The ultimate purpose of environmental studies is the forecast of its natural evolution. A prerequisite before a prediction is to retrieve at best the state of the environment. Data assimilation is the ensemble of techniques which, starting from heterogeneous information, permit to retrieve the initial state of a flow. In the first part, the mathematical models governing geophysical flows are presented together with the networks of observations of the atmosphere and of the ocean. In variational methods, we seek for the minimum of a functional estimating the discrepancy between the solution of the model and the observation. The derivation of the optimality system, using the adjoint state, permits to compute a gradient which is used in the optimization. The definition of the cost function permits to take into account the available statistical information through the choice of metrics in the space of observation and in the space of the initial condition. Some examples are presented on simplified models, especially an application in oceanography. Among the tools of optimal control, the adjoint model permits to carry out sensitivity studies, but if we look for the sensitivity of the prediction with respect to the observations, then a second-order analysis should be considered. One of the first methods used for assimilating data in oceanography is the nudging method, adding a forcing term in the equations. A variational variant of nudging method is described and also a so-called "back and forth" nudging method. The proper orthogonal decomposition method is introduced in order to reduce the cost of the variational method. For assimilating data, stochastic methods can be considered, being based on the Kalman filter extended to nonlinear problems, but the inconvenience of this method consists in the difficulty of handling huge covariance matrices. The dimension of the systems used for operational purposes (several hundred of millions of variables) requires to work with reduced variable techniques. The ensemble Kalman filter method, which is a Monte-Carlo implementation of the Bayesian update problem, is described. A considerable amount of information on geophysical flows is provided by satellites displaying images of their evolution, the assimilation of images into numerical models is a challenge for the future: variational methods are successfully considered in this perspective. © 2009 Elsevier B.V. All rights reserved.
Blum, J., Le Dimet, F. X., & Navon, I. M. (2009). Data Assimilation for Geophysical Fluids. In Handbook of Numerical Analysis (Vol. 14, pp. 385–441). Elsevier. https://doi.org/10.1016/S1570-8659(08)00209-3