# Reverse time migration for inverse acoustic scattering by locally rough
surfaces

Jianliang Li Key Laboratory of Computing and Stochastic Mathematics, School of
Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081,
China<EMAIL_ADDRESS><EMAIL_ADDRESS>Hao Wu School of Mathematics
and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China
<EMAIL_ADDRESS>Jiaqing Yang School of Mathematics and Statistics,
Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China
<EMAIL_ADDRESS>
###### Abstract

Consider the inverse scattering of time-harmonic acoustic scattering by an
infinite rough surface which is supposed to be a local perturbation of a
plane. A novel version of reverse time migration (RTM) is proposed to
reconstruct the shape and location of the rough surface. The method is based
on a modified Helmholtz-Kirchhoff identity associated with a special rough
surface, leading to a modified imaging functional which uses the near-field
data generated by point sources as measurements. The modified imaging
functional always reaches a peak on the boundary of the rough surface for
sound-soft case and penetrable case, and hits a nadir on the boundary of the
rough surface for sound-hard case. Furthermore, we also establish the RTM
method associated with the far-field data generated by plane waves. As far as
we know, this is the first result for the RTM method with the far-filed data.
Numerical experiments are presented to show the powerful imaging quality.

Keywords: inverse acoustic scattering, locally rough surface, reverse time
migration.

##  1 Introduction

Motivated by significant applications in medical imaging [1] and exploration
geophysics [3], we consider the two-dimensional inverse acoustic scattering of
time-harmonic acoustic scattering by a locally rough surface, which aims to
reconstruct the shape and location of the rough surface.

The scattering surface is described by a curve

$\displaystyle\Gamma:=\\{(x_{1},x_{2})\in\mathbb{R}^{2}:x_{2}=f(x_{1})\\}$
(1.1)

where $f$ is assumed to be a Lipschitz continuous function with compact
support. This means that the surface $\Gamma$ is a local perturbation of the
plane $\Gamma_{0}:=\\{(x_{1},x_{2})\in\mathbb{R}^{2}:x_{2}=0\\}.$ The rough
surface $\Gamma$ separates the whole space into two half-spaces denoted by

$\displaystyle\Omega_{1}:=\\{(x_{1},x_{2})\in\mathbb{R}^{2}:x_{2}>f(x_{1})\\}\quad{\rm
and}\quad\Omega_{2}:=\\{(x_{1},x_{2})\in\mathbb{R}^{2}:x_{2}<f(x_{1})\\},$

which are filled with homogeneous mediums described by wave numbers
$\kappa_{1}>0$ and $\kappa_{2}>0$, respectively.

For impenetrable cases, the incident wave $u^{i}$ is induced by the point
source, which means

$\displaystyle u^{i}(x)=\Phi_{\kappa_{1}}(x,x_{s}):=\frac{\rm
i}{4}H_{0}^{(1)}(\kappa_{1}|x-x_{s}|)\quad{\rm for}\;\;x_{s}\in\Omega_{1}.$
(1.2)

Here, $H_{0}^{(1)}$ is the Hankel function of the first kind of order zero,
and $\Phi_{\kappa_{1}}$ is the fundamental solution of the Helmholtz equation
satisfying
$\Delta\Phi_{\kappa_{1}}(\cdot,x_{s})+\kappa_{1}^{2}\Phi_{\kappa_{1}}(\cdot,x_{s})=-\delta_{x_{s}}(\cdot)$
in $\mathbb{R}^{2}$, where $\delta$ is the Kronecker delta distribution. Then
the scattering of $u^{i}$ by the rough surface $\Gamma$ can be modelled by

$\left\\{\begin{aligned} &\Delta
u_{\alpha}+\kappa_{1}^{2}u_{\alpha}=-\delta_{x_{s}}\qquad\qquad\textrm{in}\;\;\Omega_{1},\\\
&\mathcal{B}u_{\alpha}=0\qquad\qquad\qquad\qquad\;\;\;\;\textrm{on}\;\;\Gamma,\\\
&\lim_{|x|\rightarrow\infty}|x|^{\frac{1}{2}}\left(\partial_{|x|}u_{\alpha}^{s}-{\rm
i}\kappa_{1}u_{\alpha}^{s}\right)=0,\quad\end{aligned}\right.$ (1.3)

where $u_{\alpha}:=u^{i}+u_{\alpha}^{s}$ for $\alpha\in\\{D,N\\}$ denotes the
total field which is the sum of the incident field $u^{i}$ and the scattered
field $u_{\alpha}^{s}$. ${\mathcal{B}}$ stands for the boundary condition on
$\Gamma$ satisfying ${\mathcal{B}}u_{D}:=u_{D}$ if $\Gamma$ is a sound-soft
rough surface, and ${\mathcal{B}}u_{N}=\partial_{\nu}u_{N}$ if $\Gamma$ is a
sound-hard rough surface. Here and throughout, $\nu=\nu(x)$ is the upward
normal vector directing into $\Omega_{1}$ for $x\in\Gamma$, and
$\partial_{\nu}$ stands for the normal derivative. The last condition in (1.3)
is the well-known Sommerfeld radiation condition which holds uniformly for all
directions
$\hat{x}:=x/|x|\in{\mathbb{S}}_{+}:=\\{x\in{\mathbb{R}}^{2}:|x|=1,x_{2}>0\\}$.

For the penetrable case, consider an incoming wave induced by the point source
(1.2) to be incident on the scattering interface $\Gamma$ from the domain
$\Omega_{1}$. Then the scattering of $u^{i}$ by $\Gamma$ can be modelled by

$\displaystyle\left\\{\begin{array}[]{lll}\Delta
u^{s}+\kappa_{1}^{2}u^{s}=0&\textrm{in}\;\Omega_{1}\\\\[5.69054pt] \Delta
u^{s}+\kappa_{2}^{2}u^{s}=0&\textrm{in}\;\Omega_{2}\\\\[5.69054pt]
u^{s}|_{+}-u^{s}|_{-}=-u^{i}&{\rm on\;}\Gamma\\\\[5.69054pt]
\partial_{\nu}u^{s}|_{+}-\partial_{\nu}u^{s}|_{-}=-\partial_{\nu}u^{i}&{\rm
on\;}\Gamma\\\\[5.69054pt]
\lim\limits_{|x|\rightarrow\infty}|x|^{\frac{1}{2}}\left(\partial_{|x|}u^{s}-{\rm
i}\kappa u^{s}\right)=0.\end{array}\right.$ (1.9)

Here the notation $\cdot|_{\pm}$ represents the limits of $\cdot$ approaching
$\Gamma$ from $\Omega_{1}$ and $\Omega_{2}$, respectively, $\kappa$ is the
wavenumber defined by $\kappa:=\kappa_{1}$ in $\Omega_{1}$ and
$\kappa:=\kappa_{2}$ in $\Omega_{2}$. The last condition in (1.9) is the
Sommerfeld radiation condition which holds uniformly for all directions
$\hat{x}:=x/|x|\in{\mathbb{S}}:=\\{x\in{\mathbb{R}}^{2}:|x|=1\\}$. It is worth
pointing out that the scattered field $u^{s}$ satisfying the Sommerfeld
radiation condition has the asymptotic behavior of an outgoing spherical wave

$\displaystyle u^{s}(x)=\frac{e^{{\rm
i}\kappa|x|}}{|x|^{\frac{1}{2}}}\left\\{u^{\infty}(\hat{x})+O\left(\frac{1}{|x|}\right)\right\\}\quad{\rm
for}\;|x|\to\infty$

uniformly in all direction $\hat{x}\in{\mathbb{S}}$, where
$u^{\infty}(\hat{x})$ is known as the far field pattern of $u^{s}$.

Given the incident wave, the rough surface, and the boundary condition, the
direct scattering problem is to determine the distribution of the scattered
wave, which is extensively studied by the variational method [19, 42] and the
integral equation approach [30, 39, 44] with employing a generalized Fredholm
theory [21, 22]. Recently, a novel technique was proposed to prove the well-
posedness of (1.3) for sound-soft case in [23], based on transferring the
unbounded, locally rough surface scattering problem into an equivalent
boundary value problem with the compactly supported boundary data, whose well-
posedness follows from the classical Fredholm theory. The novel technique has
been extended to deal with penetrable, locally rough surface in [33], and it
has been extended in [34] to investigate penetrable, locally rough surface
with embedded obstacles in the lower half-space. While the inverse scattering
problem is to determine the rough surface from the measured scattered field in
some certain domain. For the time-harmonic case, there exists a large number
of references on inversion methods such as Newton-type approaches [5, 8, 9,
20, 40, 41, 46], the Kirsch-Kress schemes [10, 32], nonlinear integral
equation methods [26, 31], reconstruction algorithms based on transformed
field expansions [6, 7], the factorization method [24, 25], linear sampling
methods [23, 33, 45], and the direct imaging methods [29, 37, 38, 36]. For the
time-domain case, a singular source method has been extended to solve the
inverse rough surfae scattering problem[28].

The RTM method is a sample-type method which are widely applied in exploration
geophysics [3, 4, 11] and seismic imaging [4]. The main idea of the RTM method
consists of two steps. The first step is to back-propagate the complex
conjugated data into the back-ground medium, and the second step is to compute
the cross-correlation between the incident field and the back propagated
field. Thus we can define the imaging functional as the imaginary part of the
cross-correlation, which always peak on the boundary of the scatterer. Since
the RTM method can provide an effective, stable and powerful reconstruction of
the scatterer, it has gained considerable attention and has been extensively
investigated by mathematicians and engineers. Mathematically, the
justification of the RTM method has been proved rigorously for the inverse
obstacle scattering problem for acoustic waves [12, 27, 16, 15], elastic waves
[14], and electromagnetic waves [13]. It is worth mentioning that the RTM
method in these references requires the full scattering data (both the
intensity and phase information). On the other hand, in a variety of realistic
applications, only the phaseless data is available. In this case, the RTM
approach have been developed for acoustic waves [18] and electromagnetic waves
[17]. It is shown in [17, 18] that the imaging functional with phaseless data
has essentially the same asymptotic behavior as the case of full data. It is
noticed that the almost all references about the RTM method are related to the
inverse scattering associated with bounded obstacles. However, it is
challenging to develop the RTM method for reconstructing an infinite rough
surface since the general Helmholtz-Kirchhoff identity is not valid in this
case. No such a result is available so far.

In this paper, we aim to investigate the RTM scheme to reconstruct infinite,
locally rough surfaces where the key difficulty is that the usual Helmholtz-
Kirchhoff identity presented in [12] is not applicable. The first novelty of
this paper is to establish a modified Helmholtz-Kirchhoff identity by the
Green’s function associated with a special locally rough surface. Thus, a
modified imaging functional is proposed from the near-field data generated by
point sources, where its mathematical justification is proved rigorously.
Specifically, we demonstrate that the modified imaging functional enjoys the
nice feature that it always peaks on the boundary of the sound-soft and
penetrable rough surfaces, and reaches a nadir on the boundary of the sound-
hard rough surface. The second novelty is associated with the RTM with the
far-field measurements generated by plane waves. An imaging functional is
first proposed based on a novel mixed reciprocity relation, which is indeed
the limit of the imaging functional with the near-field data generated by
point sources. It is further shown that the imaging functional has the same
asymptotic property with the case of the near-filed measurements, and can be
thus used to recover the the location and shape of the rough surfaces. To the
best of our knowledge, this is the first result for the RTM method with using
far-field measurements, especially for the reconstruction of infinite rough
surfaces. The numerical experiments shows that the two modified imaging
functionals can provide a stable and powerful reconstruction for the rough
surfaces.

The rest of the paper is organized as follows. In section 2, we develop the
RTM method for the locally sound-soft and sound-hard rough surface, which
includes the near-field reconstruction in the first subsection and the far-
field reconstruction in the second subsection. Section 3 is devoted to the RTM
approach for penetrable locally rough surfaces which consists of the near-
field and far-field reconstructions, respectively. In section 4, we present
some numerical experiments to demonstrate the validity of the RTM method. This
paper concludes with some general remarks and discussions on the future work
in section 5.

##  2 The RTM for impenetrable locally rough surfaces

In this section, we investigate the RTM method for inverse acoustic scattering
by impenetrable locally rough surfaces with the Dirichlet or Neumann boundary
conditions. This section consists of two subsections. The first subsection
will discuss the RTM method based on the near-field data which corresponds to
the point source incidence. Based on a new mixed reciprocity relation, we will
present the RTM method based on the far-field data generated by the plane wave
incidence.

###  2.1 The near-field reconstruction

Figure 1: The setting of RTM method for sound-soft and sound-hard cases.

As shown in Figure 1, let $\Gamma$ be the locally rough surface defined by
(1.1), whose local perturbation is contained in a rectangle sampling domain
$S$. We choose a large enough $R$ and define a special locally rough surface
$\Gamma_{R}$ as

$\displaystyle\Gamma_{R}:=\\{x\in\mathbb{R}^{2}:x_{2}=0\;{\rm
for}\;|x_{1}|\geq R\;{\rm and}\;x_{2}=-\sqrt{R^{2}-x_{1}^{2}}\;{\rm
for}\;|x_{1}|\leq R\\}$ (2.1)

such that the sampling domain $S$ lies totally above $\Gamma_{R}$. We assume
that there are $N_{s}$ point sources $x_{s}$ uniformly distributed on
$\Gamma_{s}$ and $N_{r}$ receivers $x_{r}$ uniformly distributed on
$\Gamma_{r}$. Here, $\Gamma_{s}$ and $\Gamma_{r}$ denote the upper semicircle
with the origin as the center and $R_{s}$, $R_{r}$ as the radius,
respectively. We assume a priori that $R<R_{s}\leq R_{r}$.

We first consider the scattering of the incident point source
$\Phi_{\kappa_{1}}(x,x_{s})$ given by (1.2) by the special locally rough
surface $\Gamma_{R}$, which reads

$\displaystyle\left\\{\begin{array}[]{lll}\Delta
G_{\alpha}^{s}(x,x_{s})+\kappa_{1}^{2}G_{\alpha}^{s}(x,x_{s})=0&\textrm{in}\;\;\Omega_{R},\\\\[8.53581pt]
\mathcal{B}G_{\alpha}^{s}(x,x_{s})=-\mathcal{B}\Phi_{\kappa_{1}}(x,x_{s})&\textrm{on}\;\;\Gamma_{R},\\\\[8.53581pt]
\lim\limits_{|x|\rightarrow\infty}|x|^{\frac{1}{2}}\left(\partial_{|x|}G_{\alpha}^{s}(x,x_{s})-{\rm
i}\kappa_{1}G_{\alpha}^{s}(x,x_{s})\right)=0.\end{array}\right.$ (2.5)

Here $\alpha\in\\{D,N\\}$, $\alpha=D$ represents the Dirichlet boundary
condition imposed on $\Gamma_{R}$ which means
$G_{D}^{s}(x,x_{s})=-\Phi_{\kappa_{1}}(x,x_{s})$ on $\Gamma_{R}$, and
$\alpha=N$ represents the Neumann boundary condition imposed on $\Gamma_{R}$
which means
$\partial_{\nu}G_{N}^{s}(x,x_{s})=-\partial_{\nu}\Phi_{\kappa_{1}}(x,x_{s})$
on $\Gamma_{R}$. $G^{s}_{\alpha}(x,x_{s})$ is the scattered field,
$G_{\alpha}(x,x_{s}):=G_{\alpha}^{s}(x,x_{s})+\Phi_{\kappa_{1}}(x,x_{s})$
denotes the total field, and $\Omega_{R}$ is the upper half-space separated by
$\Gamma_{R}$, which means

$\displaystyle\Omega_{R}:=\\{(x_{1},x_{2})\in\mathbb{R}^{2}:x_{2}>0\;\;{\rm
for}\;|x_{1}|\geq R,\;\;{\rm and}\;\;x_{2}>-\sqrt{R^{2}-x_{1}^{2}}\;\;{\rm
for}\;|x_{1}|\leq R\\}.$

It follows from [8, 23, 40] that Problem (2.5) is well-posed in a standard
Sobolev space.

For $x_{s}\in\Gamma_{s}$, recall that $u_{\alpha}^{s}(x,x_{s})$ is the
solution of Problem (1.3) when the incident source is located at $x_{s}$.
Define

$\displaystyle
V_{\alpha}(x,x_{s}):=u_{\alpha}^{s}(x,x_{s})-G^{s}_{\alpha}(x,x_{s})$ (2.6)

then it is easily checked that it solves

$\displaystyle\left\\{\begin{aligned} &\Delta
V_{\alpha}(x,x_{s})+\kappa_{1}^{2}V_{\alpha}(x,x_{s})=0\qquad\qquad\qquad\;\;\textrm{in}\;\;\Omega_{1},\\\
&\;\mathcal{B}V_{\alpha}(x,x_{s})=-\mathcal{B}G_{\alpha}(x,x_{s})\qquad\qquad\qquad\;\;\;\;\;\textrm{on}\;\;\Gamma\setminus\Gamma_{R},\\\
&\;\mathcal{B}V_{\alpha}(x,x_{s})=0\qquad\qquad\qquad\qquad\qquad\qquad\;\;\textrm{on}\;\;\Gamma\cap\Gamma_{R},\\\
&\lim\limits_{|x|\rightarrow\infty}|x|^{\frac{1}{2}}\left(\partial_{|x|}V_{\alpha}(x,x_{s})-{\rm
i}\kappa_{1}V_{\alpha}(x,x_{s})\right)=0.\end{aligned}\right.$ (2.7)

Noting that we can compute the scattered field $G_{\alpha}^{s}(x_{r},x_{s})$
by solving Problem (2.5) using Nyström method or finite element method. Thus,
we can obtain $V_{\alpha}(x_{r},x_{s})$ from the measurement
$u_{\alpha}^{s}(x_{r},x_{s})$ and (2.6).

