q 2 + r 2 ) − γ ( q 2 − r 2 ) − 2 γ ( 0 ) . = ⇒ One can get a general characterization of the fourth-order moment [1]. [1] J. Garnier et al., ARMA 220 (2016) 37.

Scintillation � � − | x | 2 Assume that f ( x ) = exp . 2 r 2 o • The scintillation index: �� � 4 � �� � 2 � 2 � � � ˆ � ˆ φ ( r , z ) − E φ ( r , z ) E S ( r , z ) := �� � 2 � 2 � � ˆ φ ( r , z ) E satisfies (in the paraxial regime): 1 S ( r , z ) = 1 − 2 . � � � � � � z � � � � dz ′ − | u | 2 r o + | r | 2 ω 2 u c o z ′ � 1 + i u · r R 2 exp 0 γ d u � 4 π 4 c 2 ωr o 4 r 2 o o The physical conjecture is that S ≃ 1 when the propagation distance is larger than the scattering mean free path, as it should be for a (complex) Gaussian process. HKUST May 2019

Scintillation 1 0.8 scintillation index 0.6 Z c / Z sca =0.125 0.4 Z c / Z sca =0.25 Z c / Z sca =0.5 0.2 Z c / Z sca =1 0 0 1 2 3 4 5 6 z / Z sca Scintillation index at the beam center S ( 0 , z ) as a function of the propagation 8 c 2 ω 2 γ ( 0 ) and Z c = ωr o ℓ c distance for different values of Z sca = c o . Here o γ ( x ) = γ ( 0 ) exp( −| x | 2 /ℓ 2 c ). → The physical conjecture that S ≃ 1 when z ≫ Z sca is true in the random paraxial regime. HKUST May 2019

Stability of the Wigner transform of the field • The Wigner transform � � � ˆ � � ˆ � � r + q r − q W ( r , ξ , z ) := R 2 exp − i ξ · q φ 2 , z φ 2 , z d q is not statistically stable (i.e. standard deviation > mean). • Let us consider the smoothed Wigner transform (for r s , ξ s > 0): �� � � − | ξ ′ | 2 − | r ′ | 2 1 R 2 × R 2 W ( r − r ′ , ξ − ξ ′ , z ) exp d r ′ d ξ ′ . W s ( r , ξ , z ) = (2 π ) 2 r 2 s ξ 2 2 r 2 2 ξ 2 s s s Its coefficient of variation: � E [ W s ( r , ξ , z ) 2 ] − E [ W s ( r , ξ , z )] 2 C s ( r , ξ , z ) := E [ W s ( r , ξ , z )] determines its statistical stability. → Analysis of high-order moments of ˆ ֒ φ [1]. [1] J. Garnier et al., ARMA 220 (2016) 37.

Stability of the Wigner transform of the field 2 0.33 0.75 1 0 4 2 . 1.5 5 1.25 1.5 0.5 1 r s 0.75 1 1.5 1.25 0 . 2 7 5 4 0.5 1 0 0 0.5 1 1.5 2 ξ s Contour levels of the coefficient of variation of the smoothed Wigner transform. Here r s = r s /ρ , ξ s = ξ s ρ , and ρ = ρ ( z ; ω, r o , ℓ c , Z sca ). → This result makes it possible to achieve optimal trade-off between stability and resolution for correlation-based imaging [1,2]. Example: ultrasonic non-destructive testing in concrete. [1] L. Borcea et al., Inverse Problems 27 (2011) 085004. [2] J. Garnier et al., ARMA 220 (2016) 37.

Wave propagation in random waveguides x d/ 2 Wave propagation in two-dimensional waveguides: � � ( ∂ 2 x + ∂ 2 z ) + k 2 n 2 ( x, z ) u ( x, z ) = δ ( z ) f s ( x ) ˆ − d/ 2 z • Ideal waveguide:   n if x ∈ ( − d/ 2 , d/ 2) , n (0) ( x ) =  1 otherwise, where n > 1 is the relative index of the core and d > 0 is its diameter. • Type I perturbation: the index of refraction within the core is randomly perturbed:   if x ∈ ( − d/ 2 , d/ 2) and z ∈ (0 , L/ε 2 ) , n + εν ( x , z ) n ( ε ) ( x, z ) =  1 otherwise . • Type II perturbation: the boundaries of the core are randomly perturbed:  � �  and z ∈ (0 , L/ε 2 ) , n if x ∈ − d/ 2 + εdν − ( z ) , d/ 2 + εdν + ( z ) n ( ε ) ( x, z ) =  1 otherwise. HKUST May 2019