Localization of the Discontinuity Lines of the Bottom Scattering Coefficient According to Acoustic Sounding Data

Abstract

The paper considers the mathematical problems of constructing sonar images of the seabed from the data of measurements of a multibeam side-scan sonar. For the nonstationary radiative transfer equation describing the process of acoustic sounding in the ocean, we investigate the inverse problem of finding the discontinuity lines of the bottom scattering coefficient. A numerical algorithm for solving the inverse problem is developed, and the analysis of the quality of localization of the boundaries of inhomogeneities of the seabed depending on the number of angles and the sounding range is carried out.

INTRODUCTION

The radiative transfer equation describes a wide variety of physical processes and is used in the design of nuclear reactors, in astrophysics and atmospheric optics, in X-ray and optical tomography, and in gas dynamics and acoustics. We consider the radiative transfer equation in the context of modeling the process of high-frequency acoustic sounding in a fluctuating ocean [1,2,3,4,5]. Our interest in the model is due to the specific goal of improving the quality of sonar images of the seabed according to measurements obtained from a side-scan sonar (SSS) installed on autonomous unmanned underwater vehicles (AUV) [6,7,8,9].

The mathematical model includes a monochromatic integro-differential transfer equation, an initial condition, and a boundary condition describing the diffuse reflection from the bottom surface [9,10,11,12,13,14,15,16]. For simplicity, it is assumed that the carrier of the receiving-transmitting antenna emitting a pulsed signal moves at a constant speed in a certain straight line. The inverse problem is to find the bottom scattering coefficient under some additional conditions for redefining the solution of the radiative transfer equation, the physical meaning of which is to measure the backward reflected signal.

For a narrow radiation pattern in the single-scattering approximation, an explicit solution was obtained in [16] to find the diffuse reflection coefficient, the use of which leads to defocusing bottom objects on sonar images as the width of the radiation pattern increases. Attempts to eliminate this defect in single-beam probing are often unsuccessful, because the resulting system of linear algebraic equations has an ill-conditioned matrix when the problem is discretized and the solution of the problem becomes sensitive to errors in the initial data [17]. To overcome the difficulties that arise, one resorts to the use of multibeam echo sounders or to an increase in the number of tacks when monitoring water areas [7, 8]. The papers [18, 19] propose an approximate method for finding the bottom scattering coefficient from multibeam sounding data, which, despite its efficiency, has a number of significant drawbacks and requires additional a priori information about the acoustic characteristics of the medium.

In the present paper, we suggest to modify the statement of the original problem assuming that the desired characteristic is not the reflection coefficient but only the lines where the coefficient undergoes a discontinuity of the first kind. Information about the discontinuity lines of the coefficient is often sufficient to determine the location and shape of the desired object. The interest in inverse problems in which it is required to find the discontinuity surfaces or the singular support of a function is quite high and is connected, first of all, with integral geometry problems [21,22,23,24,25]. Generalized statements of such problems were considered in [26, 27] and consisted in determining the discontinuity surfaces of the coefficients of the stationary radiative transfer equation from information about the radiation emerging from the medium.

In practice, the number of angles of sounding the medium is relatively low, and we are dealing with the so-called problems of few-view tomography [28]. The main part of the known results in this area deals with the problem of inverting the Radon transform based on incomplete data. More complex problems concerning the partial localization of the discontinuity surfaces of the coefficients of the stationary radiative transfer equation for one- and two-angle probing were considered in the relatively recent papers [29,30,31,32].

1. DIRECT AND INVERSE PROBLEMS FOR NONSTATIONARY RADIATIVE TRANSFER EQUATION

The mathematical model under consideration describing the propagation of high-frequency acoustic wave fields in scattering media is based on a nonstationary radiative transfer equation of the form [6,7,8,9,10,11,12,13,14,15,16,17,18,19]

$$ \bigg (\frac {1}{c}\frac {\partial }{\partial t} + {\mathbf {k}} \cdot \nabla _r + \mu \bigg ) I({\mathbf {r}}, {\mathbf {k}}, t) = \frac {\sigma }{4\pi } \int \limits _{\Omega } I({\mathbf {r}},{\mathbf {k}}^{\prime }, t)\thinspace d {\mathbf {k}}^{\prime } + J({\mathbf {r}},{\mathbf {k}}, t), $$

