The shallow water equations (SWE) model a variety of geophysical flows. Flows in channels with rectangular cross sections may be modelled 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 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. 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.

翻译：浅水方程(SWE) 模型为各种地球物理流。 带有矩形交叉段的通道流动可能以简化的单维SWE为模型,宽度不一。 在其他模型参数中,需要关于测深和摩擦系数的信息,以便正确和准确地预测水流。虽然模型参数的合成值可能足以测试数字方案,但测深和其他参数的近似值可能是应用所需的。通过实验方法可能获得估计值,但其中一些技术可能是昂贵的、耗时的,而且并不总是可用。在这项工作中,我们提议从中流速度数据中解决测深和曼宁摩擦系数的反向问题。这需要利用成本功能的帮助,包括SWE,通过拉格朗乘数乘数计算。解决办法是通过连续的下降方法解决受限制的优化问题。直接和相交错的问题都是用二阶精确的Roe型上风方案在数字上解决的。包括数字测试,以显示算法的优点。