Particle filters are a class of sequential importance sampling methods for estimating the Bayesian posterior of an imperfectly observed dynamical system. Particle filters converge, under mild conditions, to the true posterior as the number of samples increases, but they are afflicted with a curse of dimensionality: the number of samples required to accurately represent the posterior grows catastrophically with the effective dimension of the system. The number of samples required to accurately represent a posterior on the state of the atmosphere is astronomical, so methods currently used to provide probabilistic estimates of the atmosphere have unknown and uncontrolled errors compared to the true Bayesian posterior. We propose a method to reduce the effective dimensionality in spatially-extended problems by replacing the true likelihood with a likelihood that has increased variance at small scales, equivalent to smoothing the data. The observation errors are thus modelled as a generalized random field. This effectively reduces the dimensionality, but it also changes the posterior. However, the character of the errors in the posterior is known and controllable.