Abstract
This article shows that increasing the observation variance at small scales can reduce the ensemble size required to avoid collapse in particle filtering of spatially extended dynamics and improve the resulting uncertainty quantification at large scales. Particle filter weights depend on how well ensemble members agree with observations, and collapse occurs when a few ensemble members receive most of the weight. Collapse causes catastrophic variance underestimation. Increasing small-scale variance in the observation error model reduces the incidence of collapse by de-emphasizing small-scale differences between the ensemble members and the observations. Doing so smooths the posterior mean, though it does not smooth the individual ensemble members. Two options for implementing the proposed observation error model are described. Taking a discretized elliptic differential operator as an observation error covariance matrix provides the desired property of a spectrum that grows in the approach to small scales. This choice also introduces structure exploitable by scalable computation techniques, including multigrid solvers and multiresolution approximations to the corresponding integral operator. Alternatively the observations can be smoothed and then assimilated under the assumption of independent errors, which is equivalent to assuming large errors at small scales. The method is demonstrated on a linear stochastic partial differential equation, where it significantly reduces the occurrence of particle filter collapse while maintaining accuracy. It also improves continuous ranked probability scores by as much as 25%, indicating that the weighted ensemble more accurately represents the true distribution. The method is compatible with other techniques for improving the performance of particle filters.
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