A numerical method to solve an acoustic inverse scattering problem involving ghost obstacles
Lorella Fatone
Dipartimento di Matematica Pura ed Applicata
Università di Modena e Reggio Emilia
Via Campi 213/b, 41100 Modena, Italy
Ph. N. +39-059-2055589, FAX N. +39-059-370513
e-mail : fatone.lorella@unimo.it
Maria Cristina Recchioni
Dipartimento di Scienze Sociali "D. Serrani"
Università Politecnica delle Marche
Piazza Martelli 8, 60121 Ancona, Italy
Ph. N. +39-071-2207066, FAX N. +39-071-2207150
e-mail : m.c.recchioni@univpm.it
Francesco Zirilli
Dipartimento di Matematica "G. Castelnuovo"
Università di Roma "La Sapienza"
Piazzale Aldo Moro 2, 00185 Roma, Italy
Ph. N. +39-06-49913282, FAX N. +39-06-44701007
e-mail : f.zirilli@caspur.it
1. Abstract
In [8] we study a time harmonic inverse acoustic scattering problem involving an obstacle that when hit by an incoming acoustic wave tries to appear in a location in space different from its actual location eventually with a shape and an acoustic boundary impedance different from its actual ones. We refer to this problem as inverse ghost obstacle problem and to the scatterers of this type as a special class of smart obstacles. The term "ghost" comes from the fact that the smart obstacle, when hit by the incoming wave, tries to appear as a ghost obstacle, that is as a virtual obstacle in a location in space where no real obstacle is present. We assume that the intersection between the obstacle and the ghost obstacle is empty. The obstacle tries to produce this virtual image of itself circulating on its boundary a suitable pressure current. A pressure current is a quantity whose physical dimension is pressure divided by time. The following time harmonic inverse ghost obstacle scattering problem is considered: from the knowledge of several far fields generated by the smart obstacle when hit by known time harmonic waves, of its acoustic boundary impedance and of the acoustic boundary impedance of the ghost obstacle find the location of the smart obstacle, that is find a point in the interior of the smart obstacle and eventually find the shape of the smart obstacle. The method proposed to solve this problem is a generalization of the well known Herglotz function method. We present some numerical experiments based on synthetic data that are used to test the numerical method introduced here. This website contains some auxiliary material that helps the understanding of the paper [8].
2. The inverse scattering problem
We consider a time harmonic acoustic inverse scattering problem involving a smart obstacle that when hit by an incoming wave pursues the goal of appearing in a location in space different from its actual location eventually with a shape and an acoustic boundary impedance different from its actual ones. The obstacle pursues this goal circulating a pressure current on its boundary. A pressure current is a quantity whose physical dimension is pressure divided by time. This type of obstacles is a special class of smart obstacles. Mathematical models of time dependent and time harmonic acoustic and electromagnetic direct scattering problems involving smart obstacles have been proposed in [1]-[8]. In the time dependent acoustic case these models are optimal control problems for the wave equation where the pressure current mentioned above plays the role of the control variable. In the time harmonic acoustic case these models reduce to a constrained optimization problem with the constraints given by an exterior boundary value problem for the Helmholtz equation with the space dependent part of a time harmonic pressure current as independent variable of the optimization problem. The solution of the optimal control problem is obtained formulating the first order necessary optimality condition via the Pontryagin maximum principle while the solution of the constrained optimization problem is obtained formulating the first order necessary optimality condition using the Lagrangian functional. The first order optimality conditions obtained are solved numerically.
We consider the following inverse problem:
Time Harmonic Ghost Obstacle Inverse Scattering Problem: From the knowledge of several far fields generated by the smart obstacle when hit by known incident acoustic waves, the knowledge of the boundary acoustic impedances of the smart obstacle and of the ghost obstacle find the location of the smart obstacle, that is find a point in the interior of the smart obstacle and, eventually find its shape.
Note that we assume that the intersection between the smart obstacle and the ghost obstacle is empty. Obviously the solution of an inverse problem involving a smart obstacle is more difficult than the solution of the analogous problem involving a passive obstacle, that is an obstacle that does not react pursuing an assigned goal when hit by an incoming wave. The degree of difficulty of the inverse problem for the smart obstacle increases with the ability of the obstacle of pursuing its goal.
The classical inverse problem for a passive obstacle can be formulated as follows:
Acoustic Time Harmonic (Passive) Inverse Scattering Problem: From the knowledge of the far fields generated by the (passive) obstacle when hit by several known incoming time harmonic waves and of the acoustic boundary impedance of the obstacle, determine the boundary of the obstacle.
