Monthly Weather Review
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Review of the Ensemble Kalman Filter for Atmospheric Data Assimilation Abstract 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. Contents Introduction ...4490Popular flavors of the EnKF algorithm ...4491General description...4491 Stochastic and deterministic filters...4492 The stochastic filter...4492 The deterministic filter...4492 Sequential or local filters...4493 Sequential ensemble Kalman filters...4493 The local ensemble transform Kalman filter...4494 Extended state vector...4494 Issues for the development of algorithms...4495 Use of small ensembles ...4495Monte Carlo methods...4495 Validation of reliability...4497 Use of group filters with no inbreeding...4498 Sampling error due to limited ensemble size: The rank problem...4498 Covariance localization...4499 Localization in the sequential filter...4499 Localization in the LETKF...4499 Issues with localization...4500 Summary...4501 Methods to increase ensemble spread ...4501Covariance inflation...4501 Additive inflation...4501 Multiplicative inflation...4502 Relaxation to prior ensemble information...4502 Issues with inflation...4503 Diffusion and truncation...4503 Error in physical parameterizations...4504 Physical tendency perturbations...4504 Multimodel, multiphysics, and multiparameter approaches...4505 Future directions...4505 Realism of error sources...4506 Balance and length of the assimilation window ...4506The need for balancing methods...4506 Time-filtering methods...4506 Toward shorter assimilation windows...4507 Reduction of sources of imbalance...4507 Regional data assimilation ...4508Boundary conditions and consistency across multiple domains...4509 Initialization of the starting ensemble...4510 Preprocessing steps for radar observations...4510 Use of radar observations for convective-scale analyses...4511 Use of radar observations for tropical cyclone analyses...4511 Other issues with respect to LAM data assimilation...4511 The assimilation of satellite observations ...4512Covariance localization...4512 Data density...4513 Bias-correction procedures...4513 Impact of covariance cycling...4514 Assumptions regarding observational error...4514 Recommendations regarding satellite observations...4515 Computational aspects ...4515Parameters with an impact on quality...4515 Overview of current parallel algorithms...4516 Evolution of computer architecture...4516 Practical issues...4517 Approaching the gray zone...4518 Summary...4518 Hybrids with variational and EnKF components ...4519Hybrid background error covariances...4519 E4DVar with the α control variable...4519 Not using linearized models with 4DEnVar...4520 The hybrid gain algorithm...4521 Open issues and recommendations...4521 Summary and discussion ...4521Stochastic or deterministic filters...4522 The nature of system error...4522 Going beyond the synoptic scales...4522 Satellite observations...4523 Hybrid systems...4523 Future of the EnKF...4523 APPENDIX A ...4524Types of Filter Divergence ...4524Classical filter divergence...4524 Catastrophic filter divergence...4524 APPENDIX B ...4524Systems Available for Download ...4524References ...4525
Monthly Weather Review - Tập 144 Số 12 - Trang 4489-4532 - 2016
E4DVar: Coupling an Ensemble Kalman Filter with Four-Dimensional Variational Data Assimilation in a Limited-Area Weather Prediction Model A hybrid data assimilation approach that couples the ensemble Kalman filter (EnKF) and four-dimensional variational (4DVar) methods is implemented for the first time in a limited-area weather prediction model. In this coupled system, denoted E4DVar, the EnKF and 4DVar systems run in parallel while feeding into each other. The multivariate, flow-dependent background error covariance estimated from the EnKF ensemble is used in the 4DVar minimization and the ensemble mean in the EnKF analysis is replaced by the 4DVar analysis, while updating the analysis perturbations for the next cycle of ensemble forecasts with the EnKF. Therefore, the E4DVar can obtain flow-dependent information from both the explicit covariance matrix derived from ensemble forecasts, as well as implicitly from the 4DVar trajectory. The performance of an E4DVar system is compared with the uncoupled 4DVar and EnKF for a limited-area model by assimilating various conventional observations over the contiguous United States for June 2003. After verifying the forecasts from each analysis against standard sounding observations, it is found that the E4DVar substantially outperforms both the EnKF and 4DVar during this active summer month, which featured several episodes of severe convective weather. On average, the forecasts produced from E4DVar analyses have considerably smaller errors than both of the stand-alone EnKF and 4DVar systems for forecast lead times up to 60 h.
