Simultaneous reconstruction of material and transient source parameters using the invariant imbedding method

This paper extends the time domain wave splitting and invariant imbedding method to an inhomogeneous wave equation with a source term: uxx−utt+A(x)ux=2D(x)i′(t). The direct scattering and inverse source problems of this equation are studied. Operators J̃± that map the source function into the scattered waves at the edges of the slab are defined. A system of coupled nonlinear integrodifferential equations for these scattering operator kernels is obtained. The direct scattering problem is to obtain the scattering operator kernels J± and R+ when parameters A and D are given. The inverse problem is to simultaneously reconstruct A(x) and D(x) from the scattering operator kernels R+(0,t), 0≤t≤2 and J−(0,t), 0≤t≤1. Both numerical inversion algorithms and the small time approximate reconstruction method are presented. A Green’s function technique is used to derive Green’s operator kernel equations for the calculation of the internal field. It provides an alternative effective and fast way to compute the scattering kernels J±. For constant A and D the Green’s operator kernels and source scattering kernels are expressed in closed form. Several numerical examples are given.