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Chengsi Liu and Ming Xue

Abstract

Ensemble–variational data assimilation algorithms that can incorporate the time dimension (four-dimensional or 4D) and combine static and ensemble-derived background error covariances (hybrid) are formulated in general forms based on the extended control variable and the observation-space-perturbation approaches. The properties and relationships of these algorithms and their approximated formulations are discussed. The main algorithms discussed include the following: 1) the standard ensemble 4DVar (En4DVar) algorithm incorporating ensemble-derived background error covariance through the extended control variable approach, 2) the 4DEnVar neglecting the time propagation of the extended control variable (4DEnVar-NPC), 3) the 4D ensemble–variational algorithm based on observation space perturbation (4DEnVar), and 4) the 4DEnVar with no propagation of covariance localization (4DEnVar-NPL). Without the static background error covariance term, none of the algorithms requires the adjoint model except for En4DVar. Costly applications of the tangent linear model to localized ensemble perturbations can be avoided by making the NPC and NPL approximations. It is proven that En4DVar and 4DEnVar are mathematically equivalent, while 4DEnVar-NPC and 4DEnVar-NPL are mathematically equivalent. Such equivalences are also demonstrated by single-observation assimilation experiments with a 1D linear advection model. The effects of the non-flow-following or stationary localization approximations are also examined through the experiments.

All of the above algorithms can include the static background error covariance term to establish a hybrid formulation. When the static term is included, all algorithms will require a tangent linear model and an adjoint model. The first guess at appropriate time (FGAT) approximation is proposed to avoid the tangent linear and adjoint models. Computational costs of the algorithms are also discussed.

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Chengsi Liu and Qingnong Xiao

Abstract

A four-dimensional ensemble-based variational data assimilation (4DEnVar) algorithm proposed in Part I of the 4DEnVar series (denoted En4DVar in Part I, but here we refer to it as 4DEnVar according to WMO conference recommendation to differentiate it from En4DVar algorithm using adjoint model) uses a flow-dependent background error covariance calculated from ensemble forecasts and performs 4DVar optimization based on an incremental approach and a preconditioning algorithm. In Part II, the authors evaluated 4DEnVar with observing system simulation experiments (OSSEs) using the Advanced Research Weather Research and Forecasting Model (ARW-WRF, hereafter WRF). The current study extends the 4DEnVar to assimilate real observations for a cyclone in the Antarctic and the Southern Ocean in October 2007. The authors performed an intercomparison of four different WRF variational approaches for the case, including three-dimensional variational data assimilation (3DVar), first guess at the appropriate time (FGAT), and ensemble-based three-dimensional (En3DVar) and four-dimensional (4DEnVar) variational data assimilations. It is found that all data assimilation approaches produce positive impacts in this case. Applying the flow-dependent background error covariance in En3DVar and 4DEnVar yields forecast skills superior to those with the homogeneous and isotropic background error covariance in 3DVar and FGAT. In addition, the authors carried out FGAT and 4DEnVar 3-day cycling and 72-h forecasts. The results show that 4DEnVar produces a better performance in the cyclone prediction. The inflation factor on 4DEnVar can effectively improve the 4DEnVar analysis. The authors also conducted a short period (10-day lifetime of the cyclone in the domain) of analysis/forecast intercomparison experiments using 4DEnVar, FGAT, and 3DVar. The 4DEnVar scheme demonstrates overall superior and robust performance.

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Chengsi Liu, Ming Xue, and Rong Kong

Abstract

Despite the well-known importance of background error covariance in data assimilation, not much study has been focused on its impact on the assimilation of radar reflectivity within a three-dimensional variational (3DVar) framework. In this study, it is shown that unphysical analysis increments of hydrometeors are produced when using vertically homogeneous background error variance. This issue cannot be fully solved by using the so-called hydrometeor classification in the reflectivity observation operator. Alternatively, temperature-dependent background error profiles for hydrometeor control variables are proposed. With such a treatment, the vertical background error profiles are specified to be temperature dependent, allowing for more physical partitioning of radar-observed precipitation information among the liquid and ice hydrometeors. The 3DVar analyses using our treatment are compared with those using constant background error or “hydrometeor classification” through observing system simulation experiments with a simulated supercell storm. Results show that 1) 3DVar with constant hydrometeor background errors produces unphysical rainwater at the high levels and unphysical snow at the low levels; 2) the hydrometeor classification approach reduces unphysical rainwater and snow at those levels, but the analysis increments are still unphysically spread in the vertical by the background error covariance when the vertically invariant background errors are used; and 3) the temperature-dependent background error profiles enable physically more reasonable analyses of liquid and ice hydrometeors from reflectivity assimilation.