(1)

where ${\mathbf {r}} \in G \subset \mathbb R^3$, $t \in [0, T]$, and the wave vector $\mathbf {k}$ belongs to the unit sphere $\Omega = \{ {\mathbf {k}} \in \mathbb R^3\mid |{\mathbf {k}}|=1\}$. The function $I({\mathbf {r}}, {\mathbf {k}}, t)$ is interpreted as the energy flux density of a wave propagating in the direction $\mathbf {k}$ with velocity $c$ at time $t$ at point $\mathbf {r}$. The numbers $\mu$ and $\sigma$ have the meaning of attenuation and scattering coefficients, and the function $J$ describes the sound field sources.

The domain $G$ is the upper half-space bounded by the horizontal plane $\gamma =\{{\mathbf {r}}=(r_1,r_2,r_3) \in \mathbb R^3 \mid r_3=-l\}$, $l>0$. Equation (1) is supplemented with the initial and boundary conditions [13, 14]

$$ I^-({\mathbf {r}},{\mathbf {k}}, t)|_{t<0} = 0, \quad ({\mathbf {r}},{\mathbf {k}}) \in G\times \Omega , $$

(2)

$$ I^-({\mathbf {y}}, {\mathbf {k}}, t) = \dfrac {\sigma _d({\mathbf {y}})}{\pi } \int \limits _{\Omega _+} |{\mathbf {n}} \cdot {\mathbf {k}}^{\prime }| I^+({\mathbf {y}}, {\mathbf {k}}^{\prime }, t)({\mathbf {y}}, {\mathbf {k}}^{\prime }, t)\thinspace d{\mathbf {k}}, \quad ({\mathbf {y}}, {\mathbf {k}}, t) \in \Gamma ^- . $$

(3)

In relations (2), (3), we use the notation

$$ \begin {aligned} I^{\pm }({\mathbf {y}}, {\mathbf {k}}, t) &= \lim \limits _{\varepsilon \to -0}I({\mathbf {y}} \pm \varepsilon {\mathbf {k}}, {\mathbf {k}}, t\pm \varepsilon /c),\\ \Gamma ^{\pm } &= \big \{ ({\mathbf {y}}, {\mathbf {k}}, t) \in \gamma \times \Omega _{\pm } \times (0, T)\big \}, \\ \Omega _{\pm } &= \big \{{\mathbf {k}} \in \Omega \mid \mathrm{sgn}\, ({\mathbf {n}} \cdot {\mathbf {k}}) = \pm 1\big \},\end {aligned} $$

where ${\mathbf {n}}=(0,0,-1)$ is the unit outward normal vector on the boundary of the domain $G$. Condition (2) means that there is no radiation in the medium at the initial time, and the boundary condition (3) describes the effects of diffuse reflection from the seabed by the Lambert law with reflection coefficient $\sigma _d({\mathbf {y}})$. The function $\sigma _d({\mathbf {y}})$ varying on the interval from zero to one in the $\gamma$-plane is also called the bottom scattering coefficient. It is assumed that the $\gamma$-plane on which the function $\sigma _d({\mathbf {y}})$ is defined can be represented as the union of disjoint two-dimensional subdomains $\gamma _i$, $i=0,\dots ,p$, with piecewise smooth boundaries $\partial \gamma _i$ such that $\sigma _d({\mathbf {y}})$ is constant in each of the domains $\gamma _i$.

Problem 1.

Equation (1) with the initial and boundary conditions (2), (3) and with given $\mu$, $\sigma$, $\sigma _d$, $J$, and $c$ forms an initial–boundary value problem of finding the unknown function $I$ on the set $G \times \Omega \times (0,T)$.

The papers [13,14,15,16,17,18,19] deal with the well-posedness of Problem 1, also known as the direct problem for the radiative transfer equation.

The function $J$, which describes a point pulsed sound source moving at a constant velocity $V$ in the positive direction of the $r_2$-axis, has the form

$$ J({\mathbf {r}},{\mathbf {k}}, t) = \delta ({\mathbf {r}} -{\mathbf {V}} t) \sum \limits _{i=1}^{m}\delta (t-t_i), \quad {\mathbf {V}}=(0,V,0), \quad t_i>0, $$

(4)

where $\delta$ is the Dirac delta function. Let us supplement the system of relations (1)–(3) with the relation

$$ \int \limits _{\Omega } S_{j}({\mathbf {k}}) I^{+}({\mathbf {V}} t, {\mathbf {k}}, t)\thinspace d{\mathbf {k}} = P_{j}(t), \quad j=1,\dots , q, $$

(5)

where the function $S_j({\mathbf {k}})$ is integrable, nonnegative, and distinct from zero only in some subdomain $\Omega _j \subset \Omega$. We state the following inverse problems.

