## 1. Introduction

During the past decade many studies have been performed aiming to use Doppler radar data to initialize cloud-resolving models. The three-dimensional variational data assimilation (3DVAR) methods can easily use mass continuity equations and other appropriate model equations as weak constraints, and are efficient for implementation in convective-scale nonhydrostatic flows (Gao et al. 2004; Barker et al. 2004; Xiao et al. 2005; Hu et al. 2006; Hu and Xue 2007; Ge and Gao 2007; Ge et al. 2010; Stensrud and Gao 2010; Xie et al. 2011). However, the major shortcoming of 3DVAR is that the background error covariances are stationary and isotropic, and error covariances related to using model equations as weak constraints cannot be simply defined. The more advanced four-dimensional variational data assimilation (4DVAR) technique incorporates the full prediction model into the assimilation system and implicitly includes the effects of flow-dependent error covariances through the forward and backward models. Storm-scale radar data assimilation (DA) using the 4DVAR method has been applied to a number of storm events by Sun and Crook (1997, 1998, 2001), Sun (2005), Wang et al. (2013), and Sun and Wang (2013). Despite some encouraging results, 4DVAR for convective-scale applications has been limited to simple microphysics in almost all cases because the strong nonlinearity within sophisticated microphysics schemes is difficult to handle in the minimization process.

Encouraging results also have been obtained in recent years using the ensemble Kalman filter method to assimilate radar data for convective storms (Snyder and Zhang 2003; Zhang et al. 2004; Tong and Xue 2005; Dowell et al. 2004, 2011; Yussouf and Stensrud 2010). For convective-scale weather, and considering the nature of radar data, flow-dependent background error covariances—such as derived from the EnKF method—are needed. One of the advantages of the EnKF method over variational methods is that it can dynamically evolve the background error covariances throughout the assimilation cycles. However, a major shortcoming with ensemble-based DA is rank deficiency or sampling error as a result of relatively small ensemble sizes. This problem may be even more severe with storm-scale DA because the total degrees of freedom of the system may be significantly larger than the practical ensemble size.

The relative advantages and disadvantages of variational and EnKF approaches to DA were also extensively discussed by Lorenc (2003) and Kalnay et al. (2007). To blend the advanced features of both variational and EnKF methods, and to overcome their respective shortcomings, they suggested that a hybrid ensemble and variational framework could be a best choice. For large-scale DA, such an approach was initially demonstrated for a quasigeostrophic system by Hamill and Snyder (2000) and further suggested by Lorenc (2003), Buehner (2005), and Zupanski (2005) with different formulations. Wang et al. (2007) showed that the formulations proposed by Hamill and Snyder (2000), Lorenc (2003), and Buehner (2005), though different in their implementations and computational cost, are all mathematically equivalent. In the hybrid method, the flow-dependent background error covariance estimated from ensemble members is used to fully or partially replace the static background error covariance commonly used in a variational framework (either 3DVAR or 4DVAR). Further case studies have demonstrated the potential advantages of the hybrid method (Wang et al. 2008a,b; Buehner et al. 2010a,b). Barker et al. (2012), Li et al. (2012), and Zhang et al. (2013) recently reported the capability of the hybrid system for mesoscale weather systems using the Weather Research and Forecasting (WRF) Model.

Separately, Liu et al. (2008, 2009) developed another hybrid ensemble and variational formulation in which the four-dimensional background error covariances are estimated from an ensemble of forecasts and used in a variational framework. Compared to 4DVAR algorithms, this method gets rid of the need for developing and maintaining the tangent-linear or adjoint versions of the forecast model; Compared to some of the EnKF algorithms (e.g., Whitaker and Hamill 2002), this method avoids the localization of background error covariances in observation space. This relatively new approach was recently coined four-dimensional ensemble–variational data assimilation (4DEnVAR) by Bowler et al. (2013).

However, whether these hybrid methods can be effectively extended to storm-scale DA has not been widely studied. Gao et al. (2013) demonstrate the potential usefulness of a hybrid 3DVAR and EnKF method for convective scale DA. The algorithm uses the extended control variable approach to combine the static and ensemble-derived flow-dependent forecast error covariances, in which the EnKF provides the ensemble perturbations. Similar to that suggested by Bowler et al. (2013), this method can be called the hybrid three-dimensional ensemble–variational data assimilation (3DEnVAR). The method is applied to the assimilation of simulated radar data for a supercell storm, with results indicating that the hybrid method provides the best analyses among the 3DVAR, EnKF, and 3DEnVAR for hydrometeor-related state variables in term of root-mean-squared errors. But for other state variables, the performance of the hybrid 3DEnVAR is very close to that of EnKF and both are much better than that of 3DVAR. However, there are still a number of questions that have not been answered. 1) What do the extended control variables represent in storm-scale radar DA? 2) What is the optimal choice for the relative weighting of the static and flow-dependent covariances for storm-scale radar DA? 3) How is the overall performance of the hybrid method influenced by ensemble size? Single-observation and sensitivity experiments for storm-scale DA when ingesting radar data are performed to answer these questions in this paper.

