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Abstract Many data assimilation methods require knowledge of the first two moments of the background and observation errors to function optimally. To ensure the effective performance of such methods, it is often advantageous to estimate the second moment of the observation errors directly. We examine three different strategies for doing so, focusing specifically on the case of a single scalar observation error variance parameterr. The first method is the well-known Desroziers et al. “diagnostic check” iteration (DBCP). The second method, described in Karspeck, adapts the “spread–error” diagnostic—used for assessing ensemble reliability—to observations and generates a point estimate ofrby taking the expectation of various observation-space statistics and using an ensemble to model background error statistics explicitly. The third method is an approximate Bayesian scheme that uses an inverse-gamma prior and a modified Gaussian likelihood. All three methods can recover the correct observation error variance when both the background and observation errors are Gaussian and the background error variance is well specified. We also demonstrate that it is often possible to estimatereven when the observation error is not Gaussian or when the forward operator mapping model states into observation space is nonlinear. The DBCP method is found to be most robust to these complications; however, the other two methods perform similarly well in most cases and have the added benefit that they can be used to estimaterbefore data assimilation. We conclude that further investigation is warranted into the latter two methods, specifically into how they perform when extended to the multivariate case. Significance StatementObservations of the Earth system (e.g., from satellites, radiosondes, aircraft, etc.,) each have some associated uncertainty. To use observations to improve model forecasts, it is important to understand the size of that uncertainty. This study compares three statistical methods for estimating observation errors, all of which can be continuously implemented whenever new observations are used to correct a model. Our results suggest that all three methods can improve forecast outcomes, but that, if observations are believed to have highly biased or skewed errors, care should be taken in choosing which to use and interpreting its results. Future studies should investigate robust methods for estimating more complicated types of errors. 

Abstract Estimating and predicting the state of the atmosphere is a probabilistic problem for which an ensemble modeling approach often is taken to represent uncertainty in the system. Common methods for examining uncertainty and assessing performance for ensembles emphasize pointwise statistics or marginal distributions. However, these methods lose specific information about individual ensemble members. This paper explores contour band depth (cBD), a method of analyzing uncertainty in terms of contours of scalar fields. cBD is fully nonparametric and induces an ordering on ensemble members that leads to box-and-whisker-plot-type visualizations of uncertainty for two-dimensional data. By applying cBD to synthetic ensembles, we demonstrate that it provides enhanced information about the spatial structure of ensemble uncertainty. We also find that the usefulness of the cBD analysis depends on the presence of multiple modes and multiple scales in the ensemble of contours. Finally, we apply cBD to compare various convection-permitting forecasts from different ensemble prediction systems and find that the value it provides in real-world applications compared to standard analysis methods exhibits clear limitations. In some cases, contour boxplots can provide deeper insight into differences in spatial characteristics between the different ensemble forecasts. Nevertheless, identification of outliers using cBD is not always intuitive, and the method can be especially challenging to implement for flow that exhibits multiple spatial scales (e.g., discrete convective cells embedded within a mesoscale weather system). Significance StatementPredictions of Earth’s atmosphere inherently come with some degree of uncertainty owing to incomplete observations and the chaotic nature of the system. Understanding that uncertainty is critical when drawing scientific conclusions or making policy decisions from model predictions. In this study, we explore a method for describing model uncertainty when the quantities of interest are well represented by contours. The method yields a quantitative visualization of uncertainty in both the location and the shape of contours to an extent that is not possible with standard uncertainty quantification methods and may eventually prove useful for the development of more robust techniques for evaluating and validating numerical weather models.