Abstract
To improve the quantity and impact of observations used in data assimilation, it is necessary to take into account the full, potentially correlated, observation error statistics. A number of methods for estimating correlated observation errors exist, but a popular method is a diagnostic that makes use of statistical averages of observation‐minus‐background and observation‐minus‐analysis residuals. The accuracy of the results it yields is unknown as the diagnostic is sensitive to the difference between the exact background and exact observation error covariances and those that are chosen for use within the assimilation. It has often been stated in the literature that the results using this diagnostic are only valid when the background and observation error correlation length‐scales are well separated. Here we develop new theory relating to the diagnostic. For observations on a 1D periodic domain we are able to the show the effect of changes in the assumed error statistics used in the assimilation on the estimated observation error covariance matrix. We also provide bounds for the estimated observation error variance and eigenvalues of the estimated observation error correlation matrix. We demonstrate that it is still possible to obtain useful results from the diagnostic when the background and observation error length‐scales are similar. In general, our results suggest that when correlated observation errors are treated as uncorrelated in the assimilation, the diagnostic will underestimate the correlation length‐scale. We support our theoretical results with simple illustrative examples. These results have potential use for interpreting the derived covariances estimated using an operational system.
Highlights
Data assimilation techniques combine model states, known as forecasts or backgrounds, with observations, weighted by their respective errors, to provide a best estimate of the state, known as the analysis
Proposed as a consistency check, the diagnostic uses the statistical average of observation-minus-background and observation-minus-analysis residuals to provide an estimate of the observation error covariance matrix
Journal of the Royal Meteorological Society published by John Wiley & Sons Ltd on behalf of the Royal Meteorological Society
Summary
Data assimilation techniques combine model states, known as forecasts or backgrounds, with observations, weighted by their respective errors, to provide a best estimate of the state, known as the analysis. Despite the limitations the diagnostic has been successfully used in some studies to estimate observation error variances and correlations It has been used in simple model experiments in both variational (Stewart, 2010) and ensemble (Li et al, 2009; Miyoshi et al, 2013) data assimilation systems and to estimate time varying observation errors (Waller et al, 2014a). Theoretical results relating to the diagnostic under some simplifying assumptions have been previously published, both in the original manuscript of Desroziers et al (2005) and in workshop proceedings (Menard et al, 2009; Desroziers et al, 2009) These results relate to scalar cases or consider the estimation of variances or the convergence of the method under iteration.
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