A Simple Finite Difference‐Based Approximation for Biogeochemical Tangent Linear and Adjoint Models

We present a technique that accurately approximates tangent linear and adjoint models for data assimilation applications using only evaluations of the nonlinear model. The approximation offers a simple way to create tangent linear and adjoint model codes that are easily maintainable, as only major changes to the nonlinear model formulation necessitate modifications of the tangent linear or adjoint model code. The approach is particularly well suited to marine biogeochemical models and takes advantage of typical features of these types of models to be computationally viable. We illustrate the approximation in a realistic application, using a three‐dimensional coupled physical‐biogeochemical 4D‐Var data assimilation system, set in the California Current system, in which the approximation is only applied to the 11 state variable biogeochemical model. In this application, the approximation‐based model solution tracks the reference solution accurately over thirty 4‐day assimilation cycles but leads to a ∼10% increase in the computational cost compared to the hand‐coded reference. Plain Language Summary Data assimilation refers to the process by which computer simulations of the Earth's atmosphere or ocean are constrained by measured data to be closer to the natural system. For example, in weather forecasting, measurements of the atmosphere are used to improve the computer model‐generated forecast. Many data assimilation applications require extra computer code to calculate the appropriate adjustments shifting the model toward observations. This data assimilation code is time consuming to build; it must be updated whenever changes are made to the model code. We describe a relatively easy way to automatically generate the data assimilation code using only the underlying model code. Using this method, changes to the model code are automatically propagated to the data assimilation code. Although this approach is simple to construct and maintain, it comes at a cost of increased computer execution time. It is therefore not sensible for all computer models. We show that our approach is useful for marine biological models that simulate biological and chemical interactions in the ocean. For our biological model of the U.S. west coast, the automatically generated data assimilation code takes 10% longer to run than the assimilation code created by hand and results in functionally equivalent results. Key Points A technique that accurately approximates tangent linear and ad