Problem 2.

Find the function $\sigma _d({\mathbf {y}})$ from relations (1), (2), (3), (5) for given $\mu$, $\sigma$, $c$, $V$, $J$, $l$ and $S_j$, and $P_j(t)$.

Problem 3.

Find the lines of discontinuity $\partial \gamma _i$, $i=0,\dots ,p$, of the function $\sigma _d({\mathbf {y}})$ from relations (1)–(3), (5) for given $V$, $J$ and $S_j$, and $P_j(t)$.

As was already mentioned, the physical meaning of the inverse Problem 2 is to reconstruct the reflection coefficient $\sigma _d$ during acoustic sounding of the seabed by a side-scan sonar that moves rectilinearly at a constant velocity $V$ and sounds the surrounding space with pulsed signals. There are antennas on the carrier that measure the total intensity $P_{j}(t)$ in the field of view $\Omega _j$ at time $t$, and the functions $S_j({\mathbf {k}})$ characterize the radiation pattern of the $j$th antenna. If $q=2$, $\Omega _1=\{{\mathbf {k}} \in \Omega \mid k_1 < 0 \}$, and $\Omega _2=\{{\mathbf {k}} \in \Omega \mid k_1 > 0 \}$, then we are dealing with the simplest case of a side-scan sonar with one receiving antenna on each side of the carrier [9].

Problem 3 is to determine not the function $\sigma _d$ itself but only its discontinuity lines $\partial \gamma _i$ with much less information about the initial data of the problem. In particular, the quantities $\mu$ and $\sigma$ are not supposed to be known, but they cannot be determined in such a statement of the problem either. Problem 2 was studied in [18, 19], and so the main attention will be focused on Problem 3. As applied to X-ray tomography problems, similar problems for the stationary transport equation were considered in [26, 27, 29,30,31,32]. These papers propose original methods based on constructing an inhomogeneity indicator, which permit efficiently reconstructing the discontinuity surfaces of the coefficients of the radiative transfer equation. Note that in Problem 3 we determine the lines of discontinuity of the coefficient $\sigma _d({\mathbf {y}})$, which occurs in the boundary condition (3) rather than in Eq. (1).

2. EXPRESSIONS FOR THE FUNCTIONS $P_{j}(t)$ IN THE SINGLE-SCATTERING APPROXIMATION

As a rule, the antenna carrier velocity $V$, the medium parameters $\mu$, $\sigma$, and $c$, and the sounding periods $t_{i+1}-t_i$ are such that $(\sigma /\mu )^2 \ll 1$, $V/c \ll 1$, and $\exp (-\mu c |t-t_i|) \ll 1$, $t \not \in (t_i,t_ {i+1})$. This gives grounds to apply the single-scattering approximation to find a solution of the initial–boundary value problem and to neglect echolocation signals from the other sounding periods $(t_j,t_{j+ 1})$, $j\not = i$, at the current reception interval $t\in (t_i,t_{i+1})$ [10].

Under the above-indicated conditions, the approximate solution of the initial–boundary value problem (1)–(3) has the form [18, 19]

$$ I({\mathbf {r}}, {\mathbf {k}}, t) = I_0({\mathbf {r}},{\mathbf {k}},t) + I_{\gamma }({\mathbf {r}},{\mathbf {k}},t)+I_{G}({\mathbf {r}},{\mathbf {k}},t), $$

(6)

where

$$ I_0({\mathbf {r}},{\mathbf {k}},t)=\int \limits ^{d({\mathbf {r}}, -{\mathbf {k}}, t)}_0 \exp (-\mu \tau ) J ({\mathbf {r}}-\tau {\mathbf {k}}, {\mathbf {k}}, t - {\tau }/{c} )\thinspace d\tau ,\qquad \qquad \qquad \qquad \quad $$