This last problem and several related problems have been widely investigated and many efficient computational methods have been developed for their solution (see for example [9], [10], [11] and the references therein) while the formulation of mathematical models for inverse scattering problems involving smart obstacles and the development of numerical methods to solve them are rather unexplored research fields. Recently the authors and a co-author have formulated mathematical models for inverse problems and developed numerical methods to solve inverse scattering problems when smart obstacles that pursue the goal of being undetectable (i.e.:furtive obstacles) or of appearing with a shape and/or boundary impedance different from their actual ones (i.e.: masked obstacles) are considered (see [4], [7]). An inverse scattering problem involving an obstacle that tries to appear in a location different from its actual one (i.e.: ghost obstacle) has been proposed in [5] where the solution of the inverse problem has been obtained with an iterative procedure that minimizes a suitable functional. The evaluation of this functional implies the solution of a direct scattering problem for the smart obstacle.
Here we develop a new approach to solve this last inverse problem in particular we develop an ad hoc Herglotz wave function method that generalizes the Herglotz wave function method introduced in [7] to solve inverse problems involving furtive or masked obstacles. The method proposed to solve the inverse ghost problem is an iterative procedure that does not require the solution of a direct scattering problem at each iteration and that it is based on the minimization of a suitable functional defined via the vector Herglotz wave function. The method proposed here when compared to the numerical method proposed in [7] has the advantage of being computationally much more efficient. The main difficulty that we must overcome in developing this method consists in the fact that the far field measures are taken in a coordinate system with origin in a point that we choose in the interior of the ghost obstacle and not with origin in a point in the interior of the smart obstacle. Hence the new formulation of the Herglotz wave function method must take into account this fact and it must consider as unknowns the coordinates of a point in the interior of the ghost obstacle (i.e. for example the center of mass of the boundary of the ghost obstacle) in a coordinate system having the origin in a point in the interior of the smart obstacle (i.e. for example the center of mass of the boundary of the smart obstacle). We assume that the obstacle and the ghost obstacle are such that they contain the centers of mass of their boundaries. The choice of taking the center of mass of the boundary of the obstacle as origin is made only for simplicity and any other point in the interior of the obstacle can be used without substantial changes. We note that we assume that the smart obstacle is a starlike set with respect to the center of mass of its boundary. This assumption is due to the particular mathematical formulation of the direct ghost obstacle scattering problem used and can be substituted with other assumptions.
3. A Numerical Experiment
Let (r, q, n)
be the canonical spherical coordinate system with pole in the orgin of the Cartesian coordinate system
shown in Figure 1.
The setting of the numerical experiment consists in an acoustically soft (i.e. c=0) smart
sphere
of radius one (i.e. the smart obstacle W) (see Figure 1) that tries to
appear outside a shere of radius 2.4 that will be called W_{a}^{*}
(see Figure 1) as an acoustically soft
(i..e. c_{G}=0) sphere of radius 0.5 (i.e. the ghost obstacle W_{G}) in a location
in space different from its actual one (see Figure 1). The smart sphere is hit by an incoming plane wave
of the form:
| (1) |
where k=1, c=1, a=(sinqcosn, sinqsinn,cosq)^{T} where 0 ≤ q ≤ p, 0 ≤ n < 2p, for some choices of the incoming direction a.
In particular the ghost sphere has center in the point of cartesian coordinates (1.7,0,0)^{T} in the frame that has the origin the center of mass of the smart obstacle. The boundary of the sphere of radius 2.4 will be denoted in the following with ¶ W_{e}, e =1.4.
