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Dick P. Dee

Abstract

A simple scheme is presented for on-line estimation of covariance parameters in statistical data assimilation systems. The scheme is based on a maximum-likelihood approach in which estimates are produced on the basis of a single batch of simultaneous observations. Single-sample covariance estimation is reasonable as long as the number of available observations exceeds the number of tunable parameters by two or three orders of magnitude.

Not much is known at present about model error associated with actual forecast systems. Our scheme can be used to estimate some important statistical model error parameters such as regionally averaged variances or characteristic correlation length scales. The advantage of the single-sample approach is that it does not rely on any assumptions about the temporal behavior of the covariance parameters: time-dependent parameter estimates can be continuously adjusted on the basis of current observations. This is of practical importance since it is likely to be the case that both model error and observation error strongly depend on the actual state of the atmosphere.

The single-sample estimation scheme can he incorporated into any four-dimensional statistical data assimilation system that involves explicit calculation of forecast error covariances, including optimal interpolation (OI) and the simplified Kalman filter (SKF). The computational cost of the scheme is high but not prohibitive: online estimation of one or two covariance parameters in each analysis box of an operational boxed-OI system is currently feasible.

A number of numerical experiments performed with an adaptive SKF and an adaptive version of OI, using a linear two-dimensional shallow-water model and artificially generated model error are described. The performance of the nonadaptive versions of these methods turns out to depend rather strongly on the correct specification of model error parameters. The parameters are estimated under a variety of conditions, including uniformly distributed model error and time-dependent model error statistics.

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Dick P. Dee and Ricardo Todling

Abstract

The authors describe the application of the unbiased sequential analysis algorithm developed by Dee and da Silva to the Goddard Earth Observing System moisture analysis. The algorithm estimates the slowly varying, systematic component of model error from rawinsonde observations and adjusts the first-guess moisture field accordingly. Results of two seasonal data assimilation cycles show that moisture analysis bias is almost completely eliminated in all observed regions. The improved analyses cause a sizable reduction in the 6-h forecast bias and a marginal improvement in the error standard deviations.

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Dick P. Dee and Arlindo M. da Silva

Abstract

The maximum-likelihood method for estimating observation and forecast error covariance parameters is described. The method is presented in general terms but with particular emphasis on practical aspects of implementation. Issues such as bias estimation and correction, parameter identifiability, estimation accuracy, and robustness of the method, are discussed in detail. The relationship between the maximum-likelihood method and generalized cross-validation is briefly addressed.

The method can be regarded as a generalization of the traditional procedure for estimating covariance parameters from station data. It does not involve any restrictions on the covariance models and can be used with data from moving observers, provided the parameters to be estimated are identifiable. Any available a priori information about the observation and forecast error distributions can be incorporated into the estimation procedure. Estimates of parameter accuracy due to sampling error are obtained as a by-product.

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Patrick Laloyaux, Jean-Noël Thépaut, and Dick Dee

Abstract

The European Centre for Medium-Range Weather Forecasts (ECMWF) has developed a coupled assimilation system that ingests simultaneously ocean and atmospheric observations in a coupled ocean–atmosphere model. Employing the coupled model constraint in the analysis implies that assimilation of an ocean observation has immediate impact on the atmospheric state estimate, and, conversely, assimilation of an atmospheric observation affects the ocean state. In this context, observing system experiments have been carried out withholding scatterometer surface wind data over the period September–November 2013. Impacts in the coupled assimilation system have been compared to the uncoupled approach used in ECMWF operations where atmospheric and ocean analyses are computed sequentially. The assimilation of scatterometer data has reduced the background surface wind root-mean-square error in the coupled and uncoupled assimilation systems by 3.7% and 2.5%, respectively. It has been found that the ocean temperature in the mixed layer is improved in the coupled system, while the impact is neutral in the uncoupled system. Further investigations have been conducted over a case of a tropical cyclone when strong interactions between atmospheric wind and ocean temperature occur. Cyclone Phailin in the Bay of Bengal has been selected since the conventional observing system has measured surface wind speed and ocean temperature at a high frequency. In this case study, the coupled assimilation system outperforms the uncoupled approach, being able to better use the scatterometer measurements to estimate the cold wake after the cyclone.

