NWP is an initial/boundary value problem: given an estimate of the present state of the atmosphere (initial conditions) and appropriate boundary conditions, the model simulates (forecasts) the atmospheric evolution. More accurate estimates of initial conditions lead to better forecasts. Currently, operational NWP centers produce initial conditions through a statistical combination of observations and short-range forecasts that account for the uncertainty associated with each source of information. This approach has become known as “data assimilation.” In this chapter, we review early attempts at data assimilation and then introduce the statistical estimation methods that provide a solid foundation for data assimilation. Examples using toy models are provided to illustrate the principles of data assimilation. We then discuss in detail all state-of-the-art data assimilation methods adopted in operational centers, including optimal interpolation, 3D-Var, 4D-Var, ensemble Kalman filter, and hybrid methods. Specifically, we discuss several improvements for the ensemble Kalman filter that make it competitive with 4D-Var. We also discuss Ensemble Forecast Sensitivity to Observations (EFSO), a powerful tool that can estimate the impact of any observations on short-range forecasts, and then we discuss the proactive quality control (PQC) built upon EFSO. We also briefly introduce the non-Gaussian assimilation method particle filters.

In this chapter the data assimilation problem is introduced as a control theory problem for partial differential equations, with initial conditions, model error, and empirical model parameters as optional control variables. An alternative interpretation of data assimilation as a processing of information in a dynamic-stochastic system is also introduced. Both approaches will be addressed in more detail throughout this book. The historical development of data assimilation has been documented, starting from the early nineteenth-century works by Legendre, Gauss, and Laplace, to optimal interpolation and Kalman filtering, to modern data assimilation based on variational and ensemble methods, and finally to future methods such as particle filters. This suggests that data assimilation is not a very new concept, given that it has been of scientific and practical interest for a long time. Part of the chapter focuses on introducing the common terminologies and notation used in data assimilation, with special emphasis on observation equation, observation errors, and observation operators. Finally, a basic linear estimation problem based on least squares is presented.