In section 2, we briefly introduce the three DA methods while section 3 describes the DA experiment design. In section 4, single-observation experiments will be performed to investigate the general behavior of the DA systems. Sensitivity experiments and quantitative performance are assessed in section 5. We conclude in section 6 with a summary and outlook for future work.

## 2. Data assimilation methods

In the following, we briefly describe the three DA methods being used: 3DVAR, EnKF and 3DEnVAR. The last one is a kind of combination of the first two. Each method has been tested for storm-scale DA with radar data in various applications.

### a. 3DVAR method

*J*used in the 3DVAR may be written as the sum of the background and observational terms plus a penalty or equation constraint term

*J*

_{c}:

**x**and

**x**

^{b}are the analysis and background state vectors;

**y**

^{o}is the observation vector;

*H*(

**x**) is the observation operator; and the term

*J*

_{c}in (1) includes any penalty or dynamic equation constraint terms that may be added to serve the important role of correlating the desired analysis variables. The mass continuity equation is imposed as a weak constraint in most of our previous applications (Gao et al. 1999; Hu et al. 2006). To effectively precondition the minimization problem, we follow Derber and Rosati (1989), and Courtier (1997), define an alternative control variable

**v**, such that

The square root of matrix

The analysis vector **x** contains the three wind components (*u*, *υ*, and *w*), potential temperature *θ*, pressure *p*, water vapor mixing ratio *q*_{υ}, and the hydrometeor-related model variables, including the mixing ratios for rainwater *q*_{r}, snow *q*_{s}, and hail *q*_{h}, which are added to the analysis vector to assimilate reflectivity directly in a variational framework (Gao and Stensrud 2012). The observation term in the cost function includes both radial velocity and reflectivity; their respective forward operators will be discussed in the next section.

### b. EnKF method

*a*refers to the analysis (posteriori estimate), where

*H*defined in (1), and

*n*represents the

*n*th ensemble member, the overbar denotes the ensemble mean,

*β*is a covariance inflation factor that is usually slightly larger than 1, and

*α*, a factor that reduces the gain matrix, is given in the EnKF algorithm by Whitaker and Hamill (2002) as

### c. Hybrid 3DEnVAR scheme

**v**to (

**v**,

**w**) within the 3DVAR cost function of (2), which then can be rewritten as

**x**. The control variable

**v**is defined in association with

**w**is the augmented control variable associated with

**v**and

**w**, instead of

*β*

_{1}and

*β*

_{2}are used to determine relative weights for the static background error covariance and the ensemble error covariance, and the following rule should be applied:

The above notation is a little bit different from the two similar coefficients used in either Buehner (2005) or Wang et al. (2008a,b) for the purpose of easy discussion of the analysis results. This approach for combining two covariance matrices to form a hybrid covariance provides flexibility since it allows for different relative contributions from the two covariance matrices. When *β*_{1} = 1 the analysis is back to a 3DVAR analysis scheme and when *β*_{2} = 1 the analysis is mathematically equivalent to an EnKF scheme, as the scheme is essentially a variational formulation with full ensemble-derived covariance. In between when both *β* values are nonzero we have a hybrid scheme that incorporates a mixture of both static and flow-dependent error covariances. Though the dimension of the control variables is increased by including extended control variables, the form of the background term of the cost function has not changed too much from that of 3DVAR [(2) vs. (8)] so that codes from an existing 3DVAR system can readily be utilized (Lorenc 2003).

The localization of the ensemble covariance in a variational system is necessary to reduce sampling errors. The procedure of localization with preconditioning is fully discussed in Lorenc (2003), Buehner (2005), and Wang et al. (2007) and is not repeated here. There are several ways to do covariance localization (Bishop et al. 2011). Similar to Wang et al. (2008a), we use the recursive filter for covariance localization in model space. How to properly choose the number of extended control variables (the size of **w**) is key for successful implementation of the hybrid scheme. This has not been fully explored and will be investigated in this study for convective-scale DA.

