Correctly placing hydropower plants in a river is one of many examples where good knowledge of the bottom topography, also called bathymetry, is needed. While direct measurement of the bathymetry is possible, for example with a side scan sonar operated by a boat or an underwater remotely operated vehicle, this is very time consuming and expensive. Therefore, methods that can infer the bathymetry from the easier to measure surface height of the water are attractive.
Mathematically speaking, this is an inverse problem where unknown parameters of a system are reconstructed from typically incomplete and noisy measurements of the system state. One approach to solve such inverse problems is so-called partial differential equation constrained optimisation, where system parameters are computed that reproduce the measurements but also satisfy physical constraints like mass or momentum conservation.
Researchers from TUHH’s Institute of Mathematics (E-10) and Institute of Mechanics and Ocean Engineering (M-13) as well as from the the Department of Mathematics at the University of Hamburg (UHH) have published a joint paper that provides the first demonstration that this approach can reconstruct a real-world bathymetry. In their experiment, they placed a small hill, manufactured from skate board ramps, at the bottom of a 12 m long wave flume. The water at rest had a depth of 30cm and waves were being generated by a wave flap. Four sensors were installed that measure wave heights.
This measured data was used to reconstruct the manufactured bathymetry by numerically solving a minimisation problem with the shallow water equations as constraints. The mathematical algorithm was implemented in Python using the Dedalus software. It could generate a qualitative reconstruction of the hill, even though the change in wave height caused by the bathymetry was only in the range of a few millimetres.
Contact:
Judith Angel
judith.angel(a)tuhh.de
Prof. Daniel Ruprecht
ruprecht(a)tuhh.de
Institute of Mathematics (E-10)
