Abstract: |
The Ensemble Kalman Filter (EnKF) is a Monte Carlo method for solving non-linear spatiotemporal inverse problems in high dimensions. Applications include numerical weather prediction, oceanography, hydrology and petroleum reservoir characterisation .The EnKF is based on the traditional Kalman Filter (KF), which provides an analytical solution for the posterior probability density function (pdf) of interest, assuming linear system dynamics and linear, Gaussian assumptions, termed the Gausslinear model. One known problem with the standard EnKF algorithm, however, is that the updated ensemble members will fail to correctly represent the statistical properties of the posterior distribution.The reason for this behaviour, however, has not been properly understood. In the current paper we present theoretical results for the bias and covariances in the forecast based on the classical EnKF updating scheme, taking into account the uncertainty of the unknown Kalman gain matrix. Furthermore, we have formulated an alternative EnKF updating scheme based on Bayesian regression techniques. In this scheme, each ensemble member is updated based on a Kalman gain matrix independently generated from a matrix variate distribution, rather than using one common plug-in estimate. We consider both conjugate and non-informative prior distributions, and an approximate dimension reducing scheme for high dimensional models is suggested. Synthetic examples inspired by petroleum reservoir evaluation problems are used to empirically evaluate the performance of the suggested procedures. The results reveal that we can dramatically improve the accuracy of the forecast and predictions intervals, especially for small ensemble sizes. |