(7)

$$ \begin {aligned} I_{\gamma }({\mathbf {r}},{\mathbf {k}},t) &=\dfrac {\sigma _d \big ({\mathbf {r}} - d({\mathbf {r}}, -{\mathbf {k}},t){\mathbf {k}}\big )}{\pi }\exp \big (-\mu d ({\mathbf {r}}, -{\mathbf {k}},t)\big )\\ &\qquad \qquad {}\times \int \limits _{\Omega _+} |{\mathbf {n}} \cdot {\mathbf {k}}^{\prime }| I_0^+ \bigg ( {\mathbf {r}} - d({\mathbf {r}}, -{\mathbf {k}},t){\mathbf {k}}, {\mathbf {k}}^{\prime }, t - \frac {d({\mathbf {r}}, -{\mathbf {k}},t)}{c} \bigg ) d {\mathbf {k}}^{\prime }, \end {aligned} $$

(8)

$$ I_{G}({\mathbf {r}},{\mathbf {k}},t) = \int \limits ^{d({\mathbf {r}}, -{\mathbf {k}},t)}_0 \exp (-\mu \tau ) \dfrac {\sigma }{4\pi } \int \limits _{\Omega } I_0\bigg ({\mathbf {r}}-\tau {\mathbf {k}}, {\mathbf {k}}^{\prime }, t - \frac {\tau }{c} \bigg ) d {\mathbf {k}}^{\prime } d\tau .\qquad \qquad \; $$

(9)

In the expressions (7)–(9), we use the notation $d({\mathbf {r}}, -{\mathbf {k}},t)=\min \{d({\mathbf {r}}, -{\mathbf {k}}), c t\}$, where $d({\mathbf {r}}, -{\mathbf {k}})$ is the distance from a point ${\mathbf {r}} \in G$ in the direction $-{\mathbf {k}}$ to the boundary of the domain $G$. If ${\mathbf {r}} - d({\mathbf {r}}, -{\mathbf {k}},t){\mathbf {k}} \not \in \partial G$, then $I_{\gamma }=0$. The simple structure of the domain $G$ allows one to write an explicit representation $d({\mathbf {r}}, -{\mathbf {k}},t)=\min \{l/k_3, c t\}$ of the function $d({\mathbf {r}}, -{\mathbf {k}},t)$ at the points belonging to the plane $r_3=0$ for $k_3>0$ and $d({\mathbf {r}}, -{\mathbf {k}},t)=c t$ for $k_3\leq 0$. By substituting $I_0$, $I{\gamma }$, and $I_{G}$ into (5), we obtain

$$ \quad \; P_j (t) = P_{j,0} (t)+ P_{j,\gamma }(t)+P_{j,G}(t), $$

(10)

where

$$ P_{j,0}(t)=\int \limits _{\Omega } S_j({\mathbf {k}}) I^{+}_{0} ({\mathbf {V}} t,{\mathbf {k}},t)\thinspace d{\mathbf {k}}, $$

(11)

$$ P_{j,\gamma }(t)=\int \limits _{\Omega } S_j({\mathbf {k}}) I^{+}_{\gamma } ({\mathbf {V}} t,{\mathbf {k}},t)\thinspace d{\mathbf {k}}, $$

(12)

$$ P_{j,G}(t)=\int \limits _{\Omega } S_j({\mathbf {k}}) I^{+}_{G} ({\mathbf {V}} t,{\mathbf {k}},t)\thinspace d{\mathbf {k}}. $$

(13)

Taking into account the structure of the function $J$ and the paper [18], we obtain

$$ P_{j,0}(t)=0,\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \; $$

$$ P_{j,\gamma }(t)= \dfrac {c l^2 }{2\pi } \dfrac {\exp \big (-\mu c(t- t_i)\big )}{\big (c (t-t_i)/2\big )^5}\int \limits ^{2\pi }_{0} S_j \big ({\mathbf {k}}(\varphi ,\theta _i)\big ) \sigma _d \big ({\mathbf {y}}(\varphi ,\theta _i)\big )\thinspace d \varphi , $$

(14)

$$ P_{j,G}(t)= \dfrac {\sigma c \exp \big (-\mu c (t-t_i)\big )}{8\pi \big (c(t-t_i)/2\big )^2} \int \limits _{0}^{2\pi } \int \limits _{\theta _i}^{\pi } S_j \big ({\mathbf {k}}(\theta ,\varphi )\big ) \sin \theta \thinspace d \theta d \varphi , $$

(15)

where ${\mathbf {k}}(\theta ,\varphi )=(-\sin \varphi \sin \theta ,\cos \varphi \sin \theta ,\cos \theta )$, the angle $\theta$ varies from $\theta _i=\arccos \bigg ( \dfrac {2 l}{c (t-t_i)}\bigg )$ to $\pi$, and $\varphi \in [0,2\pi )$.

