Local Bathymetry Estimation Using Variational Inverse Modeling: A Nested Approach

Abstract:

Estimation of subreach river bathymetry from remotely-sensed surface velocity data is presented using variational inverse modeling applied to the 2D depth-averaged, shallow-water equations (SWEs). A nested approach is adopted to focus on obtaining an accurate estimate of bathymetry over a small region of interest within a larger complex hydrodynamic system. This approach reduces computational cost significantly. We begin by constructing a minimization problem with a cost function defined by the error between observed and estimated surface velocities, and then apply the SWEs as a constraint on the velocity field. An adjoint SWE model is developed through the use of Lagrange multipliers, converting the unconstrained minimization problem into a constrained one. The adjoint model solution is used to calculate the gradient of the cost function with respect to bathymetry. The gradient is used in a descent algorithm to determine the bathymetry that yields a surface velocity field that is a best-fit to the observational data. In this application of the algorithm, the 2D depth-averaged flow is computed within a nested framework using Delft3D-FLOW as the forward computational model. First, an outer simulation is generated using discharge rate and other measurements from USGS and NOAA, assuming a uniform bottom-friction coefficient. Then a nested, higher resolution inner model is constructed using open boundary condition data interpolated from the outer model (see figure). Riemann boundary conditions with specified tangential velocities are utilized to ensure a near seamless transition between outer and inner model results. The initial guess bathymetry matches the outer model bathymetry, and the iterative assimilation procedure is used to adjust the bathymetry only for the inner model. The observation data was collected during the ONR Rivet II field exercise for the mouth of the Columbia River near Hammond, OR. A dual beam squinted along-track-interferometric, synthetic-aperture radar (ATI-SAR) system was used which provides high-resolution nearly bank-to-bank vector-surface-velocity observations of the river mouth. Data and bathymetry estimation results are presented, and the algorithm results are compared to contemporaneous USGS measured bathymetry data, with favorable results.