Bathymetry reconstruction based on the zero-inertia shallow water approximation

Abstract

The knowledge of the channel bed topography is paramount in modeling the hydrodynamics of open channel flows. Indeed, flow models based on the Shallow Water Approximation require prior information on the channel bed topography to accurately capture the flow features in natural rivers, estuaries, and flood plains. We present here a numerical technique for reconstructing the channel bed topography from given free surface elevation data for steep open channel flows for which the zero-inertia shallow water approximation holds. In this context, the shallow water equations are modified by neglecting inertia terms while retaining the effects of the bed slope and friction terms. We show in this work that by algebraic manipulation, we can recast the governing equations into a single first-order partial differential equation which describes the inverse problem which consists in finding the bed topography from known free surface elevation data. Interestingly, the analysis shows that the inverse problem does not require the knowledge of the bed roughness. The forward problem is solved using MacCormack’s explicit numerical scheme by considering unsteady modified shallow water equations. However, the inverse problem is solved using the method of characteristics. The results of the inverse and the forward problem are successfully tested against each other on two different test cases.