If the receiving antenna pattern $S_j$ is narrowly directed in planes perpendicular to the bottom surface $r_3=-l$,

$$ S_j\big ({\mathbf {k}}(\theta ,\varphi )\big )= \delta (\varphi -\varphi _j), $$

where $\delta (\varphi -\varphi _j)$ is the Dirac delta function, then formulas (14) and (15) for $t \in ( t_i+2l/c, t_{i+1})$ acquire the very simple form

$$ P_{j,\gamma }(t)= \dfrac {c l^2 c}{2\pi } \dfrac { \exp \big (-\mu c(t-t_i)\big )}{\big (c(t-t_i)/2\big )^5}\sigma _d\big ({\mathbf {y}}(\varphi _j,\theta _i)\big ), $$

(16)

$$ P_{j,G}(t)=\dfrac {\sigma c\exp \big (-\mu c(t-t_i)\big )}{8\pi \big (c(t-t_i)/2\big )^2}\bigg (1+ \dfrac {2l}{c(t-t_i)}\bigg ). $$

(17)

In this case, for determining the function $\sigma _d$, from (10), (14), and (15) for any $j$ we obtain the explicit formula [18]

$$ \sigma _{d,j} ({\mathbf {y}}) = \bigg ( P_{j}\bigg ( t\bigg ) - \dfrac {\sigma c \exp \big (-2\mu |{\mathbf {y}} -{\mathbf {V}} t|\big )}{8 \pi |{\mathbf {y}} -{\mathbf {V}} t|^2} \bigg (1+ \dfrac {l}{|{\mathbf {y}} -{\mathbf {V}} t|} \bigg )\bigg ) \bigg ( \dfrac {c l^2}{2\pi } \dfrac { \exp \big (-2\mu |{\mathbf {y}}-{\mathbf {V}} t|\big )}{|{\mathbf {y}} -{\mathbf {V}} t|^5} \bigg )^{-1}, $$

(18)

where $t=(y_2 + y_1 \text {cot} \varphi _j)/V$. Thus, for a narrowly collimated (in the angle $\varphi$) radiation pattern, formula (18) gives an explicit solution of Problem 2 in the single-scattering approximation. Obviously, in this case it suffices to carry out measurements using two receiving antennas located on different sides of the carrier, say, with $S_1 = \delta (\varphi -\pi /2)$ and $S_2 = \delta (\varphi -3\pi /2)$; this corresponds to the widespread method of constructing sonar images successively strip by strip perpendicular to the direction of motion of the antenna carrier.

With an increase in the width of the radiation pattern, the calculation of the function $\sigma _d$ by formula (18) leads to an increase in the error, especially when reconstructing high-contrast structures. On sonar images, the so-called effect of “smearing,” or defocusing, of the image appears. To eliminate such defects, various focusing methods are used, which in mathematical terms are reduced to solving an integral equation of the first kind. When the desired function is discretized, the solution of the integral equation is equivalent to the solution of an (as a rule, ill-conditioned) system of linear algebraic equations; this considerably complicates determining the desired function [17].

Another, in our opinion, the most important drawback of the algorithm for reconstructing the bottom scattering coefficient by using formula (18) is due to the fact that additional information about the attenuation and volume-scattering coefficients is required. As a rule, the coefficients of Eq. (1) that describe the interaction of radiation in the ocean are known only approximately; therefore, the reconstruction of the bottom scattering coefficient by formula (18) leads to large distortions of tomographic images.

Thus, the construction of numerical algorithms for solving Problem 3 that require much less information about the initial data and are free from most of the indicated drawbacks is really important for the development of modern methods of acoustic sounding of the seabed.