In the current study, the hybrid system assimilates both radar reflectivity and radial velocity data (Gao et al. 2013) and because of this, flow-dependent background error covariances, in particular cross covariances between microphysical and dynamic variables, are more reliable. Different from other hybrid systems (Wang et al. 2008a,b), for this hybrid method an extra model integration for the length of the analysis cycle is needed to produce a control forecast and analysis cycle, similar to the dual-resolution EnKF scheme implemented in Gao and Xue (2008). The EnKF analyses are performed to update analysis perturbations for each ensemble member. Then, the cost function in (8) is minimized to obtain optimal analyses of control vectors **v** and **w**, and the optimal analysis increment

## 3. Model and experimental design

### a. Prediction model and truth simulation for observing system simulation experiments (OSSEs)

The hybrid 3DEnVAR algorithm described above is tested with simulated data from a classic supercell storm of 20 May 1977 near Del City, Oklahoma (Ray et al. 1981). The Advanced Regional Prediction System (ARPS) model is used in a 3D cloud model mode and the prognostic variables include three velocity components *u*, *υ*, *w*; perturbation potential temperature *θ*; perturbation pressure *p*; and six categories of water substances (i.e., water vapor specific humidity *q*_{υ}, and mixing ratios for cloud water *q*_{c}, rainwater *q*_{r}, cloud ice *q*_{i}, snow *q*_{s}, and hail *q*_{h}).The microphysical processes are parameterized using the single-moment, three-category ice scheme of Lin et al. (1983). More details on ARPS can be found in Xue et al. (2000, 2001).

For our experiments, the model domain is 57 × 57 × 16 km^{3}. The horizontal grid spacing is 1 km and the mean vertical grid spacing is 500 m, resulting in 35 vertical levels. The truth simulation run is initialized from a modified real sounding plus a 4-K ellipsoidal thermal bubble centered at *x* = 48, *y* = 16, and *z* = 1.5 km, with radii of 10 km in *x* and *y* and 1.5 km in the *z* direction. Open conditions are used at the lateral boundaries. The length of simulation is 2 h. A constant wind of *u* = 3 and *υ* = 14 m s^{−1} is subtracted from the observed sounding to keep the primary storm cell near the center of model grid. The evolution of simulated storms is similar to those documented in Xue et al. (2000, 2001). During the truth simulation, the initial convective cell develops over the first 30 min, with clouds starting to form at about 10 min, and rainwater appearing at about 15 min. Ice phase fields appear at about 20 min into the run. The strength of the cell then decreases between 30 and 60 min, associated with the splitting of the cell at around 55 min. The right-moving (relative to the storm motion vector that is toward north-northeast) cell tends to dominate the system after this point, with its updraft reaching a peak value of over 40 m s^{−1} at 90 min. A similar truth simulation was also used in Gao et al. (1999, 2001, 2004), Tong and Xue (2005), and Gao and Xue (2008).

### b. Simulation of radar observations

The assimilation of reflectivity observations is somewhat complicated because the reflectivity factor is a function of all three hydrometeor variables (rainwater, snow, and hail). This likely leads to the DA solution being underdetermined. For example, it is possible to obtain a nonzero snow mixing ratio in the low levels of the model where only rainwater is expected because of the very warm temperatures at these levels. To solve this problem, a new forward reflectivity operator from Gao and Stensrud (2012), which uses the background temperature from an NWP model for hydrometer classification, is used in this study.

*υ*

_{r}is calculated from

*μ*is the elevation angle and

*ϕ*is the azimuthal angle of radar beams, and

*u*,

*υ*, and

*w*are the model-simulated velocities interpolated to the scalar points of the staggered model grid.

For reflectivity, random errors drawn from a normal distribution with a mean of 0 dB*Z* and a standard deviation of 3 dB*Z* are added to the simulated data. For radial velocity, random errors drawn from a normal distribution with a mean of 0 m s^{−1} and a standard deviation of 1 m s^{−1} are added to the simulated data. Since *υ*_{r} is sampled directly from the model velocity fields, hydrometeor sedimentation is not involved. The radial velocity data are assimilated and are only available where the truth reflectivity is greater than zero in the analysis domain. We also use only the data at every other horizontal grid point from the 1-km truth simulation so that the total data used are one fourth of total model grid points.

### c. Design of assimilation experiments

We start the initial ensemble forecast at 30 min of the model integration time when the storm cell is well developed. To initialize the ensemble members, random noise is first added to the initially horizontally homogeneous first guess throughout the domain defined using the environmental sounding. The random noise is sampled from Gaussian distributions with zero mean and standard deviations of 5 m s^{−1} for *u*, *υ*, and *w*, and 3 K for potential temperature. A 2D five-point smoother is applied to the resultant fields, similar to a method used by Zupanski et al. (2006). The initial perturbation variances are somewhat larger than those used in Tong and Xue (2005) but the standard deviation of the final perturbations is not necessarily larger because of the smoothing. Other model variables, including the microphysical variables, are not perturbed at the initial time. The radial velocity and reflectivity observations are calculated and assimilated using a 5-min cycle in all DA experiments with the first analysis performed at 30 min. A cutoff radius of 4 km is used in all of the EnKF runs, while in the 3DEnVAR experiments the correlation scale is 4 km in the horizontal and 1 km in the vertical. The localization scale for the recursive filter is a little bit larger in 3DEnVAR with 6 km horizontally and 1 km vertically.