Now, we are able to introduce the RTM method which consists of two steps. The
first step is to back-propagate the complex conjugated data
$\overline{V_{\alpha}(x_{r},x_{s})}$ into the domain $\Omega_{R}$; the second
step is to calculate the imaginary part of the cross-correlation of
$G_{\alpha}(\cdot,x_{s})$ and the back-propagation field. More precisely, we
summarize it in the following algorithm.

Algorithm 1 (RTM for impenetrable locally rough surfaces): Given the data
$V_{\alpha}(x_{r},x_{s})$ for $r=1,2,...,N_{r}$ and $s=1,2,...,N_{s}$.

  * •

Back-propagation: for $s=1,2,...,N_{s}$, solve the problem

$\displaystyle\left\\{\begin{aligned} &\Delta
W_{\alpha}(x,x_{s})+\kappa_{1}^{2}W_{\alpha}(x,x_{s})=\frac{|\Gamma_{r}|}{N_{r}}\sum_{r=1}^{N_{r}}\overline{V_{\alpha}(x_{r},x_{s})}\delta_{x_{r}}(x)\;\textrm{in}\;\;\Omega_{R},\\\
&\;\mathcal{B}W_{\alpha}(x,x_{s})=0\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\textrm{on}\;\;\Gamma_{R},\\\
&\;\lim\limits_{|x|\rightarrow\infty}|x|^{\frac{1}{2}}\left(\partial_{|x|}W_{\alpha}(x,x_{s})-{\rm
i}\kappa_{1}W_{\alpha}(x,x_{s})\right)=0,\end{aligned}\right.$ (2.8)

to address the solution $W_{\alpha}$.

  * •

Cross-correlation: for each sampling point $z\in S$, calculate the indicator
function

$\displaystyle{\rm Ind}_{\alpha}(z)=\kappa_{1}^{2}{\rm
Im}\left\\{\frac{|\Gamma_{s}|}{N_{s}}\sum_{s=1}^{N_{s}}G_{\alpha}(z,x_{s})W_{\alpha}(z,x_{s})\right\\}$

and then plot the mapping ${\rm Ind}_{\alpha}(z)$ against $z$.

It follows from the linearity that the solution of Problem (2.8) can be
represented by

$\displaystyle
W_{\alpha}(x,x_{s})=-\frac{|\Gamma_{r}|}{N_{r}}\sum_{r=1}^{N_{r}}\overline{V_{\alpha}(x_{r},x_{s})}G_{\alpha}(x,x_{r}),$

which leads to

$\displaystyle{\rm Ind}_{\alpha}(z)=-\kappa_{1}^{2}{\rm
Im}\left\\{\frac{|\Gamma_{s}|}{N_{s}}\frac{|\Gamma_{r}|}{N_{r}}\sum_{s=1}^{N_{s}}\sum_{r=1}^{N_{r}}G_{\alpha}(z,x_{s})G_{\alpha}(z,x_{r})\overline{V_{\alpha}(x_{r},x_{s})}\right\\}\;\;z\in
S.$ (2.9)

Observing that the sampling point $z\in S$, the source location
$x_{s}\in\Gamma_{s}$, the receiver $x_{r}\in\Gamma_{r}$, and $S$ is inside in
$B_{R}$, we have $G_{\alpha}(z,x_{s})$ and $G_{\alpha}(z,x_{r})$ are smooth.
Combining of the smoothness of $V_{\alpha}(x_{r},x_{s})$ and the trapezoid
quadrature formula yields that ${\rm Ind}_{\alpha}(z)$ given by (2.9) is a
discrete formula of the following continuous function:

$\displaystyle\widetilde{{\rm Ind}}_{\alpha}(z)=-\kappa_{1}^{2}{\rm
Im}\int_{\Gamma_{r}}\int_{\Gamma_{s}}G_{\alpha}(z,x_{s})G_{\alpha}(z,x_{r})\overline{V_{\alpha}(x_{r},x_{s})}{\rm
d}s(x_{s}){\rm d}s(x_{r}),\quad z\in S.$

The remaining part of this subsection aims to give a resolution analysis of
the function $\widetilde{{\rm Ind}}_{\alpha}(z)$. Our destination is to show
that $\widetilde{{\rm Ind}}_{\alpha}(z)$ will have contrast at the rough
surface $\Gamma$ and decay away from $\Gamma$. To this end, we introduce the
following Lemmas 2.1-2.3 and Theorem 2.4 and, for simplicity, just give the
proof for the Dirichlet boundary condition case, the results for the Neumann
boundary condition case can be proven by similar arguments. We first introduce
the following modified Helmholtz-Kirchhoff identity.

######  Lemma 2.1.

Let $G_{\alpha}$ be the total field of the scattering problem (2.5), and $\nu$
be the unit upward normal to $\Gamma_{p}$ for $p\in\\{r,s\\}$, then we have

$\displaystyle\int_{\Gamma_{p}}\left(\overline{G_{\alpha}(\xi,x)}\frac{\partial
G_{\alpha}(\xi,z)}{\partial\nu(\xi)}-\frac{\partial\overline{G_{\alpha}(\xi,x)}}{\partial\nu(\xi)}G_{\alpha}(\xi,z)\right){\rm
d}s(\xi)=2{\rm i}{\rm Im}G_{\alpha}(x,z)$

for any $x,z\in B_{R_{s}}\cap\Omega_{R}$.

###### Proof.

For $\alpha=D$, $\Gamma_{p}=\Gamma_{s}$ and any $x,z\in
B_{R_{s}}\cap\Omega_{R}$, we choose a sufficient small $\varepsilon>0$ such
that the circles $B_{\varepsilon}(x)$, $B_{\varepsilon}(z)$ with $x,z$ as the
center and $\varepsilon$ as the radius contains in the domain
$B_{R_{s}}\cap\Omega_{R}$. A direct application of the Green theorem to
$\overline{G_{D}(\cdot,x)}$ and $G_{D}(\cdot,z)$ in the domain
$(B_{R_{s}}\cap\Omega_{R})\setminus(\overline{B_{\varepsilon}(x)}\cup\overline{B_{\varepsilon}(z)})$
yields

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{(B_{R_{s}}\cap\Omega_{R})\setminus(\overline{B_{\varepsilon}(x)}\cup\overline{B_{\varepsilon}(z)})}\left(\overline{G_{D}(\xi,x)}\Delta
G_{D}(\xi,z)-\Delta\overline{G_{D}(\xi,x)}G_{D}(\xi,z)\right){\rm d}\xi$
(2.10) $\displaystyle=$
$\displaystyle\int_{\Gamma_{s}\cup(B_{R_{s}}\cap\Gamma_{R})\cup\partial
B_{\varepsilon}(x)\cup\partial
B_{\varepsilon}(z)}\left(\overline{G_{D}(\xi,x)}\frac{\partial
G_{D}(\xi,z)}{\partial\nu(\xi)}-\frac{\partial\overline{G_{D}(\xi,x)}}{\partial\nu(\xi)}G_{D}(\xi,z)\right){\rm
d}s(\xi)$ $\displaystyle=$ $\displaystyle I_{1}+I_{2}+I_{3}+I_{4},$

where $\nu(\xi)$ denotes the unit downward normal to $B_{R_{s}}\cap\Gamma_{R}$
when $\xi\in B_{R_{s}}\cap\Gamma_{R}$, and $\nu(\xi)$ denotes the unit normal
to $\partial B_{\varepsilon}(x)$, $\partial B_{\varepsilon}(z)$ into the
interior of $B_{\varepsilon}(x)$, $B_{\varepsilon}(z)$, respectively. Since
$G_{D}(\xi,x)$ and $G_{D}(\xi,z)$ vanish on $B_{R_{s}}\cap\Gamma_{R}$, we have
$I_{2}=0$. For the item $I_{3}$, using
$G_{D}(\xi,x)=G_{D}^{s}(\xi,x)+\Phi_{\kappa_{1}}(\xi,x)$ gives that

$\displaystyle I_{3}$ $\displaystyle=$ $\displaystyle\int_{\partial
B_{\varepsilon}(x)}\left(\overline{\Phi_{\kappa_{1}}(\xi,x)}\frac{\partial
G_{D}(\xi,z)}{\partial\nu(\xi)}-\frac{\partial\overline{\Phi_{\kappa_{1}}(\xi,x)}}{\partial\nu(\xi)}G_{D}(\xi,z)\right){\rm
d}s(\xi)$ $\displaystyle+\int_{\partial
B_{\varepsilon}(x)}\left(\overline{G^{s}_{D}(\xi,x)}\frac{\partial
G_{D}(\xi,z)}{\partial\nu(\xi)}-\frac{\partial\overline{G^{s}_{D}(\xi,x)}}{\partial\nu(\xi)}G_{D}(\xi,z)\right){\rm
d}s(\xi)$ $\displaystyle\to$ $\displaystyle-G_{D}(x,z),\quad{\rm
as}\;\;\varepsilon\to 0.$

Similarly, we can obtain

$\displaystyle\lim_{\varepsilon\to 0}I_{4}=\overline{G_{D}(z,x)}.$ (2.11)

Combining (2.10)–(2.11) implies that

$\displaystyle\int_{\Gamma_{s}}\left(\overline{G_{D}(\xi,x)}\frac{\partial
G_{D}(\xi,z)}{\partial\nu(\xi)}-\frac{\partial\overline{G_{D}(\xi,x)}}{\partial\nu(\xi)}G_{D}(\xi,z)\right){\rm
d}s(\xi)=2{\rm i}{\rm Im}G_{D}(x,z),$

where we use the reciprocity $G_{D}(x,z)=G_{D}(z,x)$ for any
$x,z\in\Omega_{R}$, which is proven in the Lemma 3.1 of [23]. We can obtain
the result for the case $\Gamma_{p}=\Gamma_{r}$ by a similar argument and we
omit it here. The proof is completed. ∎

With the help of the above Helmholtz-Kirchhoff identity, we can address the
following lemma which plays a key role in the analysis of $\widetilde{{\rm
Ind}}_{\alpha}(z)$.

######  Lemma 2.2.

For any $x,z\in S$, we have

$\displaystyle\kappa_{1}\int_{\Gamma_{s}}\overline{G_{\alpha}(x,\xi)}G_{\alpha}(\xi,z){\rm
d}s(\xi)={\rm Im}G_{\alpha}(x,z)+\zeta_{\alpha,s}(x,z)$ (2.12)
$\displaystyle\kappa_{1}\int_{\Gamma_{r}}\overline{G_{\alpha}(x,\xi)}G_{\alpha}(\xi,z){\rm
d}s(\xi)={\rm Im}G_{\alpha}(x,z)+\zeta_{\alpha,r}(x,z)$ (2.13)

where $|\zeta_{\alpha,s}(x,z)|+|\nabla_{x}\zeta_{\alpha,s}(x,z)|\leq
CR_{s}^{-1}$, $|\zeta_{\alpha,r}(x,z)|+|\nabla_{x}\zeta_{\alpha,r}(x,z)|\leq
CR_{r}^{-1}$ uniformly for any $x,z\in S$.

###### Proof.

For $\alpha=D$ and any $x,z\in S$, it follows from Lemma 2.1 that

$\displaystyle 2{\rm i}{\rm Im}G_{D}(x,z)$ $\displaystyle=$
$\displaystyle\int_{\Gamma_{s}}\left(\overline{G_{D}(\xi,x)}\frac{\partial
G_{D}(\xi,z)}{\partial\nu(\xi)}-\frac{\partial\overline{G_{D}(\xi,x)}}{\partial\nu(\xi)}G_{D}(\xi,z)\right){\rm
d}s(\xi)$ $\displaystyle=$
$\displaystyle\int_{\Gamma_{s}}\Bigg{\\{}\overline{G_{D}(\xi,x)}\left[\frac{\partial
G_{D}(\xi,z)}{\partial\nu(\xi)}-{\rm i}\kappa_{1}G_{D}(\xi,z)\right]$
$\displaystyle\qquad\;\;\;-G_{D}(\xi,z)\left[\frac{\overline{\partial
G_{D}(\xi,x)}}{\partial\nu(\xi)}+{\rm
i}\kappa_{1}\overline{G_{D}(\xi,x)}\right]\Bigg{\\}}{\rm d}s(\xi)$
$\displaystyle\qquad\;\;\;+2{\rm
i}\kappa_{1}\int_{\Gamma_{s}}\overline{G_{D}(\xi,x)}G_{D}(\xi,z){\rm
d}s(\xi).$

Thus, a direct application of the reciprocity $G_{D}(\xi,x)=G_{D}(x,\xi)$ for
$\xi\in\Gamma_{s}$ and $x\in S$ yields

$\displaystyle\kappa_{1}\int_{\Gamma_{s}}\overline{G_{D}(x,\xi)}G_{D}(\xi,z){\rm
d}s(\xi)={\rm Im}G_{D}(x,z)+\zeta_{D,s}(x,z)\quad{\rm for}\;\;{\forall x,z\in
S},$

with

$\displaystyle\zeta_{D,s}(x,z)$ $\displaystyle=$ $\displaystyle\frac{\rm
i}{2}\int_{\Gamma_{s}}\Bigg{\\{}\overline{G_{D}(\xi,x)}\left[\frac{\partial
G_{D}(\xi,z)}{\partial\nu(\xi)}-{\rm i}\kappa_{1}G_{D}(\xi,z)\right]$
$\displaystyle\qquad\quad-G_{D}(\xi,z)\left[\frac{\overline{\partial
G_{D}(\xi,x)}}{\partial\nu(\xi)}+{\rm
i}\kappa_{1}\overline{G_{D}(\xi,x)}\right]\Bigg{\\}}{\rm d}s(\xi).$

Thus, the inequality (2.12) holds. Due to

$\displaystyle G_{D}(\xi,y)=O(|\xi|^{-\frac{1}{2}}),\qquad\frac{\partial
G_{D}(\xi,y)}{\partial\nu(\xi)}-{\rm
i}\kappa_{1}G_{D}(\xi,y)=O(|\xi|^{-\frac{3}{2}})$

for $y\in\\{x,z\\}$ and $x,z\in S$, it follows that

$\displaystyle|\zeta_{D,s}(x,z)|\leq CR_{s}^{-1}$

uniformly for $x,z\in S$. Since

$\displaystyle\frac{\partial G_{D}(\xi,x)}{\partial
x_{j}}=O(|\xi|^{-\frac{1}{2}}),\qquad\frac{\partial}{\partial
x_{j}}\left[\frac{\partial G_{D}(\xi,x)}{\partial\nu(\xi)}-{\rm
i}\kappa_{1}G_{D}(\xi,x)\right]=O(|\xi|^{-\frac{3}{2}})$

for $j=1,2$ and $x\in S$, it follows that

$\displaystyle|\nabla_{x}\zeta_{D,s}(x,z)|\leq CR_{s}^{-1}$

uniformly for $x,z\in S$. Thus, we conclude that
$|\zeta_{D,s}(x,z)|+|\nabla_{x}\zeta_{D,s}(x,z)|\leq CR_{s}^{-1}$ holds. It is
obvious that we can prove (2.13) similarly. The proof is completed. ∎

######  Lemma 2.3.

Let $V_{\alpha}(x,x_{s})$ be the solution to Problem (2.7), then we have the
Green’s formula

$\displaystyle
V_{\alpha}(x,x_{s})=\int_{\Gamma\setminus\Gamma_{R}}\left[\frac{\partial
G_{\alpha}(\xi,x)}{\partial\nu(\xi)}V_{\alpha}(\xi,x_{s})-\frac{\partial
V_{\alpha}(\xi,x_{s})}{\partial\nu(\xi)}G_{\alpha}(\xi,x)\right]{\rm
d}s(\xi)\qquad{\rm for}\;\;x\in\Omega_{1}.$

###### Proof.

For $\alpha=D$ and any $x\in\Omega_{1}$, we choose a sufficient large $\rho>0$
and a sufficiently small $\varepsilon>0$ such that the disc
$B_{\varepsilon}(x)$ with $x$ as the center and $\varepsilon$ as the radius
contains in $\Omega_{\rho}:=B_{\rho}\cap\Omega_{1}$, where $B_{\rho}$ denotes
the disc with the origin as the center and $\rho$ as the radius. We now apply
Green’s theorem to the functions $V_{D}(\cdot,x_{s})$ and $G_{D}(\cdot,x)$ in
the domain $\Omega_{\rho}\setminus B_{\varepsilon}(x)$ to obtain

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\int_{\Omega_{\rho}\setminus
B_{\varepsilon}(x)}\left[\Delta V_{D}(\xi,x_{s})G_{D}(\xi,x)-\Delta
G_{D}(\xi,x)V_{D}(\xi,x_{s})\right]{\rm d}\xi$ (2.14) $\displaystyle=$
$\displaystyle\left\\{\int_{\partial B_{\rho}^{+}}-\int_{\Gamma\cap
B_{\rho}}+\int_{\partial B_{\varepsilon}(x)}\right\\}\left[\frac{\partial
V_{D}(\xi,x_{s})}{\partial\nu(\xi)}G_{D}(\xi,x)-\frac{\partial
G_{D}(\xi,x)}{\partial\nu(\xi)}V_{D}(\xi,x_{s})\right]{\rm d}s(\xi)$
$\displaystyle:=$ $\displaystyle I_{1}-I_{2}+I_{3},$

where $\nu(\xi)$ denotes the unit normal which directs into the exterior of
$B_{\rho}$ for $\xi\in\partial B_{\rho}^{+}$, and directs into the interior of
$B_{\varepsilon}(x)$ for $\xi\in\partial B_{\varepsilon}(x)$. For the item
$I_{1}$, our task is to show

