Maximum-likelihood estimation of forecast and observation error covarianceparameters. Part II: Applications

Three different applications of maximum-likelihood estimation of error cova riance parameters for atmospheric data assimilation are described. Height e rror standard deviations, vertical correlation coefficients, and isotropic decorrelation length scales are estimated from rawinsonde height observed-m inus-forecast residuals. Sea level pressure error standard deviations and d ecorrelation length scales are obtained from ship reports, and wind observa tion error standard deviations and forecast error stream function and veloc ity potential decorrelation length scales are estimated from aircraft data. These applications serve to demonstrate the ability of the method to estim ate covariance parameters using multivariate data from moving observers. Estimates of the parameter uncertainty due to sampling error can be obtaine d as a by-product of the maximum-likelihood estimation. By bounding this so urce of error ii is found that many statistical parameters that are usually presumed constant in operational data assimilation systems in fact vary si gnificantly with time. This may well reflect the use of overly simplistic c ovariance models that cannot adequately describe state-dependent error comp onents such as representativeness error: The sensitivity of the parameter e stimates to the treatment of bias, and to the choice of the model represent ing spatial correlations, is examined in detail. Several experiments emulat e an online covariance parameter estimation procedure using a sliding windo w of data, and it is shown that such a procedure is both desirable and comp utationally feasible.