Scattering amplitude and algorithm for solving the inverse scattering problem for a class of non-convex obstacles
Let B be a bounded, simply connected domain in \({\mathbb{R}}^ 3\) with smooth boundary \(\Gamma\). We define the scattering amplitude \(f(s,s_ 0,k)\) by \[ (\nabla^ 2+k^ 2)u=0\quad in\quad {\mathbb{R}}^ 3\setminus \bar B;\quad u=0\quad on\quad \Gamma, \] \[ u=u_ 0+v,\quad u_ 0=\exp (iks_ 0\cdot x),\quad | s_ 0| =1, \] \[ v\sim r^{-1} \exp (ikr)f(s,s_ 0,k),\quad r=| x|,\quad s=x/| x|. \] The inverse scattering problem consists in finding the surface \(\Gamma\) from the scattering data. In this paper, as the idea, the values of \(f(s,s_ 0,k)\) at a fixed k and for s and \(s_ 0\) are used, where s and \(s_ 0\) are running through small solid angles around the directions \(\{\pm s_ j\}^ N_{j=1}\), which are uniformly distributed on the unit sphere. The main results of the paper are the calculation of the high frequency asymptotics of the scattering amplitude and an algorithm for recovering the surface from the scattering data for a class \(R_ 1\) of non-convex obstacles. It is shown that the values of v and f converge asymptotically in the \(L^ 2\)-sense to the values predicted by the Kirchhoff approximation. Using this result some properties of the scattering amplitude are given. The high frequency limit of the total cross section is calculated. It is also shown that the ''diffraction peak'' in forward scattering direction is given by the Fourier transformation of the shadow projection of the obstacle. Moreover, the analytical formula predicted by the Kirchhoff approximation is used for recovering the surface \(\Gamma\) of the obstacle.
scientific article; zbMATH DE number 4008965
scientific article