Characterizing and forecasting the state of the ocean is essential for various scientific, management, commercial, and recreational applications. This is, however, a challenging problem due to the large, multiscale and nonlinear nature of the ocean state dynamics and the limited amount of observations.
Combining all available information from numerical models describing the ocean dynamics, observations, and prior information has proven to be the most viable approach to determine the best estimates of the ocean state, a process called data assimilation (DA). DA is becoming widespread in many ocean applications; stimulated by continuous advancement in modeling, observational, and computational capabilities. This chapter offers a comprehensive presentation of the theory and methods of ocean DA, outlining its current status and recent developments, and discussing new directions and open questions. Casting DA as a Bayesian state estimation problem, the chapter will gradually advance from the basic principles of DA to its most advanced methods. Three-dimensional DA methods, 3DVAR and Optimal Interpolation, are first derived, before incorporating time and present the most popular, Gaussian-based DA approaches:4DVAR, Kalman filters and smoothers methods, which exploit past and/or future observations. Ensemble Kalman methods are next introduced in their stochastic and deterministic formulations as a stepping-stone toward the more advanced nonlinear/non-Gaussian DA methods, Particle and Gaussian Mixture filters. Other sophisticated hybrid extensions aimed at exploiting the advantages of both ensemble and variational methods are also presented. The chapter then concludes with a discussion on the importance of properly addressing the uncertainties in the models and the data, and available approaches to achieve this through parameters estimation, model errors quantification, and coupled DA.