$\displaystyle\lim_{\rho\to\infty}I_{1}=0.$ (2.15)

To accomplish this, using the Sommerfeld radiation condition gives that

$\displaystyle\int_{\partial B_{\rho}^{+}}\left[\left|\frac{\partial
V_{D}(\xi,x_{s})}{\partial\nu(\xi)}\right|^{2}+\kappa_{1}^{2}|V_{D}(\xi,x_{s})|^{2}+2\kappa_{1}{\rm
Im}\left(V_{D}(\xi,x_{s})\frac{\partial\overline{V_{D}(\xi,x_{s})}}{\partial\nu(\xi)}\right)\right]{\rm
d}s(\xi)$ $\displaystyle=\int_{\partial B_{\rho}^{+}}\left|\frac{\partial
V_{D}(\xi,x_{s})}{\partial\nu(\xi)}-{\rm
i}\kappa_{1}V_{D}(\xi,x_{s})\right|^{2}{\rm d}s(\xi)\to 0\quad{\rm
as}\;\;\rho\to\infty.$ (2.16)

A direct application of Green’s theorem leads to

$\displaystyle\int_{\partial
B_{\rho}^{+}}V_{D}(\xi,x_{s})\frac{\partial\overline{V_{D}(\xi,x_{s})}}{\partial\nu(\xi)}{\rm
d}s(\xi)$
$\displaystyle=\int_{B_{\rho}\cap\Gamma}V_{D}(\xi,x_{s})\frac{\partial\overline{V_{D}(\xi,x_{s})}}{\partial\nu(\xi)}{\rm
d}s(\xi)+\int_{B_{\rho}\cap\Omega_{1}}\left[|\nabla
V_{D}(\xi,x_{s})|^{2}-\kappa_{1}^{2}|V_{D}(\xi,x_{s})|^{2}\right]{\rm d}\xi.$
(2.17)

We now insert the imaginary part of (2.17) into (2.16) and find that

$\displaystyle\lim_{\rho\to\infty}\int_{\partial
B_{\rho}^{+}}\left(\left|\frac{\partial
V_{D}(\xi,x_{s})}{\partial\nu(\xi)}\right|^{2}+\kappa_{1}^{2}|V_{D}(\xi,x_{s})|^{2}\right){\rm
d}s(\xi)$ $\displaystyle=-2\kappa_{1}{\rm
Im}\int_{\Gamma\setminus\Gamma_{R}}V_{D}(\xi,x_{s})\frac{\partial\overline{V_{D}(\xi,x_{s})}}{\partial\nu(\xi)}{\rm
d}s(\xi),$

where we use the fact $V_{D}(\xi,x_{s})=0$ for $\xi\in\Gamma\cap\Gamma_{R}$.
Thus, we conclude

$\displaystyle\int_{\partial B_{\rho}^{+}}|V_{D}(\xi,x_{s})|^{2}{\rm
d}s=O(1),\quad\rho\to\infty.$ (2.18)

Similarly, we have

$\displaystyle\int_{\partial B_{\rho}^{+}}|G_{D}(\xi,x)|^{2}{\rm
d}s=O(1),\quad\rho\to\infty.$ (2.19)

The item $I_{1}$ can be rewritten as

$\displaystyle I_{1}$ $\displaystyle=$ $\displaystyle\int_{\partial
B_{\rho}^{+}}\left[\frac{\partial V_{D}(\xi,x_{s})}{\partial\nu(\xi)}-{\rm
i}\kappa_{1}V_{D}(\xi,x_{s})\right]G_{D}(\xi,x){\rm d}s(\xi)$
$\displaystyle-\int_{\partial B_{\rho}^{+}}\left[\frac{\partial
G_{D}(\xi,x)}{\partial\nu(\xi)}-{\rm
i}\kappa_{1}G_{D}(\xi,x)\right]V_{D}(\xi,x_{s}){\rm d}s(\xi)$

which, combining (2.18), (2.19), the Sommerfeld radiation condition, and
Cauchy-Schwartz inequality, shows that (2.15) holds true.

For the item $I_{3}$, by
$G_{D}(\xi,x)=G^{s}_{D}(\xi,x)+\Phi_{\kappa_{1}}(\xi,x)$, we have

$\displaystyle I_{3}$ $\displaystyle=$ $\displaystyle\int_{\partial
B_{\varepsilon}(x)}\left[\frac{\partial
V_{D}(\xi,x_{s})}{\partial\nu(\xi)}G^{s}_{D}(\xi,x)-\frac{\partial
G^{s}_{D}(\xi,x)}{\partial\nu(\xi)}V_{D}(\xi,x_{s})\right]{\rm d}s(\xi)$
(2.20) $\displaystyle+\int_{\partial B_{\varepsilon}(x)}\left[\frac{\partial
V_{D}(\xi,x_{s})}{\partial\nu(\xi)}\Phi_{\kappa_{1}}(\xi,x)-\frac{\partial\Phi_{\kappa_{1}}(\xi,x)}{\partial\nu(\xi)}V_{D}(\xi,x_{s})\right]{\rm
d}s(\xi)$ $\displaystyle:=$ $\displaystyle I_{31}+I_{32}.$

Applying the Green’s theorem in $B_{\varepsilon}(x)$ shows that $I_{31}=0$. A
straightforward calculation with using the mean value theorem shows that

$\displaystyle\lim_{\varepsilon\to 0}I_{32}=-V_{D}(x,x_{s}).$ (2.21)

Thus, from (2.14), (2.15),(2.20), and (2.21), we have the following Green’s
formula

$\displaystyle
V_{D}(x,x_{s})=\int_{\Gamma\setminus\Gamma_{R}}\left[\frac{\partial
G_{D}(\xi,x)}{\partial\nu(\xi)}V_{D}(\xi,x_{s})-\frac{\partial
V_{D}(\xi,x_{s})}{\partial\nu(\xi)}G_{D}(\xi,x)\right]{\rm d}s(\xi).$

The proof is completed. ∎

Now, we are in position to present the resolution result of the RTM method for
recovering an impenetrable locally rough surface.

######  Theorem 2.4.

For any $z\in S$, let $\psi_{\alpha}(\xi,z)$ solve

$\displaystyle\left\\{\begin{aligned}
&\Delta\psi_{\alpha}(\xi,z)+\kappa_{1}^{2}\psi_{\alpha}(\xi,z)=0\qquad\qquad\qquad\;\;\;{\rm
in}\;\;\Omega_{1},\\\ &\;\mathcal{B}\psi_{\alpha}(\xi,z)=-\mathcal{B}{\rm
Im}G_{\alpha}(\xi,z)\qquad\qquad\qquad\;\;{\rm
on}\;\;\Gamma\setminus\Gamma_{R},\\\
&\;\mathcal{B}\psi_{\alpha}(\xi,z)=0\qquad\qquad\qquad\qquad\qquad\qquad\;{\rm
on}\;\;\Gamma\cap\Gamma_{R},\\\
&\lim_{|\xi|\rightarrow\infty}|\xi|^{\frac{1}{2}}\left(\partial_{|\xi|}\psi_{\alpha}(\xi,z)-{\rm
i}\kappa_{1}\psi_{\alpha}(\xi,z)\right)=0,\end{aligned}\right.$ (2.22)

and $\psi_{\alpha}^{\infty}(\hat{\xi},z)$ be the far-field pattern of
$\psi_{\alpha}(\xi,z)$. Then for the indicator function $\widetilde{{\rm
Ind}}_{\alpha}(z)$, we have

$\displaystyle\widetilde{{\rm
Ind}}_{\alpha}(z)=\kappa_{1}\int_{{\mathbb{S}}_{+}}|\psi_{\alpha}^{\infty}(\hat{\xi},z)|^{2}{\rm
d}s(\hat{\xi})+\eta_{\alpha}(z),\qquad\forall z\in S,$

where $\|\eta_{\alpha}(z)\|_{L^{\infty}(S)}\leq C(R_{s}^{-1}+R_{r}^{-1})$ with
some constant $C$ depending on $R$.

###### Proof.

For the case $\alpha=D$, recall that

$\displaystyle\widetilde{{\rm Ind}}_{D}(z)$ $\displaystyle=$
$\displaystyle-\kappa_{1}^{2}{\rm
Im}\int_{\Gamma_{r}}\int_{\Gamma_{s}}G_{D}(z,x_{s})G_{D}(z,x_{r})\overline{V_{D}(x_{r},x_{s})}{\rm
d}s(x_{s}){\rm d}s(x_{r})$ (2.23) $\displaystyle=$
$\displaystyle-\kappa_{1}{\rm
Im}\int_{\Gamma_{s}}G_{D}(z,x_{s})\widetilde{W}_{D}(z,x_{s}){\rm d}s(x_{s}),$

where

$\displaystyle\widetilde{W}_{D}(z,x_{s}):=\kappa_{1}\int_{\Gamma_{r}}G_{D}(z,x_{r})\overline{V_{D}(x_{r},x_{s})}{\rm
d}s(x_{r}).$ (2.24)

Substituting the Green’s formula presented by Lemma 2.3 into (2.24) and
exchanging the order of integration leads to

$\displaystyle\widetilde{W}_{D}(z,x_{s})$ $\displaystyle=$
$\displaystyle\int_{\Gamma\setminus\Gamma_{R}}\bigg{\\{}\overline{V_{D}(\xi,x_{s})}\frac{\partial}{\partial\nu(\xi)}\left[\kappa_{1}\int_{\Gamma_{r}}G_{D}(z,x_{r})\overline{G_{D}(\xi,x_{r})}{\rm
d}s(x_{r})\right]$ (2.25)
$\displaystyle\qquad\qquad-\left[\kappa_{1}\int_{\Gamma_{r}}G_{D}(z,x_{r})\overline{G_{D}(\xi,x_{r})}ds(x_{r})\right]\frac{\partial\overline{V_{D}(\xi,x_{s})}}{\partial\nu(\xi)}\bigg{\\}}{\rm
d}s(\xi)$ $\displaystyle=$
$\displaystyle\int_{\Gamma\setminus\Gamma_{R}}\bigg{\\{}\overline{V_{D}(\xi,x_{s})}\frac{\partial}{\partial\nu(\xi)}\left[{\rm
Im}G_{D}(\xi,z)+\zeta_{D,r}(\xi,z)\right]$
$\displaystyle\qquad\qquad-\left[{\rm
Im}G_{D}(\xi,z)+\zeta_{D,r}(\xi,z)\right]\frac{\partial\overline{V_{D}(\xi,x_{s})}}{\partial\nu(\xi)}\bigg{\\}}{\rm
d}s(\xi).$

Substituting (2.25) into (2.23) gives

$\displaystyle\widetilde{{\rm Ind}}_{D}(z)$ $\displaystyle=-{\rm
Im}\int_{\Gamma\setminus\Gamma_{R}}\bigg{\\{}\left[\kappa_{1}\int_{\Gamma_{s}}G_{D}(z,x_{s})\overline{V_{D}(\xi,x_{s})}{\rm
d}s(x_{s})\right]\frac{\partial}{\partial\nu(\xi)}\left[{\rm
Im}G_{D}(\xi,z)+\zeta_{D,r}(\xi,z)\right]$ $\displaystyle\qquad-\left[{\rm
Im}G_{D}(\xi,z)+\zeta_{D,r}(\xi,z)\right]\frac{\partial}{\partial\nu(\xi)}\left[\kappa_{1}\int_{\Gamma_{s}}G_{D}(z,x_{s})\overline{V_{D}(\xi,x_{s})}{\rm
d}s(x_{s})\right]\bigg{\\}}{\rm d}s(\xi)$ $\displaystyle=-{\rm
Im}\int_{\Gamma\setminus\Gamma_{R}}\bigg{\\{}\phi_{D}(\xi,z)\frac{\partial}{\partial\nu(\xi)}\left[{\rm
Im}G_{D}(\xi,z)+\zeta_{D,r}(\xi,z)\right]$
$\displaystyle\qquad\qquad-\left[{\rm
Im}G_{D}(\xi,z)+\zeta_{D,r}(\xi,z)\right]\frac{\partial\phi_{D}(\xi,z)}{\partial\nu(\xi)}\bigg{\\}}{\rm
d}s(\xi),$

where $\phi_{D}(\xi,z)$ is defined by

$\displaystyle\phi_{D}(\xi,z):=\kappa_{1}\int_{\Gamma_{s}}G_{D}(z,x_{s})\overline{V_{D}(\xi,x_{s})}{\rm
d}s(x_{s}).$

Since $V_{D}(\xi,x_{s})$ satisfies Problem (2.7), it follows from the
linearity that $\overline{\phi_{D}}(\xi,z)$ solves the following problem

$\displaystyle\left\\{\begin{aligned}
&\Delta\overline{\phi_{D}}(\xi,z)+\kappa_{1}^{2}\overline{\phi_{D}}(\xi,z)=0\qquad\qquad\qquad\;\;{\rm
in}\;\;\Omega_{1},\\\ &\;\overline{\phi_{D}}(\xi,z)=-{\rm
Im}G_{D}(\xi,z)-\zeta_{D,s}(\xi,z)\qquad\;\;{\rm
on}\;\;\Gamma\setminus\Gamma_{R},\\\
&\;\overline{\phi_{D}}(\xi,z)=0\qquad\qquad\qquad\qquad\qquad\qquad\;\;\;\;{\rm
on}\;\;\Gamma\cap\Gamma_{R},\\\
&\lim_{|\xi|\rightarrow\infty}|\xi|^{\frac{1}{2}}\left(\partial_{|\xi|}\overline{\phi_{D}}(\xi,z)-{\rm
i}\kappa_{1}\overline{\phi_{D}}(\xi,z)\right)=0,\end{aligned}\right.$ (2.26)

where we use Lemma 2.2 to derive the boundary condition on
$\Gamma\setminus\Gamma_{R}$. By the linearity, it is easy to obtain the
following decomposition

$\displaystyle\overline{\phi_{D}}(\xi,z)=\psi_{D}(\xi,z)+\varphi_{D}(\xi,z),$

where $\psi_{D}(\xi,z)$ and $\varphi_{D}(\xi,z)$ solve Problem (2.26) with the
boundary data $-{\rm Im}G_{D}(\xi,z)$ and $-\zeta_{D,s}(\xi,z)$ on
$\Gamma\setminus\Gamma_{R}$, respectively. Thus, we conclude

$\displaystyle\widetilde{{\rm Ind}}_{D}(z)=-{\rm
Im}\int_{\Gamma\setminus\Gamma_{R}}\bigg{\\{}\left[\overline{\psi_{D}}(\xi,z)+\overline{\varphi_{D}}(\xi,z)\right]\frac{\partial}{\partial\nu(\xi)}\left[{\rm
Im}G_{D}(\xi,z)+\zeta_{D,r}(\xi,z)\right]$
$\displaystyle\qquad\qquad-\left[{\rm
Im}G_{D}(\xi,z)+\zeta_{D,r}(\xi,z)\right]\frac{\partial}{\partial\nu(\xi)}\left[\overline{\psi_{D}}(\xi,z)+\overline{\varphi_{D}}(\xi,z)\right]\bigg{\\}}{\rm
d}s(\xi)$ $\displaystyle=-{\rm
Im}\int_{\Gamma\setminus\Gamma_{R}}\left[\overline{\psi_{D}}(\xi,z)\frac{\partial{\rm
Im}G_{D}(\xi,z)}{\partial\nu(\xi)}-{\rm
Im}G_{D}(\xi,z)\frac{\partial\overline{\psi_{D}}(\xi,z)}{\partial\nu(\xi)}\right]{\rm
d}s(\xi)+\eta_{D}(z)$ $\displaystyle=-{\rm
Im}\int_{\Gamma\setminus\Gamma_{R}}\left[-{\rm
Im}G_{D}(\xi,z)\frac{\partial{\rm
Im}G_{D}(\xi,z)}{\partial\nu(\xi)}+\psi_{D}(\xi,z)\frac{\partial\overline{\psi_{D}}(\xi,z)}{\partial\nu(\xi)}\right]{\rm
d}s(\xi)+\eta_{D}(z)$ $\displaystyle=-{\rm
Im}\int_{\Gamma\setminus\Gamma_{R}}\psi_{D}(\xi,z)\frac{\partial\overline{\psi_{D}}(\xi,z)}{\partial\nu(\xi)}{\rm
d}s(\xi)+\eta_{D}(z).$ (2.27)

Here, $\eta_{D}(z)$ is defined by

$\displaystyle\eta_{D}(z)=-{\rm
Im}\int_{\Gamma\setminus\Gamma_{R}}\left[\overline{\psi_{D}}(\xi,z)\frac{\partial\zeta_{D,r}(\xi,z)}{\partial\nu(\xi)}-\zeta_{D,r}(\xi,z)\frac{\partial\overline{\psi_{D}}(\xi,z)}{\partial\nu(\xi)}\right]{\rm
d}s(\xi)$ $\displaystyle-{\rm
Im}\int_{\Gamma\setminus\Gamma_{R}}\left[\overline{\varphi_{D}}(\xi,z)\frac{\partial\zeta_{D,r}(\xi,z)}{\partial\nu(\xi)}-\zeta_{D,r}(\xi,z)\frac{\partial\overline{\varphi_{D}}(\xi,z)}{\partial\nu(\xi)}\right]{\rm
d}s(\xi)$ $\displaystyle-{\rm
Im}\int_{\Gamma\setminus\Gamma_{R}}\left[\overline{\varphi_{D}}(\xi,z)\frac{\partial{\rm
Im}G_{D}(\xi,z)}{\partial\nu(\xi)}-{\rm
Im}G_{D}(\xi,z)\frac{\partial\overline{\varphi_{D}}(\xi,z)}{\partial\nu(\xi)}\right]{\rm
d}s(\xi).$ (2.28)

