Two-dimensional topography estimation from noisy free surface synthetic data

The inverse topography problem involves estimating the bottom topography from knowledge of a free surface. In this paper, we present a weakly nonlinear approach to address this problem using the forced Korteweg–de Vries equation for steady two-dimensional free surface flow of an inviscid incompressible fluid corrupted by measurement noise. Owing to the presence of noise, there is a compounding inverse solution error as the grid resolution is refined due to numerical differentiation and so a linear least squares regression method is proposed as a way to mitigate noise on the surface. A closed-form expression for the mean and variance of the inverse solutions is derived to examine the uncertainty of our topographic predictions. We demonstrate the adaptability of our contributions to six classical flow types, providing a systematic method for quantifying the uncertainty in topographic estimates that is insightful and interpretable.