This study presents the first convective-scale 1,000-member ensemble simulation over central Europe, which provides a unique data set for various applications. A comparison with the operational regional 40-member ensemble of Deutscher Wetterdienst shows that the 1,000-member simulation exhibits realistic spread properties overall. Based on this, we discuss two potential applications. First, we quantify the sampling error of spatial covariances of smaller subsets compared with the 1,000-member simulation. Knowledge about sampling errors and their dependence on ensemble size is crucial for ensemble and hybrid data assimilation and for developing better approaches for localization in this context. Secondly, we present an approach for estimating the relative potential impact of different observable quantities using ensemble sensitivity analysis. This will provide the basis for consecutive studies developing future observation and data assimilation strategies. Sensitivity studies on the ensemble size indicate that about 200 ensemble members are required to estimate the potential impact of observable quantities with respect to precipitation forecasts. K E Y W O R D Sconvective-scale, covariance, data assimilation, ensemble sensitivity analysis, localization, observing system, sampling error
The errors in numerical weather forecasts resulting from limited ensemble size are explored using 1,000‐member forecasts of convective weather over Germany at 3‐km resolution. A large number of forecast variables at different lead times were examined, and their distributions could be classified into three categories: quasi‐normal (e.g., tropospheric temperature), highly skewed (e.g. precipitation), and mixtures (e.g., humidity). Dependence on ensemble size was examined in comparison to the asymptotic convergence law that the sampling error decreases proportional to N−1/2 for large enough ensemble size N, independent of the underlying distribution shape. The asymptotic convergence behavior was observed for the ensemble mean of all forecast variables, even for ensemble sizes less than 10. For the ensemble standard deviation, sizes of up to 100 were required for the convergence law to apply. In contrast, there was no clear sign of convergence for the 95th percentile even with 1,000 members. Methods such as neighborhood statistics or prediction of area‐averaged quantities were found to improve accuracy, but only for variables with random small‐scale variability, such as convective precipitation.
Over the past 15 years, adjoint-based, ensemble-based and hybrid methods have been developed for estimating observation impact based on the forecast sensitivity to observation impact (FSOI). These methods are now commonly used in global modelling systems. However, little attention has been given to assessing observation impact in regional convection-permitting modelling systems. This study presents the first evaluation of ensemble-based estimates of observation impact over an extended period of six weeks in such a convection-permitting modelling system, namely the regional ensemble system of Deutscher Wetterdienst. Another aspect that has received little attention is the choice of the forecast-error verification metric. Nearly all previous studies used the difference between the forecast and a subsequent analysis of the same modelling system expressed in terms of energy (total energy norm). While such a self-verification generally needs to be treated with caution, it appears unsuitable for convection-permitting regional forecasts. Firstly, total energy does not really reflect parameters that forecast users are interested in, and important forecast quantities such as surface wind gusts and precipitation are not even part of the analysis. Secondly, systematic analysis and forecast errors are non-negligible in the presence of convection, especially for important variables that are related to convection. To overcome this issue, we introduce the use of independent radar observations for the verification of observation impact and compare results for a variety of different observation-based metrics for a six-week high-impact weather period in summer 2016. This revealed a particular sensitivity of the estimated impact to model as well as observation biases and sensitivity studies indicated that even small biases can have an influence on the estimated impact. Additionally, we demonstrate that FSOI can be used to identify biases through comparison of results for different metrics.
State-of-the-art ensemble prediction systems usually provide ensembles with only 20–250 members for estimating the uncertainty of the forecast and its spatial and spatiotemporal covariance. Given that the degrees of freedom of atmospheric models are several magnitudes higher, the estimates are therefore substantially affected by sampling errors. For error covariances, spurious correlations lead to random sampling errors, but also a systematic overestimation of the correlation. A common approach to mitigate the impact of sampling errors for data assimilation is to localize correlations. However, this is a challenging task given that physical correlations in the atmosphere can extend over long distances. Besides data assimilation, sampling errors pose an issue for the investigation of spatiotemporal correlations using ensemble sensitivity analysis. Our study evaluates a statistical approach for correcting sampling errors. The applied sampling error correction is a lookup table–based approach and therefore computationally very efficient. We show that this approach substantially improves both the estimates of spatial correlations for data assimilation as well as spatiotemporal correlations for ensemble sensitivity analysis. The evaluation is performed using the first convective-scale 1000-member ensemble simulation for central Europe. Correlations of the 1000-member ensemble forecast serve as truth to assess the performance of the sampling error correction for smaller subsets of the full ensemble. The sampling error correction strongly reduced both random and systematic errors for all evaluated variables, ensemble sizes, and lead times.
Abstract. The success of ensemble data assimilation systems substantially depends on localization, which is required to mitigate sampling errors caused by modeling background error covariances with undersized ensembles. However, finding an optimal localization is highly challenging, as covariances, sampling errors, and appropriate localization depend on various factors. Our study investigates vertical localization based on a unique convection-permitting 1000-member ensemble simulation; 1000-member ensemble correlations serve as truth for examining vertical correlations and their sampling error. We discuss requirements for vertical localization by deriving an empirical optimal localization (EOL) that minimizes the sampling error in 40-member subsample correlations with respect to the 1000-member reference. Our analysis covers temperature, specific humidity, and wind correlations on various pressure levels. Results suggest that vertical localization should depend on several aspects, such as the respective variable, vertical level, or correlation type (self- or cross-correlations). Comparing the empirical optimal localization with common distance-dependent localization approaches highlights that finding suitable localization functions bears substantial room for improvement. Furthermore, we examine approaches for achieving positive semi-definiteness for covariance localization that hardly affect the sampling error reduction. Finally, we discuss the gain of combining different localization approaches with an adaptive statistical sampling error correction.
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