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P. L. Houtekamer

Abstract

A three-level quasigeostrophic model of the atmosphere is used to refine a method, based on an ensemble of perturbations, for medium-range forecasting. Using as a hypothesis that it is most efficient to use for the perturbations those structures that have maximal growth, given a constraint on the initial perturbation, one arrives at the optimal perturbation method. Optimal perturbations are constructed to have a maximum forecast error in the short range. It is hoped that the perturbations continue to grow up to the medium-range forecast time.

Information on vertical covariances, spectral variances, and horizontal variances is accounted for in the constraint. The optimal perturbations are thus by construction consistent with these statistical properties of the initial error.

The experiments have been repeated with different durations of the short-range forecast If this duration is sufficiently long, and the ensemble sufficiently large, then the ensemble shows optimal growth for both the short and medium range. With smaller ensembles, the spread in the forecast error is shown to be systematically underestimated as a consequence of neglecting some important components of the error.

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P. L. Houtekamer

Abstract

A skill forecast gives the probability distribution for the error in a forecast. Statistically, Well-founded skill forecasting methods have so far only been applied within the context of simple models. In this paper, the growth of analysis errors is studied. This means that errors that are already present in the estimate of the initial state can grow only in accordance with the dynamics of a model. Errors in the description of the model itself are neglected. This paper uses a three-level quasigeotrophic spectral model of the atmospheric circulation, truncated at T21. It is shown that a linear theory for the evolution of errors can be used for the first three days of a forecast. For the description of the global error, Monte Carlo methods are more efficient that methods based on the use of the adjoint of the tangent linear equations. The limitation to spatially local errors dramatically reduces the dimension of the error vector. In that case, adjoins methods are the most efficient ones. Local skill forecasts for three days ahead am computed for a period of 24 consecutive days, using the T21 model and the adjoint of its tangent linear equations. The variability in the predicted distributions for the local errors is fitted with a two-parameter stochastic model. Within the context of a perfect model assumption providing perfect skill forecasts the variability in the distribution of the error at day 3 is such that for equal quality forecasts the maximum extension of the forecast length is two days.

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P. L. Houtekamer

Abstract

A skill forecast gives the probability distribution for the error in the forecast. The purpose of this paper is to develop a skill-forecasting method. The method is applied to a spectral two-layer quasigeostrophic atmospheric model with a triangular truncation at wavenumber 5. The analysis is restricted to internal error growth. It is investigated how observational errors lead to errors in the analysis. It appears that climatological distributions can be used for the errors in the analysis. In the forecast run the evolution of these distributions is computed. For that purpose the tangent-linear equations for the errors are used. Because of this linearization, the results are valid for short-range skill forecasts only. The Lanczos algorithm is used to find the structures that dominate the forecast error. This algorithm is intended to be applicable in a realistic model.

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P. L. Houtekamer

Abstract

No abstract available.

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P. L. Houtekamer
and
Herschel L. Mitchell

Abstract

No abstract available.

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Herschel L. Mitchell
and
P. L. Houtekamer

Abstract

This paper examines ensemble Kalman filter (EnKF) performance for a number of different EnKF configurations. The study is performed in a perfect-model context using the logistic map as forecast model. The focus is on EnKF performance when the ensemble is small. In accordance with theory, it is found that those configurations that maintain an appropriate ensemble spread are indeed those with the smallest ensemble mean error in a data assimilation cycle. Thus, the deficient ensemble spread produced by the single-ensemble EnKF results in increased ensemble mean error for this configuration. This problem with the conceptually simplest EnKF motivates an examination of a variety of other configurations. These include the configuration with a pair of ensembles and several configurations with overlapping ensembles, such as the four-subensemble configuration (used operationally at the Canadian Meteorological Centre) and the configuration in which observations are assimilated into each member using a gain computed from all of the other members. Also examined is a configuration that uses the jackknife estimator to obtain an estimate of the gain and an estimate of its uncertainty. Using these estimates, a different perturbed gain is then produced for each ensemble member. In general, it is found that these latter configurations outperform both the single-ensemble EnKF and the configuration with a pair of ensembles. In addition to these “stochastic” filters, the performance of a “deterministic” filter (which does not use perturbed observations) is also examined.

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P. L. Houtekamer
and
Herschel L. Mitchell

Abstract

The possibility of performing data assimilation using the flow-dependent statistics calculated from an ensemble of short-range forecasts (a technique referred to as ensemble Kalman filtering) is examined in an idealized environment. Using a three-level, quasigeostrophic, T21 model and simulated observations, experiments are performed in a perfect-model context. By using forward interpolation operators from the model state to the observations, the ensemble Kalman filter is able to utilize nonconventional observations.

In order to maintain a representative spread between the ensemble members and avoid a problem of inbreeding, a pair of ensemble Kalman filters is configured so that the assimilation of data using one ensemble of short-range forecasts as background fields employs the weights calculated from the other ensemble of short-range forecasts. This configuration is found to work well: the spread between the ensemble members resembles the difference between the ensemble mean and the true state, except in the case of the smallest ensembles.

