## 1. Introduction

The assimilation of radial-velocity and reflectivity observations of severe weather events like thunderstorms and convective systems from Doppler radars using the ensemble Kalman filter (EnKF) approach can provide important information for the initialization of storm-scale numerical prediction models (Snyder and Zhang 2003; Zhang et al. 2004; Dowell et al. 2004a; Tong and Xue 2005; Xue et al. 2006; Aksoy et al. 2009; Yussouf and Stensrud 2010). Unfortunately, while the assimilation of Doppler radar data in storm-scale models yields high-quality analyses, reliable short-term forecasts from these analyses remains a challenge (Snyder and Zhang 2003; Dowell et al. 2004b; Tong and Xue 2005; Yussouf and Stensrud 2010; Aksoy et al. 2010).

One major source of error in storm-scale data assimilation and forecasts is the error introduced into the model from various parameterization schemes, such as the microphysics, turbulence mixing, planetary boundary layer, and radiation schemes. In particular, thunderstorm simulations are known to be very sensitive to the microphysics parameterization scheme (Gilmore et al. 2004b; van den Heever and Cotton 2004; Wicker and Dowell 2004; Snook and Xue 2008; Tong and Xue 2008a). Microphysics schemes represent a number of different phase changes of water species and a number of different interactions between cloud and precipitation particles, requiring many assumptions to make these schemes both realistic and computationally affordable (Stensrud 2007). The most commonly used type of microphysical scheme in storm-scale modeling is a single-moment bulk microphysics scheme that predicts only the hydrometeor mixing ratios (Lin et al. 1983; Tao and Simpson 1993; Schultz 1995; Straka and Mansell 2005; Hong and Lim 2006). However, even single-moment bulk microphysics schemes tend to differ from one another in the number of precipitation particles represented and the interactions between the particles that are included.

Single-moment schemes typically use constant values for the intercept parameters and the densities of hydrometeors in the calculation of hydrometeor size distributions, and these intercept and density values are defined somewhat arbitrarily since in situ observations of these parameters are rare. Observational studies indicate that the particle densities and the intercept parameters of hydrometeor distributions can vary widely among storms and even within a single storm (Joss and Waldvogel 1969; Pruppacher and Klett 1978; Knight et al. 1982; Ziegler et al. 1983; Cheng et al. 1985; Cifelli et al. 2000). The impact of the variations of these hydrometeor parameters on convective storm dynamics is demonstrated in several pure model sensitivity studies (Gilmore et al. 2004b; van den Heever and Cotton 2004; Cohen and McCaul 2006; Snook and Xue 2008). Gilmore et al. (2004b) show that reasonable selections of intercept parameters and density for hail/graupel yield substantial and operationally important differences in simulations of thunderstorms in terms of storm structure, intensity, and precipitation characteristics. Large sensitivity in simulated supercell storm characteristics due to changes in the mean diameter of the gamma hail distribution is also found by van den Heever and Cotton (2004). Cohen and McCaul (2006) show that the low-level downdraft cooling varies considerably as a result of changes in hydrometeor sizes and distributions in the simulation of deep convection. Snook and Xue (2008) further indicate that varying the intercept and density parameters within their typical observational range yields a wide range of solutions to tornadogenesis in a simulated supercell storm. In addition, Tong and Xue (2008a) conduct sensitivity experiments of the individual microphysical parameters (intercept parameters of rain, snow, and hail/graupel and densities of snow and hail/graupel) using EnKF data assimilation technique for a supercell thunderstorm. They show that the microphysical parameters have a larger impact on the forecast reflectivity than radial velocity and the reflectivity forecast is more sensitive to intercept and density parameters of hail/graupel than for the intercepts of rain and intercepts and density of snow.

Since the slope of hydrometeor drop size distribution is proportional to the intercept parameters, increasing the intercept parameters shifts the hydrometeor distribution toward smaller drops. When the density of hail/graupel and the intercept parameters of hail/graupel and rain are larger, for a given mixing ratio, the mean diameter of these hydrometeors is smaller, resulting in slower mass-weighted fall speed and a stronger cold pool due to enhanced evaporative cooling/melting. Changes in cold pool strength influence the storm updraft and, hence, storm evolution (Brooks et al. 1994; Gilmore and Wicker 1998). The storm structure and dynamics are less sensitive to the variations in snow intercept and density parameters where the changes obtained in storm structure are minimal (Gilmore et al. 2004b; Tong and Xue 2008a). In combination, these studies suggest that a single set of parameters for precipitation particles within a single-moment microphysical parameterization will be unlikely to adequately represent the highly uncertain storm precipitation characteristics and could lead to significant errors in the analyses and forecasts of severe storms. Thus, this study is focused on the impact of perturbing these microphysical parameters across the ensemble members on the analyses and prediction of a supercell thunderstorm using the EnKF data assimilation technique.

