Two-point or multiple-point statistics? A comparison between the ensemble Kalman filtering and the ensemble pattern matching inverse methods

• The EnKF and EnPAT are compared for inverse modeling in a channelized aquifer.
• The EnPAT is extended to handle the continuous variable.
• Rejection sampling is applied to explore the full posterior uncertainty.

The Ensemble Kalman Filter (EnKF) has been commonly used to assimilate real time dynamic data into geologic models over the past decade. Despite its various advantages such as computational efficiency and its capability to handle multiple sources of uncertainty, the EnKF may not be used to reliably update models that are characterized by curvilinear geometries such as fluvial deposits where the permeable channels play a crucial role in the prediction of solute transport. It is well-known that the EnKF performs optimally for updating multi-Gaussian distributed fields, basically because it uses two-point statistics (i.e., covariances) to represent the relationship between the model parameters and between the model parameters and the observed response, and this is the only statistic necessary to fully characterize a multiGaussian distribution. The Ensemble PATtern matching (EnPAT) is an alternative ensemble based method that shows significant potential to condition complex geology such as channelized aquifers to dynamic data. The EnPAT is an evolution of the EnKF, replacing, in the analysis step, two-point statistics with multiple-point statistics. The advantages of EnPAT reside in its capability to honor the complex spatial connectivity of geologic structures as well as the measured static and dynamic data. In this work, the performance of the classical EnKF and the EnPAT are compared for modeling a synthetic channelized aquifer. The results reveal that the EnPAT yields a better prediction of transport characteristics than the EnKF because it characterizes the conductivity heterogeneity better. Issues such as uncertainty of multiple variables and the effect of measurement errors on EnPAT results will be discussed.