Applying Green’s theorem to the functions $\psi_{D}(\xi,z)$ and
$\overline{\psi_{D}}(\xi,z)$ in the domain $\Omega\cap B_{\rho}$ yields

$\displaystyle-{\rm
Im}\int_{\Gamma\setminus\Gamma_{R}}\psi_{D}(\xi,z)\frac{\partial\overline{\psi_{D}}(\xi,z)}{\partial\nu(\xi)}{\rm
d}s(\xi)=-{\rm Im}\int_{\partial
B^{+}_{\rho}}\psi_{D}(\xi,z)\frac{\partial\overline{\psi_{D}}(\xi,z)}{\partial\nu(\xi)}{\rm
d}s(\xi)$ $\displaystyle+{\rm Im}\int_{\Omega\cap
B_{\rho}}\left[\psi_{D}(\xi,z)\Delta\overline{\psi_{D}}(\xi,z)+|\nabla\psi_{D}(\xi,z)|^{2}\right]{\rm
d}\xi$ $\displaystyle=-{\rm Im}\int_{\partial
B^{+}_{\rho}}\psi_{D}(\xi,z)\left[\frac{\partial\overline{\psi_{D}}(\xi,z)}{\partial\nu(\xi)}+{\rm
i}\kappa_{1}\overline{\psi_{D}}(\xi,z)\right]{\rm d}s(\xi)+{\rm
Im}\int_{\partial B^{+}_{\rho}}{\rm i}\kappa_{1}|\psi_{D}(\xi,z)|^{2}{\rm
d}s(\xi)$ (2.29)

Note that

$\displaystyle\psi_{D}(\xi,z)=O(|\xi|^{-\frac{1}{2}})\quad{\rm
and}\quad\frac{\partial\psi_{D}(\xi,z)}{\partial\nu(\xi)}-{\rm
i}\kappa_{1}\psi_{D}(\xi,z)=o(|\xi|^{-\frac{1}{2}}),$

we have

$\displaystyle\lim_{\rho\to\infty}\int_{\partial
B^{+}_{\rho}}\psi_{D}(\xi,z)\left[\frac{\partial\overline{\psi_{D}}(\xi,z)}{\partial\nu(\xi)}+{\rm
i}\kappa_{1}\overline{\psi_{D}}(\xi,z)\right]{\rm d}s(\xi)=0.$

Since $\psi_{D}(\xi,z)$ satisfies the Sommerfeld radiation condition, it
admits the following asymptotic behavior

$\displaystyle\psi_{D}(\xi,z)=\frac{e^{{\rm
i}\kappa_{1}|\xi|}}{|\xi|^{\frac{1}{2}}}\left[\psi^{\infty}_{D}(\hat{\xi},z)+O(|\xi|^{-1})\right]$

which implies

$\displaystyle\lim_{\rho\to\infty}\int_{\partial
B_{\rho}^{+}}\kappa_{1}|\psi_{D}(\xi,z)|^{2}{\rm
d}s(\xi)=\kappa_{1}\int_{{\mathbb{S}}_{+}}|\psi_{D}^{\infty}(\hat{\xi},z)|^{2}{\rm
d}s(\hat{\xi}).$ (2.30)

Hence, with the help of (2.27) and (2.29)–(2.30), we arrive at

$\displaystyle\widetilde{{\rm
Ind}}_{D}(z)=\kappa_{1}\int_{{\mathbb{S}}_{+}}|\psi_{D}^{\infty}(\hat{\xi},z)|^{2}{\rm
d}s(\hat{\xi})+\eta_{D}(z).$

The remaining part of the proof is the estimate of $\eta_{D}(z)$. For
$\xi\in\Gamma\setminus\Gamma_{R}$ and $z\in S$, it follows from Lemma 2.2 that

$\displaystyle|\zeta_{D,r}(\xi,z)|+|\nabla_{\xi}\zeta_{D,r}(\xi,z)|\leq
CR_{r}^{-1}.$ (2.31)

Observing that

$\displaystyle{\rm Im}G_{D}(\xi,z)={\rm
Im}G^{s}_{D}(\xi,z)+\frac{1}{4}J_{0}(\kappa_{1}|\xi-z|),$

where $J_{0}$ stands for the Bessel function of order zero, it follows from
the smoothness of $G_{D}^{s}(\xi,z)$ and $J_{0}(\kappa_{1}|\xi-z|)$ that

$\displaystyle\left|{\rm Im}G_{D}(\xi,z)\right|+\left|\frac{\partial{\rm
Im}G_{D}(\xi,z)}{\partial\nu(\xi)}\right|\leq C.$ (2.32)

Since $\psi_{D}(\xi,z)$ and $\varphi_{D}(\xi,z)$ solve Problem (2.26) with the
boundary data $-{\rm Im}G_{D}(\xi,z)$ and $-\zeta_{D,s}(\xi,z)$ on
$\Gamma\setminus\Gamma_{R}$, respectively, a direct application of the well-
posedness of Problem (2.22) (cf. [23, Theorem 2.1]) and the trace theorem
shows that

$\displaystyle\|\psi_{D}(\xi,z)\|_{H^{\frac{1}{2}}(\Gamma\setminus\Gamma_{R})}+\left\|\frac{\partial\psi_{D}(\xi,z)}{\partial\nu(\xi)}\right\|_{H^{-\frac{1}{2}}(\Gamma\setminus\Gamma_{R})}\lesssim\|{\rm
Im}G_{D}(\xi,z)\|_{H^{\frac{1}{2}}(\Gamma\setminus\Gamma_{R})}\leq C,$ (2.33)

and

$\displaystyle\|\varphi_{D}(\xi,z)\|_{H^{\frac{1}{2}}(\Gamma\setminus\Gamma_{R})}+\left\|\frac{\partial\varphi_{D}(\xi,z)}{\partial\nu(\xi)}\right\|_{H^{-\frac{1}{2}}(\Gamma\setminus\Gamma_{R})}\lesssim\|\zeta_{D,s}(\xi,z)\|_{H^{\frac{1}{2}}(\Gamma\setminus\Gamma_{R})}\lesssim
R_{s}^{-1},$ (2.34)

where the notation $a\lesssim b$ means $a\leq Cb$ for some generic constant
$C>0$, which may change step by step. Thus, with the aid of (2.28) and
(2.31)–(2.34), we can easily obtain

$\displaystyle\|\eta_{D}(z)\|_{L^{\infty}(S)}\leq C(R_{r}^{-1}+R_{s}^{-1}),$

with $C$ depending on $R$. The proof is finished. ∎

###  2.2 The far-field reconstruction

This subsection is devoted to the RTM method with the far-field measurement.
To this end, we need to establish a mixed reciprocity relation. Let
$w^{i}(x,d)=e^{{\rm i}\kappa_{1}x\cdot d}$ with
$d=(d_{1},d_{2})^{\rm\top}\in{\mathbb{S}}_{-}:=\\{x\in{\mathbb{R}}^{2}:|x|=1,x_{2}<0\\}$
be the plane wave, then the reflected wave of $w^{i}$ by the infinite plane
$\Gamma_{0}$ is given by $w_{\alpha}^{r}(x,d)=-e^{{\rm i}\kappa_{1}x\cdot
d^{r}}$ for $\alpha=D$ and $w_{\alpha}^{r}(x,d)=e^{{\rm i}\kappa_{1}x\cdot
d^{r}}$ for $\alpha=N$ with $d^{r}=(d_{1},-d_{2})^{\rm\top}$. We define

$w_{0,\alpha}(x,d):=w^{i}(x,d)+w_{\alpha}^{r}(x,d)=\left\\{\begin{array}[]{l}e^{{\rm
i}\kappa_{1}x\cdot d}-e^{{\rm i}\kappa_{1}x\cdot
d^{r}}\qquad\qquad\;\textrm{for}\;\;\;\alpha=D,\\\\[2.84526pt] e^{{\rm
i}\kappa_{1}x\cdot d}+e^{{\rm i}\kappa_{1}x\cdot
d^{r}}\qquad\qquad\;\textrm{for}\;\;\;\alpha=N,\end{array}\right.$

which satisfies $w_{0,D}(x,d)=0$ on $\Gamma_{0}$ and
$\partial_{\nu}w_{0,N}(x,d)=0$ on $\Gamma_{0}$. Then the scattering of
$w_{0,\alpha}(x,d)$ by the locally rough surface $\Gamma$ can be modelled by

$\left\\{\begin{aligned} &\Delta
w_{\alpha}^{s}(x,d)+\kappa_{1}^{2}w_{\alpha}^{s}(x,d)=0\qquad\qquad\qquad\;\textrm{in}\;\;\Omega_{1},\\\
&\mathcal{B}w_{\alpha}^{s}(x,d)=-\mathcal{B}w_{0,\alpha}(x,d)\qquad\qquad\qquad\;\;\;\;\textrm{on}\;\;\Gamma,\\\
&\lim_{|x|\rightarrow\infty}|x|^{\frac{1}{2}}\left(\partial_{|x|}w_{\alpha}^{s}(x,d)-{\rm
i}\kappa_{1}w_{\alpha}^{s}(x,d)\right)=0,\quad\end{aligned}\right.$

where $w_{\alpha}^{s}$ denotes the scattered field and the Sommerfeld
radiation condition holds uniformly for all directions
$\hat{x}\in{\mathbb{S}}_{+}$.

Let $v_{0,\alpha}(x,x_{s})$ be the Dirichlet or Neumann Green’s function with
respect to $\Gamma_{0}$, which is given by

$v_{0,\alpha}(x,x_{s})=\left\\{\begin{array}[]{l}\Phi(x,x_{s})-\Phi(x,x_{s}^{\prime})\qquad\qquad\;\textrm{for}\;\;\alpha=D,\\\\[2.84526pt]
\Phi(x,x_{s})+\Phi(x,x_{s}^{\prime})\qquad\qquad\;\textrm{for}\;\;\alpha=N.\end{array}\right.$

Here, $x_{s}^{\prime}$ is the image point of $x_{s}$ with respect to
$\Gamma_{0}$. We have $v_{0,D}(x,x_{s})=0$ on $\Gamma_{0}$ and
$\partial_{\nu}v_{0,N}(x,x_{s})=0$ on $\Gamma_{0}$. Let
$v_{0,\alpha}^{\infty}(\hat{x},x_{s})$ be the far-field of
$v_{0,\alpha}(x,x_{s})$, it is easy to see that

$\displaystyle
v_{0,\alpha}^{\infty}(\hat{x},x_{s})=\gamma_{1}w_{0,\alpha}(x_{s},-\hat{x})\qquad{\rm
for}\;\;\hat{x}\in{\mathbb{S}}_{+},x_{s}\in{\mathbb{R}}^{2}_{+},$ (2.35)

where $\gamma_{1}:=\frac{e^{\frac{\pi}{4}{\rm i}}}{\sqrt{8\kappa_{1}\pi}}$.
Define
$v_{\alpha}^{s}(x,x_{s}):=u_{\alpha}^{s}(x,x_{s})-v_{0,\alpha}(x,x_{s})+\Phi(x,x_{s})$
where $u_{\alpha}^{s}(x,x_{s})$ is the scattered field of Problem (1.3), then
we have $v_{\alpha}^{s}$ solves

$\left\\{\begin{aligned} &\Delta
v_{\alpha}^{s}(x,x_{s})+\kappa_{1}^{2}v_{\alpha}^{s}(x,x_{s})=0\qquad\qquad\qquad\;\;\textrm{in}\;\;\Omega_{1},\\\\[2.84526pt]
&\mathcal{B}v_{\alpha}^{s}(x,x_{s})=-\mathcal{B}v_{0,\alpha}(x,x_{s})\qquad\qquad\qquad\;\;\;\;\;\textrm{on}\;\;\Gamma,\\\\[2.84526pt]
&\lim_{|x|\rightarrow\infty}|x|^{\frac{1}{2}}\left(\partial_{|x|}v_{\alpha}^{s}(x,x_{s})-{\rm
i}\kappa_{1}v_{\alpha}^{s}(x,x_{s})\right)=0,\quad\end{aligned}\right.$

where the Sommerfeld radiation condition holds uniformly for all directions
$\hat{x}\in{\mathbb{S}}_{+}$.

######  Theorem 2.5.

For acoustic scattering of plane waves $w_{0,\alpha}(\cdot,-\hat{x})$,
$\hat{x}\in{\mathbb{S}}_{+}$ and point sources $v_{0,\alpha}(\cdot,x_{s})$,
$x_{s}\in\Omega_{1}$ from a locally rough surface $\Gamma$ we have

$\displaystyle
v_{\alpha}^{\infty}(\hat{x},x_{s})=\gamma_{1}w_{\alpha}^{s}(x_{s},-\hat{x})$

for $\alpha\in\\{D,N\\}$, $\hat{x}\in{\mathbb{S}}_{+},x_{s}\in\Omega_{1}$ and
$x_{s}^{\prime}\notin\Omega_{1}$.

###### Proof.

Let $D_{1}:=\Omega_{2}\cap{\mathbb{R}}^{2}_{+}$ and
$D_{2}:=\Omega_{1}\cap{\mathbb{R}}^{2}_{-}$. For $\alpha=D$,
$x_{s}\in\Omega_{1}$ and $x_{s}^{\prime}\notin\Omega_{1}$, a direct
application of Green formula yields

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{D_{1}}\left[v_{0,D}(\xi,x_{s})\Delta
w_{0,D}(\xi,-\hat{x})-w_{0,D}(\xi,-\hat{x})\Delta
v_{0,D}(\xi,x_{s})\right]{\rm d}\xi$ (2.36) $\displaystyle=$
$\displaystyle\int_{\partial
D_{1}\setminus\Gamma_{0}}\left[v_{0,D}(\xi,x_{s})\frac{\partial
w_{0,D}(\xi,-\hat{x})}{\partial\nu(\xi)}-w_{0,D}(\xi,-\hat{x})\frac{\partial
v_{0,D}(\xi,x_{s})}{\partial\nu(\xi)}\right]{\rm d}s(\xi)$

and

$\displaystyle\int_{\partial
D_{2}\setminus\Gamma_{0}}\left[v_{0,D}(\xi,x_{s})\frac{\partial
w_{0,D}(\xi,-\hat{x})}{\partial\nu(\xi)}-w_{0,D}(\xi,-\hat{x})\frac{\partial
v_{0,D}(\xi,x_{s})}{\partial\nu(\xi)}\right]{\rm d}s(\xi)=0.$ (2.37)

Combining (2.36), (2.37) and the fact $v_{0,D}$ and $w_{0,D}$ vanish on
$\Gamma_{0}$ gives that

$\displaystyle\int_{\Gamma}\left[v_{0,D}(\xi,x_{s})\frac{\partial
w_{0,D}(\xi,-\hat{x})}{\partial\nu(\xi)}-w_{0,D}(\xi,-\hat{x})\frac{\partial
v_{0,D}(\xi,x_{s})}{\partial\nu(\xi)}\right]{\rm d}s(\xi)=0.$ (2.38)

It follows from the Green’s formula, the Sommerfeld radiation condition and
the Cauchy-Schwartz inequality that

$\displaystyle\int_{\Gamma}\left[v_{D}^{s}(\xi,x_{s})\frac{\partial
w^{s}_{D}(\xi,-\hat{x})}{\partial\nu(\xi)}-w^{s}_{D}(\xi,-\hat{x})\frac{\partial
v^{s}_{D}(\xi,x_{s})}{\partial\nu(\xi)}\right]{\rm d}s(\xi)=0.$ (2.39)

For $x\in\Omega_{1}$, we choose a sufficient large $\rho>0$ and a sufficient
small $\varepsilon>0$ such that
$B_{\varepsilon}(x)\subset\Omega_{\rho}:=B_{\rho}\cap\Omega_{1}$. By the
Green’s formula, we obtain

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{\Omega_{\rho}\setminus\overline{B_{\varepsilon}(x)}}\left[v_{0,D}(\xi,x)\Delta
v^{s}_{D}(\xi,x_{s})-v^{s}_{D}(\xi,x_{s})\Delta v_{0,D}(\xi,x)\right]{\rm
d}\xi$ $\displaystyle=$ $\displaystyle\left(\int_{\partial
B_{\rho}^{+}}+\int_{\partial B_{\varepsilon}(x)}-\int_{\Gamma\cap
B_{\rho}}\right)\left[v_{0,D}(\xi,x)\frac{\partial
v_{D}^{s}(\xi,x_{s})}{\partial\nu(\xi)}-v_{D}^{s}(\xi,x_{s})\frac{\partial
v_{0,D}(\xi,x)}{\partial\nu(\xi)}\right]{\rm d}s(\xi)$ $\displaystyle=$
$\displaystyle I_{1}+I_{2}-I_{3}.$

By a similar argument with (2.15) and (2.21), we have

$\displaystyle\lim_{\rho\to\infty}I_{1}=0\qquad{\rm
and}\qquad\lim_{\varepsilon\to 0}I_{2}=-v_{D}^{s}(x,x_{s}).$

