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

TL;DR

This work tackles the inverse tsunami run-up problem for the 1D nonlinear shallow water equations on inclined power-shaped bays. By applying the Carrier–Greenspan hodograph transform, the nonlinear system is linearized in a transformed plane, allowing shoreline data $R(t)$ to encode both the initial displacement $η_0(x)$ and initial velocity $u_0(x)$ through Abel-type transforms. A key result is the separation of initial conditions at the shoreline: the even component of the shoreline pressure depends on $η_0$ while the odd component depends on $u_0$, enabling exact recovery from $R(t)$ when the transform is invertible. Numerical experiments with Gaussian, soliton, and N-wave inputs confirm accurate reconstruction and illustrate computational efficiency via FFT/FHT-based transforms. Overall, the paper uncovers a surprising asymmetry between the direct and inverse problems and offers a practical pathway for tsunami forecasting using shoreline records.

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 $η_0(x)$ 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 $η_0$ 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 $η_0(x$) 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 $η_0$ 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.

Figures (4)

• Figure 1: A sketch of the parabolic bay geometry. The bathymetry $z (x, y) = - x + \left\lvert y\right\rvert^m$ is in red, the unperturbed water level is in yellow, an the water level is in blue. The total perturbed water depth is given by $H(x,t)=h(x)+\eta(x,t)$.
• Figure 2: Comparison of the original initial conditions to the recovered initial conditions for a Gaussian wave.
• Figure 3: Comparison of the original initial conditions to the recovered initial conditions for a soliton wave.
• Figure 4: Comparison of the original initial conditions to the recovered initial conditions for an N-wave.

Theorems & Definitions (2)

• Proposition 1
• Proposition 2