4D‐Var Using Hessian Approximation and Backpropagation Applied to Automatically Differentiable Numerical and Machine Learning Models

Abstract Constraining a numerical weather prediction (NWP) model with observations via 4D variational (4D‐Var) Data assimilation (DA) is often difficult to implement due to the need to develop and maintain a software‐based tangent linear model and adjoint model. One of the most common 4D‐Var algorithms uses an incremental update procedure, which has been shown to be an approximation of the Gauss‐Newton method. Here we demonstrate that when using a forecast model that supports flexible automatic differentiation, an efficient and in some cases more accurate alternative approximation of the Gauss‐Newton method can be applied by combining backpropagation of errors with a Hessian approximation. This approach can be used with either a conventional physical model implemented with automatic differentiation or a machine learning (ML) based surrogate model. We test the new approach on a variety of Lorenz‐96 and quasi‐geostrophic models. The results indicate potential for a deeper integration of modeling, DA, and new technologies in a next‐generation of operational forecast systems that leverage weather models designed to support flexible, on‐the‐fly automatic differentiation.