3. NUMERICAL ALGORITHM FOR SOLVING INVERSE PROBLEM 3

In this section, we present a numerical scheme for solving the inverse Problem 3. As we have already noted, the problem under consideration can be viewed as a problem of few-view tomography, and the algorithm for solving it is close to the methods described in the papers [29,30,31,32].

Let the radiation pattern $S_j ({\mathbf {k}}(\varphi ,\theta ))$ be equal to

$$ S_j ({\mathbf {k}}(\varphi ,\theta ))=\begin {cases} 1/2\beta , &\varphi \in (\varphi _j-\beta ,\varphi _j +\beta )\\ 0, & \varphi \notin (\varphi _j-\beta ,\varphi _j +\beta ), \end {cases} $$

with the support of each of the functions $S_j$ concentrated either in the interval $0< \pi /2 - \delta /2 < \varphi < \pi /2 + \delta /2$ or in the interval $0< 3\pi /2 - \delta /2 < \varphi < 3 \pi /2 + \delta /2$, where $\delta <\pi$. The latter restriction is typical of scanning the seabed with a side-scan sonar and makes it possible to recover the function $\sigma _d$ separately for $r_1>0$ and $r_1<0$ [8–10]. When constructing the algorithm and carrying out numerical experiments, we consider only the half-plane $r_3=-l$, $r_1>0$, and the value of the exponent $q$ corresponds not to the total number of sounding angles but only to one of the “shipboards” of antenna carrier motion.

Taking into account the stated constraints and the relation

$$ |{\mathbf {V}} t -{\mathbf {y}}|^2=y_1^2/\sin ^2 \varphi _j+l^2\quad \mbox {for}\quad t=(y_2 + y_1 \cot \varphi _j)/V, $$

the expressions for the functions $P_{j,\gamma }((y_2 + y_1 \cot \varphi _j)/V)$ and $P_{j,G}((y_2 + y_1 \cot \varphi _j)/V)$ can be written in the form

$$ \begin {aligned} &{}P_{j,\gamma }\big ((y_2 + y_1 \text {cot} \varphi _j)/V\big ) \\ &\qquad {}=\dfrac {c\thinspace l^2 c}{2\pi }\dfrac { \exp \left (- 2 \mu \sqrt {y_1^2/\sin ^2 \varphi _j+l^2} \right )}{(y_1^2/\sin ^2 \varphi _j+l^2)^{5/2}} \\ &\qquad \qquad \quad {}\times \dfrac {1}{2\beta } \int \limits ^{\varphi _j+\beta }_{\varphi _j-\beta } \sigma _d \big ( |y_1|\sin \varphi / |\sin \varphi _j| , y_2 + y_1 \cot \varphi _j - |y_1|\cos \varphi / |\sin \varphi _j|\big )\thinspace d \varphi \\ &\qquad {}= \dfrac {c l^2 c}{2\pi } \dfrac { \exp \left (- 2 \mu \sqrt {y_1^2/\sin ^2 \varphi _j+l^2} \right )} {(y_1^2/\sin ^2 \varphi _j+l^2)^{5/2}} \\ &\quad \qquad \qquad {}\times \dfrac {1}{2\beta } \int \limits ^{\varphi _j+\beta }_{\varphi _j-\beta } \sigma _d \big ( |y_1|\sin \varphi /|\sin \varphi _j|, y_2 +|y_1| (\cos \varphi _j - \cos \varphi )/|\sin \varphi _j|\big )\thinspace d \varphi , \end {aligned} $$

(19)

$$ \begin {aligned} &{}P_{j,G}\big ((y_2 + y_1 \text {cot} \varphi _j)/V\big ) \\ &\qquad {}=\dfrac {c \sigma }{8\pi } \dfrac { \exp \left (- 2 \mu \sqrt {y_1^2/\sin ^2 \varphi _j+l^2}\right )} {(y_1^2/\sin ^2 \varphi _j+l^2)}\left (1+\dfrac {l}{\sqrt {y_1^2/\sin ^2 \varphi _j+l^2}}\right ).\qquad \qquad \qquad \end {aligned} $$

(20)