Hence, we obtain

$\displaystyle
v_{D}^{s}(x,x_{s})=\int_{\Gamma}\left[v_{D}^{s}(\xi,x_{s})\frac{\partial
v_{0,D}(\xi,x)}{\partial\nu(\xi)}-v_{0,D}(\xi,x)\frac{\partial
v_{D}^{s}(\xi,x_{s})}{\partial\nu(\xi)}\right]{\rm d}s(\xi),$

which implies

$\displaystyle
v_{D}^{\infty}(\hat{x},x_{s})=\gamma_{1}\int_{\Gamma}\left[v_{D}^{s}(\xi,x_{s})\frac{\partial
w_{0,D}(\xi,-\hat{x})}{\partial\nu(\xi)}-w_{0,D}(\xi,-\hat{x})\frac{\partial
v_{D}^{s}(\xi,x_{s})}{\partial\nu(\xi)}\right]{\rm d}s(\xi),$ (2.40)

where we use (2.35). Similarly, we have

$\displaystyle
w_{D}^{s}(x_{s},-\hat{x})=\int_{\Gamma}\left[w_{D}^{s}(\xi,-\hat{x})\frac{\partial
v_{0,D}(\xi,x_{s})}{\partial\nu(\xi)}-v_{0,D}(\xi,x_{s})\frac{\partial
w_{D}^{s}(\xi,-\hat{x})}{\partial\nu(\xi)}\right]{\rm d}s(\xi),$ (2.41)

Combining (2.38),(2.39),(2.40) and (2.41) leads to

$\displaystyle\gamma_{1}^{-1}v_{D}^{\infty}(\hat{x},x_{s})-w_{D}^{s}(x_{s},-\hat{x})$
$\displaystyle=\int_{\Gamma}\left[v_{D}(\xi,x_{s})\frac{\partial
w_{D}(\xi,-\hat{x})}{\partial\nu(\xi)}-w_{D}(\xi,-\hat{x})\frac{\partial
v_{D}(\xi,x_{s})}{\partial\nu(\xi)}\right]{\rm d}s(\xi)$ $\displaystyle=0,$

where the fact has been used that the total fields
$v_{D}(\xi,x_{s}):=v_{0,D}(\xi,x_{s})+v_{D}^{s}(\xi,x_{s})$ and
$w_{D}(\xi,-\hat{x}):=w_{0,\alpha}(\xi,-\hat{x})+w_{D}^{s}(\xi,-\hat{x})$
vanish on $\Gamma$. The proof is thus complete. ∎

With the above mixed reciprocity relation, we are in a position to present the
RTM method based on far-field measurements. To this end, we first introduce
some notations. Note that the wave fields $w_{\alpha}^{s}(x,d)$,
$w_{\alpha}^{\infty}(\hat{x},d)$, $w_{\alpha}(x,d)$,
$v_{\alpha}^{s}(x,x_{s})$, $v_{\alpha}^{\infty}(\hat{x},x_{s})$,
$v_{\alpha}(x,x_{s})$ depend on the rough surface $\Gamma$. For clarity, we
write $w_{\alpha}^{s}(x,d,\Gamma)$, $w_{\alpha}^{\infty}(\hat{x},d,\Gamma)$,
$w_{\alpha}(x,d,\Gamma)$, $v_{\alpha}^{s}(x,x_{s},\Gamma)$,
$v_{\alpha}^{\infty}(\hat{x},x_{s},\Gamma)$, $v_{\alpha}(x,x_{s},\Gamma)$ to
express explicitly the dependence of the wave fields on $\Gamma$.

######  Theorem 2.6.

For the indicator function $\widetilde{{\rm Ind}}_{\alpha}(z)$ with
$\alpha\in\\{D,N\\}$, we have the following limit identity

$\displaystyle\lim_{R_{s}\to\infty}\lim_{R_{r}\to\infty}\widetilde{{\rm
Ind}}_{\alpha}(z)=-\kappa_{1}^{2}|\gamma_{1}|^{2}{\rm
Im}\bigg{\\{}\gamma_{1}\int_{{\mathbb{S}_{+}}}\int_{{\mathbb{S}_{+}}}w_{\alpha}(z,-\hat{x}_{r},\Gamma_{R})w_{\alpha}(z,-\hat{x}_{s},\Gamma_{R})$
$\displaystyle\qquad\times\left[\overline{w_{\alpha}^{\infty}(\hat{x}_{s},-\hat{x}_{r},\Gamma)}-\overline{w_{\alpha}^{\infty}(\hat{x}_{s},-\hat{x}_{r},\Gamma_{R})}\right]{\rm
d}s(\hat{x}_{r}){\rm d}s(\hat{x}_{s})\bigg{\\}}:=\widehat{{\rm
Ind}}_{\alpha}(z).$ (2.42)

###### Proof.

For $\alpha=D$ and $x\in\\{x_{r},x_{s}\\}$, it follows from the Sommerfeld
radiation condition, the mixed reciprocity relation, and (2.35) that

$\displaystyle G_{D}(z,x)$ $\displaystyle=$ $\displaystyle
v_{0,D}(x,z)+v_{D}^{s}(x,z,\Gamma_{R})$ (2.43) $\displaystyle=$
$\displaystyle\frac{e^{{\rm
i}\kappa_{1}|x|}}{|x|^{\frac{1}{2}}}\left[v_{0,D}^{\infty}(\hat{x},z)+v_{D}^{\infty}(\hat{x},z,\Gamma_{R})+O\left(\frac{1}{|x|}\right)\right]$
$\displaystyle=$ $\displaystyle\gamma_{1}\frac{e^{{\rm
i}\kappa_{1}|x|}}{|x|^{\frac{1}{2}}}\left[w_{0,D}(z,-\hat{x})+w_{D}^{s}(z,-\hat{x},\Gamma_{R})+O\left(\frac{1}{|x|}\right)\right]$
$\displaystyle=$ $\displaystyle\gamma_{1}\frac{e^{{\rm
i}\kappa_{1}|x|}}{|x|^{\frac{1}{2}}}\left[w_{D}(z,-\hat{x},\Gamma_{R})+O\left(\frac{1}{|x|}\right)\right]$

where we use the reciprocity $G_{D}(z,x)=G_{D}(x,z)$ for $x,z\in\Omega_{1,R}$.
For the data $V_{D}(x_{r},x_{s})$, we have

$\displaystyle V_{D}(x_{r},x_{s})$ $\displaystyle=$ $\displaystyle
u_{D}^{s}(x_{r},x_{s},\Gamma)-G_{D}^{s}(x_{r},x_{s},\Gamma_{R})$ (2.44)
$\displaystyle=$ $\displaystyle
v_{D}^{s}(x_{r},x_{s},\Gamma)-v_{D}^{s}(x_{r},x_{s},\Gamma_{R})$
$\displaystyle=$ $\displaystyle\frac{e^{{\rm
i}\kappa_{1}|x_{r}|}}{|x_{r}|^{\frac{1}{2}}}\left[v_{D}^{\infty}(\hat{x}_{r},x_{s},\Gamma)-v_{D}^{\infty}(\hat{x}_{r},x_{s},\Gamma_{R})+O\left(\frac{1}{|x_{r}|}\right)\right]$
$\displaystyle=$ $\displaystyle\gamma_{1}\frac{e^{{\rm
i}\kappa_{1}|x_{r}|}}{|x_{r}|^{\frac{1}{2}}}\left[w_{D}^{s}(x_{s},-\hat{x}_{r},\Gamma)-w_{D}^{s}(x_{s},-\hat{x}_{r},\Gamma_{R})+O\left(\frac{1}{|x_{r}|}\right)\right].$

By (2.43) and (2.44), we obtain

$\displaystyle\lim_{R_{r}\to\infty}\int_{\Gamma_{r}}G_{D}(z,x_{r})\overline{V_{D}(x_{r},x_{s})}{\rm
d}s(x_{r})$
$\displaystyle=|\gamma_{1}|^{2}\lim_{R_{r}\to\infty}\int_{\Gamma_{r}}\frac{1}{|x_{r}|}\left[w_{D}(z,-\hat{x}_{r},\Gamma_{R})+O\left(\frac{1}{|x_{r}|}\right)\right]$
$\displaystyle\times\left[\overline{w_{D}^{s}(x_{s},-\hat{x}_{r},\Gamma)}-\overline{w_{D}^{s}(x_{s},-\hat{x}_{r},\Gamma_{R})}+O\left(\frac{1}{|x_{r}|}\right)\right]{\rm
d}s(x_{r})$
$\displaystyle=|\gamma_{1}|^{2}\int_{{\mathbb{S}}_{+}}w_{D}(z,-\hat{x}_{r},\Gamma_{R})\left[\overline{w_{D}^{s}(x_{s},-\hat{x}_{r},\Gamma)}-\overline{w_{D}^{s}(x_{s},-\hat{x}_{r},\Gamma_{R})}\right]{\rm
d}s(\hat{x}_{r})$ (2.45)

Since

$\displaystyle
w_{D}^{s}(x_{s},-\hat{x}_{r},\Gamma)-w_{D}^{s}(x_{s},-\hat{x}_{r},\Gamma_{R})$
$\displaystyle=\frac{e^{{\rm
i}\kappa_{1}|x_{s}|}}{|x_{s}|^{\frac{1}{2}}}\left[w_{D}^{\infty}(\hat{x}_{s},-\hat{x}_{r},\Gamma)-w_{D}^{\infty}(\hat{x}_{s},-\hat{x}_{r},\Gamma_{R})+O\left(\frac{1}{|x_{s}|}\right)\right],$

combining with (2.43) yields that

$\displaystyle\lim_{R_{s}\to\infty}\int_{\Gamma_{s}}G_{D}(z,x_{s})\left[\overline{w_{D}^{s}(x_{s},-\hat{x}_{r},\Gamma)}-\overline{w_{D}^{s}(x_{s},-\hat{x}_{r},\Gamma_{R})}\right]{\rm
d}s(x_{s})$
$\displaystyle=\gamma_{1}\lim_{R_{s}\to\infty}\int_{\Gamma_{s}}\frac{1}{|x_{s}|}\left[w_{D}(z,-\hat{x}_{s},\Gamma_{R})+O\left(\frac{1}{|x_{s}|}\right)\right]$
$\displaystyle\quad\times\left[\overline{w_{D}^{\infty}(\hat{x}_{s},-\hat{x}_{r},\Gamma)}-\overline{w_{D}^{\infty}(\hat{x}_{s},-\hat{x}_{r},\Gamma_{R})}+O\left(\frac{1}{|x_{s}|}\right)\right]{\rm
d}s(x_{s})$
$\displaystyle=\gamma_{1}\int_{{\mathbb{S}}_{+}}w_{D}(z,-\hat{x}_{s},\Gamma_{R})\left[\overline{w_{D}^{\infty}(\hat{x}_{s},-\hat{x}_{r},\Gamma)}-\overline{w_{D}^{\infty}(\hat{x}_{s},-\hat{x}_{r},\Gamma_{R})}\right]{\rm
d}s(\hat{x}_{s}).$ (2.46)

A direct application of (2.45) and (2.46) shows that the limit identity (3.26)
holds for $\alpha=D$. This completes the proof. ∎

##  3 The RTM for penetrable locally rough interfaces

The destination of this section is to develop the RTM method for penetrable,
locally rough interfaces. This section consists of two subsections. In the
first subsection, we will introduce the RTM method based on near-field
associated with point sources incidence, and in the second subsection we will
present a mixed reciprocity relation which leads to the RTM method based on
far-field data which corresponds to the plane wave incidence.

###  3.1 The near-filed reconstruction

As shown in Figure 2, let $\Gamma$ be the locally rough interface and $S$ be
the sampling domain which contains the local perturbation of $\Gamma$. We
choose a sufficient large $R$ and define $\Gamma_{R}$ by (2.1) so that the
sampling domain $S$ lies totally above $\Gamma_{R}$. We suppose that there are
$N_{s}$ point sources $x_{s}$ uniformly distributed on $\Gamma_{s}$ and
$N_{r}$ receivers $x_{r}$ uniformly distributed on $\Gamma_{r}$. Here,
$\Gamma_{s}$ and $\Gamma_{r}$ denote the circle with the origin as the center
and $R_{s}$, $R_{r}$ as the radius, respectively. We suppose a priori that
$R<R_{s}\leq R_{r}$.

Figure 2: The setting of RTM method for the penetrable case.

To establish the mathematic justification of the RTM method, we first
introduce the Green’s function $G_{P}$ associated with the two-dimensional
Helmholtz equation in a two-layered medium separated by $\Gamma_{R}$, which
satisfies

$\displaystyle\left\\{\begin{array}[]{lll}\Delta
G_{P}(x,x_{s})+\kappa_{P}^{2}(x)G_{P}(x,x_{s})=-\delta_{x_{s}}(x)&\textrm{in}\;\;{\mathbb{R}}^{2},\\\\[5.69054pt]
\lim\limits_{|x|\rightarrow\infty}|x|^{\frac{1}{2}}\left(\partial_{|x|}G_{P}(x,x_{s})-{\rm
i}\kappa_{P}(x)G_{P}(x,x_{s})\right)=0\end{array}\right.$ (3.3)

in the distributional sense and the Sommerfeld radiation condition uniformly
for all directions $\hat{x}\in\mathbb{S}$. Here,
$x_{s}\in{\mathbb{R}}^{2}\setminus\Gamma_{R}$ and the wave number
$\kappa_{P}(x)$ is defined by $\kappa_{P}(x):=\kappa_{1}$ in $\Omega_{1,R}$
and $\kappa_{P}(x):=\kappa_{2}$ in $\Omega_{2,R}$, with $\Omega_{1,R}$ and
$\Omega_{2,R}$ being the upper and lower half-space separated by $\Gamma_{R}$,
respectively. We refer to Theorem 2.1 and Theorem 2.2 in [43] for the well-
posedness of the background Green’s function $G_{P}(x,x_{s})$ for
$x_{s}\in{\mathbb{R}}^{2}\setminus\Gamma_{R}$.

Define

$\displaystyle V_{P}(x,x_{s}):=u(x,x_{s})-G_{P}(x,x_{s}),$

which, from (1.9) and (3.3), satisfies

$\displaystyle\left\\{\begin{array}[]{lll}\Delta
V_{P}(x,x_{s})+\kappa^{2}(x)V_{P}(x,x_{s})=g(x,x_{s})&\textrm{in}\;\;{\mathbb{R}}^{2},\\\\[8.53581pt]
\lim\limits_{|x|\rightarrow\infty}|x|^{\frac{1}{2}}\left(\partial_{|x|}V_{P}(x,x_{s})-{\rm
i}\kappa(x)V_{P}(x,x_{s})\right)=0.\end{array}\right.$ (3.6)

Here $g(x,x_{s})$ is a function with compact support and given by
$g(x,x_{s}):=\sigma G_{P}(x,x_{s})$ in $D_{R}$ and $g(x,x_{s})=0$ in
${\mathbb{R}}^{2}\setminus\overline{D_{R}}$, where
$\sigma:=\kappa_{1}^{2}-\kappa_{2}^{2}$ and
$D_{R}:=\Omega_{2}\cap\Omega_{1,R}$.

The main idea of the RTM algorithm is to break up the reconstruction of
$\Gamma$ into two parts: the first part is to back-propagate the complex
conjugated data $\overline{V_{P}(x_{r},x_{s})}$ and the second part is to
calculate the imaginary part of the cross-correlation of $G_{P}(z,x_{s})$ and
the back-propagation field. We summarize it in the following algorithm.

Algorithm 2 (RTM for penetrable locally rough interface): Given the data
$V_{P}(x_{r},x_{s})$ for $r=1,2,...,N_{r}$ and $s=1,2,...,N_{s}$.

  * •

Back-propagation: for $s=1,2,...,N_{s}$, solve the problem

$\displaystyle\left\\{\begin{aligned} &\Delta
W_{P}(x,x_{s})+\kappa_{P}^{2}(x)W_{P}(x,x_{s})=\frac{|\Gamma_{r}|}{N_{r}}\sum_{r=1}^{N_{r}}\overline{V_{P}(x_{r},x_{s})}\delta_{x_{r}}(x)\;\;\textrm{in}\;\;{\mathbb{R}}^{2},\\\
&\;\lim_{|x|\rightarrow\infty}|x|^{\frac{1}{2}}\left(\partial_{|x|}W_{P}(x,x_{s})-{\rm
i}\kappa_{P}(x)W_{P}(x,x_{s})\right)=0,\end{aligned}\right.$ (3.7)

to obtain the solution $W_{P}$.

  * •

Cross-correlation: for each sampling point $z\in S$, calculate the indicator
function

$\displaystyle{\rm Ind}_{P}(z)=\kappa(x_{r}){\rm
Im}\left\\{\frac{|\Gamma_{s}|}{N_{s}}\sum_{s=1}^{N_{s}}\kappa(x_{s})G_{P}(z,x_{s})W_{P}(z,x_{s})\right\\}$

and then plot the mapping ${\rm Ind}_{P}(z)$ against $z$.

Combination of (3.3) and the linearity shows that the solution of (3.7) can be
expressed by

$\displaystyle
W_{P}(x,x_{s})=-\frac{|\Gamma_{r}|}{N_{r}}\sum_{r=1}^{N_{r}}G_{P}(x,x_{r})\overline{V_{P}(x_{r},x_{s})}.$

Hence, we obtain

$\displaystyle{\rm Ind}_{P}(z)=-{\rm
Im}\left\\{\frac{|\Gamma_{s}|}{N_{s}}\frac{|\Gamma_{r}|}{N_{r}}\sum_{s=1}^{N_{s}}\sum_{r=1}^{N_{r}}\kappa(x_{r})\kappa(x_{s})G_{P}(z,x_{s})G_{P}(z,x_{r})\overline{V_{P}(x_{r},x_{s})}\right\\}\quad
z\in S,$

which is a discrete formula of the following continuous function

$\displaystyle\widetilde{{\rm Ind}}_{P}(z)=-{\rm
Im}\int_{\Gamma_{r}}\int_{\Gamma_{s}}\kappa(x_{r})\kappa(x_{s})G_{P}(z,x_{s})G_{P}(z,x_{r})\overline{V_{P}(x_{r},x_{s})}{\rm
d}s(x_{s}){\rm d}s(x_{r}),\quad z\in S.$

In the remaining part of this section, we restrict us to show that the
function $\widetilde{{\rm Ind}}_{P}(z)$ will have contrast at the rough
interface $\Gamma$ and decay away from $\Gamma$. To this end, we first
introduce the following modified Helmholtz-Kirchhoff identity. It can be shown
by a direct application of the Green’s theorem along with the continuity of
$G_{P}$ and its normal derivative across $\Gamma_{R}$, which is similar to the
proof of Lemma 2.1 and we omit it here.