We perform seven sets of experiments. The first experiment is performed with the weighting for ensemble covariance set to zero, so the assimilation is equivalent to a pure 3DVAR. The second to fifth sets of experiments use different combinations of ensemble size and weighting for ensemble covariance as listed in Table 1. In most of the experiments, the number of extended control variables is proportional to the degrees of freedom in each dimension and the ensemble size. For example if the ensemble size is 50, the number of extended control variables is 57 × 57 × 35 × 50 = 5 685 750. The number of control variables needed for each type of experiment is listed in Table 1. For sensitivity experiments with different weightings for ensemble covariance and static covariance, the values of *β*_{1} and *β*_{2} define the relative weights placed on the static background error and ensemble covariances in (9). For convenience, we mention only one of them, the weighting for ensemble-estimated covariance *β*_{2}. When the weighting for ensemble covariance is set to 0.2, 20% of the covariance comes from the ensemble and 80% comes from the static estimation. Generally four types of experiments with weighting values of 0.2, 0.5, 0.8, and 1.0 are conducted. A value of 1.0 means only the ensemble-estimated error covariance is used in 3DEnVAR experiment. This option is quite similar to the pure EnKF method, except that the minimization of 3DVAR cost function in (8) is used to solve the problem instead of (5). Sensitivity experiments with different ensemble size will be discussed in combination with different weightings for ensemble covariance and static covariance (see Table 1).

List of data assimilation experiments.

The final two sets of experiments further test the performance of the 3DEnVAR with the different size of extended control variables and 50 ensemble members. For these two sensitivity experiments, the number of extended control variables is either increased by a factor of 11 (the number of model variables) or significantly reduced by a factor of 35 (no vertical variation allowed, see Table 1). For comparison purposes, all DA experiments are performed with 12 DA cycles where each cycle has a 5-min analysis-prediction interval. The total assimilation period is 60 min.

Before any of the DA experiments, several single-observation experiments are performed to test the behavior of the extended control variables and assess how a single observation influences nearby grid points in the analysis.

## 4. Results of single-observation experiments

In the single-observation experiments, we place only one radial velocity observation with *υ*_{r} = 33.94 m s^{−1} within the analysis domain at a selected gridpoint location (43, 21, 16) and exclude the mass continuity constraint from the minimization process (*J*_{c} = 0.0). Two types of experiments are performed. For the first experiment, we test the impact of this single radial velocity on different model variables with full ensemble covariance calculated from 50 ensemble forecasts. For the second experiment, the impact of differences in weightings of ensemble-estimated covariance is examined. The purpose of this single-observation experiment is to clearly show the behaviors of the analysis methods for the control experiments listed in Table 1.

### a. Impact on extended control variables and different model variables

Figure 1 illustrates the impact of this single observation on the analysis increments of extended control variables from four arbitrarily chosen ensemble members. It can be seen that the structure of extended control variables all exhibit isotropic Gaussian shapes. As expected, the ranges of the analysis increments for each ensemble member are completely different, with contours of extended control variables related to selected ensemble members 1 and 3 all have negative values and ensemble members 7 and 15 have positive values. The structures of extended control variables for all other ensemble members have similar behavior, showing a Gaussian shape but with different value ranges. When these extended variables are used to calculate analysis increments for model variables using (9), the results are different because of the incorporation of both spatial correlations and cross-variable correlations estimated directly from the forecast ensembles. Figure 2 provides analysis increments resulting from the spread of the information from this single radial velocity observation for several selected model variables. It clearly shows that the analysis increments for different model variables all have flow-dependent features. For model variables *u* and *p*, the analysis increments display a positive center and a negative center around the single observation, respectively; for model variables *υ* and *q*_{υ}, they show a more complicated spatial structure.

### b. Impact of differences in weightings for ensemble covariance

In the second single-observation experiment, we still use 50 ensemble members but the weightings for ensemble covariance are increased gradually from 0.2, 0.5, 0.8, to 1.0 (similar to control experiments in Table 1). Figure 3 shows the impact of this single observation on the covariance structure of extended control variables for different relative weights for randomly picked ensemble member 11. With the increased weight for ensemble covariance, the patterns are not changed much, but the maximum value increases gradually from 0.206 in Fig. 3a to 0.922 in Fig. 3d. Correspondingly, for vertical velocity *w*, the results of analysis increments are changed from nearly isotropic (Fig. 4a) to more flow dependent, where a negative center gradually appears near the southeastern side of larger positive center near the single observation (Fig. 4d). It is also worth pointing out that the wind vectors also change gradually from Fig. 4a to Fig. 4d, from a more unified, symmetrical structure to a structure with multiple wind directions and curved structures. The changes in perturbation potential temperature *θ* are quite different from the behavior of vertical velocity *w*. The analysis increment patterns change only slightly as the ensemble covariance weights are increased from Figs. 5a to 5d. The increments all illustrate a flow-dependent dipole structure for *θ*, though with different magnitudes. This is because no cross-variable contribution exists from static part of background error covariance [i.e., the first part of (9) is zero].

