Bathymetry reconstruction from experimental data using PDE-constrained optimisation

TL;DR

This work addresses inferring bathymetry from surface-wave data by formulating a PDE-constrained inverse problem using the nonlinear shallow water equations $(SWE)$ and a continuous adjoint for gradient computation. The authors implement a gradient-descent reconstruction with boundary data from a single sensor and observations from additional sensors, validating the forward model against measurements and then applying the method to real wave-flume data. They demonstrate that a Gaussian-shaped bathymetry can be recovered qualitatively using two to three sensors, achieving an $NRMSE$ around $14\%$, and provide open-source code and data for reproducibility. The study highlights the feasibility and limitations of this approach, notably sensitivity to sensor placement and the SWE’s inability to capture dispersive effects, suggesting future work on sensor optimization and model enhancements. The results offer a practical pathway for bathymetry estimation from surface measurements and establish a baseline for integrating PDE-constrained optimisation with real-world hydrographic data.

Abstract

Knowledge of the bottom topography, also called bathymetry, of rivers, seas or the ocean is important for many areas of maritime science and civil engineering. While direct measurements are possible, they are time consuming and expensive. Therefore, many approaches have been proposed how to infer the bathymetry from measurements of surface waves. Mathematically, this is an inverse problem where an unknown system state needs to be reconstructed from observations with a suitable model for the flow as constraint. In many cases, the shallow water equations can be used to describe the flow. While theoretical studies of the efficacy of such a PDE-constrained optimisation approach for bathymetry reconstruction exist, there seem to be few publications that study its application to data obtained from real-world measurements. This paper shows that the approach can, at least qualitatively, reconstruct a Gaussian-shaped bathymetry in a wave flume from measurements of the water height at up to three points. Achieved normalized root mean square errors (NRMSE) are in line with other approaches.

Figures (8)

• Figure 1: Free surface elevation $H$ (dotted blue), bathymetry $b$ (brown) and water height $h$. Our aim is to reconstruct $b$ from point measurements of $H$.
• Figure 2: Sketch of used experimental setup using the wave flume at the Institute of Mechanics and Ocean Engineering at Hamburg University of Technology. Waves are generated at the left by a wave flap. Four sensors are positioned throughout the flume at increasing distance from the flap. The bathymetry is positioned between sensor 2 and sensor 3. At the right, a beach is installed to minimise reflections. In all simulations, we use the readings from sensor 1 to define the left boundary condition while readings from sensors 2, 3, 4 are used to reconstruct the bathymetry.
• Figure 3: Measured points (top) and Dedalus representation (bottom) of the bathymetry that was used in the experiment. The Dedalus representation in the lower figure was generated by manually copying the coordinates from the data points in the upper sketch and connecting them using SciPy's CubicSpline function.
• Figure 4: Measured (dashed) free surface elevation $H$ (dashed) with $95\%$ confidence interval (grey) and numerical simulation (dotted).
• Figure 5: Difference between the averages of measurements with and without bathymetry (dashed) and $95\%$ confidence interval (grey).