Data Assimilation

Data assimilation refers to the statistical techniques used to combine numerical and statistical models with observations to give an improved estimate of the state of a system or process. Typically a data assimilation problem has a sequential aspect where data as it becomes available over time is used to update the state or parameters of a dynamical system. Data assimilation is usually distinguished from more traditional statistical time series applications because the system can have complicated nonlinear dynamical behavior and the state vector and the number of observations may be large. One of its primary roles is in estimating the state of a physical process when applied to geophysical models and physical measurements. Data assimilation has its roots in Bayesian inference and the restriction to linear dynamics and Gaussian distributions fits within the methods associated with the Kalman filter. Because data assimilation also involves estimating an unknown state based on possibly irregular, noisy or indirect observations it also has an interpretation as solving an inverse problem (e.g. [26]). One goal of this article is to tie these concepts back to a general Bayesian framework. One of the most successful applications of data assimilation is in numerical weather prediction where a large and heterogeneous set of observations are combined with a sophisticated physical model for the evolution of the atmosphere to produce detailed and high resolution forecasts of weather (see e.g. [18] for an introduction). The application to weather forecasting and in general to assimilation of atmospheric and oceanographic observations has a distinctly spatial aspect as the processes of interest are typically three dimensional fields. For this reason it is important to include this topic in this handbook. Although there are other applications of assimilation, such as target tracking or process control, and a more general class of Bayesian filtering methods (see [30], [6]) such as particle filters, these topics tend not to emphasize spatial processes and so are not as relevant to this handbook. The reader is referred to more statistical treatments of state space models in [28] and [14] but again these general texts do not focus on the large spatial fields typical in geophysical data assimilation. Spatial methods in data assimilation typically involve non-Gaussian fields and infer the spatial structure dynamically from a physical model. In this way the dynamical model and a statistical model are connected more closely than in a standard application of spatial statistics. In addition the sheer size of data assimilation problems requires approximate solutions that are not typical for smaller spatial data sets. In this article these differences will be highlighted by reviewing the principles behind current methods. We also point out some new areas where more standard space-time statistical models might be helpful in handling model error. A large scale example for the global atmosphere is included at the end of this article to illustrate some of the details of practical data assimilation for atmospheric prediction.