Data assimilation is the process of combining predictions from a model of a dynamical system with some limited observations of the state of the system to obtain an estimate of the state of the entire system. This is a problem that arises naturally in weather forecasting, where observations of pressure, temperature, and wind speed are available from weather stations but estimates of the current weather and predictions of future weather must be made for points where no observations are available. Data assimilation problems also occur in many other areas of the earth sciences, where data is limited in scope and computer models may have limited resolution due to both limited data on initial and boundary conditions and practical limits on the resolution of the computations.

Historically, methods for data assimilation have been developed by scientists and engineers working in various areas of application. One of the most widely used methods, the Kalman filter, comes out of the field of engineering control systems. The other commonly used family of variational methods for data assimilation were developed in the fields of weather forecasting and oceanography.

The first half of this book presents these widely used methods for data assimilation in discrete time dynamical systems. The authors begin by formulating two versions of the data assimilation problem in a unified Bayesian framework. In the filtering version of the problem, the goal is to estimate the current state of the system using data from the current point in time and previous times. In the smoothing version of the problem the goal is to estimate the state of the system over a period of time using all available data. The system dynamics can be linear or nonlinear. Random perturbations to the system state and observations can occur, but these have Gaussian distributions. The resulting output distributions are Gaussian under the assumption that the system dynamics are linear, and approximately Gaussian if the system dynamics are not too strongly nonlinear. The mean and covariance of the posterior distribution of the system can be computed. The Kalman filter, Extended Kalman filter, Ensemble Kalman filter, 3DVAR, and 4DVAR methods all developed within this framework.

The second half of the book can be thought of as a monograph on the authors’ approach to extending data assimilation methods to continuous time dynamical systems. The filtering and smoothing problems are reformulated in terms of stochastic differential equations. The authors go on to derive continuous time versions of the methods previously derived for discrete time dynamical systems.

The authors have used a collection of dynamical systems examples throughout the book. They have included MATLAB codes for the analysis of these systems, and it is possible for the reader to reconstruct all of the computational examples and plots presented in the text. Exercises are also given at the end of each chapter. The first half of this book would be very suitable as a graduate level textbook and concise reference on discrete time approaches to the data assimilation problem from a Bayesian point of view. The second half of the book is written at a much more sophisticated level and will primarily be of interest to researchers working in this area.

Brian Borchers is a professor of Mathematics at the New Mexico Institute of Mining and Technology. His interests are in optimization and applications of optimization in parameter estimation and inverse problems.