As shown above, the impact of this single observation on the analysis increments of extended control variables for different ensemble members all exhibit isotropic Gaussian structures, centered at the same location, but with different value ranges given the different degrees of influence from each ensemble member on the analysis. So the role of extended control variables is to provide weighting functions for each ensemble perturbations given each observation. In other words, extended control variables provide a link, or a bridge between ensemble perturbations and observations, so that each ensemble perturbation can make contributions to the analysis as constrained by observations during the minimization process of the cost function.

The results for analysis increments with each model variable in response to the single observation are different. For model variables that are partially observed by radar, including the two horizontal wind components (*u*, *υ*) and the mixing ratios for rainwater *q*_{r}, snow *q*_{s}, and hail *q*_{h}, the analysis increments are quite similar to those of *w*, and change from nearly isotropic to having more flow-dependent structures as the weights for ensemble covariance are increased. The reason is that the static-estimated covariances (being Gaussian shaped) also make contributions to the analysis increments for these variables. But for the model variables that are not observed by radars, including perturbation potential temperature *θ*, perturbation pressure *p*, water vapor mixing ratio *q*_{υ}, and mixing ratios for cloud water *q*_{c}, cloud ice *q*_{i}, and others, the hybrid 3DEnVAR provides flow-dependent structures regardless of the relative weights for static and ensemble covariance in the analysis scheme. This occurs because only ensemble-estimated covariance contributes to the analysis increments for these variables.

By analyzing only the results of single-observation experiments, it is very hard to say how and when the hybrid 3DEnVAR may provide the best analysis results given different weightings for ensemble covariance and static covariance, different ensemble size, etc. Several OSSEs are performed in the following sections to further answer these questions.

## 5. Results of sensitivity experiments

### a. Sensitivity experiments with different ensemble size and covariance weightings

Most ensemble-based DA schemes are sensitive to ensemble size, but these sensitivities may be quite different for different cases and/or different dynamics of systems involved or even different types of observations. In this subsection, we examine the sensitivity of 3DEnVAR to ensemble size combined with different weighting factors for ensemble covariance and static covariance. This will involve four groups of experiments with ensemble size increasing from 5 to 100 (groups 2–5 as listed in Table 1). In each group of experiments, the values of weighting for ensemble covariance used are 0.2, 0.5, 0.8, and 1.0.

To evaluate the analyses quantitatively, the root-mean-square (rms) errors of the analyzed fields are calculated against the truth. Similar to Gao and Xue (2008), the rms errors are averaged over those grid points where the reflectivity is greater than 10 dB*Z* in the truth simulation. First we examine the variations of rms errors for perturbation potential temperature *θ.* Results show that the analysis is significantly worse when only ensemble-estimated covariance (*β*_{2} = 1.0) is used with only five members (Fig. 6a). However, the rms errors of 3DEnVAR with mixed covariance of *β*_{2} = 0.2 and *β*_{2} = 0.5 are lower than the other methods including the 3DVAR method and 3DEnVAR with *β*_{2} = 0.8 and *β*_{2} = 1.0. At the end of assimilation, the analysis rms errors are smallest with an equal weighting of static and ensemble covariance (*β*_{2} = 0.5). Figure 7 shows the final assimilation results after 12 assimilation cycles. The low-level flow, reflectivity patterns, and the strength of the cold pool from the 3DVAR method and 3DEnVAR with less ensemble covariance weighting values (Figs. 7b–e) agree very well with the simulated truth (Fig. 7a) and are significantly better than the result using full ensemble-estimated covariance (Fig. 7f). While the 3DEnVAR with full ensemble covariance can establish reasonable storm structures there are significant errors in cold pool and reflectivity patterns. The most obvious difference appears in the reflectivity field in the center of model domain. The area of reflectivity greater than 45 dB*Z* for 3DEnVAR with full ensemble covariance is overextended without any values greater than 55 dB*Z* in this area (Fig. 7f).