Figure 1: The numerical experiment: W=sphere of radius one, W_{G}=sphere of radius 0.5, W^{*}_{a}=W_{e}=sphere of radius 2.4. |
Let q_{i} = ip/10, i = 1,2,¼,9, f_{j} = 2jp/10, j = 1,2,¼,9 and let us denote with x_{e}(q,f) a point of ¶W_{e}. Table 1 shows how the field scattered by the smart obstacle resembles the one scattered by the ghost obstacle. That is shows how much the smart obstacle has achieved its goal. We compute the fields u^{s}_{k, a}( x_{e}(q_{i},f_{j})), u^{s}_{G,k, a}( x_{e}(q_{i},f_{j})), u^{s}_{p,k, a}( x_{e}(q_{i},f_{j})), x_{e}(q_{i},f_{j}) = (1+e)(sinq_{i}cosf_{j},sinq_{i}sinf_{j},cosq_{i})^{T} Î ¶W_{e}, e = 1.4, i = 1,2,¼, 9, j = 1,2,¼,9 scattered by the smart obstacle, by the ghost obstacle and by the passive obstacle respectively when hit by a harmonic plane wave (1) with c = 1, k = 1, a = (sinq^{*}cosf^{*},sinq^{*}sinf^{*},cosq^{*})^{T} for several choices of q^{*}, f^{*} as shown in Table 1. We consider the following quantities:
| (2) |
| (3) |
l = 0.5, V = 1 | |||
q^{*} | f^{*} | E_{Smart} | E_{Passive} |
p/5 | p/4 | 0.630 | 1.861 |
2p/5 | p/4 | 0.582 | 1.871 |
p/2 | p/2 | 0.593 | 1.801 |
l = 0.8, V = 1 | |||
q^{*} | f^{*} | E_{Smart} | E_{Passive} |
p/5 | p/4 | 0.332 | 1.861 |
2p/5 | p/4 | 0.341 | 1.871 |
p/2 | p/2 | 0.321 | 1.801 |
Table 1: Comparison between the acoustic fields scattered by the ghost obstacle and by the smart obstacle (E_{Smart}) and comparison between the acoustic fields scattered by the ghost obstacle and by the passive obstacle (E_{Passive})
Table 1 shows that the quantity E_{smart} is smaller than the quantity E_{passive} and the quantity E_{smart} decreases when l increases, that is shows that the smart obstacle is achieving its goal. As discussed in [8] the parameter l is such that 0≤ l ≤ 1 and when l increases the ability of the smart obstacle of pursuing its goal increases.
The following virtual reality animation shows the approximation of the point in the interior of the smart obstacle generated by the iterative procedure described in [8].
The setting of the experiment is shown in Figure 1. We work in a spherical coordinate system with pole in a point G in the interior of the ghost obstacle. This assumption is made for simplicity and we take G to be the center of mass of the ghost obstacle (i.e. the sphere of radius 0.5). The aim of the inverse procedure is to determine a point in the interior of the smart obstacle, in particular we want to determine the center of mass S of the boundary of the smart obstacle. Hence the unknown that we want to determine consists in the vector p that joins the point G to the point S. The solution of the inverse problem is the vector p=(-1.7,0,0)^{T}. The initial guess p^{*} for the location of the center of mass of the smart obstacle is chosen to be the center of mass of the boundary of the ghost obstacle that is p^{*}=0. The initial guess of the boundary of the smart obstacle has been chosen to be a sphere of radius 2 having its center in the approximation of the point S given by G+p^{*}.=G Moreover we choose c = 1, l = 0.9, m = 1-l, V = 1 (see [8] ). The far field measurements are affected by a noise of 1% that is we consider the measures F_{0,u, p}( x,k, a)(1+E_{r}( x,k, a)), x=(sinqcosn, sinqsinn,cosq)^{T} where 0 ≤ q ≤ p, 0 ≤ n < 2p, where E_{r} is a random variable uniformly distributed in [-10^{-2},10^{-2}], and F_{0,u, p}(x,k, a) is computed solving numerically the direct problem.
Table 2 shows from left to right the number of iteration i, the cartesian components p_{1}^{*}, p_{2}^{*}, p_{3}^{*} of the current approximation p^{*} of the vector p. The sequence of the p^{*} computed in the iteration is shown in Figure 2. . Note that from the seventh iteration (i.e. the point numbered with 3 in Figure 2) the point G+p^{*} is in the interior of the smart obstacle.
iteration i | p_{1}^{*} | p_{2}^{*} | p_{3}^{*} |
0 | 0 | 0 | 0 |
5 | -0.776 | -0.0007 | -0.0008 |
6 | -0.776 | -0.203 | -0.203 |
9 | -1.089 | -0.146 | -0.148 |
12 | -1.204 | -0.443 | -0.372 |
Figure 2: Sequence of points generated by the inversion algorithm. The initial point used by the algorithm is the center of the sphere of radius 0.05 numbered with 0 that is the center of mass of the ghost obstacle. Some of the points generated by the algorithm are shown as spheres of radius 0.05. These points are numbered progressively to show the order in which they are generated by the algorithm.
Finally we present a virtual reality animation of the numerical experiment discussed above to help the understanding of the behaviour of the reconstruction procedure. The virtual reality animation consists in four windows. Three windows are static windows showing the ghost obstacle, the smart obstacle and the setting of the experiment. The remaining window is a dynamic window showing how the reconstruction of the point in the interior of the smart obstacle takes place.
Virtual reality animation of the reconstruction of the location of the smart obstacle: Animated reconstruction of the location of the smart obstacle (Virtual scene: VRML file)
4. References
5. Useful Links