Recovery of an infinite rough surface by a nonlinear integral equation method from phaseless near-field data

A.1 Appendix A

In this appendix, we will look at a combination of the Nyström method and the truncation developed in [44] to solve the boundary integral equation (3.1), and give the convergence analysis result.

For x , y ∈ Γ f , setting x = ( s , f ⁢ ( s ) ) and y = ( t , f ⁢ ( t ) ) , equation (3.1) can be rewritten in the form

where ψ ~ ⁢ ( s ) := ψ ⁢ ( s , f ⁢ ( s ) ) , g ~ ⁢ ( s ) = g ⁢ ( s , f ⁢ ( s ) ) , and

with

The truncation equation of (A.1) is given by

with

For the truncation equation (A.2), we will apply the Nyström method to obtain its numerical solution. Let h = π N be the step length. Then the Nyström method will be written as

Here

with

and 𝒜 ⁢ ( s , t ) and ℬ ⁢ ( s , t ) are given explicitly in [39]. Thus, we have the following convergence analysis, see [44, Theorem 8.2] for details.

Now we illustrate the feasibility of the Nyström method by a numerical example. We consider the scattering by a rough surface given by

The incident wave is chosen to be u i ⁢ ( x ) = Φ ⁢ ( x , z ) with z = ( 0 , - 4 ) . Since the source z is located below the surface, it follows from the well-posedness of problem (2.1)–(2.4) that the related scattered wave has the explicit expression u e s ⁢ ( x ) = - Φ ⁢ ( x , z ) . For the numerical solution u n s ⁢ ( x ) , we solve equation (A.3) for A = 8 ⁢ π , 16 ⁢ π , 32 ⁢ π and h = π N with N = 15 , 30 , 45 , the wavenumber κ = 3 , and the observation x = ( 0 , 2 ) . Thus, we can calculate the relative error | u n s ⁢ ( x ) - u e s ⁢ ( x ) | / | u e s ⁢ ( x ) | and present the result in Table 1, which shows that the relative error between the numerical solution and the exact solution decreases as A and N increase.

A.2 Appendix B

This appendix is devoted to introducing a uniqueness result established in [60], which aims to determine a locally rough surface from the modulus of the near-field data due to superpositions of point sources.

Let the scattering surface be described by Γ := { x ∈ ℝ 2 : x 2 = f ⁢ ( x 1 ) } , where f ∈ C 2 ⁢ ( ℝ ) has a compact support. Thus, Γ can be represented by Γ = Γ c ∪ Γ p , where Γ c := { x ∈ Γ : x 2 = 0 } denotes the plane part and Γ p := { x ∈ Γ : x 2 ≠ 0 } denotes the local perturbation part. We denote the upper half-space separated by Γ by Ω := { x ∈ ℝ 2 : x 2 > f ⁢ ( x 1 ) } . Consider the incident field u i ⁢ ( x , z ) to be generated by a point source

Then the scattering of u i ⁢ ( x , z ) by Γ can be reduced to the problem of seeking a scattered field u s ⁢ ( x , z ) in Ω satisfying that

where u = u i + u s stands for the total field, the last condition in (A.4) is the well-known Sommerfeld radiation condition which holds uniformly for all directions

ℬ c and ℬ p are the boundary operators defined by ℬ c ⁢ u = u for sound-soft case and ℬ c ⁢ u = ∂ ⁡ u ∂ ⁡ ν for sound-hard case, and ℬ p ⁢ u = u on Γ p , D , ℬ p ⁢ u = ∂ ⁡ u ∂ ⁡ ν + λ ⁢ u on Γ p , I , where Γ p , D ∪ Γ p , I = Γ p , Γ p , D ∩ Γ p , I = ∅ . Here, ν is the upward unit normal on Γ directing into Ω, λ ∈ C ⁢ ( Γ p , I ) is the impedance function satisfying Im ⁡ λ ≥ 0 .

To present the uniqueness result, we first introduce the following definition of admissible curves.

Now we are in a position to formulate the uniqueness theorem, we refer to [60] for its proof.

A.3 Appendix C

In this appendix, we give the formulation of ( A ⁢ [ f , ψ ] ⁢ A ⁢ [ f , ψ ] ¯ ) ⁢ ( x ) and ( A ⁢ [ f , ψ ] ⁢ A ⁢ [ f , ψ ] ¯ ) ′ ⁢ f δ ⁢ ( x ) in (3.6). For convenience, we denote

Therefore,

In view of

we obtain

where r = | x - y | and r ′ = | x - y ′ | . The Fréchet derivative of ( A ⁢ [ f , ψ ] ⁢ A ⁢ [ f , ψ ] ¯ ) ⁢ ( x ) at f with respect to the direction f δ is presented by

which can be obtained by differentiating the kernel with respect to f (see [48]). Hence, the derivative T ′ ⁢ [ f , ψ ] ⁢ f δ ⁢ ( x ) is given by

for f δ ∈ B , where ψ ~ ⁢ ( y 1 ) := ψ ⁢ ( y 1 , f ⁢ ( y 1 ) ) .