Some properties of the inverse operator for a tsunami source recovery

The application of an inversion method to the problem of recovering the initial water elevation field generating a tsunami is considered. This article addresses the usually ill-posed problem of the hydrodynamical inversion of remote measurements of water-level data without any a priori information on a source but its very general spatial localization. The tsunami wave propagation is considered within the scope of the linear shallow-water theory. The ill-posed inverse problem at hand is regularized by means of the least square inversion using the truncated $SVD$ approach. Some properties of inverting operator in the context of retrieving a tsunami source are studied numerically. The goodness of the inversion is defined by the relative errors of the tsunami source reconstruction. As the result, we find the dependence of the goodness of the inversion in terms of the number of receivers, their azimuthal coverage and the frequency band. The applied approach allows one to control the instability of the numerical solution and to obtain an acceptable result in spite of the ill-posedness of the problem and makes it possible to predict a probable quality of the inversion by a certain observational system. This result should be kept in mind when designing a tide-gauge network to study a tsunami source.