Exploring High-Dimensional Applicability of the Particle Filter in Increasingly Complex Systems - A Case Study for Compacting Reservoir and Subsidence.

Subsidence arises from a complex entanglement of subsurface processes, where factors like varying hydrocarbon production rates or spatio-temporal changes in precipitation and evaporation result in nonlinear geomechanical processes. Consequently, parameter estimation of the subsidence signal requires large-scale studies of high-dimensional systems. The nonlinearity and the non-Gaussianity of these systems suggest that particle methods are well-suited for parameter estimation using data assimilation. In the specific case of gas extraction in Groningen, the Netherlands, processes from shallow and deep subsurface contribute to the observed subsidence. Estimating the combined effect of these processes increases the degree of complexity of the data assimilation problem.

This work aims to assess the performance of a particle method in models with increasing degree of complexity, in the case of surface deformation induced by reservoir compaction. The particle method performs well for low-dimensional problems but in high dimensions it may experience a collapse of the weighted posterior probability density function. A simple example with a Gaussian distribution for both prior and likelihood demonstrates that with increasing state- and observational-space dimensions, the so-called effective sample size (ESS, the Euclidean distance between the posterior and a uniform distribution) decreases and that the maximum weight (w_max) increases.

Here we explore the limits of the applicability of an importance sampling particle filter, and quantify the effectiveness of the method. The empirical behavior of w_max and of the ESS describes how degeneracy evolves. Resampling may help to avoid degeneracy by removing particles with a very low weight. Information about the rate of degeneracy can help to choose effective criteria for the application of resampling. Based on our results, we propose a further refinement of the generally used criterion for resampling that the effective sample size should be half the number of particles. In our approach, the acceptable lowest value of the ESS does not only depend on the required ensemble size but also on the dimension and on the root mean square error of the update.