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Chengsi Liu, Qingnong Xiao, and Bin Wang

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.

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Rong Kong, Ming Xue, and Chengsi Liu

Abstract

A hybrid ensemble–3DVar (En3DVar) system is developed and compared with 3DVar, EnKF, “deterministic forecast” EnKF (DfEnKF), and pure En3DVar for assimilating radar data through perfect-model observing system simulation experiments (OSSEs). DfEnKF uses a deterministic forecast as the background and is therefore parallel to pure En3DVar. Different results are found between DfEnKF and pure En3DVar: 1) the serial versus global nature and 2) the variational minimization versus direct filter updating nature of the two algorithms are identified as the main causes for the differences. For 3DVar (EnKF/DfEnKF and En3DVar), optimal decorrelation scales (localization radii) for static (ensemble) background error covariances are obtained and used in hybrid En3DVar. The sensitivity of hybrid En3DVar to covariance weights and ensemble size is examined. On average, when ensemble size is 20 or larger, a 5%–10% static covariance gives the best results, while for smaller ensembles, more static covariance is beneficial. Using an ensemble size of 40, EnKF and DfEnKF perform similarly, and both are better than pure and hybrid En3DVar overall. Using 5% static error covariance, hybrid En3DVar outperforms pure En3DVar for most state variables but underperforms for hydrometeor variables, and the improvement (degradation) is most notable for water vapor mixing ratio q υ (snow mixing ratio q s). Overall, EnKF/DfEnKF performs the best, 3DVar performs the worst, and static covariance only helps slightly via hybrid En3DVar.

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Chengsi Liu, Ming Xue, and Rong Kong

Abstract

Radar reflectivity (Z) data are either directly assimilated using 3DVar, 4DVar, or ensemble Kalman filter, or indirectly assimilated using, for example, cloud analysis that preretrieves hydrometeors from Z. When directly assimilating radar data variationally, issues related to the highly nonlinear Z operator arise that can cause nonconvergence and bad analyses. To alleviate the issues, treatments are proposed in this study and their performances are examined via observing system simulation experiments. They include the following: 1) When using hydrometeor mixing ratios as control variables (CVq), small background Z can cause extremely large cost function gradient. Lower limits are imposed on the mixing ratios (qLim treatment) or the equivalent reflectivity (ZeLim treatment) in Z observation operator. ZeLim is found to work better than qLim in terms of analysis accuracy and convergence speed. 2) With CVq, the assimilation of radial velocity (V r) is ineffective when assimilated together with Z data due to the much smaller cost function gradient associated with V r. A procedure (VrPass) that assimilates V r data in a separate pass is found very helpful. 3) Using logarithmic hydrometeor mixing ratios as control variables (CVlogq) can also avoid extremely large cost function gradient, and has much faster convergence. However, spurious analysis increments can be created when transforming the analysis increments back to mixing ratios. A background smoothing and a lower limit are applied to the background mixing ratios, and are shown to be effective. Using CVlogq with associated treatments produces better reflectivity analysis that is much closer to the observation without resorting to multiple analysis passes, and the cost function minimization also converges faster. CVlogq is therefore recommended for variational radar data assimilation.

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Chengsi Liu, Qingnong Xiao, and Bin Wang

Abstract

An ensemble-based four-dimensional variational data assimilation (En4DVAR) algorithm and its performance in a low-dimension space with a one-dimensional shallow-water model have been presented in Part I. This algorithm adopts the standard incremental approach and preconditioning in the variational algorithm but avoids the need for a tangent linear model and its adjoint so that it can be easily incorporated into variational assimilation systems. The current study explores techniques for En4DVAR application in real-dimension data assimilation. The EOF decomposed correlation function operator and analysis time tuning are formulated to reduce the impact of sampling errors in En4DVAR upon its analysis. With the Advanced Research Weather Research and Forecasting (ARW-WRF) model, Observing System Simulation Experiments (OSSEs) are designed and their performance in real-dimension data assimilation is examined. It is found that the designed En4DVAR localization techniques can effectively alleviate the impacts of sampling errors upon analysis. Most forecast errors and biases in ARW are reduced by En4DVAR compared to those in a control experiment. En3DVAR cycling experiments are used to compare the ensemble-based sequential algorithm with the ensemble-based retrospective algorithm. These experiments indicate that the ensemble-based retrospective assimilation, En4DVAR, produces an overall better analysis than the ensemble-based sequential algorithm, En3DVAR, cycling approach.