The minimum of surface potential temperature in 3DVAR is slightly colder than the truth simulation (Fig. 7a vs Fig. 7b). Further inspection shows that the analysis that produces the strength of the cold pool closest to the truth run is the 3DEnVAR with partial ensemble covariance. These results reveal that even for small ensemble size, when the estimated ensemble covariance is not stable or reliable and may have significant rank deficiency or sampling error, the added covariance information is still beneficial if properly used in a hybrid method with a smaller weighting factor. This result in principle agrees with the finding of Wang et al. (2008a).

When the ensemble size is increased to 10, the performance of 3DEnVAR with full ensemble-estimated covariance improves (Fig. 6b vs. Fig. 6a). The rms errors for 3DEnVAR with partial ensemble covariance are reduced in some respects in comparison with the 3DVAR method. The greatest rms error reductions between ensemble size of 5 and 10 happen for weighting values equal to 0.5 and 0.8 (Fig. 6b vs. Fig. 6a). When the ensemble size increases to 50, the performance of all hybrid 3DEnVAR with partial or full ensemble-estimated covariance become much better than the 3DVAR method. The 3DEnVAR with weighting for an ensemble-estimated covariance of 0.8 outperforms the other combinations near the end of DA period. In contrast, the rms errors for 3DEnVAR with full ensemble covariance are lowest during the early stages of DA. When ensemble size is further increased to 100, the results of all the hybrid schemes are significantly better than the 3DVAR method and it is obvious that the 3DEnVAR with full ensemble-estimated covariance is the best throughout most of DA period. But at the end of assimilation window, 3DEnVAR with a weighting value of 0.8 has the lowest rms errors. This point is more clearly demonstrated in Fig. 8. With weighting for ensemble covariance increasing from 0.2 to 1.0, all the storm structures displayed in Figs. 8a–d are very close to those in the truth simulation in Fig. 8a. But the strength of the cold pool in 3DEnVAR with a weighting of 0.8 for ensemble covariance (Fig. 8c), as indicated by minimum perturbation potential temperature of −8.38 K, is closer to the truth simulation (−8.37 K) than other hybrid schemes.

For the control experiments with ensemble size equal to 50, the rms errors of vertical velocity *w*, perturbation pressure *p*, water vapor mixing ratio *q*_{υ}, and reflectivity *Z* (derived from the mixing ratios of rainwater, snow, and hail) are shown in Fig. 9. It is obvious for all variables that the rms errors tend to decrease gradually starting from the first analysis. Wild zigzag patterns in the rms error are present for variables directly related to radial velocity or reflectivity, especially for vertical velocity *w*. This is in contrast with the rms errors for variables not directly observed by radar, where the zigzag patterns are discernable but not obvious. This occurs because during the analysis step, the variables directly related to observations incur larger changes than variables not directly observed by radar. Then during the 5-min forecast step, the errors for these variables grow faster than the errors in those variables not directly related to the observations in the analysis step. Not surprisingly, all 3DEnVAR analyses substantially outperform the analyses with the 3DVAR method for all variables. This clearly demonstrates the benefit of incorporating flow-dependent ensemble covariance into a variational analysis in a hybrid scheme. As shown in the single-observation experiments, the ensemble backgrounds provide a way of estimating covariances for model variables not directly related to radial velocity and reflectivity; these covariances are nonexistent in the static covariance used by the 3DVAR method.

The 3DEnVAR with a weighting value of 0.8 provides the lowest rms errors among all variables, though the performance of the 3DEnVAR with a weighting value of 0.5 also looks pretty good (Fig. 9). The rms errors of the vertical velocity *w* for almost all experiments are relatively volatile. Since updrafts are small-scale discrete features, we hypothesize that small errors in structure and position lead to these rather large rms errors. Even so, the rms errors for *w* in experiments with weighting values 0.5 and 0.8 yield the best results with rms errors being reduced from 6 m s^{−1} at the beginning to 1.5 m s^{−1} at the end of the assimilation window (Fig. 6a). For perturbation pressure *p*, the rms errors drop fastest with weighting value 1.0 for several assimilation cycles, but in general the best results are given by experiment with a weighting value of 0.8 (Fig. 9b). For relative humidity *q*_{υ}, the variation of errors over time is relatively smooth. Similar to *p*, the rms errors drop fastest with a weighting value of 1.0 only for the first several cycles of assimilation. After that the 3DEnVAR with middle weighting values of 0.5 and 0.8 give the best results (Fig. 9c). Interestingly, the differences among the rms errors for reflectivity *Z* in the various experiments are smallest (Fig. 9d). All the experiments produce a close fit to observations with the rms errors dropping from 40 to about 5 dB*Z* after the first assimilation cycle. The variation of rms errors for *Z* is relatively volatile for 3DVAR throughout the whole assimilation period. The method decreases the rms errors from 40 to near 5 dB*Z* in one DA cycle, but the errors quickly increase to above 18 dB*Z* after the 5-min model integration step. In contrast, the rms errors for all 3DEnVAR schemes decrease more smoothly throughout the DA window. Perhaps the advantage of the hybrid 3DEnVAR is most obvious for reflectivity. Although the evolution of rms errors is volatile for the first 10 min of assimilation, they quickly settle down and remain at low levels for most of assimilation window. For other model variables, the rms errors are seen to decrease gradually. In general, the best performance of the 3DEnVAR is given by the experiments with a weighting value of 0.8 for the ensemble-estimated covariance when 50 ensemble members are used. Going to 100 members significantly increases the computational cost while the improvement in accuracy is rather moderate (not shown).