Monthly Weather Review - Tập 140 Số 2 - Trang 587-600 - 2012
An Ensemble-Based Four-Dimensional Variational Data Assimilation Scheme. Part I: Technical Formulation and Preliminary Test Abstract
Applying a flow-dependent background error covariance (𝗕 matrix) in variational data assimilation has been a topic of interest among researchers in recent years. In this paper, an ensemble-based four-dimensional variational (En4DVAR) algorithm, designed by the authors, is presented that uses a flow-dependent background error covariance matrix constructed by ensemble forecasts and performs 4DVAR optimization to produce a balanced analysis. A great advantage of this En4DVAR design over standard 4DVAR methods is that the tangent linear and adjoint models can be avoided in its formulation and implementation. In addition, it can be easily incorporated into variational data assimilation systems that are already in use at operational centers and among the research community.
A one-dimensional shallow water model was used for preliminary tests of the En4DVAR scheme. Compared with standard 4DVAR, the En4DVAR converges well and can produce results that are as good as those with 4DVAR but with far less computation cost in its minimization. In addition, a comparison of the results from En4DVAR with those from other data assimilation schemes [e.g., 3DVAR and ensemble Kalman filter (EnKF)] is made. The results show that the En4DVAR yields an analysis that is comparable to the widely used variational or ensemble data assimilation schemes and can be a promising approach for real-time applications.
In addition, experiments were carried out to test the sensitivities of EnKF and En4DVAR, whose background error covariance is estimated from the same ensemble forecasts. The experiments indicated that En4DVAR obtained reasonably sound analysis even with larger observation error, higher observation frequency, and more unbalanced background field.
Monthly Weather Review - Tập 136 Số 9 - Trang 3363-3373 - 2008
Systematic Comparison of Four-Dimensional Data Assimilation Methods With and Without the Tangent Linear Model Using Hybrid Background Error Covariance: E4DVar versus 4DEnVar Abstract
Two ensemble formulations of the four-dimensional variational (4DVar) data assimilation technique are examined for a low-dimensional dynamical system. The first method, denoted E4DVar, uses tangent linear and adjoint model operators to minimize a cost function in the same manner as the traditional 4DVar data assimilation system. The second method, denoted 4DEnVar, uses an ensemble of nonlinear model trajectories to replace the function of linearized models in 4DVar, thus improving the parallelization of the data assimilation. Background errors for each algorithm are represented using a hybrid error covariance, which includes climatological errors as well as ensemble-estimated errors from an ensemble Kalman filter (EnKF). Numerical experiments performed over a range of scenarios suggest that both methods provide similar analysis accuracy for dense observation networks, and in perfect model experiments with large ensembles. Nevertheless, E4DVar has clear benefits over 4DEnVar when substantial covariance localization is required to treat sampling error. The greatest advantage of the tangent-linear approach is that it implicitly propagates a localized, full-rank ensemble covariance in time, thus avoiding the need to localize a time-dependent ensemble covariance. The tangent linear and adjoint model operators also provide a means of evolving flow-dependent information from the climate-based error component, which is found to be beneficial for treating model error. Challenges that need to be overcome before adopting a pure ensemble framework are illustrated through experiments estimating time covariances with four-dimensional ensembles and comparing results with those estimated with a tangent linear model.
