Recovery of a Time-Dependent Bottom Topography Function from the Shallow Water Equations via an Adjoint Approach

TL;DR

This paper presents a robust adjoint-based method for recovering riverbed topography from noisy boundary data in shallow water models, incorporating regularization and demonstrating effectiveness through numerical tests.

Contribution

It introduces a novel adjoint approach with regularization for topography recovery in shallow water equations, capable of handling noisy data and discontinuities.

Findings

Method robust to initial guess and noise

Effective in presence of solution discontinuities

Regularization improves recovery accuracy

Abstract

We develop an adjoint approach for recovering the topographical function included in the source term of one-dimensional hyperbolic balance laws. We focus on a specific system, namely the shallow water equations, in an effort to recover the riverbed topography. The novelty of this work is the ability to robustly recover the bottom topography using only noisy boundary data from one measurement event and the inclusion of two regularization terms in the iterative update scheme. The adjoint scheme is determined from a linearization of the forward system and is used to compute the gradient of a cost function. The bottom topography function is recovered through an iterative process given by a three-operator splitting method which allows the feasibility to include two regularization terms. Numerous numerical tests demonstrate the robustness of the method regardless of the choice of initial guess and in the presence of discontinuities in the solution of the forward problem.