Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reconstruction of the non-linear wave at a buoy from shoreline data and applications to the tsunami inverse problem for piece-wise sloping bathymetry

Oleksandr Bobrovnikov, Madison Jones, Shriya Prasanna, Josiah Smith, Alexei Rybkin, Efim Pelinovsky

TL;DR

The paper tackles the inverse problem of reconstructing the initial tsunami shape from shoreline run-up within 1+1D dynamics governed by the NSWE, and extends this framework to piecewise sloping bathymetry by incorporating dispersion through the Boussinesq equation. It derives an analytical inversion based on the Carrier-Greenspan transform to recover buoy boundary data $(u(L,t), \eta(L,t))$ from the shoreline function $R(t)$, validated by numerical experiments and a cost analysis that emphasizes feasibility in 1D. Beyond NSWE, the authors present a stitching approach with a dispersive Boussinesq model on a flat region and explore a linear SWE variant on the half-line, using soliton- and traveling-wave-based boundary inputs to test the methodology. The work offers practical implications for optimizing buoy placement (e.g., DART-like systems) and provides a path toward dispersion-aware tsunami inversion on more complex bathymetries, with future work on admissible shoreline data and multi-slope configurations.

Abstract

We discuss the following inverse problem: given the run-up data of a tsunami wave, can we recover its initial shape? We study this problem within the framework of the non-linear shallow water equations, a model widely used to study tsunami propagation and inundation. Previously, it has been demonstrated that in the case of infinite sloping bathymetry, it is possible to recover the initial water displacement and velocity from shoreline readings \cite{Rybkin23,Rybkin24,Rybkin25}. We consider a finite sloping bathymerty. We show that it is possible to recover boundary conditions (water displacement and velocity) on a virtual buoy from the shoreline data. Further, we discuss stitching together the shallow water equations and the Boussinesq equation in a more complex piece-wise sloping bathymetry in order to recover the initial conditions, while incorporating the dispersion to our model.

Reconstruction of the non-linear wave at a buoy from shoreline data and applications to the tsunami inverse problem for piece-wise sloping bathymetry

TL;DR

The paper tackles the inverse problem of reconstructing the initial tsunami shape from shoreline run-up within 1+1D dynamics governed by the NSWE, and extends this framework to piecewise sloping bathymetry by incorporating dispersion through the Boussinesq equation. It derives an analytical inversion based on the Carrier-Greenspan transform to recover buoy boundary data from the shoreline function , validated by numerical experiments and a cost analysis that emphasizes feasibility in 1D. Beyond NSWE, the authors present a stitching approach with a dispersive Boussinesq model on a flat region and explore a linear SWE variant on the half-line, using soliton- and traveling-wave-based boundary inputs to test the methodology. The work offers practical implications for optimizing buoy placement (e.g., DART-like systems) and provides a path toward dispersion-aware tsunami inversion on more complex bathymetries, with future work on admissible shoreline data and multi-slope configurations.

Abstract

We discuss the following inverse problem: given the run-up data of a tsunami wave, can we recover its initial shape? We study this problem within the framework of the non-linear shallow water equations, a model widely used to study tsunami propagation and inundation. Previously, it has been demonstrated that in the case of infinite sloping bathymetry, it is possible to recover the initial water displacement and velocity from shoreline readings \cite{Rybkin23,Rybkin24,Rybkin25}. We consider a finite sloping bathymerty. We show that it is possible to recover boundary conditions (water displacement and velocity) on a virtual buoy from the shoreline data. Further, we discuss stitching together the shallow water equations and the Boussinesq equation in a more complex piece-wise sloping bathymetry in order to recover the initial conditions, while incorporating the dispersion to our model.
Paper Structure (12 sections, 38 equations, 16 figures)

This paper contains 12 sections, 38 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Inverse Problem for NSWE on the Finite Interval
  3. Numerical Verification
  4. Computational Cost
  5. Boussinesq Equation
  6. Heuristic Understanding
  7. Implementation
  8. Linear Shallow Water Equations
  9. Inversion Implementation
  10. Comparison
  11. Conclusions
  12. Justification of swapping integration and summation

Figures (16)

  • Figure 1: Sketch of the bay geometry. Cross-sectional view; bathymetry is in red, unperturbed water level is in yellow, and water level is in blue.The total perturbed water depth is given by $H(x,t)=h(x)+\eta(x,t)$.
  • Figure 2: Initial displacements as defined in \ref{['eq:ics_eta0']}.
  • Figure 3: Run-ups corresponding to IC in \ref{['eq:ics_eta0']}.
  • Figure 4: Comparison of $\psi_{\text{b}}(\tau)$ and $\psi_{\text{b}}^{\text{e}}(\tau)$ corresponding to IC in \ref{['eq:ics_eta0']}.
  • Figure 5: Comparison of $\varphi_{\text{b}}(\tau)$ and $\varphi_{\text{b}}^{\text{e}}(\tau)$ corresponding to IC in \ref{['eq:ics_eta0']}.
  • ...and 11 more figures