A series of 30-day data assimilation cycles is performed using ensembles of different sizes. The results indicate that (i) as the size of the ensembles increases, correlations are estimated more accurately and the root-mean-square analysis error decreases, as expected, and (ii) ensembles having on the order of 100 members are sufficient to accurately describe local anisotropic, baroclinic correlation structures. Due to the difficulty of accurately estimating the small correlations associated with remote observations, a cutoff radius beyond which observations are not used, is implemented. It is found that (a) for a given ensemble size there is an optimal value of this cutoff radius, and (b) the optimal cutoff radius increases as the ensemble size increases.

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Herschel L. Mitchell
and
P. L. Houtekamer

Abstract

To the extent that model error is nonnegligible in numerical models of the atmosphere, it must be accounted for in 4D atmospheric data assimilation systems. In this study, a method of estimating and accounting for model error in the context of an ensemble Kalman filter technique is developed. The method involves parameterizing the model error and using innovations to estimate the model-error parameters. The estimation algorithm is based on a maximum likelihood approach and the study is performed in an idealized environment using a three-level, quasigeostrophic, T21 model and simulated observations and model error.

The use of a limited number of ensemble members gives rise to a rank problem in the estimate of the covariance matrix of the innovations. The effect of this problem on the two terms of the log-likelihood function is that the variance term is underestimated, while the χ 2 term is overestimated. To permit the use of relatively small ensembles, a number of strategies are developed to deal with these systematic estimation problems. These include the imposition of a block structure on the covariance matrix of the innovations and a Richardson extrapolation of the log-likelihood value to infinite ensemble size. It is shown that with the use of these techniques, estimates of the model-error parameters are quite acceptable in a statistical sense, even though estimates based on any single innovation vector can be poor.

It is found that, with temporal smoothing of the model-error parameter estimates, the adaptive ensemble Kalman filter produces fairly good estimates of the parameters and accounts rather well for the model error. In fact, its performance in a data assimilation cycle is almost as good as that of a cycle in which the correct model-error parameters are used to increase the spread in the ensemble.

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P. L. Houtekamer
and
Herschel L. Mitchell

Abstract

An ensemble Kalman filter may be considered for the 4D assimilation of atmospheric data. In this paper, an efficient implementation of the analysis step of the filter is proposed. It employs a Schur (elementwise) product of the covariances of the background error calculated from the ensemble and a correlation function having local support to filter the small (and noisy) background-error covariances associated with remote observations. To solve the Kalman filter equations, the observations are organized into batches that are assimilated sequentially. For each batch, a Cholesky decomposition method is used to solve the system of linear equations. The ensemble of background fields is updated at each step of the sequential algorithm and, as more and more batches of observations are assimilated, evolves to eventually become the ensemble of analysis fields.

A prototype sequential filter has been developed. Experiments are performed with a simulated observational network consisting of 542 radiosonde and 615 satellite-thickness profiles. Experimental results indicate that the quality of the analysis is almost independent of the number of batches (except when the ensemble is very small). This supports the use of a sequential algorithm.

A parallel version of the algorithm is described and used to assimilate over 100 000 observations into a pair of 50-member ensembles. Its operation count is proportional to the number of observations, the number of analysis grid points, and the number of ensemble members. In view of the flexibility of the sequential filter and its encouraging performance on a NEC SX-4 computer, an application with a primitive equations model can now be envisioned.

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P. L. Houtekamer
and
Jacques Derome

Abstract

It is desirable to filter the unpredictable components from a medium-range forecast. Such a filtered forecast can be obtained by averaging an ensemble of predictions that started from slightly different initial atmospheric states. Different strategies have been proposed to generate the initial perturbations for such an ensemble. “Optimal” perturbation give the largest error at a prespecified forecast time. “Bred” perturbations have grown during a period prior to the analysis. “OSSE-MC” perturbations are obtained using a Monte Carlo-like observation system simulation experiment (OSSE).

In the current pilot study, the properties of the different strategies are compared. A three-level quasigeostrophic model is used to describe the evolution of the errors. The tangent linear version of this model and its adjoint version are used to generate the optimal perturbations, while bred perturbations are generated using the full nonlinear model. In the OSSE-MC method, random perturbations of model states are used in the simulation of radiosonde and satellite observations. These observations are then assimilated using an optimal interpolation (OI) assimilation system. A large OSSE-MC ensemble is obtained using such input and the OI system, which then provides the ground truth for the other ensembles. Its observed statistical properties are also used in the construction of the optimal and the bred perturbations.

The quality of the different ensemble mean medium-range forecasts is compared for forecast lengths of up to 15 days and ensembles of 2, 8, and 32 members. Before 6 days the control performs almost as well as any ensemble mean. Bred and OSSE-MC ensembles of only two members are of marginal quality. For all three methods an ensemble size of 8 is sufficient to obtain the main part of the possible improvement over the control, and all perform well for 32-member ensembles. Still better results are obtained from a weighted mean of the climate and the ensemble mean.

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