On diagnosing observation‐error statistics with local ensemble data assimilation

• Published in: Quarterly journal of the Royal Meteorological Society Vol. 143; no. 708; pp. 2677 - 2686
• Main Authors: Waller, J. A., Dance, S. L., Nichols, N. K.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 01-10-2017; Wiley Subscription Services, Inc
• Subjects: Analysis; correlated observation errors; covariance matrix; Data assimilation; data assimilation diagnostic; Data collection; domain localization; Errors; local ensemble assimilation; Localization; Mathematical models; residuals; Statistical analysis; Statistical methods; Statistics
• ISSN: 0035-9009; 1477-870X
• DOI: 10.1002/qj.3117

Recent research has shown that the use of correlated observation errors in data assimilation can lead to improvements in analysis accuracy and forecast skill. As a result, there is increased interest in characterizing, understanding and making better use of correlated observation errors. A simple diagnostic for estimating observation‐error statistics makes use of statistical averages of observation‐minus‐background and observation‐minus‐analysis residuals. This diagnostic is derived assuming that the analysis is calculated using a best linear unbiased estimator. In this work, we consider whether the diagnostic is still applicable when the analysis is calculated using ensemble assimilation schemes with domain localization. We show that the diagnostic equations no longer hold: the statistical averages of observation‐minus‐background and observation‐minus‐analysis residuals no longer result in an estimate of the observation‐error covariance matrix. Nevertheless, we are able to show that, under certain circumstances, some elements of the observation‐error covariance matrix can be recovered. Furthermore, we provide a method to determine which elements of the observation‐error covariance matrix can be estimated correctly. In particular, the correct estimation of correlations is dependent on both the localization radius and the observation operator. We provide numerical examples that illustrate these mathematical results.
Estimates of observation‐error statistics are required for use in data assimilation. In this work, we consider whether a simple diagnostic for estimating observation‐error statistics is still applicable when the analysis is calculated using local ensemble assimilation schemes. We show that the diagnostic equations no longer hold: the statistical averages of observation‐minus‐background and observation‐minus‐analysis residuals no longer result in an estimate of the observation‐error covariance matrix. Nevertheless, we show that, under certain circumstances, some elements of the observation‐error covariance matrix can be recovered.