Separation of the Initial Conditions in the Inverse Problem for One-Dimensional Nonlinear Tsunami Wave Run-Up Theory

Abstract

We investigate the inverse tsunami wave problem within the framework of the one-dimensional (1D) nonlinear shallow water equations (SWE). Specifically, we show that the initial displacement ${\eta }_0\left(x\right)$ and velocity $u_0\left(x\right)$ of the wave can be recovered, given the known motion of the shoreline $R\left(t\right)$ (the wet/dry free boundary), in terms of the Abel transform. We demonstrate that for power-shaped inclined bathymetries, this problem admits a complete solution for any ${\eta }_0$ and $u_0$, provided the wave does not break.

It is important to note that, in contrast to the direct problem (also known as the tsunami wave run-up problem), where $R\left(t\right)$ can be computed exactly only for $u_0\left(x\right)=0$, our algorithm can recover ${\eta }_0$ and $u_0$ exactly for any non-zero $u_0$. This highlights an interesting asymmetry between the direct and inverse problems. Our results extend the work presented earlier, where the inverse problem was solved for $u_0\left(x\right)=0$. As in previous work, our approach utilizes the Carrier–Greenspan transformation, which linearizes the SWE for inclined bathymetries. Extensive numerical experiments confirm the efficiency of our algorithms.