### b. Sensitivity experiments with different numbers of extended control variables

As we know, in most hybrid DA schemes, the ensemble-estimated covariance is incorporated into variational DA schemes by the extended control variable method (Lorenc 2003; Buehner 2005; Wang et al. 2007, 2008a). However, what kind of the degrees of freedom for extended control variables should be used has not been well investigated. In Buehner (2005), the independent extended control variable fields are used for each model variable. That means that the number of extended control variables is equal to the dimension of the model space for all analysis variables times the number of ensemble members. To reduce the extra cost of minimization due to the increased number of control variables, Wang et al. (2008a) used a two-dimensional field for each analysis variable. The same two-dimensional field of coefficients was applied for all levels and all variables. Thus, only horizontal recursive filters were used to model the ensemble covariance and there is no vertical covariance localization. In Clayton et al. (2013), the same three-dimensional fields are applied for all five analysis variables used in their analysis. This means that the cross-variable covariance calculated from the ensemble is used but is not modified by the covariance localization. However, which option is best has not been fully explored, especially for radar data.

In this section, two more sensitivity experiments are performed and compared with the control experiments to investigate the effectiveness of the above options with the ensemble covariance weighing value set to 0.8 and 50 ensemble members. In Table 1, the sixth type of experiment is performed with a much larger number of extended control variables by using independent extended fields for each model variable (BigCV experiment). This experiment is very computationally expensive because it represents a significant increase in the number of control variables and the ratio of increased control variables to the original 3DVAR method without incorporating any ensemble information is about 50 times (see Table 1). The seventh type of experiment is performed with a much smaller number of extended control variables (reduced by a factor of 35, no vertical dimension) by using a common extended field for all model variables, but in two-dimensional space (2dCV experiment). For this experiment, the ratio of the increased number of extended control variables to the original 3DVAR method is only 0.13. Figure 10 shows that the performance of the control experiment (CtrCV) with a common 3D field for all analysis variables gives the best results with the lowest rms errors during most of the assimilation period. For vertical velocity *w*, the rms errors drop fastest during the first two assimilation cycles for the BigCV experiment, although the rms errors for CtrCV experiment also decrease fast and stay at a low level after four assimilation cycles (Fig. 10a). For perturbation pressure *p*, the rms errors for the 2dCV experiment increase by nearly 20 hPa during the first two cycles before decreasing afterward (Fig. 10b). This behavior indicates that using only two-dimensional extended control variables without vertical localization may yield severe sampling errors for some model variables. For cloud water mixing ratio *q*_{c}, the rms errors drop more smoothly for all three experiments with CtrCV giving the best results (Fig. 10c). The rms errors for a hydrometeor-related field, the rainwater mixing ratio *q*_{r}, again show that experiments with a small number (2dCV) and a big number of extended control variables (BigCV) have large rms errors at later assimilation cycles (Fig. 10d).

Ensemble spread is used to examine analysis uncertainty for several selected variables. The ensemble spread is defined as the square root of the ensemble variance calculated only for points at which the reflectivity is greater than 10 dB*Z*. Overall, the ensemble spread decreases with time throughout the analysis period (Fig. 11) with the spread magnitudes smaller than the rms errors (cf. Figs. 10 and 11). This comparison suggests that the ensemble is underdispersive, a common problem for storm-scale radar data assimilation (Aksoy et al. 2009; Snook et al. 2011; Yussouf et al. 2013). Spread decreases after the hybrid analysis and then increases during the short model ensemble integration. While the spread values for CtrCV are slightly larger than calculated from 2dCV and BigCV in the first several DA cycles, they are slightly smaller than calculated from 2dCV and BigCV in the ensuing DA cycles. The spread magnitude is closest to the rms error for vertical motion and pressure and is farthest from the rms error for the hydrometeor variables. At the very beginning the spread for rainwater mixing ratio is zero because there is no precipitation at the beginning of DA cycles (Fig. 11d). This result is also consistent with other studies showing that radar data assimilation performs best for the dynamic variables and has challenges with variables related to radar reflectivity (Aksoy et al. 2009; Yussouf et al. 2013). The horizontal variation in ensemble spread is examined using reflectivity at 1 km above the ground at the end of data assimilation cycle (Fig. 12). The spread for all three experiments is similar in both areal coverage and amplitude. Larger values of spread are seen encircling the ensemble-mean thunderstorm position (not shown, but similar to the two storm cells shown in Fig. 8), with the largest values of spread found just outside of where the left-moving storm is located. Thus, the analysis uncertainty for reflectivity is largest near convective boundaries.