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Dick P. Dee and Arlindo Moraes Da Silva

Abstract

The implementation of a technique for locating programming errors in shallow-water codes, establishing the correctness of the code, and assessing the performance of the numerical model under various flow conditions is described. The right-hand side of the differential equations is modified in such a way that the exact solution of the nonlinear initial-value problem is known, so that the truncation error of the numerical scheme can be studied in detail. The exact solution is prescribed to be any linear combination of Hough harmonies which propagate in time according to their natural frequencies.

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Dick P. Dee and Arlindo M. da Silva

Abstract

The implications of using different control variables for the analysis of moisture observations in a global atmospheric data assimilation system are investigated. A moisture analysis based on either mixing ratio or specific humidity is prone to large extrapolation errors, due to the high variability in space and time of these parameters and to the difficulties in modeling their error covariances. Using the logarithm of specific humidity does not alleviate these problems, and has the further disadvantage that very dry background estimates cannot be effectively corrected by observations. Relative humidity is a better choice from a statistical point of view, because this field is spatially and temporally more coherent and error statistics are therefore easier to obtain. If, however, the analysis is designed to preserve relative humidity in the absence of moisture observations, then the analyzed specific humidity field depends entirely on analyzed temperature changes. If the model has a cool bias in the stratosphere this will lead to an unstable accumulation of excess moisture there.

A pseudo-relative humidity can be defined by scaling the mixing ratio by the background saturation mixing ratio. A univariate pseudo-relative humidity analysis will preserve the specific humidity field in the absence of moisture observations. A pseudo-relative humidity analysis is shown to be equivalent to a mixing ratio analysis with flow-dependent variance specifications. In the presence of multivariate (temperature–moisture) observations it produces analyzed relative humidity values that are nearly identical to those produced by a relative humidity analysis. Based on a time series analysis of radiosonde observed-minus-background differences it appears to be more justifiable to neglect specific humidity–temperature correlations (in a univariate pseudo-relative humidity analysis) than to neglect relative humidity–temperature correlations (in a univariate relative humidity analysis). A pseudo-relative humidity analysis can be implemented in an existing moisture analysis system simply by scaling the observed-minus-background residuals prior to solving the analysis equation, and rescaling the analyzed increments afterward.

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Gennady A. Chepurin, James A. Carton, and Dick Dee

Abstract

Numerical models of ocean circulation are subject to systematic errors resulting from errors in model physics, numerics, inaccurately specified initial conditions, and errors in surface forcing. In addition to a time-mean component, the systematic errors include components that are time varying, which could result, for example, from inaccuracies in the time-varying forcing. Despite their importance, most assimilation algorithms incorrectly assume that the forecast model is unbiased. In this paper the authors characterize the bias for a current assimilation scheme in the tropical Pacific. The characterization is used to show how relatively simple empirical bias forecast models may be used in a two-stage bias correction procedure to improve the quality of the analysis.

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Dick P. Dee, Greg Gaspari, Chris Redder, Leonid Rukhovets, and Arlindo M. da Silva

Abstract

Three different applications of maximum-likelihood estimation of error covariance parameters for atmospheric data assimilation are described. Height error standard deviations, vertical correlation coefficients, and isotropic decorrelation length scales are estimated from rawinsonde height observed-minus-forecast residuals. Sea level pressure error standard deviations and decorrelation length scales are obtained from ship reports, and wind observation error standard deviations and forecast error stream function and velocity potential decorrelation length scales are estimated from aircraft data. These applications serve to demonstrate the ability of the method to estimate covariance parameters using multivariate data from moving observers.

Estimates of the parameter uncertainty due to sampling error can be obtained as a by-product of the maximum-likelihood estimation. By bounding this source of error, it is found that many statistical parameters that are usually presumed constant in operational data assimilation systems in fact vary significantly with time. This may well reflect the use of overly simplistic covariance models that cannot adequately describe state-dependent error components such as representativeness error. The sensitivity of the parameter estimates to the treatment of bias, and to the choice of the model representing spatial correlations, is examined in detail. Several experiments emulate an online covariance parameter estimation procedure using a sliding window of data, and it is shown that such a procedure is both desirable and computationally feasible.

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