
Contents
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7.1 Error equations and their simulation 7.1 Error equations and their simulation
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7.1.1 Analysis and forecast error equations in general 7.1.1 Analysis and forecast error equations in general
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7.1.2 Analysis and forecast perturbation equations in EnDA 7.1.2 Analysis and forecast perturbation equations in EnDA
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Schematic description of EnDA Schematic description of EnDA
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Analysis and forecast perturbation equations Analysis and forecast perturbation equations
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Some features of EnDA Some features of EnDA
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7.1.3 Comparison with the NMC method 7.1.3 Comparison with the NMC method
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7.2 Innovation-based estimations 7.2 Innovation-based estimations
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7.2.1 Innovation auto-covariances 7.2.1 Innovation auto-covariances
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7.2.2 Covariances of analysis residuals 7.2.2 Covariances of analysis residuals
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7.2.3 Practical use of innovation-based estimates 7.2.3 Practical use of innovation-based estimates
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7.3 Diagnosis of background error covariances 7.3 Diagnosis of background error covariances
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7.3.1 Variances and their spatio-temporal variations 7.3.1 Variances and their spatio-temporal variations
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7.3.2 Horizontal correlations 7.3.2 Horizontal correlations
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7.3.3 Vertical correlations 7.3.3 Vertical correlations
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7.3.4 Cross-covariances 7.3.4 Cross-covariances
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7.4 Modelling and filtering covariances 7.4 Modelling and filtering covariances
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7.4.1 Modelling with sparse operators 7.4.1 Modelling with sparse operators
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7.4.2 Cross-covariances and multivariate couplings 7.4.2 Cross-covariances and multivariate couplings
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7.4.3 Sampling noise in ensemble-based variances 7.4.3 Sampling noise in ensemble-based variances
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7.4.4 Filtering of ensemble-based variances 7.4.4 Filtering of ensemble-based variances
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7.4.5 Modelling of spatial correlations in spectral space 7.4.5 Modelling of spatial correlations in spectral space
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7.4.6 Modelling of spatial correlations in wavelet space 7.4.6 Modelling of spatial correlations in wavelet space
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7.4.7 Modelling of spatial correlations in gridpoint space 7.4.7 Modelling of spatial correlations in gridpoint space
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7.4.8 Modelling of spatial covariances in ensemble space 7.4.8 Modelling of spatial covariances in ensemble space
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7.5 Conclusions 7.5 Conclusions
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References References
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7 Background error covariances: estimation and specification
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Published:October 2014
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Abstract
This chapter deals with the estimation and specification of realistic background error covariances, which is a key issue in data assimilation, since these covariances are used to filter and propagate observations. The underlying equations of error evolution are summarized, and associated simulation techniques are also presented, based on either ensemble techniques or the so-called NMC method. Another important source of information on background error covariances corresponds to innovation-based estimates, which may be combined with ensemble data assimilation to estimate model error covariances, for instance. Moreover, because the covariance matrix is huge, diagnostics of its main components and salient features have to be employed. The possibility of modelling this matrix using a sequence of sparse operators is then reviewed, in addition to filtering methods that account for the finite ensemble size and associated sampling noise effects.
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