Time domain direct and inverse problems for a nonuniform LCRG line with internal sources

In this paper, direct and inverse problems for a nonuniform LCRG line with internal transient voltage and current sources are considered. In the inverse source problem, the compact Green functions approach based on wave splitting is used to reconstruct the transient voltage and current sources from the transient signals that are received at the two endpoints of the nonuniform line. The location of the sources is assumed to be known or possible to estimate from the arrival times of the signals. The compact Green functions are convolution kernels in the map from the received signals (at the two endpoints) to the internal split voltages. The partial differential equations for the compact Green functions are given together with the initial and boundary conditions. The present approach to the inverse source problem is exact and noniterative. To obtain independent synthetic input data for the inverse problem, the direct problem is solved by a finite difference scheme. Numerical results for reconstructions using both clean and noisy data are presented.