Monthly Weather Review - Tập 143 Số 5 - Trang 1601-1621 - 2015
The Hybrid Local Ensemble Transform Kalman Filter Abstract
Hybrid data assimilation methods combine elements of ensemble Kalman filters (EnKF) and variational methods. While most approaches have focused on augmenting an operational variational system with dynamic error covariance information from an ensemble, this study takes the opposite perspective of augmenting an operational EnKF with information from a simple 3D variational data assimilation (3D-Var) method. A class of hybrid methods is introduced that combines the gain matrices of the ensemble and variational methods, rather than linearly combining the respective background error covariances. A hybrid local ensemble transform Kalman filter (Hybrid-LETKF) is presented in two forms: 1) a traditionally motivated Hybrid/Covariance-LETKF that combines the background error covariance matrices of LETKF and 3D-Var, and 2) a simple-to-implement algorithm called the Hybrid/Mean-LETKF that falls into the new class of hybrid gain methods. Both forms improve analysis errors when using small ensemble sizes and low observation coverage versus either LETKF or 3D-Var used alone. The results imply that for small ensemble sizes, allowing a solution to be found outside of the space spanned by ensemble members provides robustness in both hybrid methods compared to LETKF alone. Finally, the simplicity of the Hybrid/Mean-LETKF design implies that this algorithm can be applied operationally while requiring only minor modifications to an existing operational 3D-Var system.
Monthly Weather Review - Tập 142 Số 6 - Trang 2139-2149 - 2014
Calibrated Probabilistic Forecasting Using Ensemble Model Output Statistics and Minimum CRPS Estimation Abstract
Ensemble prediction systems typically show positive spread-error correlation, but they are subject to forecast bias and dispersion errors, and are therefore uncalibrated. This work proposes the use of ensemble model output statistics (EMOS), an easy-to-implement postprocessing technique that addresses both forecast bias and underdispersion and takes into account the spread-skill relationship. The technique is based on multiple linear regression and is akin to the superensemble approach that has traditionally been used for deterministic-style forecasts. The EMOS technique yields probabilistic forecasts that take the form of Gaussian predictive probability density functions (PDFs) for continuous weather variables and can be applied to gridded model output. The EMOS predictive mean is a bias-corrected weighted average of the ensemble member forecasts, with coefficients that can be interpreted in terms of the relative contributions of the member models to the ensemble, and provides a highly competitive deterministic-style forecast. The EMOS predictive variance is a linear function of the ensemble variance. For fitting the EMOS coefficients, the method of minimum continuous ranked probability score (CRPS) estimation is introduced. This technique finds the coefficient values that optimize the CRPS for the training data. The EMOS technique was applied to 48-h forecasts of sea level pressure and surface temperature over the North American Pacific Northwest in spring 2000, using the University of Washington mesoscale ensemble. When compared to the bias-corrected ensemble, deterministic-style EMOS forecasts of sea level pressure had root-mean-square error 9% less and mean absolute error 7% less. The EMOS predictive PDFs were sharp, and much better calibrated than the raw ensemble or the bias-corrected ensemble.
Monthly Weather Review - Tập 133 Số 5 - Trang 1098-1118 - 2005
Variogram-Based Proper Scoring Rules for Probabilistic Forecasts of Multivariate Quantities* Abstract
Proper scoring rules provide a theoretically principled framework for the quantitative assessment of the predictive performance of probabilistic forecasts. While a wide selection of such scoring rules for univariate quantities exists, there are only few scoring rules for multivariate quantities, and many of them require that forecasts are given in the form of a probability density function. The energy score, a multivariate generalization of the continuous ranked probability score, is the only commonly used score that is applicable in the important case of ensemble forecasts, where the multivariate predictive distribution is represented by a finite sample. Unfortunately, its ability to detect incorrectly specified correlations between the components of the multivariate quantity is somewhat limited. In this paper the authors present an alternative class of proper scoring rules based on the geostatistical concept of variograms. The sensitivity of these variogram-based scoring rules to incorrectly predicted means, variances, and correlations is studied in a number of examples with simulated observations and forecasts; they are shown to be distinctly more discriminative with respect to the correlation structure. This conclusion is confirmed in a case study with postprocessed wind speed forecasts at five wind park locations in Colorado.
