It is often desirable to estimate the distribution over quantities arising from inverse problems, but extant methods for estimating the state and uncertainty of dynamical systems are either computationally intractable for problem sizes in the domain of fluid dynamics, or they are provably inconsistent for nonlinear problems. Strategies to improve upon the bias those methods experience, while balancing for computational scalability, would therefore be helpful in obtaining good uncertainty estimates in inverse problems for nonlinear fluids. This research introduces an approximation to the sequential state estimation problem for spatially-extended dynamics that improves the dimensional scaling of the provably-consistent sequential importance resampling algorithm (SIR): we assume that observation errors are correlated with a strange spatial structure, having a spectrum that grows in the progression toward small scales. This decreases the ensemble size required to attain an accurate distributional estimate with SIR. Next we develop a fast implementation of our error model that is compatible with scattered observations, as required for numerical weather forecast, using a multiresolution approximation to the inverse of an elliptic differential operator with properties we prescribe of a covariance operator to make SIR more tractable. Finally, we combine our error model with a hybrid of SIR and the ensemble square root filter (ESRF) and apply it to a toy model in a class widely used to test meteorological data assimilation methods. The hybrid still suffers from the ESRF's inconsistency, but the SIR step helps by presenting ESRF with a prior that is closer to Gaussian. Our error model allows SIR to do more of that mitigation. Relative to a state-of-the-art ESRF, this improved the continuous ranked probability score by 15% and the root mean squared error by 10% in both the posterior and forecast. This improvement is substantial, comparable to the last 15 years of improvement in operational 6-day weather forecasts.