Set

$$ \begin {aligned} \widehat {P}_{j,\gamma }({\mathbf {y}}) &\equiv P_{j,\gamma }\big ((y_2 + y_1 \cot \varphi _j)/V\big ), \\ \widehat {P}_{j,G}({\mathbf {y}}) &\equiv P_{j,G}\big ((y_2 + y_1 \cot \varphi _j)/V\big ),\end {aligned} $$

where ${\mathbf {y}}=(y_1,y_2,-l)$ and the functions $P_{j,\gamma }$ and $P_{j,G}$ are defined in (19) and (20). An approximate method for solving the inverse Problem 3 is based on preliminary calculation of functions of the form

$$ \widehat {\sigma }_{d,j} ({\mathbf {y}}) = \Big |\nabla \big (\widehat {P}_{j}({\mathbf {y}}) A_j ({\mathbf {y}})\big ) \cdot {\mathbf {k}}_j\Big | $$

(21)

for each ${\mathbf {k}}_j=(\cos \varphi _j,\sin \varphi _j,0)$ and subsequent construction of the indicator function

$$ \widehat {\sigma }_{d} ({\mathbf {y}}) = \sum \limits _{j=1}^q \widehat {\sigma }_{d,j} ({\mathbf {y}}). $$

(22)

In Eq. (21), the function $A_j ({\mathbf {y}})$ is some continuously differentiable weight function selected so as to compensate for a strong decay of the function $P$ as $|y_1| \to \infty$. For example, if we know the quantities $\mu$ and $l$ or at least their approximate values, then for the function $A_j ({\mathbf {y}})$ we can take an expression of the form

$$ \left ( \dfrac { \exp (- 2 \mu \sqrt {y_1^2/\sin ^2 \varphi _j+l^2} )}{(y_1^2/\sin ^2 \varphi _j+l^2)^{5/2}} \right )^{-1}. $$

If, however, there is no a priori information, then we can set $A=1$. One can readily see that the functions $\nabla \widehat {P}_{j,G}({\mathbf {y}}) \cdot {\mathbf {k}}_j$ are bounded on the entire range of their arguments, and the functions $\nabla \widehat {P}_{j,\gamma }({\mathbf {y}}) \cdot {\mathbf {k}}_j$ may grow unboundedly only for the case in which for $\varphi \in [\varphi _j,-\beta , \varphi _j +\beta ]$ the line of integration in (19),

$$ \begin {aligned} z_1&=|y_1|\sin \varphi /|\sin \varphi _j|,\\ z_2&=y_2 + |y_1| (\cos \varphi _j - \cos \varphi )/|\sin \varphi _j|,\\ z_3&=-l, \end {aligned} $$

(23)

intersects the line of discontinuity of the function $\sigma _d({\mathbf {y}})$, ${\mathbf {y}}=(y_1,y_2,-l)$. As the point $\mathbf {y}$ tends to the line of discontinuity $\partial \gamma _i$, the number of terms $\widehat {\sigma }_{d,j} ({\mathbf {y}})$ in the sum (22) with the indicated property will grow; this leads to the growth of the function $\widehat {\sigma }_{d} ({\mathbf {y}})$. The degree of growth of the functions $\widehat {\sigma }_{d,j} ({\mathbf {y}})$ depends on the angle between the curve (23) and the line of discontinuity at the point of their intersection, and the growth is maximal if the angle is zero, i.e., the curves touch each other. A rigorous substantiation of these facts is hampered by the restriction associated with the discreteness of the set of sounding directions; moreover, it is rather cumbersome and goes beyond the scope of this article. In the next section, we give a numerical confirmation of the efficiency of the algorithm based on the construction of the indicator function (22).

4. RESULTS OF NUMERICAL SIMULATION

When monitoring water areas by autonomous unmanned underwater vehicles, the speed of the carrier, the height of its trajectory above the bottom of the reservoir, and the sounding interval usually depend on the mission objectives, the size of the basin under study, and the features of the equipment. In numerical experiments, the following values of sounding parameters were chosen: $V = 1$ m/s, $l=12$ m, $t_{i+1}-t_i=0{.}4$ s. The receiving antenna beam width in the horizontal plane was $2$ deg ($\beta =1/\pi$), which is typical for most modern side-scan sonars [7, 8]. The remaining quantities took the following values, typical for acoustic sounding in the oceanic medium at frequencies of the order of $100$ kHz [5, 9, 11]: $\mu = 0{.}018$ m$^{ -1}$, $\sigma = 0{.}1\mu$, $c = 1500$ m/s.

