On-line Estimation of Error Covariance Parameters for Atmospheric Data Assimilation

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  • 1 Delft Hydraulics, Emmeloord, the Netherlands, and Data Assimilation Office, NASA/Goddard Space Flight Center, Greenbelt, Maryland
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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.

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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