Application of $r$-solutions to reconstructing the initial tsunami waveform

A new approach to reconstructing the initial tsunami waveform in a tsunami source area is proposed. The approach is based on the inversion of remote measurements of water-level data. The tsunami wave propagation is considered within the scope of the linear shallow-water theory. The numerical simulation is based on the finite-difference technique and the method of splitting. The ill-posed inverse problem of reconstructing initial tsunami waveforms is regularized by means of a least-square inversion using the truncated SVD approach. As a result of the numerical process, an $r$-solution is obtained. This method 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. The algorithm was verified by the numerical simulating with real bathymetry of the Peru subduction zone and synthetic data. It is shown that the accuracy of the tsunami source reconstruction strongly depends on the signal-to-noise ratio, the azimuthal and temporal coverage of assimilated tide gauge stations relative to the target area, and the bathymetric features along the wave path. The results thus obtained show that the initial tsunami waveform reconstruction by the technique presented in this paper is successful.