Separation of the initial conditions in the inverse problem for 1D non-linear tsunami wave run-up theory

TL;DR

This paper presents a method to uniquely recover initial wave displacement and velocity in 1D nonlinear tsunami models from shoreline motion data, extending previous work to include non-zero initial velocities.

Contribution

The authors develop an algorithm that solves the inverse tsunami problem for inclined bathymetries with non-zero initial velocities, using Abel transform and Carrier-Greenspan transformation.

Findings

Exact recovery of initial conditions for non-zero velocities.

Algorithm validated by extensive numerical experiments.

Extension of previous inverse problem solutions to more general initial states.

Abstract

We investigate the inverse tsunami wave problem within the framework of the 1D nonlinear shallow water equations (SWE). Specifically, we focus on determining the initial displacement $\eta_0(x)$ and velocity $u_0(x)$ of the wave, given the known motion of the shoreline $R(t)$ (the wet/dry free boundary). 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. In particular, we show that the knowledge of $R(t)$ enables the unique recovery of both $\eta_0(x$) and $u_0(x)$ in terms of the Abel transform. It is important to note that, in contrast to the direct problem (also known as the tsunami wave run-up problem), where $R(t)$ can be computed exactly only for $u_0(x)=0$, 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 in \cite{Rybkin23,Rybkin24}, where the inverse problem was solved for $u_0(x)=0$. 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.