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Rong Kong, Ming Xue, Chengsi Liu, and Youngsun Jung

Abstract

In this study, a hybrid En3DVar data assimilation (DA) scheme is compared with 3DVar, EnKF, and pure En3DVar for the assimilation of radar data in a real tornadic storm case. Results using hydrometeor mixing ratios (CVq) or logarithmic mixing ratios (CVlogq) as the control variables are compared in the variational DA framework. To address the lack of radial velocity impact issues when using CVq, a procedure that assimilates reflectivity and radial velocity data in two separate analysis passes is adopted. Comparisons are made in terms of the root-mean-square innovations (RMSIs) as well as the intensity and structure of the analyzed and forecast storms. For pure En3DVar that uses 100% ensemble covariance, CVlogq and CVq have similar RMSIs in the velocity analyses, but errors grow faster during forecasts when using CVlogq. Introducing static background error covariance B at 5% in hybrid En3DVar (with CVlogq) significantly reduces the forecast error growth. Pure En3DVar produces more intense reflectivity analyses than EnKF that more closely match the observations. Hybrid En3DVar with 50% B outperforms other weights in terms of the RMSIs and forecasts of updraft helicity and is thus used in the final comparison with 3DVar and EnKF. The hybrid En3DVar is found to outperform EnKF in better capturing the intensity and structure of the analyzed and forecast storms and outperform 3DVAR in better capturing the intensity and evolution of the rotating updraft.

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Jonathan Labriola, Youngsun Jung, Chengsi Liu, and Ming Xue

Abstract

In an effort to improve radar data assimilation configurations for potential operational implementation, GSI EnKF data assimilation experiments based on the operational system employed by the Center for Analysis and Prediction of Storms (CAPS) real-time Spring Forecast Experiments are performed. These experiments are followed by 6-h forecasts for an MCS on 28–29 May 2017. Configurations examined include data thinning, covariance localization radii and inflation, observation error settings, and data assimilation frequency for radar observations. The results show experiments that assimilate radar observations more frequently (i.e., 5–10 min) are initially better at suppressing spurious convection. However, assimilating observations every 5 min causes spurious convection to become more widespread with time, and modestly degrades forecast skill through the remainder of the forecast window. Ensembles that assimilate more observations with less thinning of data or use a larger horizontal covariance localization radius for radar data predict fewer spurious storms and better predict the location of observed storms. Optimized data thinning and horizontal covariance localization radii have positive impacts on forecast skill during the first forecast hour that are quickly lost due to the growth of forecast error. Forecast skill is less sensitive to the ensemble spread inflation factors and observation errors tested during this study. These results provide guidance toward optimizing the configuration of the GSI EnKF system. Among the DA configurations tested, the one employed by the CAPS Spring Forecast Experiment produces the most skilled forecasts while remaining computationally efficient for real-time use.

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Lianglyu Chen, Chengsi Liu, Ming Xue, Gang Zhao, Rong Kong, and Youngsun Jung

Abstract

When directly assimilating radar data within a variational framework using hydrometeor mixing ratios (q) as control variables (CVq), the gradient of the cost function becomes extremely large when background mixing ratio is close to zero. This significantly slows down minimization convergence and makes the assimilation of radial velocity and other observations ineffective because of the dominance of reflectivity observation term in the cost function gradient. Using logarithmic hydrometeor mixing ratios as control variables (CVlogq) can alleviate the problem but the high nonlinearity of logarithmic transformation can introduce spurious analysis increments into mixing ratios.

In this study, power transform of hydrometeors is proposed to form new control variables (CVpq) where the nonlinearity of transformation can be adjusted by tuning exponent or power parameter p. The performance of assimilating radar data using CVpq is compared with those using CVq and CVlogq for the analyses and forecasts of five convective storm cases from spring of 2017. Results show that CVpq with p = 0.4 (CVpq0.4) gives the best reflectivity forecasts in terms of root mean square error and equitable threat score. Furthermore, CVpq0.4 has faster convergence of cost function minimization than CVq and produces less spurious analysis increment than CVlogq. Compared to CVq and CVlogq, CVpq0.4 have better skills of 0-3h composite reflectivity forecasts, and the updraft helicity tracks for the 16 May 2017 Texas and Oklahoma tornado outbreak case are more consistent with observations when using CVpq0.4.

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