Monthly Weather Review - Tập 143 Số 4 - Trang 1321-1334 - 2015
Distinct Modes of the East Asian Winter Monsoon Abstract
Two distinct modes of the East Asian winter monsoon (EAWM) have been identified, and they correspond to real and imaginary parts of the leading mode of the EAWM, respectively. Analyses of these modes used the National Centers for Environment Prediction (NCEP) and National Center for Atmospheric Research (NCAR) monthly mean reanalysis datasets for the period 1968–2003, as well as the Southern Oscillation index (SOI), North Atlantic Oscillation index, and eastern equatorial Pacific sea surface temperature (SST) data. Results were obtained by resolving a complex Hermite matrix derived from 850-hPa anomalous wind fields, and determining the resulting modes’ associations with several climate variables. The first distinct mode (M1) is characterized by an anomalous meridional wind pattern over East Asia and the western North Pacific. Mode M1 is closely related to several features of the atmospheric circulation, including the Siberian high, East Asian trough, East Asian upper-tropospheric jet, and local Hadley circulation over East Asia. Thus, M1 reflects the traditional EAWM pattern revealed in previous studies. The second distinct EAWM mode (M2), which was not identified previously, displays dominant zonal wind anomalies over the same area. Mode M2 exhibits a closer relation than M1 to sea level pressure anomalies over the northwestern Pacific southeast of Japan and with the SOI and equatorial eastern Pacific SST. Unlike M1, M2 does not show coherent relationships with the Siberian high, East Asian trough, and East Asian upper-tropospheric jet. Since atmospheric circulation anomalies relevant to M2 exhibit a quasi-barotropic structure, its existence cannot simply be attributed to differential land–sea heating. El Niño events tend to occur in the negative phase of M1 and the positive phase of M2, both corresponding to a weakened EAWM. The Arctic Oscillation does not appear to impact the EAWM on interannual time scales. Although the spatial patterns for the two modes are very different, the two distinct modes are complementary, with the leading EAWM mode being a linear combination of the two. The results herein therefore demonstrate that a single EAWM index may be inappropriate for investigating and predicting the EAWM.
Monthly Weather Review - Tập 134 Số 8 - Trang 2165-2179 - 2006
Assimilation of Precipitable Water Measurements into a Mesoscale Numerical Model
Monthly Weather Review - Tập 121 Số 4 - Trang 1215-1238 - 1993
Mesoscale Convective Systems over Western Equatorial Africa and Their Relationship to Large-Scale Circulation Abstract
This study examines mesoscale convective systems (MCSs) over western equatorial Africa using data from the Tropical Rainfall Measuring Mission (TRMM) satellite. This region experiences some of the world’s most intense thunderstorms and highest lightning frequency, but has low rainfall relative to other equatorial regions. The analyses of MCS activity include the frequency of occurrence, diurnal and annual cycles, and associated volumetric and convective rainfall. Also evaluated is the lightning activity associated with the MCSs. Emphasis is placed on the diurnal cycle and on the continental-scale motion fields in this region. The diurnal cycle shows a maximum in MCS count around 1500–1800 LT, a morning minimum, and substantial activity during the night; there is little seasonal variation in the diurnal cycle, suggesting stationary influences such as orography. Our analysis shows four maxima in MCS activity, three of which are related to local geography (two orographic and one over Lake Victoria). The fourth coincides with a midtropospheric convergence maximum in the right entrance quadrant of the African easterly jet of the Southern Hemisphere (AEJ-S). This maximum is substantially stronger in the September–November rainy season, when the jet is well developed, than in the March–May rainy season, when the jet is absent. Lightning frequency and flashes per MCS are also greatest during September–November; maxima occur in the right entrance quadrant of the AEJ-S. The lightning maximum is somewhat south of the MCS maximum and coincides with the low-lying areas of central Africa. Overall, the results of this study suggest that large-scale topography plays a critical role in the spatial and diurnal patterns of convection, lightning, and rainfall in this region. More speculative is the role of the AEJ-S, but this preliminary analysis suggests that it does play a role in the anomalous intensity of convection in western equatorial Africa.
Monthly Weather Review - Tập 137 Số 4 - Trang 1272-1294 - 2009
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