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.

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The study of open channel flow modelling often requires an accurate representation of the channel bed topography to accurately predict the flow hydrodynamics. Experimental techniques are the most widely used approaches to measure the bed topographic elevation of open channels. However, they are usually cost and time consuming. Free surface measurement is, on the other hand, relatively easy to obtain using airborne photographic techniques. We present in this work an easy to implement and fast to solve numerical technique to identify the underlying bedrock topography from given free surface elevation data in shallow open channel flows. The main underlying idea is to derive explicit partial differential equations which govern this inverse reconstruction problem. The technique described here is a "one-shot technique" in the sense that the solution of the partial differential equation provides the solution to the inverse problem directly. The idea is tested on a set of artificial data obtained by first solving the forward problem governed by the shallow-water equations. Numerical results show that the channel bed topographic elevation can be reconstructed with a level of accuracy less than 3%. The method is also shown to be robust when noise is present in the input data.