Convective-scale weather events usually contain complicated 3D structures, so using only 2D extended control variables without considering vertical localization does not benefit the quality of the analysis. With a much larger number of extended control variables, we have the opportunity to use different weighting coefficients and localization scales for different fields. However, the cross-variable relationships are lost when the number of extended control variables is too large. Given the nature of radar DA in which most of the model variables cannot be observed directly and have to instead be retrieved, the control experiment which considers the cross-variable correlations is the best choice and provides the best results. While the cross-variable covariance localization is not considered in this approach and the impact of sampling errors may exist as pointed out by Wang et al. (2013), cross-variable relationships are seen to be very important for convective radar DA.

## 6. Summary and conclusions

A hybrid 3DEnVAR DA system has been developed based on existing 3DVAR and ensemble Kalman filter (EnKF) programs within the ARPS model. The algorithm uses the extended control variable approach to combine the static and ensemble-derived flow-dependent forecast error covariances (Lorenc 2003; Buehner 2005; Wang et al. 2007). The method is applied to the assimilation of radar data from a simulated supercell storm.

First, single-observation experiments have been performed. The impact of a single observation on the covariance structure of the extended control variables for different ensemble members all exhibit isotropic Gaussian structures, but with different value ranges given the different degrees of influence for each ensemble member on the resulting analysis. This result indicates that the role of extended control variables is to provide weighting functions for each ensemble perturbation given the observations. Results for the analysis increments with each model variable in response to a single observation are different. For model variables partially observed by radar, response structures to the single observation change from being nearly isotropic to being more flow dependent as the weightings for ensemble covariance are increased. But for model variables not observed by radar, the hybrid 3DEnVAR provides only flow-dependent structures regardless how the relative weightings for static and ensemble covariance are used in the analysis scheme.

Results from sensitivity experiments clearly demonstrate the benefit of incorporating flow-dependent ensemble covariance into a variational analysis using the hybrid 3DEnVAR method. With small ensemble size, the ensemble-estimated covariance may contain significant sampling errors, but it is still useful if properly applied in a hybrid method with a smaller weighting factor. For the control experiments with relatively larger ensemble sizes of 50 or more, all 3DEnVAR analyses substantially outperform the analysis that only uses static covariance (the 3DVAR method) for all variables. The best performance of the 3DEnVAR is given to the experiments with a weighting value of 0.8 for the ensemble-estimated covariance when 50 or 100 ensemble members are used. The improvements seen when going from 50 to 100 members are moderate, yet this increase in ensemble size significantly increases the computational cost.

In hybrid DA schemes, choosing the number of the augmented extended control vectors for each ensemble member remains a challenging problem. Our sensitivity experiments also indicate that the best results are obtained when the number of the augmented control variables is a function of three spatial dimensions and ensemble members, and is the same for all analysis variables. Results also suggest that the ensemble is underdispersive, suggesting that future work should focus on methods to increase ensemble spread.

In this study, we demonstrated that the incorporation of ensemble-estimated covariance into a variational method (3DVAR in this case) can significantly improve the accuracy of the assimilation of simulated radar data for a supercell storm. This result holds even when just a few ensemble members are used and the estimated covariance contains severe sampling errors. This may technically benefit the Warn-on-Forecast concept proposed by Stensrud et al. (2009), which envisions a frequently updated numerical model–based probabilistic convective-scale analysis and forecast system to support warning operations within the National Oceanic and Atmospheric Administration (NOAA). It is essential that ensemble forecasts are utilized in the Warn-On-Forecast implementation. However, because of the computationally intensive nature of ensemble DA and given the current and near-future availability of computational power, it is likely that only a few high-resolution ensemble members can be produced fast enough for real-time operations. Results from the current study suggest that even a small amount of ensemble information can still be very useful to convective DA if properly used in a hybrid 3DEnVAR system. This gives us confidence to implement a unified ensemble-based DA and prediction system without worrying about the limitations due to the needed computational power.

## Acknowledgments

This research was funded by the NOAA Warn-on-Forecast Project. The first author was partially supported by NSF Grants AGS-0802888 and AGS-1341878. This paper has benefited from the detailed and insightful comments from David Dowell and two other anonymous reviewers.

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