Bathymetry and friction estimation from transient velocity data for one-dimensional shallow water flows in open channels with varying width

Abstract

The shallow water equations (SWE) model a variety of geophysical flows. Flows in channels with rectangular cross sections may be modeled with a simplified one-dimensional SWE with varying width. Among other model parameters, information about the bathymetry and friction coefficient is needed for the correct and precise prediction of the flow. Although synthetic values of the model parameters may suffice for testing numerical schemes, approximations of the bathymetry and other parameters may be required for specific applications. Estimations may be obtained by experimental methods, but some of those techniques may be expensive, time consuming, and not always available. In this work, we propose to solve the inverse problem to estimate the bathymetry and the Manning's friction coefficient from transient velocity data. This is done with the aid of a cost functional, which includes the SWE through Lagrange multipliers. We prove that the velocity data determine uniquely the derivative of the bathymetry in a linearized shallow water system. That is, the inverse problem is identifiable. The solution is obtained by solving the constrained optimization problem by a continuous descent method. The direct and the adjoint problems are both solved numerically using a second-order accurate Roe-type upwind scheme. Numerical tests are included to show the merits of the algorithm.