Calculation of the initial elevation of the water surface at the tsunami source in the basin with arbitrary bottom topography

Within the framework of the potential theory of an incompressible fluid, under the instantaneous sea-bottom deformation assumption, a two-dimensional ($0xz$) numerical model is developed, which makes it possible to calculate the initial elevation of the water surface in the tsunami source in a basin of variable depth. Through the use of the $\sigma$-coordinate, the model makes it possible to take into account the contribution of the horizontal component of the bottom deformation and the “smoothing effect” of the water layer. To test the numerical model, we obtained an analytical solution to the problem of the initial elevation in a basin with a flat sloping bottom with bottom deformation of a triangular shape. The results of the test show that for a spatial step typical for numerical tsunami models, there is a good agreement between the numerical and analytical solutions. Using the developed $\sigma$-model, we calculated the initial elevations of the water surface during the Kuril earthquake on January 13, 2007 and the Great East Japan Tohoku earthquake on March 11, 2011 (along selected 2D sections). The results obtained are used to test an approximate method for calculating the initial elevation, known as the Kajiura filter, in which the ocean depth is assumed to be constant throughout the entire tsunami source area.