Ray Statement of the Acoustic Tomography Problem

The ray statement of the inverse problem of determining three unknown variable coefficients in the linear acoustic equation is studied. These coefficients are assumed to differ from given constants only inside some bounded domain. There are point pulse sources and acoustic receivers on the boundary of this domain. Acoustic signals are measured by a receiver near the moment of time at which the signal from a source arrives at the receiver. It is shown that this information makes it possible to uniquely determine all the three desired coefficients. Algorithmically, the original inverse problem splits into three subproblems solved successively. One of them is a well-known inverse kinematic problem (of determining the speed of sound), while the other two lead to the same integral geometry problem for a family of geodesic lines determined by the speed of sound.

Acoustic tomography problems were posed rather long ago (see, for example, [1][2][3][4]). Initially, they were motivated by the use of sound waves for early diagnosis of breast cancer. Recently, due to the spread of COVID-19 infection, the use of acoustic tomography methods has been proposed for diagnosing lung diseases as well. Computational modeling of acoustic tomography problems was carried out in [5][6][7][8][9] (see also numerous references therein). Below, we consider one of the possible variants for posing an acoustic tomography problem. In this problem, point pulse sources and receivers of acoustic signals are located outside the domain in which the variable coefficients in the acoustic equation are sought. Acoustic signals are measured in a small neighborhood of the moment of time at which the signal from a source arrives at the corresponding receiver. This problem belongs to the class of ray inverse problems introduced by the author in [10]. A characteristic feature of problems of this type is that the original problem of determining several coefficients is split into several successively solved problems of determining one of the required coefficients. In our case, these sought coefficients are the speed of sound c(x), the sound absorption coefficient , and the medium density . Determining c(x) reduces to solving a well-known inverse kinematic problem, whereas determining and leads to solving some integral geometry problem for a family of geodesic lines for the conformal Riemannian metric specified by the coefficient c(x). In the case when , this problem is a common X-ray tomography problem.
We consider the Cauchy problem (1) where is the sound pressure, is the speed of sound, is the sound absorption, and > 0 is the medium density. Equation (1) describes the propagation of acoustic waves in an inhomogeneous absorbing medium. Acoustic tomography is based on this equation. The main acoustic tomography problem is to solve an inverse problem for Eq. (1), namely, to construct the coefficients , , and inside a bounded domain based on information given outside this domain, namely, the solutions of Eq. (1) on some interval [0, T] for a set of point sources y. For definiteness, the following model is considered below.
Let be the ball of radius R centered at the origin, and let be its boundary. We assume that the parameters are continuously differentiable functions everywhere in (see below for additional conditions (9)); they are unknown inside a ball , , and are given constants outside it: Let denote the projection of the ball onto the sphere around the source located at , that is, = .
We denote the solution of problem (1) by , thus underlying its dependence on the parameter y. We fix and . Let T 0 (x, y) := sup{τ > 0 | , . We consider the following statement of the acoustic tomography (AT) problem.
AT problem. Assume that is a solution of problem (1) and the function (3) where is fixed and arbitrarily small, is known. We need to find , , and inside given .
Introducing the new function u(x, t, y) = , we reduce problem (1) for to the form (4) where the coefficient q(x) is given by the formula (5) If the coefficient q(x) is given, then, as follows from (5), the function can be obtained inside as a solution of the Dirichlet problem (6) where . In addition, condition (2) on the function and its continuous differentiability everywhere in imply that the solution of problem (6) must satisfy the requirement at . This additional condition guarantees the uniqueness of determining m(x) and, consequently, from the given q(x). Therefore, instead of the problem of determining , , and in , it is convenient to consider the inverse problem of finding the coefficients , , and q(x) for (4) in the same domain based on information similar to (3), namely, Here, and ε is fixed and arbitrarily small.
Similar problems were considered in [10] for in the following cases: , while and q(x) are unknown [10, , while and q(x) are unknown [10,Subsection 5.3].
