River Bathymetry Estimation Using Surface Elevation Data

Abstract:

Estimation of river bathymetry from remotely sensed water surface elevation data is presented using variational inverse modeling applied to the 2D depth-averaged, shallow-water equations (SWEs). We pose the problem as a one-dimensional inversion of the depth along the thalweg, utilizing a strong-constraint formalism. The optimization problem is constructed around a cost function defined as the error between estimated and observed transverse mean water surface elevation along the river; the cost function is augmented by the SWEs (the constraints) using Lagrange multipliers. This mathematical construction allows the development of the adjoint SWEs, as well as the gradient of the cost function with respect to the maximum depth along the river, which can be computed from the forward and adjoint solutions. An iterative algorithm is used to obtain a bathymetric profile that results in water surface elevations that are a best fit to the data, as constrained by the governing equations. Defining the river cross-sectional profile as parabolic in nature, and presuming knowledge of discharge and bottom friction, the approach is shown to be effective in capturing large-scale bathymetric features. The algorithm has been vetted against synthetic data generated for two river-reaches: 2.5km of the Snohomish River in Washington State, and 17km of the Kootenai River in Idaho. These rivers were chosen due to the availability of high-resolution bathymetric data, as well as the significant difference in overall characteristics (e.g. mean slope). A range of river discharges and bottom frictions were used to generate the synthetic surface elevation datasets, using Delft3D as the hydrodynamic model. Overall, the results indicate that the approach is valid, with the algorithm performing best in shallower water (see figure). Initial results for joint estimation of river depth and discharge are also presented.