######  Lemma 3.1.

Let $G_{P}$ be the background Green’s function defined by (3.3). Then for any
$x,z\in B_{R_{s}}\setminus\Gamma_{R}$, we have

$\displaystyle\int_{\Gamma_{p}}\left(\overline{G_{P}(\xi,x)}\frac{\partial
G_{P}(\xi,z)}{\partial\nu(\xi)}-\frac{\partial\overline{G_{P}(\xi,x)}}{\partial\nu(\xi)}G_{P}(\xi,z)\right){\rm
d}s(\xi)=2{\rm i}{\rm Im}G_{P}(x,z),$

where $\Gamma_{p}$ denotes $\Gamma_{s}$ or $\Gamma_{r}$.

With the above modified Helmholtz-Kirchhoff identity and the Sommerfeld
radiation condition, it is easy to obtain the following Lemma which plays an
important role in the analysis of $\widetilde{{\rm Ind}}_{P}(z)$.

######  Lemma 3.2.

For any $x,z\in S$, we have

$\displaystyle\int_{\Gamma_{s}}\kappa(\xi)\overline{G_{P}(x,\xi)}G_{P}(\xi,z){\rm
d}s(\xi)={\rm Im}G_{P}(x,z)+\zeta_{P,s}(x,z)$ (3.8)
$\displaystyle\int_{\Gamma_{r}}\kappa(\xi)\overline{G_{P}(x,\xi)}G_{P}(\xi,z){\rm
d}s(\xi)={\rm Im}G_{P}(x,z)+\zeta_{P,r}(x,z)$ (3.9)

where $|\zeta_{P,s}(x,z)|+|\nabla_{x}\zeta_{P,s}(x,z)|\leq
CR_{s}^{-\frac{1}{4}}$ and
$|\zeta_{P,r}(x,z)|+|\nabla_{x}\zeta_{P,r}(x,z)|\leq CR_{r}^{-\frac{1}{4}}$
uniformly for any $x,z\in S$.

###### Proof.

For any $x,z\in S$, it follows from Lemma 3.1 that

$\displaystyle 2{\rm i}{\rm Im}G_{P}(x,z)$ $\displaystyle=$
$\displaystyle\int_{\Gamma_{s}}\left(\overline{G_{P}(\xi,x)}\frac{\partial
G_{P}(\xi,z)}{\partial\nu(\xi)}-\frac{\partial\overline{G_{P}(\xi,x)}}{\partial\nu(\xi)}G_{P}(\xi,z)\right){\rm
d}s(\xi)$ $\displaystyle=$
$\displaystyle\int_{\Gamma_{s}}\Bigg{\\{}\overline{G_{P}(\xi,x)}\left[\frac{\partial
G_{P}(\xi,z)}{\partial\nu(\xi)}-{\rm i}\kappa(\xi)G_{P}(\xi,z)\right]$
$\displaystyle\qquad\;\;\;-G_{P}(\xi,z)\left[\frac{\overline{\partial
G_{P}(\xi,x)}}{\partial\nu(\xi)}+{\rm
i}\kappa(\xi)\overline{G_{P}(\xi,x)}\right]\Bigg{\\}}{\rm d}s(\xi)$
$\displaystyle\qquad\;\;\;+2{\rm
i}\int_{\Gamma_{s}}\kappa(\xi)\overline{G_{P}(\xi,x)}G_{P}(\xi,z){\rm
d}s(\xi).$

Thus, a direct application of the reciprocity $G_{P}(\xi,x)=G_{P}(x,\xi)$ for
$\xi\in\partial B_{r}$ and $x\in S$ yields

$\displaystyle\int_{\Gamma_{s}}\kappa(\xi)\overline{G_{P}(x,\xi)}G_{P}(\xi,z){\rm
d}s(\xi)={\rm Im}G_{P}(x,z)+\zeta_{P,s}(x,z)\quad{\rm for}\;\;{\forall x,z\in
S},$

with

$\displaystyle\zeta_{P,s}(x,z)$ $\displaystyle=$ $\displaystyle\frac{\rm
i}{2}\int_{\Gamma_{s}}\Bigg{\\{}\overline{G_{P}(\xi,x)}\left[\frac{\partial
G_{P}(\xi,z)}{\partial\nu(\xi)}-{\rm i}\kappa(\xi)G_{P}(\xi,z)\right]$
$\displaystyle\qquad\quad-G_{P}(\xi,z)\left[\frac{\overline{\partial
G_{P}(\xi,x)}}{\partial\nu(\xi)}+{\rm
i}\kappa(\xi)\overline{G_{P}(\xi,x)}\right]\Bigg{\\}}{\rm d}s(\xi).$

Thus, the inequality (3.8) holds. By a similar argument with Theorem 1 and
Theorems 9-11 in [35], we have

$\displaystyle G_{P}(\xi,y)=O(|\xi|^{-\frac{1}{2}}),\qquad\frac{\partial
G_{P}(\xi,y)}{\partial\nu(\xi)}-{\rm
i}\kappa(\xi)G_{P}(\xi,y)=O(|\xi|^{-\frac{3}{4}})$

for $y\in\\{x,z\\}$, $x,z\in S$, which implies that

$\displaystyle|\zeta_{P,s}(x,z)|\leq CR_{s}^{-\frac{1}{4}}$

uniformly for $x,z\in S$. Since

$\displaystyle\frac{\partial G_{P}(\xi,x)}{\partial
x_{j}}=O(|\xi|^{-\frac{1}{2}}),\qquad\frac{\partial}{\partial
x_{j}}\left[\frac{\partial G_{P}(\xi,x)}{\partial\nu(\xi)}-{\rm
i}\kappa(\xi)G_{P}(\xi,x)\right]=O(|\xi|^{-\frac{3}{4}})$

for $j=1,2$ and $x\in S$, it follows that

$\displaystyle|\nabla_{x}\zeta_{P,s}(x,z)|\leq CR_{s}^{-\frac{1}{4}}$

uniformly for $x,z\in S$. Thus, we conclude that
$|\zeta_{P,s}(x,z)|+|\nabla_{x}\zeta_{P,s}(x,z)|\leq CR_{s}^{-\frac{1}{4}}$
hold. A similar argument shows that (3.9) holds. The proof is complete. ∎

To analyze the indicator function $\widetilde{{\rm Ind}}_{P}(z)$, we also need
the following Green’s formula which is shown by Theorem 2.1 in [33].

######  Lemma 3.3.

Let $V_{P}(x,x_{s})$ be the solution of Problem (3.6). Then we have

$\displaystyle V_{P}(x,x_{s})=-\sigma\int_{D_{R}}G_{P}(x,\xi)u(\xi,x_{s}){\rm
d}\xi,\quad{\rm for}\;\;x\in{\mathbb{R}}^{2}.$

Now we are in position to present the main result of this section.

######  Theorem 3.4.

For any $z\in S$, let $\psi_{P}(\xi,z)$ be the solution of

$\displaystyle\left\\{\begin{aligned}
&\Delta\psi_{P}(\xi,z)+\kappa^{2}(\xi)\psi_{P}(\xi,z)=-\sigma\chi_{D_{R}}(\xi){\rm
Im}G_{P}(\xi,z)\qquad{\rm in}\;\;{\mathbb{R}}^{2},\\\
&\;\lim_{|\xi|\rightarrow\infty}|\xi|^{\frac{1}{2}}\left(\partial_{|\xi|}\psi_{P}(\xi,z)-{\rm
i}\kappa(\xi)\psi_{P}(\xi,z)\right)=0,\end{aligned}\right.$ (3.10)

where $\chi_{D_{R}}$ is the characterization function of the domain $D_{R}$
given by $\chi_{D_{R}}=1$ in $D_{R}$ and vanishes outside $D_{R}$, and
$\psi_{P}^{\infty}(\hat{\xi},z)$ be the corresponding far field pattern. Then
we have

$\displaystyle\widetilde{{\rm
Ind}}_{P}(z)=\kappa_{1}\int_{{\mathbb{S}}_{+}}|\psi_{P}^{\infty}(\hat{\xi},z)|^{2}{\rm
d}s(\hat{\xi})+\kappa_{2}\int_{{\mathbb{S}}_{-}}|\psi_{P}^{\infty}(\hat{\xi},z)|^{2}{\rm
d}s(\hat{\xi})+\eta_{P}(z)\quad{\forall z\in S}$

where $\|\eta_{P}(z)\|_{L^{\infty}(S)}\leq
C(R_{s}^{-\frac{1}{4}}+R_{r}^{-\frac{1}{4}})$ with some constant $C$ depending
on $R$.

###### Proof.

Note that

$\displaystyle\widetilde{{\rm Ind}}_{P}(z)$ $\displaystyle=$
$\displaystyle-{\rm
Im}\int_{\Gamma_{r}}\int_{\Gamma_{s}}\kappa(x_{s})\kappa(x_{r})G_{P}(z,x_{s})G_{P}(z,x_{r})\overline{V_{P}(x_{r},x_{s})}{\rm
d}s(x_{s}){\rm d}s(x_{r})$ $\displaystyle=$ $\displaystyle-{\rm
Im}\int_{\Gamma_{s}}\kappa(x_{s})G_{P}(z,x_{s})\widetilde{W}_{P}(z,x_{s}){\rm
d}s(x_{s})$

where

$\displaystyle\widetilde{W}_{P}(z,x_{s}):=\int_{\Gamma_{r}}\kappa(x_{r})G_{P}(z,x_{r})\overline{V_{P}(x_{r},x_{s})}{\rm
d}s(x_{r}).$

With the help of Lemma 3.3 and Lemma 3.2, we can rewrite
$\widetilde{W}_{P}(z,x_{s})$ as

$\displaystyle\widetilde{W}_{P}(z,x_{s})$ $\displaystyle=$
$\displaystyle-\sigma\int_{D_{R}}\left[\int_{\Gamma_{r}}\kappa(x_{r})G_{P}(z,x_{r})\overline{G_{P}(x_{r},\xi)}{\rm
d}s(x_{r})\right]\overline{u(\xi,x_{s})}{\rm d}\xi$ $\displaystyle=$
$\displaystyle-\sigma\int_{D_{R}}\left[{\rm
Im}G_{P}(\xi,z)+\zeta_{P,r}(\xi,z)\right]\overline{u(\xi,x_{s})}{\rm d}\xi$

which leads to

$\displaystyle\widetilde{{\rm Ind}}_{P}(z)=\sigma{\rm
Im}\int_{D_{R}}\left[{\rm
Im}G_{P}(\xi,z)+\zeta_{P,r}(\xi,z)\right]\phi_{P}(\xi,z){\rm d}\xi,$

with

$\displaystyle\phi_{P}(\xi,z):=\int_{\Gamma_{s}}\kappa(x_{s})G_{P}(z,x_{s})\overline{u(\xi,x_{s})}{\rm
d}s(x_{s}).$

Due to the following Lippmann-Schwinger integral equation (cf. [33, Theorem
2.1])

$\displaystyle u(\xi,x_{s})+\sigma\int_{D_{R}}G_{P}(\xi,y)u(y,x_{s}){\rm
d}y=G_{P}(\xi,x_{s})\quad{\rm for}\;\;\xi\in{\mathbb{R}}^{2},$

we arrive at

$\displaystyle\overline{\phi_{P}(\xi,z)}=\int_{\Gamma_{s}}\kappa(x_{s})\overline{G_{P}(z,x_{s})}G_{P}(\xi,x_{s}){\rm
d}s(x_{s})-\sigma\int_{D_{R}}G_{P}(\xi,y)\overline{\phi_{P}(y,z)}{\rm d}y$
$\displaystyle={\rm
Im}G_{P}(\xi,z)+\zeta_{P,s}(\xi,z)-\sigma\int_{D_{R}}G_{P}(\xi,y)\overline{\phi_{P}(y,z)}{\rm
d}y.$

Let $\theta(\xi,z)=\overline{\phi_{P}(\xi,z)}-\left[{\rm
Im}G_{P}(\xi,z)+\zeta_{P,s}(\xi,z)\right]$, then

$\displaystyle\theta(\xi,z)=-\sigma\int_{D_{R}}G_{P}(\xi,y)\left[\theta(y,z)+{\rm
Im}G_{P}(y,z)+\zeta_{P,s}(y,z)\right]{\rm d}y.$

Hence, we conclude that $\theta(\xi,z)$ satisfies the Sommerfeld radiation
condition and

$\displaystyle\left\\{\begin{aligned}
&\Delta\theta(\xi,z)+\kappa^{2}(\xi)\theta(\xi,z)=0\qquad&&{\rm
in}\;\;{\mathbb{R}}^{2}\setminus\overline{D_{R}},\\\
&\Delta\theta(\xi,z)+\kappa_{1}^{2}\theta(\xi,z)=\sigma\left[\theta(\xi,z)+{\rm
Im}G_{P}(\xi,z)+\zeta_{P,s}(\xi,z)\right]&&{\rm
in}\;\;D_{R},\end{aligned}\right.$

which is equivalent to

$\displaystyle\left\\{\begin{aligned}
&\Delta\theta(\xi,z)+\kappa^{2}(\xi)\theta(\xi,z)=\sigma\chi_{D_{R}}(\xi)\left[{\rm
Im}G_{P}(\xi,z)+\zeta_{P,s}(\xi,z)\right]\;\;{\rm in}\;\;{\mathbb{R}}^{2},\\\
&\;\lim_{|\xi|\rightarrow\infty}|\xi|^{\frac{1}{2}}\left(\partial_{|\xi|}\theta(\xi,z)-{\rm
i}\kappa(\xi)\theta(\xi,z)\right)=0,\end{aligned}\right.$ (3.11)

Let $\psi_{P}(\xi,z)$ and $\varphi_{P}(\xi,z)$ solve the same scattering
problem (3.11) expect that the right hand term are replaced by
$\sigma\chi_{D_{R}}(\xi){\rm Im}G_{P}(\xi,z)$ and
$\sigma\chi_{D_{R}}(\xi)\zeta_{P,s}(\xi,z)$. Then by the linearity we have

$\displaystyle\theta(\xi,z)=\psi_{P}(\xi,z)+\varphi_{P}(\xi,z)$

which yields

$\displaystyle\phi_{P}(\xi,z)=\overline{\psi_{P}(\xi,z)}+\overline{\varphi_{P}(\xi,z)}+{\rm
Im}G_{P}(\xi,z)+\overline{\zeta_{P,s}(\xi,z)}.$

Hence,

$\displaystyle\widetilde{{\rm Ind}}_{P}(z)$ $\displaystyle=$
$\displaystyle\sigma{\rm Im}\int_{D_{R}}\left[{\rm
Im}G_{P}(\xi,z)+\zeta_{P,r}(\xi,z)\right]$
$\displaystyle\times\left[\overline{\psi_{P}(\xi,z)}+\overline{\varphi_{P}(\xi,z)}+{\rm
Im}G_{P}(\xi,z)+\overline{\zeta_{P,s}(\xi,z)}\right]{\rm d}\xi$
$\displaystyle=$ $\displaystyle{\rm
Im}\int_{D_{R}}\left[\Delta\psi_{P}(\xi,z)+\kappa_{2}^{2}\psi_{P}(\xi,z)\right]\overline{\psi_{P}(\xi,z)}{\rm
d}\xi+\eta_{P}(z)$ $\displaystyle=$ $\displaystyle{\rm Im}\int_{\partial
D_{R}}\frac{\partial\psi_{P}(\xi,z)}{\partial\nu(\xi)}\overline{\psi_{P}(\xi,z)}{\rm
d}s(\xi)+\eta_{P}(z)$ $\displaystyle=$
$\displaystyle\kappa_{1}\int_{{\mathbb{S}}_{+}}|\psi_{P}^{\infty}(\hat{\xi},z)|^{2}{\rm
d}s(\hat{\xi})+\kappa_{2}\int_{{\mathbb{S}}_{-}}|\psi_{P}^{\infty}(\hat{\xi},z)|^{2}{\rm
d}s(\hat{\xi})+\eta_{P}(z)$

where we use the Green’s theorem and the Sommerfeld radiation condition in the
last step, and $\eta_{P}(z)$ is defined by

$\displaystyle\eta_{P}(z)=\sigma{\rm Im}\int_{D_{R}}$
$\displaystyle\bigg{[}\zeta_{P,r}(\xi,z)\overline{\psi_{P}(\xi,z)}+\zeta_{P,r}(\xi,z)\overline{\varphi_{P}(\xi,z)}$
$\displaystyle+\zeta_{P,r}(\xi,z){\rm
Im}G_{P}(\xi,z)+\zeta_{P,r}(\xi,z)\overline{\zeta_{P,s}(\xi,z)}$
$\displaystyle+{\rm Im}G_{P}(\xi,z)\overline{\varphi_{P}(\xi,z)}+{\rm
Im}G_{P}(\xi,z)\overline{\zeta_{P,s}(\xi,z)}\bigg{]}{\rm d}\xi.$ (3.12)

