The ensemble Kalman filter for combined state and parameter estimation

This article provides a fundamental theoretical basis for understanding EnKF and serves as a useful text for future users. Data assimilation and parameter-estimation problems are explained, and the concept of joint parameter and state estimation, which can be solved using ensemble methods, is presented. KF and EKF are briefly discussed before introducing and deriving EnKF. Similarities and differences between KF and EnKF are pointed out. The benefits of using EnKF with high-dimensional and highly nonlinear dynamical models are illustrated by examples. EnKF and EnKS are also derived from Bayes theorem, using a probabilistic approach. The derivation is based on the assumption that measurement errors are independent in time and the model represents a Markov process, which allows for Bayes theorem to be written in a recursive form, where measurements are processed sequentially in time. The practical implementation of the analysis scheme isdiscussed, and it is shown that it can be computed efficiently in the space spanned by the ensemble realizations. The square root scheme is discussed as an alternative method that avoids the perturbation of measurements. However, the square root scheme has other pitfalls, and it is recommended to use the symmetric square root with or without a random rotation. The random rotation introduces a stochastic component to the update, and the quality of the scheme may then not improve compared to the original stochastic EnKF scheme with perturbed measurements.