We consider the Boussinesq-Peregrine (BP) system as described by Lannes [Lannes, D. (2013). The water waves problem: mathematical analysis and asymptotics (Vol. 188). American Mathematical Soc.], within the shallow water regime, and study the inverse problem of determining the time and space variations of the channel bottom profile, from measurements of the wave profile and its velocity on the free surface. A well-posedness result within a Sobolev framework for (BP), considering a time dependent bottom, is presented. Then, the inverse problem is reformulated as a nonlinear PDEconstrained optimization one. An existence result of the minimum, under constraints on the admissible set of bottoms, is presented. Moreover, an implementation of the gradient descent approach, via the adjoint method, is considered. For solving numerically both, the forward (BP) and its adjoint system, we derive a universal and low-dissipation scheme, which contains non-conservative products. The scheme is based on the FORCE-{\alpha} method proposed in [Toro, E. F., Saggiorato, B., Tokareva, S., and Hidalgo, A. (2020). Low-dissipation centred schemes for hyperbolic equations in conservative and non-conservative form. Journal of Computational Physics, 416, 109545]. Finally, we implement this methodology to recover three different bottom profiles; a smooth bottom, a discontinuous one, and a continuous profile with a large gradient. We compare with two classical discretizations for (BP) and the adjoint system. These results corroborate the effectiveness of the proposed methodology to recover bottom profiles.

翻译：本文考虑Lannes [Lannes, D. (2013). The water waves problem: mathematical analysis and asymptotics (Vol. 188). American Mathematical Soc.] 描述的浅水体制下Boussinesq-Peregrine (BP)系统，研究通过自由表面波剖面及其速度测量值反演河道底床时空变化的逆问题。首先给出含时变底床条件下BP系统在Sobolev框架中的适定性结果；继而将逆问题重构为非线性偏微分方程约束优化问题，提出在底床可行集约束下存在最小值解的理论证明。进一步采用伴随方法实现梯度下降法的数值求解。针对正问题(BP)及其伴随系统，我们推导出一种包含非守恒乘积项的通用低耗散格式，该格式基于Toro等人 [Toro, E. F., Saggiorato, B., Tokareva, S., and Hidalgo, A. (2020). Low-dissipation centred schemes for hyperbolic equations in conservative and non-conservative form. Journal of Computational Physics, 416, 109545] 提出的FORCE-α方法。最后通过三种典型底床形态（光滑底床、间断底床和含大梯度连续底床）的恢复实验，并与BP系统及伴随系统的两种经典离散格式进行对比，验证了该方法在底床反演中的有效性。