The algorithm for solving the inverse problem was tested for the function $\sigma _d(r)$ whose graphical representation for $r_1>0$ is given in Fig. 1. The function $\sigma _d$ takes the value $0{.}9$ in all three inclusions $\gamma _i$, $i=1,2,3$. The domains $G_i$, $i=1,2,3$, which are inclusions of the “aircraft wreckage” type, are located at distances of 50, 150, and 250 m from the axis $r_1=0$, respectively. In the complement $\gamma _0=\gamma \setminus (\gamma _1 \cup \gamma _2 \cup \gamma _3)$, the function $\sigma _d$ takes the value $0{.}1$.

Fig. 1.

Graphical representation of the bottom scattering coefficient $\sigma _{d} ({\mathbf {y}})$ (original). The linear dimensions of the surveyed bottom area are $300\times 50$ m.

Figure 2 shows the images of the functions $\widehat {\sigma }_{d,j} ({\mathbf {y}})$ calculated using formulas (21) for single-beam probing at the angles $\varphi _1=\pi /6$, $\varphi _3=\pi /2$, and $\varphi _5=5\pi /6$. The quality of the images corresponding to the angles $\varphi _1=\pi /6$ and $\varphi _5=5\pi /6$ is much lower; this is not surprising, because the sounding range at oblique angles increases. This leads to a deterioration in the resolution of the seabed image reconstruction algorithm.

Fig. 2.

Reconstructed functions $\widehat {\sigma }_{d,j} ({\mathbf {y}})$ for $j=1,3,5$ for single-beam sounding with beam width of $2$ deg: (a) $\varphi _1=\pi /6$, (b) $\varphi _1=\pi /2$, (c) $\varphi _1=5\pi /6$.

Figure 3 shows graphic representations of the function $\widehat {\sigma }_{d} ({\mathbf {y}})$ for various values of the parameter $q$. It can be seen from Fig. 3a that the first inclusion is clearly reconstructed only for $q=5$; therefore, when localizing the lines of discontinuity of the bottom scattering coefficient at a distance of up to $100$ m, it suffices to use a small number of sounding angles.

Fig. 3.

Function $\widehat {\sigma }_{d} ({\mathbf {y}})$ characterizing the set of points of discontinuity of ${\sigma }_{d} ({\mathbf {y}})$ for various numbers $q$ of sounding angles: (a) $q=5$, (b) $q=15$, (c) $q=25$.

As expected, the quality of object focusing increases with the increase in the number of probing angles. Improving the quality of the tomographic image with an increase in $q$ is especially noticeable at a probing distance of more than $100$ m. However, for a high-quality reconstruction of the discontinuity lines of the desired coefficient at a distance of approximately 250 m, an increase in the number of angles alone is not sufficient. This is due to an increase in the relative error of the approximate calculation of the function $P_{j}((y_2 + y_1 \cot \varphi _j)/V)$ with an increase in $|y_1|$.

Note that the deterioration in the seabed image reconstruction quality is also observed with distance from the transmitting-receiving antenna in real physical experiments when monitoring water areas. With an increase in the probing range, a considerable decrease in the useful reflected signal occurs, while the destructive noise in the detection devices remains at the same level. The nature of such noise can be very diverse and associated, say, with a strong instability of the trajectory of motion of an autonomous unmanned underwater vehicle or with refraction of sound rays due to a change in the density of the marine environment [6,7,8]. In the most simplified case, a timed automatic gain control algorithm is used to amplify the reflected signal in accordance with the time of its arrival. Despite the fact that today there are already many varieties of such algorithms, it is not possible to completely compensate for the loss in the image reconstruction quality [6,7,8].

CONCLUSIONS

The inverse problem of finding the discontinuity lines of the bottom scattering coefficient for the nonstationary radiative transfer equation describing the process of acoustic sounding in the ocean is studied. A numerical algorithm for solving the inverse problem has been developed and verified on model data corresponding to acoustic sounding of the seabed at frequencies of the order of $100$ kHz. It is shown that for successful localization of discontinuity lines of the bottom scattering coefficient at a distance of up to $100$ m, a small number of angles (of the order of $5$) is sufficient. With an increase in the sounding range, it is necessary not only to increase the number of sounding angles but also to reduce the error in calculating the functions $P_j$. In practice, such restrictions impose strict requirements on the transceiver equipment of side-scan sonars.