In addition, in [10, Subsection 5.4], a two-dimensional inverse problem in electrodynamics is considered. When the properties of the medium are independent of , the equation for the electric field component coincides with Eq. (1), in which the role of is played by the magnetic permeability of the medium and the coefficient specifies the electric conductivity of the medium. In this inverse problem, all three unknown coefficients are determined based on dynamical information on solutions of three Cauchy problems in which the point source is replaced by the plane wave source , where is a unit vector and k = 1, 2, 3. We consider a Riemannian metric such that the length element is defined as , where . Throughout the rest of this paper, we assume that the following assumption holds. Assumption. The Riemannian metric dτ = is simple in , that is, any two points x and y in can be connected by a unique geodesic line . Note that a sufficient condition for the simplicity of a conformal Riemannian metric in is the condition (see [11]) where . Remark 1. In fact, within the ray statement of the inverse problem, the above assumption can be weakened, namely, the Riemannian metric can be assumed to be simple only inside a fixed ball for . We have limited ourselves to a stronger assumption to avoid extra remarks in what follows.
Assume that there are finite numbers and such that (8) and that the coefficients in (4) are characterized by the following smoothness: (9) Let denote the Riemannian distance between points x and y. Physically, is the travel time of the acoustic signal between x and y. It is well known that is a solution of the following Cauchy problem for the eikonal equation: To find and derive equations of geodesic lines, we need to introduce the vector and to solve the following Cauchy problem for the Euler system of ordinary differential equations: for all possible unit vectors , cosθ). This problem can be uniquely solved due to the simplicity assumption for the Riemannian metric and the conditions imposed on c(x). Upon solving problem (11), (12), we obtain the equation of geodesic lines , the tangent vector to this geodesic line, and the Riemannian distance τ = along it. Solving the equation with respect to , we find and , that is, the correspondences between a pair of points x and y and the values of the parameters s and . The equation of the geodesic line is given by , , while the Riemannian distance , by virtue of relations (11) and (12) To obtain formulas (14) and (15), we first need to find differential equations and initial conditions for them. These equations can be derived using the common method of calculating the singular and finite amplitudes of the solution of problem (4) (see, for example, [10]). With this aim in view, we must set in (13), substitute the resulting equality into (4), and set the coefficients of and to zero. As a result, we arrive at the equations (16) (17) For a homogeneous medium with parameters , , and , it holds that Therefore, and must satisfy the following conditions as : Equations (16) and (17) are ordinary differential equations along the geodesic lines . Indeed, due to the first equation in (11) ln ( , ) div( ( ) ( , )) = 6 2 ( , ) , x y x y d ∂ ζ ζ ζ ∂ where the operator L is defined by (4) and (25) Equalities (24) and (25) and the finiteness of the speed of sound imply that for and, hence, the validity of (13). In addition, the fundamental solution for (24), that is, the solution of the problem has the form where the function is defined by (14) and is a regular continuous function of its arguments. The solution of problem (24) has the integral representation (26) It follows from (25) that for . Therefore, equality (26) can be transformed into the form (27) where is the set The second and third problems lead to an identical problem in integral geometry. Indeed, if the function c(x) is found, then the geodesic lines and the function become known. We then uniquely derive from (14) the integrals (31) where The problem of determining the function = c 2 (x)σ(x) from (31) is an integral geometry problem. It was studied in [13,14]. The stability estimate for the solution of this problem inferred in [13,14] is similar to the estimate for the solution of the inverse kinematic problem with the natural replacement of c and by and g, respectively. The solution of the integral geometry problem for a family of geodesic lines specified by a simple Riemannian metric is unique. Upon finding and, therefore, , we can calculate the function for any x and y by applying (14). Based on (15), we then uniquely calculate the integrals (32) where Consequently, to determine from Eq. (32), we have exactly the same integral geometry problem. Upon solving it, we find q(x) as well. Given this coefficient, we solve problem (6) to find m(x) and, finally, determine the density .
A result of the previous considerations is the following theorem. An algorithm for solving the inverse problem is actually described above. In computational respect, it is surely necessary to detail the scheme for constructing solutions of the inverse kinematic problem and the integral geometry problem. Unfortunately, there are currently no sufficiently substantiated efficient algorithms and software programs to solve these problems.

FUNDING
This study was carried out within the state assignment of the Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, project no. FWNF-2022-0009.

CONFLICT OF INTEREST
The author declares that he has no conflicts of interest.

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