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Mathematical Problems in Engineering
Volume 2012, Article ID 417950, 18 pages
Research Article

A Direct Solution Approach to the Inverse Shallow-Water Problem

1Department of Mechanical Engineering, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand
2School of Mechanical and Industrial Engineering, Bahir Dar University, P.O. Box 26, Bahir Dar, Ethiopia

Received 26 July 2012; Revised 28 October 2012; Accepted 29 October 2012

Academic Editor: Fatih Yaman

Copyright © 2012 Alelign Gessese and Mathieu Sellier. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.