Now we are in position to show $\|\eta_{P}(z)\|_{L^{\infty}(S)}\leq
C(R_{s}^{-\frac{1}{4}}+R_{r}^{-\frac{1}{4}})$ with $C$ depending on $R$.
Recall that $\psi_{P}(\xi,z)$ and $\varphi_{P}(\xi,z)$ solve Problem (3.11)
expect that the right hand term are replaced by $\sigma\chi_{D_{R}}(\xi){\rm
Im}G_{P}(\xi,z)$ and $\sigma\chi_{D_{R}}(\xi)\zeta_{P,s}(\xi,z)$, thus we can
write $\psi_{P}(\xi,z)$ and $\varphi_{P}(\xi,z)$ in the following form

$\displaystyle\psi_{P}(\xi,z)=-\sigma\int_{D_{R}}G_{\Gamma}(\xi,y){\rm
Im}G_{P}(y,z){\rm d}y\qquad\;{\rm for}\;\xi\in{\mathbb{R}}^{2},$ (3.13)
$\displaystyle\varphi_{P}(\xi,z)=-\sigma\int_{D_{R}}G_{\Gamma}(\xi,y)\zeta_{P,s}(y,z){\rm
d}y\qquad\qquad{\rm for}\;\xi\in{\mathbb{R}}^{2}.$ (3.14)

Here $G_{\Gamma}(\xi,y)$ is the Green’s function associated with the two-
dimensional Helmholtz equation in a two-layered medium separated by $\Gamma$,
which satisfies

$\displaystyle\left\\{\begin{array}[]{lll}\Delta_{\xi}G_{\Gamma}(\xi,y)+\kappa^{2}(\xi)G_{\Gamma}(\xi,y)=-\delta_{y}(\xi)&\textrm{in}\;\;{\mathbb{R}}^{2},\\\\[5.69054pt]
\lim\limits_{|\xi|\rightarrow\infty}|\xi|^{\frac{1}{2}}\left(\partial_{|\xi|}G_{\Gamma}(\xi,y)-{\rm
i}\kappa(\xi)G_{\Gamma}(\xi,y)\right)=0\end{array}\right.$ (3.17)

in the distributional sense and the Sommerfeld radiation condition uniformly
for all directions $\hat{\xi}\in\mathbb{S}$. We refer to Theorem 3.2 and
Theorem 3.3 in [43] for the well-posedness of Problem (3.17). It follows from
(3.13) and (3.14) that

$\displaystyle\|\psi_{P}(\cdot,z)\|_{H^{2}(D_{R})}\lesssim\|{\rm
Im}G_{P}(\cdot,z)\|_{L^{2}(D_{R})}\leq C$ (3.18)
$\displaystyle\|\varphi_{P}(\cdot,z)\|_{H^{2}(D_{R})}\lesssim\|\zeta_{P,s}(\cdot,z)\|_{L^{2}(D_{R})}\leq
CR_{s}^{-\frac{1}{4}}$ (3.19)

where we use (3.9) and $C$ depends on $R$. A direct application of the
smoothness of ${\rm Im}G_{P}(\cdot,z)$, (3.9), (3.12), (3.18), and (3.20), we
can obtain

$\displaystyle\|\eta_{P}(z)\|_{L^{\infty}(S)}\leq
C(R_{s}^{-\frac{1}{4}}+R_{r}^{-\frac{1}{4}})$

with $C$ depending on $R$. The proof is completed. ∎

###  3.2 The far-filed reconstruction

In this subsection, we present the RTM method based on far-field data to
reconstruct the penetrable, locally rough surface. It requires to develop a
mixed reciprocity relation. Throughout this subsection, for simplicity, we
restrict ourselves to the case $\kappa_{1}>\kappa_{2}$. The case
$\kappa_{1}<\kappa_{2}$ can be dealt in a similar manner. For the case
$\kappa_{1}>\kappa_{2}$, let $n:=\kappa_{2}/\kappa_{1}$ and
$\theta_{c}:=\arccos(n)\in(0,\pi)$ be the critical incident angle and
$d:=(\cos(\theta),\sin(\theta))^{\rm\top}$ be the incident direction with
$\theta$ being the incident angle. Denote by
$d^{r}:=(\cos(\theta),-\sin(\theta))^{\rm\top}$ the reflected direction and
denote by $d^{t}$ the transmitted direction which is defined by

$d^{t}=\left\\{\begin{array}[]{l}(\cos(\varphi),\sin(\varphi))^{\rm\top}\qquad\qquad\qquad\textrm{for}\;\;\;\theta\in(0,\pi)\cup(\pi+\theta_{c},2\pi-\theta_{c}),\\\\[2.84526pt]
n^{-1}(\cos(\theta),{\rm
i}p)^{\rm\top}\qquad\qquad\qquad\;\textrm{for}\;\;\;\theta\in(\pi,\pi+\theta_{c})\cup(2\pi-\theta_{c},2\pi),\end{array}\right.$

where $\varphi:=2\pi-\arccos(\kappa_{1}\cos(\theta)/\kappa_{2})$ for
$\theta\in(\pi+\theta_{c},2\pi-\theta_{c})$,
$\varphi:=\arccos(\kappa_{2}\cos(\theta)/\kappa_{1})$ for $\theta\in(0,\pi)$,
and $p:=\sqrt{\cos^{2}(\theta)-n^{2}}$. Let $w_{0}(x,d)$ be the total field of
the scattering of plane waves $w^{i}(x,d)$ from the infinite plane
$\Gamma_{0}$, it follows from the Fresnel formula and [35] that the field
$w_{0}(x,d)$ is given by

$w_{0}(x,d)=\left\\{\begin{array}[]{l}e^{{\rm i}\kappa_{1}x\cdot
d}+R(\kappa_{1},\kappa_{2},\theta)e^{{\rm i}\kappa_{1}x\cdot
d^{r}}\qquad\qquad\;\textrm{for}\;\;\;x\in{\mathbb{R}}^{2}_{+},d\in{\mathbb{S}}_{-},\\\\[2.84526pt]
T(\kappa_{1},\kappa_{2},\theta)e^{{\rm i}\kappa_{2}x\cdot
d^{t}}\qquad\qquad\;\qquad\qquad\textrm{for}\;\;\;x\in{\mathbb{R}}^{2}_{-},d\in{\mathbb{S}}_{-},\\\\[2.84526pt]
T(\kappa_{2},\kappa_{1},\theta)e^{{\rm i}\kappa_{1}x\cdot
d^{t}}\qquad\qquad\;\qquad\qquad\textrm{for}\;\;\;x\in{\mathbb{R}}^{2}_{+},d\in{\mathbb{S}}_{+},\\\\[2.84526pt]
e^{{\rm i}\kappa_{2}x\cdot d}+R(\kappa_{2},\kappa_{1},\theta)e^{{\rm
i}\kappa_{2}x\cdot
d^{r}}\qquad\qquad\;\textrm{for}\;\;\;x\in{\mathbb{R}}^{2}_{-},d\in{\mathbb{S}}_{+},\end{array}\right.$
(3.20)

which satisfies
$w_{0}(x,d)|_{+}-w_{0}(x,d)|_{-}=\partial_{\nu}w_{0}(x,d)|_{+}-\partial_{\nu}w_{0}(x,d)|_{-}=0$
on $\Gamma_{0}$. Here, the coefficients $R$ and $T$ are defined by

$R(\lambda,\mu,\theta):=\left\\{\begin{aligned}
\frac{\lambda\sin(\theta)-\mu\sin(\varphi)}{\lambda\sin(\theta)+\mu\sin(\varphi)}\qquad\qquad\qquad\;\;&\textrm{for}\;\;\theta\in(0,\pi)\cup(\pi+\theta_{c},2\pi-\theta_{c}),\\\\[5.69054pt]
\frac{{\rm i}\sin(\theta)+p}{{\rm
i}\sin(\theta)-p}\qquad\qquad\qquad\qquad\;\;\;\;\;\;\;&\textrm{for}\;\;\theta\in(\pi,\pi+\theta_{c})\cup(2\pi-\theta_{c},2\pi),\end{aligned}\right.$

and $T(\lambda,\mu,\theta):=R(\lambda,\mu,\theta)+1$. Then the scattering of
$w_{0}(x,d)$ by the locally rough surface $\Gamma$ can be modelled by

$\left\\{\begin{aligned} &\Delta
w^{s}(x,d)+\kappa^{2}w^{s}(x,d)=g(x,d)\qquad\qquad\qquad\;\textrm{in}\;\;{\mathbb{R}}^{2},\\\
&\lim_{|x|\rightarrow\infty}|x|^{\frac{1}{2}}\left(\partial_{|x|}w^{s}(x,d)-{\rm
i}\kappa w^{s}(x,d)\right)=0.\quad\end{aligned}\right.$

Where $g(x,d):=\sigma(\chi_{1}-\chi_{2})w_{0}(x,d)$ with $\chi_{j}(j=1,2)$
being the characterization function of the domain $D_{j}$ given by
$\chi_{j}=1$ in $D_{j}$ and vanishes outside $D_{j}$, the domain $D_{j}$ is
defined by $D_{1}:={\mathbb{R}}^{2}_{+}\cap\Omega_{2}$ and
$D_{2}:={\mathbb{R}}^{2}_{-}\cap\Omega_{1}$, $w^{s}$ denotes the scattered
field and the Sommerfield radiation condition holds uniformly for all
directions $\hat{x}\in{\mathbb{S}}$.

Let $v_{0}(x,x_{s})$ be the background Green’s function in a two-layered
medium seperated by $\Gamma_{0}$, which solves

$\left\\{\begin{aligned} &\Delta
v_{0}(x,x_{s})+\kappa_{0}^{2}v_{0}(x,x_{s})=-\delta_{x_{s}}(x)\qquad\qquad\qquad\;\textrm{in}\;\;{\mathbb{R}}^{2},\\\
&\lim_{|x|\rightarrow\infty}|x|^{\frac{1}{2}}\left(\partial_{|x|}v_{0}(x,x_{s})-{\rm
i}\kappa_{0}v_{0}(x,x_{s})\right)=0,\quad\end{aligned}\right.$

in the distributional sense with the Sommerfeld radiation condition uniformly
for all $\hat{x}\in{\mathbb{S}}$. Here, the wavenumber $\kappa_{0}$ is defined
by $\kappa_{0}:=\kappa_{1}$ in ${\mathbb{R}}^{2}_{+}$ and
$\kappa_{0}:=\kappa_{2}$ in ${\mathbb{R}}^{2}_{-}$. Denote by
$v_{0}^{\infty}(\hat{x},x_{s})$ the far-field of $v_{0}(x,x_{s})$, observing
from the formulation of $v_{0}^{\infty}(\hat{x},x_{s})$ in Proposition 2.1 in
[2] and Theorem1, Theorem 10 in [35], it is easily seen that

$v_{0}^{\infty}(\hat{x},x_{s})=\gamma(\hat{x})w_{0}(x_{s},-\hat{x})$ (3.21)

where $\gamma(\hat{x})=\gamma_{1}$ for $\hat{x}\in{\mathbb{S}}_{+}$ and
$\gamma(\hat{x})=\gamma_{2}:=\frac{e^{\frac{\pi}{4}{\rm
i}}}{\sqrt{8\kappa_{2}\pi}}$ for $\hat{x}\in{\mathbb{S}}_{-}$. Define
$v^{s}(x,x_{s}):=u(x,x_{s})-v_{0}(x,x_{s})$ where $u(x,x_{s})$ is the total
field of Problem (1.9), then we have $v(x,x_{s})$ solves

$\left\\{\begin{aligned} &\Delta
v^{s}(x,x_{s})+\kappa^{2}v^{s}(x,x_{s})=h(x,x_{s})\qquad\qquad\qquad\;\textrm{in}\;\;{\mathbb{R}}^{2},\\\
&\lim_{|x|\rightarrow\infty}|x|^{\frac{1}{2}}\left(\partial_{|x|}v^{s}(x,x_{s})-{\rm
i}\kappa v^{s}(x,x_{s})\right)=0,\quad\end{aligned}\right.$

with $h(x,x_{s})=\sigma(\chi_{1}-\chi_{2})v_{0}(x,x_{s})$, where the
Sommerfeld radiation condition holds uniformly for all
$\hat{x}\in{\mathbb{S}}$.

######  Theorem 3.5.

For acoustic scattering of plane waves $w_{0}(\cdot,-\hat{x})$ and point
sources $v_{0}(\cdot,x_{s})$ from a penetrable, locally rough surface $\Gamma$
we have

$v^{\infty}(\hat{x},x_{s})=\gamma(\hat{x})w^{s}(x_{s},-\hat{x})$

for all
$x,x_{s}\in(\Omega_{1}\cup\Omega_{2})\setminus(\overline{D_{1}}\cup\overline{D_{2}})$.

###### Proof.

We restrict ourselves to the proof of the case
$x,x_{s}\in\Omega_{1}\cap{\mathbb{R}}^{2}_{+}$ and the proof can be easily
extended to other cases. We choose a sufficient large $\rho>0$ and a
sufficient small $\varepsilon>0$ such that $B_{\varepsilon}(x)\subset
D^{+}_{\rho}:=B_{\rho}\cap\Omega_{1}$. Applying the Green’s formula to
$v^{s}(\xi,x_{s})$ and $v_{0}(\xi,x)$ in the domain
$D_{\rho}^{+}\setminus\overline{B_{\varepsilon}(x)}$ and in the domain
$D_{\rho}^{-}:=B_{\rho}\cap\Omega_{2}$ gives that

$v^{s}(x,x_{s})=-\sigma\left[\int_{D_{1}}v_{0}(x,\xi)u(\xi,x_{s}){\rm
d}\xi-\int_{D_{2}}v_{0}(x,\xi)u(\xi,x_{s}){\rm d}\xi\right]$ (3.22)

where we use the continuity of $v_{0}(\xi,x)$, $\partial_{\nu}v_{0}(\xi,x)$,
$v^{s}(\xi,x_{s})$, $\partial_{\nu}v^{s}(\xi,x_{s})$ across $\Gamma$, the
reciprocity $v_{0}(\xi,x)=v_{0}(x,\xi)$, and the Sommerfeld radiation
condition. It follows from (3.21) and (3.22) that the far-field
$v^{\infty}(\hat{x},x_{s})$ is given by

$\displaystyle
v^{\infty}(\hat{x},x_{s})=-\gamma_{1}\sigma\left[\int_{D_{1}}w_{0}(\xi,-\hat{x})u(\xi,x_{s}){\rm
d}\xi-\int_{D_{2}}w_{0}(\xi,-\hat{x})u(\xi,x_{s}){\rm d}\xi\right].$ (3.23)

A similar argument with (3.22) implies that

$w^{s}(x_{s},-\hat{x})=-\sigma\left[\int_{D_{1}}v_{0}(\xi,x_{s})w(\xi,-\hat{x}){\rm
d}\xi-\int_{D_{2}}v_{0}(\xi,x_{s})w(\xi,-\hat{x}){\rm d}\xi\right].$ (3.24)

Noting that $w^{s}$, $v$ satisfy the Sommerfeld radiation condition, and
$w^{s}$, $\partial_{\nu}w^{s}$, $v$, $\partial_{\nu}v$ are continuous across
$\Gamma$, using the Green’s formula for $w^{s}$ and $v$ in the domain
$D_{\rho}^{+}$ and in the domain $D_{\rho}^{-}$ yields that

$\displaystyle
0=-\sigma\left[\int_{D_{1}}w^{s}(\xi,-\hat{x})v_{0}(\xi,x_{s}){\rm
d}\xi-\int_{D_{2}}w^{s}(\xi,-\hat{x})v_{0}(\xi,x_{s}){\rm d}\xi\right]$
$\displaystyle+\sigma\left[\int_{D_{1}}w_{0}(\xi,-\hat{x})v^{s}(\xi,x_{s}){\rm
d}\xi-\int_{D_{2}}w_{0}(\xi,-\hat{x})v^{s}(\xi,x_{s}){\rm d}\xi\right].$
(3.25)

The difference between (3.24) and (3.25) yields that

$w^{s}(x_{s},-\hat{x})=-\sigma\left[\int_{D_{1}}w_{0}(\xi,-\hat{x})u(\xi,x_{s}){\rm
d}\xi-\int_{D_{2}}w_{0}(\xi,-\hat{x})u(\xi,x_{s}){\rm d}\xi\right].$

Compared with (3.23), we conclude that
$v^{\infty}(\hat{x},x_{s})=\gamma(\hat{x})w^{s}(x_{s},-\hat{x})$ for
$x,x_{s}\in\Omega_{1}\cap{\mathbb{R}}^{2}_{+}$. The proof is finished. ∎

With the above mixed reciprocity relation, we can establish the main result of
this subsection in the following theorem. Its proof is similar to Theorem 2.6
and we omit it here.

######  Theorem 3.6.

For the indicator function $\widetilde{{\rm Ind}}_{P}(z)$, we have the
following limit identity

$\displaystyle\lim_{R_{s}\to\infty}\lim_{R_{r}\to\infty}\widetilde{\rm
Ind}_{P}(z)=-{\rm
Im}\int_{{\mathbb{S}}}\int_{{\mathbb{S}}}\kappa(\hat{x}_{r})\kappa(\hat{x}_{s})|\gamma(\hat{x}_{r})|^{2}\gamma(\hat{x}_{s})w(z,-\hat{x}_{r},\Gamma_{R})w(z,-\hat{x}_{s},\Gamma_{R})$
$\displaystyle\qquad\times\left[\overline{w^{\infty}(\hat{x}_{s},-\hat{x}_{r},\Gamma)}-\overline{w^{\infty}(\hat{x}_{s},-\hat{x}_{r},\Gamma_{R})}\right]{\rm
d}s(\hat{x}_{r}){\rm d}s(\hat{x}_{s}):=\widehat{\rm Ind}_{P}(z).$ (3.26)

##  4 Numerical experiments

In this section, we first give an analysis of the indicator function
$\widetilde{\rm Ind}_{\alpha}(z)$ with $\alpha=D,N,P$ and then present several
numerical experiments to demonstrate the effectiveness of the RTM method.

