Alignment error models and ensemble-based data assimilation

The concept of alternative error models is suggested as a means to redefine estimation problems with non-Gaussian additive errors so that familiar and near-optimal Gaussian-based methods may still be applied successfully. The specific example of a mixed error model including both alignment errors and additive errors is examined. Using the specific form of a soliton, an analytical solution to the Korteweg-de Vries equation, the total (additive) errors of states following the mixed error model are demonstrably non-Gaussian for large enough alignment errors, and an ensemble of such states is handled poorly by a traditional ensemble Kalman filter, even if position observations are included. Consideration of the mixed error model itself naturally suggests a two-step approach to state estimation where the alignment errors are corrected first, followed by application of an estimation scheme to the remaining additive errors, the first step aimed at removing most of the non-Gaussianity so the second step can proceed successfully. Taking an ensemble approach for the soliton states in a perfect-model scenario, this two-step approach shows a great improvement over traditional methods in a wide range of observational densities, observing frequencies, and observational accuracies. In cases where the two-step approach is not successful, it is often attributable to the first step not having sufficiently removed the non-Gaussianity, indicating the problem strictly requires an estimation scheme that does not make Gaussian assumptions. However, in these cases a convenient approximation to the two-step approach is available, which trades obtaining a minimum variance estimate ensemble mean for more physically sound updates of the individual ensemble members.