AN INTRODUCTION TO ESTIMATION THEORY

Despite the explosive growth of activity in the field of Earth System data assimilation over the past decade or so, there remains a substant ial gap between theory and practice. The present article attempts to b ridge this gap by exposing some of the central concepts of estimation theory and connecting them with current and future data assimilation a pproaches. Estimation theory provides a broad and natural mathematical foundation for data assimilation science. Stochastic-dynamic modeling and stochastic observation modeling are described first. Optimality c riteria for linear and nonlinear state estimation problems are then ex plored, leading to conditional-mean estimation procedures such as the Kalman filter and some of its generalizations, and to conditional-mode estimation procedures such as variational methods. A detailed derivat ion of the Kalman filter is given to illustrate the role of key probab ilistic concepts and assumptions. Extensions of the Kalman filter to n onlinear observation operators and to non-Gaussian errors are then des cribed. In a simple illustrative example, rigorous treatment of repres entativeness error and model error is highlighted in finite-dimensiona l estimation procedures for continuum dynamics and observations of the continuum state.