Deep learning-based estimation of riverine bathymetry using Variational Encoder Geostatistical Approaches (VEGAs)

Estimation of the riverbed profiles, also known as bathymetry, is important in many applications, such as safe and efficient navigation, prediction of bank erosion, and flood risk management. The high cost and complex logistics of direct bathymetry surveys have encouraged the use of indirect measurements such as surface flow velocities. However, estimation of high-resolution bathymetry from indirect measurements is an ill-posed inverse problem that can be computationally challenging as it requires iterative calls to complex models to compute Jacobians and large dense matrix operations. Reduced-order modeling (ROM) is a promising tool to decrease the computational cost of these inversion problems, since it can replace calls to expensive forward models with fast, low-dimensional approximations. However, many ROMs rely on linear projection like Proper Orthogonal Decomposition (POD). Thus, they may require many modes to describe system dynamics accurately when solutions reside on a nonlinear manifold, and so lose much of their advantage over high-fidelity simulations. In this talk, we propose a ROM that is equipped with a supervised variational encoder (SVE) with a narrow layer in the middle (the reduced dimension), to compress bathymetry and flow velocity information and accelerate bathymetry inversion from flow velocity measurements. In our application, the shallow-water equations (SWEs) with appropriate boundary conditions constitute the forward problem. Then, inversion with uncertainty quantification is performed on the low-dimensional latent space in a Bayesian setting. Our Bayesian view is similar to that of the Principal Component Geostatistical Approach (PCGA). However, the forward ROM is constructed on a nonlinear manifold via SVE, and thus our algorithm is called Variational Encoder Geostatistical Approaches (VEGAs). We have tested our inversion approach on the Savannah River, GA, USA. We show that once the neural network is trained, the VEGAs can perform the inversion orders of magnitude faster than traditional inversion methods. Furthermore, we show that even with sparse measurements, the algorithm can estimate the bathymetry with good accuracy. Finally, we show that our approach can be used as a tool for the design of experiment when limited measurement resources are available.