According to Theorem 2.4 and Theorem 3.4, it is easy to see that the behavior
of the indicator function $\widetilde{\rm Ind}_{\alpha}(z)$ depends on
$\psi_{\alpha}(\xi,z)$ when the source radius $R_{s}$ and measurement radius
$R_{r}$ are large enough, where $\alpha=D,N,P$. Notice that the function
$\psi_{\alpha}(\xi,z)$ satisfies Problem (2.22) and Problem (3.10) with
boundary data $-{\rm Im}G_{D}(\xi,z)$, $-\partial_{\nu}{\rm Im}G_{N}(\xi,z)$,
and $-\sigma\chi_{D_{R}}(\xi){\rm Im}G_{P}(\xi,z)$, respectively. Observe that

$\displaystyle{\rm Im}G_{D}(\xi,z)=\frac{1}{4}J_{0}(\kappa_{1}|\xi-z|)+{\rm
Im}G_{D}^{s}(\xi,z)$ $\displaystyle\partial_{x_{j}}{\rm
Im}G_{N}(\xi,z)=\frac{\kappa}{4}J^{\prime}_{0}(\kappa_{1}|\xi-z|)\frac{\xi_{j}-z_{j}}{|\xi-z|}+\partial_{x_{j}}{\rm
Im}G^{s}_{N}(\xi,z)\quad\;{\rm for}\;\;j=1,2$ $\displaystyle{\rm
Im}G_{P}(\xi,z)=\frac{1}{4}J_{0}(\kappa_{1}|\xi-z|)+{\rm
Im}G_{P}^{s}(\xi,z)\qquad\qquad\qquad\quad\;\;{\rm
for}\;\;\xi,z\in\Omega_{1,R}$

where $G_{\alpha}^{s}(\xi,z)$ $(\alpha=D,N,P)$ denotes the corresponding
scattered fields associated with $\Gamma_{R}$. It is shown numerically that
$G_{\alpha}^{s}(\xi,z)$ can be sufficiently small for $\xi,z\in S$ when $R$ is
large enough, see [23, 33] for details. It is shown in (a) and (d) of Figure 3
that $J_{0}(\kappa_{1}|\xi-z|)$ achieves a maximum at $\xi=z$, which implies
that $J^{\prime}_{0}(\kappa_{1}|\xi-z|)=0$ when $\xi=z$. Hence, we can obtain
that the functions ${\rm Im}G_{D}(\xi,z)$ and ${\rm Im}G_{P}(\xi,z)$ will
achieve a maximum at $\xi=z$ and the function $\partial_{x_{j}}{\rm
Im}G_{N}(\xi,z)$ will achieve a minimum at $\xi=z$ for a sufficient large $R$.
This property can be easily observed in Figure 3. Based on this observation,
we can expect that $\widetilde{\rm Ind}_{D}(z)$ and $\widetilde{\rm
Ind}_{P}(z)$ will reach a peak on $\Gamma$, and $\widetilde{\rm Ind}_{N}(z)$
will hit a nadir on $\Gamma$.

(a) $J_{0}$

(b) ${\rm Im}G_{D}(x,z)$

(c) ${\rm Im}G_{P}(x,z)$

(d) $J^{\prime}_{0}$

(e) $\partial_{x_{1}}{\rm Im}G_{N}(x,z)$

(f) $\partial_{x_{2}}{\rm Im}G_{N}(x,z)$

Figure 3: The image of functions $J_{0}$, $J^{\prime}_{0}$, ${\rm
Im}G_{D}(x,z)$, ${\rm Im}G_{P}(x,z)$, $\partial_{x_{1}}{\rm Im}G_{N}(x,z)$ and
$\partial_{x_{2}}{\rm Im}G_{N}(x,z)$ with $R=95$, $x\in[-5,5]\times[-1,1.5]$
and the source $z=(0,0)$. The wavenumber $\kappa_{1}=10$ in (b),(e), (f) and
$\kappa=(10,5)$ in (c).

In all examples, we assume that the locally rough surface function $f$ is
supported in $[-5,5]$, the sample domain $S=[-5,5]\times[-1,1.5]$, and
$N_{s}=N_{r}=1024$ for impenetrable locally rough surfaces and
$N_{s}=N_{r}=2048$ for penetrable locally rough surfaces, and we set $R=95$
for the special locally rough surface $\Gamma_{R}$. In addition, we take the
wave number $\kappa_{1}=10$ for impenetrable case and $\kappa_{1}=10$,
$\kappa_{2}=5$ for penetrable case. The synthetic data is generated by
applying the Nyström method to solve the corresponding direct scattering
problem, see [30, 31] for details.

To test the stability of the RTM method, we consider the performance of this
method with noisy data. For some relative error $\tau>0$, we inject some noise
into the data by defining

$\displaystyle
u_{\tau}^{s}(x)=u^{s}(x)+\tau\frac{\beta}{\|\beta\|_{2}}\|u^{s}(x)\|_{2}$

where $\beta=\beta_{1}+{\rm i}\beta_{2}$ is complex-valued with $\beta_{1}$
and $\beta_{2}$ consisting of random numbers obeying standard normal
distribution $N(0,1)$.

Example 1. In this example, the locally rough surface $\Gamma$ is described as

$f_{1}(x_{1})=\left\\{\begin{array}[]{c}0.5+0.6\sin(0.6\pi
x_{1})\exp(16/(x_{1}^{2}-16))\qquad\;\;|x_{1}|<4,\\\\[2.84526pt]
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;\;\;0.5\;\qquad\quad|x_{1}|\geq
4.\end{array}\right.$

The reconstruction results from exact data are presented in Figure 4, where
the top row is the results from near-field data, and the bottom row is the
reconstruction from far-field data. As shown in Figure 4, the RTM approach can
present a satisfactory reconstruction.

Example 2. In this example, the locally rough surface $\Gamma$ is a multiscale
profile given by

$f_{2}(x_{1})=\left\\{\begin{array}[]{c}0.5+(0.5+0.05\sin(3\pi
x_{1}))\exp(4/(x_{1}^{2}-16))\qquad\;\;|x_{1}|<4,\\\\[2.84526pt]
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
0.5\;\qquad\;\;|x_{1}|\geq 4.\end{array}\right.$

The reconstructions with $5\%$ noise are illustrated in Figure 5, which shows
that the RTM method can provide a satisfactory imaging quality at $5\%$ noise
level.

Example 3. In the last example, the locally rough surface $\Gamma$ is
described by a piecewise continuous function given by

$f_{3}(x_{1})=\left\\{\begin{array}[]{lll}0.2\qquad\;\;|x_{1}|\leq
1,\\\\[2.84526pt] 0.3\qquad\;\;3\leq|x_{1}|\leq 4,\\\\[2.84526pt]
0.5\qquad\;\;{\rm others}.\end{array}\right.$

The numerical results are shown in Figure 6, which demonstrates that the RTM
method can provide satisfactory reconstructions for piecewise continuous
surfaces.

(a) Dirichlet

(b) Neumann

(c) Penetrable

(d) Dirichlet

(e) Neumann

(f) Penetrable

Figure 4: Reconstructions of the locally rough surface given in Example 1 from
data with no noise.The first and second rows are the reconstructions from
near-field and far-field, respectively.

(a) Dirichlet

(b) Neumann

(c) Penetrable

(d) Dirichlet

(e) Neumann

(f) Penetrable

Figure 5: Reconstructions of the locally rough surface given in Example 2 from
data with 5% noise. The first, second rows are the reconstructions from near-
field, far-field, respectively.

(a) Dirichlet

(b) Neumann

(c) Penetrable

(d) Dirichlet

(e) Neumann

(f) Penetrable

Figure 6: Reconstructions of the locally rough surface given in Example 3 from
data with 5% noise. The first, second rows are the reconstructions from near-
field, far-field, respectively.

From the above numerical experiments, it can be observed that the RTM method
proposed in Theorem 2.4 and Theorem 3.4 can provide accurate and stable
reconstructions for a variety of locally rough surfaces with the Dirichlet,
the Neumann, and the transmission boundary conditions. In addition, it is
easily seen that the RTM method could give a high quality reconstruction for
some complicated locally rough surfaces such as multiscale case and piecewise
continuous case.

##  5 Conclusion

This paper proposed extended RTM methods to recover the shape and location of
a locally rough surface with a Dirichlet, Neumann, or transmission boundary
conditions from both the near- and far-field measurements. The idea is mainly
based on constructing a modified Helmholtz-Kirchhoff identity associated with
a special locally rough surface, and a novel mixed reciprocity relation.
Numerical experiments demonstrated that the inversion algorithms can provide a
stable and satisfactory reconstruction for a variety of locally rough
surfaces. As far as we know, this is the first result for RTM approach to
recover an unbounded rough surface. However, it is more challenging to extend
the RTM method to reconstruct a diffraction grating and a non-local rough
surface. We hope to report the progress on this topic in the future.

## Acknowledgments

This work was supported by the NNSF of China grants No. 12171057, 12122114,
and Education Department of Hunan Province No. 21B0299. The authors thank
Prof. Haiwen Zhang for the valuable discussions.

## References

  * [1] S. Arridge, Optical tomography in medical imaging, Inverse Probl., 15(1999), R41-R93.
  * [2] H. Ammari, E. Iakovleva, and D. Lesselier, A music algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency, Multiscale Model. Simul.,3(2005), 597-628.
  * [3] A. Berkhout, Seismic migration: imaging of acoustic energy by wave field extrapolation, Elsevier, 1984.
  * [4] N. Bleistein, J. Cohen, and J. Stockwell, Mathematics of multidimensional seismic imaging, migration, and inversion, Springer, 2001.
  * [5] G. Bao, J. Gao, and P. Li, Analysis of direct and inverse cavity scattering problems, Numer. Math. Theory Methods Appl., 4(2011), 335-358.
  * [6] G. Bao and P. Li, Near-field imaging of infinite rough surfaces, SIAM J. Appl. Math., 73 (2013), 2162-2187.
  * [7] G. Bao and P. Li, Near-field imaging of infinite rough surfaces in dielectric media, SIAM J. Imaging Sci., 7 (2014), 867-899.
  * [8] G. Bao and J. Lin, Imaging of local surface displacement on an infinite ground plane: the multiple frequency case, SIAM J. Appl. Math., 71 (2011), 1733-1752.
  * [9] G. Bao and J. Lin, Imaging of local surface displacement on an infinite ground plane, Inverse Probl. Imag., 7(2013), 377-396.
  * [10] C. Burkard and R. Potthast, A multi-section approach for rough surface reconstruction via the Kirsch-Kress scheme, Inverse Probl., 26 (2010), 045007.
  * [11] J F. Claerbout, Imaging the Earth’s interior, Oxford: Blackwell Scientific Publication, 1985.
  * [12] J. Chen, Z. Chen, and G. Huang, Reverse time migration for extended obstacles: acoustic waves, Inverse Probl., 29 (2013), 085005.
  * [13] J. Chen, Z. Chen, and G. Huang, Reverse time migration for extended obstacles: electromagnetic waves, Inverse Probl., 29 (2013), 085006.
  * [14] Z. Chen and G. Huang, Reverse time migration for extended obstacles: elastic waves, Sci. Sin. Math., 45(2015), 1103-1114.
  * [15] Z. Chen and G. Huang, Reverse time migration for reconstructing extended obstacles in planar acoustic waveguides, Sci. China Math., 58, 1811-1834.
  * [16] Z. Chen and G. Huang, Reverse time migration for reconstructing extended obstacles in the half space, Inverse Probl., 31(2015), 055007.
  * [17] Z. Chen and G. Huang, A direct imaging method for electromagnetic scattering data without phase information, SIAM J. Imaging Sci.,9(2016), 1273-1297.
  * [18] Z. Chen and G. Huang, Phaseless imaging by reverse time migration: acoustic waves. Numer. Math. Theor. Meth. Appl., 10(2017), 1-21.
  * [19] S.N. Chandler-Wilde and J. Elschner, Variational approach in weighted Sobolev spaces to scattering by unbounded rough surfaces, SIAM J. Math. Anal., 42 (2010), 2554-2580.
  * [20] S.N. Chandler-Wilde and R. Potthast, The domain derivative in rough-surface scattering and rigorous estimates for first-order perturbation theory, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 458(2002), 2967-3001.
  * [21] S.N. Chandler-Wilde and B. Zhang, On the solvability of a class of second kind integral equations on unbounded domains, J. Math. Anal. Appl., 214 (1997), 482-502.
  * [22] S.N. Chandler-Wilde, B. Zhang and C.R. Ross, On the solvability of second kind integral equations on the real line, J. Math. Anal. Appl., 245 (2000), 28-51.
  * [23] M. Ding, J. Li, K. Liu and J. Yang, Imaging of locally rough surfaces by the linear sampling method with the near-field data, SIAM J. Imaging Sci. 10(2017), 1579-1602.
  * [24] R. Griesmaier, An asymptotic factorization method for inverse electromagnetic scattering in layered media, SIAM J. Appl. Math., 68(2008), 1378-1403.
  * [25] A. Lechleiter, Factorization Methods for Photonics and Rough Surfaces, PhD thesis, KIT, Germany, 2008.
  * [26] J. Li, Simultaneous recovery of an infinite rough surface and the impedance from near-field data, Inverse Probl. Sci. Eng., 27 (2019), 17-36.
  * [27] J. Li, Reverse time migration for inverse obstacle scattering with a generalized impedance boundary condition, Appl. Anal.,101(2022), 48-62.
  * [28] C. Lines, Inverse scattering by unbounded rough surfaces, PhD thesis, Department of Mathematics, Brunel University, UK, 2003.
  * [29] J. Li, P. Li, H. Liu and X. Liu, Recovering multiscale buried anomalies in a two-layered medium, Inverse Probl., 31 (2015), 105006.
  * [30] J. Li, G. Sun and R. Zhang, The numerical solution of scattering by infinite rough interfaces based on the integral equation method, Comput. Math. Appl., 71 (2016), 1491-1502.
  * [31] J. Li and G. Sun, A Nonlinear Integral Equation Method for the Inverse Scattering Problem by Sound-soft Rough Surfaces, Inverse Probl. Sci. Eng., 23 (2015), 557-577.
  * [32] J. Li, G. Sun and B. Zhang, The Kirsch-Kress method for inverse scattering by infinite locally rough interfaces, Appl. Anal., 96 (2017), 85-107.
  * [33] J. Li, J. Yang, and B. Zhang, A linear sampling method for inverse acoustic scattering by a locally rough interface, Inverse Probl. Imag., 15(2021), 1247-1267.
  * [34] J. Li, J. Yang, and B. Zhang, Near-field imaging of a locally rough interface and buried obstacles with the linear sampling method, J. Comput. Phys., 464(2022), 111338.
  * [35] L. Li, J. Yang, B. Zhang, and H.Zhang, Uniform far-field asymptotics of the two-layered Green function in 2D and application to wave scattering in a two-layered medium, arXiv: 2208.00456v1.
  * [36] L. Li, J. Yang, B. Zhang, and H. Zhang, Direct imaging methods for reconstructing a locally rough interface from phaseless total-field data or phased far-field data, arXiv:2305.05941v1.
  * [37] X. Liu, B. Zhang, and H. Zhang, A direct imaging method for inverse scattering by unbounded rough surfaces, SIAM J. Imaging Sci., 11(2018), 1629-1650.
  * [38] X. Liu, B. Zhang, and H. Zhang, Near-field imaging of an unbounded elastic rough surface with a direct imaging method, SIAM J. Appl. Math., 79(2019), 153-176.
  * [39] D. Natroshvili, T. Arens and S.N. Chandler-Wilde, Uniqueness, existence, and integral equation formulations for interface scattering problems, Mem. Differential Equations Math. Phys., 30 (2003), 105-146.
  * [40] F. Qu, B. Zhang, and H. Zhang, A novel integral equation for scattering by locally rough surfaces and application to the inverse problem: The Neumann case, SIAM J. Sci. Comput., 41(2019), A3763-A3702.
  * [41] D. G. Roy and S. Mudaliar, Domain derivatives in dielectric rough surface scattering, IEEE Trans. Antennas Propagation, 63(2015), 4486-4495.
  * [42] M. Thomas,Analysis of Rough Surface Scattering Problems, PhD thesis, Department of Mathematics, The University of Reading, UK, 2006.
  * [43] J. Yang, J. Li, and B. Zhang, Simultaneous recovery of a locally rough interface and the embedded obstacle with its surrounding medium, Inverse Probl., 38(2022), 045011.
  * [44] B. Zhang and S.N. Chandler-Wilde, Integral equation methods for scattering by infinite rough surfaces, Math. Methods Appl. Sci., 26 (2003), 463-488.
  * [45] T. Zhu and J. Yang, A non-iterative sampling method for inverse elastic wave scattering by rough surfaces, Inverse Probl. Imag., 16(2022), 997-1017.
  * [46] H. Zhang and B. Zhang, A novel integral equation for scattering by locally rough surfaces and application to the inverse problem, SIAM J. Appl. Math., 73 (2013), 1811-1829.