Reconstruction of river bed topography from free surface data using a direct numerical approach in one-dimensional shallow water flow

Abstract

River bed topography is of paramount importance for the study of fluvial hydraulics, flood prediction and river flow monitoring. It is therefore important to develop fast, easy-to-implement and cost-effective methods to determine underwater river topography. This paper presents a new one-step approach for reconstructing the river bed topography from known free surface data. This problem corresponds to the inverse of the classical hydrodynamic problem where the shallow water equations provide the free surface profile for a given river bed. We show in this work that instead of treating this inverse problem in the traditional partial differential equation (PDE)-constrained optimization framework, we can conveniently rearrange the governing equations for the direct problem to obtain an explicit PDE for the inverse problem. This leads to a direct solution of the inverse problem. An interesting consequence of the analysis is that the equations governing the forward and inverse problems have a very similar form, and the same discretization technique, based on an upwind conservative numerical scheme, can be used. The proposed methodology is successfully tested on a range of benchmark problems for noisy and noiseless free surface data. It was found that this solution approach creates very little amplification of noise.

Inference of effective river properties from remotely sensed observations of water surface

“…Biancamaria et al [20] improve the estimation of the bathymetry and discharge of an Arctic river through the assimilation of synthetic SWOT observations, assuming that river bathymetry and roughness are known. In [21,22] basic bathymetries (e.g. a single smooth bump) are identified from extremely dense measurements of water levels by inverting an explicit time-step scheme, while the roughness is supposed to be known.…”

Estimating the ice thickness of mountain glaciers with a shape optimization algorithm using surface topography and mass-balance

“…The recently introduced TIM, which is a particular case of SOA where the costate variables are set to zero, is an exception, since it minimizes the surface topography misfit in both steady and transient situations with a simple fixed-point procedure. Described as unreliable in the shallow water framework [15], the SOA applied to the shallow ice equation appears to be accurate, reliable, and fast. In particular, even if the initial guess for the bedrock (d) α = 10 Figure 6.…”

“…In particular, neither surface filtering with a lower slope limit nor assumptions on the basal shear stress are applied, contrary to most of the other existing methods [13,20,30,38]. The SIM was inspired by a procedure used to reconstruct river beds in the Shallow Water context [15]. While this method is more reliable and easily implementable than a Shape Optimization Algorithm (SOA) [42] in the Shallow Water Approximation, we will see that this is not the case anymore in the SIA.…”

The inverse problem in creeping film flows

“…Heining and Aksel [7] expanded the ideas and included the influence of inertia. Inspired by these studies, Gessese et al [8] applied similar methods to solve the inverse problem for river flows in the shallow water framework. A solution of the full field equations was provided by Heining [9] who developed a numerical algorithm to solve the inverse problem based on the Navier-Stokes equations.…”