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P. L. Houtekamer


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P. L. Houtekamer


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


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


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
Louis Lefaivre


An experimental ensemble forecasting system has been set up in an attempt to simulate all sources of forecast error. Errors in the observations, in the surface fields, and in the forecast model have been simulated. This has been done in different ways for different members of the ensemble. In particular, the N forecasting systems used for the N ensemble members differ in N − 1 aspects.

A model is proposed that writes the systematic component of the forecast error as the sum of the ensemble mean error and a linear combination of the impact of the N − 1 basic modifications to the forecasting system. The N − 1 coefficients of this expansion are the parameters that are to be determined from a comparison with radiosonde observations. For this purpose a merit function is defined that measures the total distance of a set of forecasts, at different days, to the verifying observations. The N − 1 coefficients, which minimize the merit function, are found using a least squares solution. The solution is the best forecasting system that can be obtained at a given truncation using a given set of parametrizations of physical processes and a given set of possibilities for the data assimilation system.

With the above system, several dependent aspects of the forecasting system have been simultaneously validated as a by-product of a daily ensemble forecast. The error bars on the validation results give information on the extent to which changes to the forecasting system are, or are not, confirmed by radiosonde measurements. As an example, results are given for the period 28 March through 17 April 1996.

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P. L. Houtekamer
Fuqing Zhang


This paper reviews the development of the ensemble Kalman filter (EnKF) for atmospheric data assimilation. Particular attention is devoted to recent advances and current challenges. The distinguishing properties of three well-established variations of the EnKF algorithm are first discussed. Given the limited size of the ensemble and the unavoidable existence of errors whose origin is unknown (i.e., system error), various approaches to localizing the impact of observations and to accounting for these errors have been proposed. However, challenges remain; for example, with regard to localization of multiscale phenomena (both in time and space). For the EnKF in general, but higher-resolution applications in particular, it is desirable to use a short assimilation window. This motivates a focus on approaches for maintaining balance during the EnKF update. Also discussed are limited-area EnKF systems, in particular with regard to the assimilation of radar data and applications to tracking severe storms and tropical cyclones. It seems that relatively less attention has been paid to optimizing EnKF assimilation of satellite radiance observations, the growing volume of which has been instrumental in improving global weather predictions. There is also a tendency at various centers to investigate and implement hybrid systems that take advantage of both the ensemble and the variational data assimilation approaches; this poses additional challenges and it is not clear how it will evolve. It is concluded that, despite more than 10 years of operational experience, there are still many unresolved issues that could benefit from further research.


  • Introduction...4490

  • Popular flavors of the EnKF algorithm...4491

    1. General description...4491

    2. Stochastic and deterministic filters...4492

      1. The stochastic filter...4492

      2. The deterministic filter...4492

    3. Sequential or local filters...4493

      1. Sequential ensemble Kalman filters...4493

      2. The local ensemble transform Kalman filter...4494

    4. Extended state vector...4494

    5. Issues for the development of algorithms...4495

  • Use of small ensembles...4495

    1. Monte Carlo methods...4495

    2. Validation of reliability...4497

    3. Use of group filters with no inbreeding...4498

    4. Sampling error due to limited ensemble size: The rank problem...4498

    5. Covariance localization...4499

      1. Localization in the sequential filter...4499

      2. Localization in the LETKF...4499

      3. Issues with localization...4500

    6. Summary...4501

  • Methods to increase ensemble spread...4501

    1. Covariance inflation...4501

      1. Additive inflation...4501

      2. Multiplicative inflation...4502

      3. Relaxation to prior ensemble information...4502

      4. Issues with inflation...4503

    2. Diffusion and truncation...4503

    3. Error in physical parameterizations...4504

      1. Physical tendency perturbations...4504

      2. Multimodel, multiphysics, and multiparameter approaches...4505

      3. Future directions...4505

    4. Realism of error sources...4506

  • Balance and length of the assimilation window...4506

    1. The need for balancing methods...4506

    2. Time-filtering methods...4506

    3. Toward shorter assimilation windows...4507

    4. Reduction of sources of imbalance...4507

  • Regional data assimilation...4508

    1. Boundary conditions and consistency across multiple domains...4509

    2. Initialization of the starting ensemble...4510

    3. Preprocessing steps for radar observations...4510

    4. Use of radar observations for convective-scale analyses...4511

    5. Use of radar observations for tropical cyclone analyses...4511

    6. Other issues with respect to LAM data assimilation...4511

  • The assimilation of satellite observations...4512

    1. Covariance localization...4512

    2. Data density...4513

    3. Bias-correction procedures...4513

    4. Impact of covariance cycling...4514

    5. Assumptions regarding observational error...4514

    6. Recommendations regarding satellite observations...4515

  • Computational aspects...4515

    1. Parameters with an impact on quality...4515

    2. Overview of current parallel algorithms...4516

    3. Evolution of computer architecture...4516

    4. Practical issues...4517

    5. Approaching the gray zone...4518

    6. Summary...4518

  • Hybrids with variational and EnKF components...4519

    1. Hybrid background error covariances...4519

    2. E4DVar with the α control variable...4519

    3. Not using linearized models with 4DEnVar...4520

    4. The hybrid gain algorithm...4521

    5. Open issues and recommendations...4521

  • Summary and discussion...4521

    1. Stochastic or deterministic filters...4522

    2. The nature of system error...4522

    3. Going beyond the synoptic scales...4522

    4. Satellite observations...4523

    5. Hybrid systems...4523

    6. Future of the EnKF...4523


Types of Filter Divergence...4524

  1. Classical filter divergence...4524

  2. Catastrophic filter divergence...4524

    APPENDIX B...4524

    Systems Available for Download...4524


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P. L. Houtekamer
Jacques Derome


Numerical experiments have been performed to determine whether it is possible to improve the quality of atmospheric forecasts by using the average of two predictions starting from slightly perturbed initial conditions. The predictions are made with a T21 quasi-nondivergent three-level model and a “perfect model” approach is used, so that all prediction errors are due to the uncertainty in the initial conditions. The two perturbed predictions are initialized by adding to and subtracting from the control initial state a small-amplitude disturbance called a “bred” mode, obtained as the fastest-growing small-amplitude perturbation of the model over a 20-day period preceding the beginning of the forecast.

The results indicate that for initial states that contain very small analysis errors the two-member ensemble yields a mean forecast of lower quality than the control forecast. For larger-amplitude analysis error fields, however, the ensemble prediction outperforms the control forecast. When a statistical distribution of possible analysis errors is considered, it is found that on average the mean of the two perturbed predictions is of higher quality than the control forecast.

The study has also shown that the spread between the two perturbed predictions is correlated with the magnitude of the forecast error for every day of the forecast period from day 1 to day 10.

The same approach has been applied to Lorenz's three-component model and similar results have been obtained.

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P. L. Houtekamer
Jacques Derome


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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Herschel L. Mitchell
P. L. Houtekamer


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
Herschel L. Mitchell


No abstract available.

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