The determination of suitable values for the microphysical parameters in storm-scale data assimilation is very difficult because of the lack of in situ microphysics observations. Since the selection of microphysical parameters in storm-scale modeling has a profound impact on the analyses and forecasts of severe weather events, and an arbitrary selection of those parameters may lead to significant error, one approach to account for this uncertainty in a storm-scale ensemble modeling system is to vary the microphysical parameters within the same microphysics scheme among the ensemble members to sample the range of microphysical parameters found in observational studies of thunderstorms. The intent is that by using a variety of realistic precipitation hydrometeor parameters, an ensemble is more likely to span the truth and maintain ensemble spread. A number of studies demonstrate that a mesoscale ensemble with multiple physical parameterizations and initial condition perturbations yields better analyses and forecasts than an ensemble with only initial condition perturbations (Stensrud et al. 2000; Fujita et al. 2007; Meng and Zhang 2007, 2008; Stensrud et al. 2009). Therefore, to explore the impact of variations in parameters within the same microphysics scheme in storm-scale forecasting, Observing System Simulation Experiments (OSSEs; Lord et al. 1997) are conducted applying initial condition variations within both a single-parameter and a multiparameter ensemble using an EnKF data assimilation technique to assimilate Doppler radar observations.

Two simulations of a splitting supercell storm, produced using different microphysics schemes, are used as reference truth runs. Synthetic radial-velocity and reflectivity observations are then constructed from these truth runs and assimilated using the EnKF technique with the same numerical model that produced the truth simulations. The first set of experiments is based on the assumption of a perfect model in which both the truth simulation and the ensemble system use the same microphysics scheme. The second set of experiments is based on an imperfect-model assumption in which the microphysics scheme for the truth simulation and the microphysics scheme for the ensemble system are different. Thus, the imperfect-model experiment includes model error from the microphysical parameterization and so is a more realistic test of the multiparameter ensemble approach. The storm-scale model, simulated radar dataset and the experimental design are described in section 2. Section 3 presents the results obtained from the EnKF analysis and forecasts, followed by a final discussion in section 4.

## 2. Assimilation system and experiment design

The Collaborative Model for Multiscale Atmospheric Simulation (COMMAS; Wicker and Skamarock 2002; Coniglio et al. 2006) used in this study is a nonhydrostatic compressible numerical cloud model. The prognostic variables for this model include the three velocity components (*u*, *υ*, and *w*), pressure in the form of the perturbation Exner function *π*, potential temperature *θ*, mixing coefficient *K _{m}*, water vapor mixing ratio

*q*, cloud water mixing ratio

_{υ}*q*, and hydrometeor mixing ratios (

_{c}*q*

_{1, … ,r}), where

*r*is the number of hydrometeor categories and varies based on the microphysics scheme used. The data assimilation scheme used is based on the ensemble square root filter (EnSRF) of Whitaker and Hamill (2002) where observations are assimilated in the filter serially. A 40-member ensemble is used, and each time an observation is assimilated the ensemble mean and each of the ensemble members are updated for each model variable at each grid point. The equations for the filter are similar to Snyder and Zhang (2003), Dowell et al. (2004a), Tong and Xue (2005), Dowell and Wicker (2009),and are shown for completeness.

*y*

^{0}, the Kalman gain

*N*is the number of ensemble members,

*n*is the index to denote a particular ensemble member,

**x**denotes a particular model variable at a particular grid point,

*f*” indicates the forecasts prior to the assimilation of

**y**

^{0}, the overbar indicates an ensemble mean, Var(

**y**

^{0}) is the observation-error variance, and

*a*” indicates the updated estimate after an observation is assimilated and

*β*is a factor to account for unperturbed observations (Whitaker and Hamill 2002) and is defined as

*π*and

*K*. Thus, these two model variables are not updated as observations are assimilated in the filter in order to reduce the computation time.

_{m}*ρ*is the density of water,

_{r}*n*

_{0r}is the rain intercept parameter, and

*q*is the rainwater mixing ratio. The snow component of reflectivity is defined as

_{r}*ρ*is the density of snow,

_{s}*n*

_{0s}is the intercept parameter for the distribution of snow, and

*q*is the snow mixing ratio. The hail component of reflectivity is defined as

_{s}*ρ*is the density of hail,

_{h}*n*

_{0h}is the intercept parameter for the distribution of hail, and

*q*is the hail mixing ratio. For simplicity, the snow and hail are assumed to be dry for all experiments in this study. The reflectivity values associated with cloud water and cloud ice are considered negligible and thus are not included in estimating reflectivity. Finally, the mean radar reflectivity factor

_{h}*Z*(

_{e}*Z*

_{e}= Z_{er}

*+ Z*

_{es}

*+ Z*

_{eh}) is converted to logarithmic radar reflectivity (in units of dB

*Z*) using

*Z*(dB

*Z*) is assimilated directly (i.e., the intercept and density parameters in the calculation of the reflectivity observation operator are consistently chosen from each ensemble members and thus do not contribute to any additional error in the forward operator calculation).

### a. The two truth simulations and synthetic radar observations

Two simulation runs of a splitting supercell thunderstorm are conducted using the Weisman–Klemp (WK) analytic sounding (Weisman and Klemp 1982) with a quarter circle hodograph for the vertical wind. The model domain for the truth runs is 100 km wide with 1-km horizontal grid spacing and is 18 km tall in the vertical direction. The domain is vertically stretched from 200-m vertical spacing at the domain bottom to 700-m vertical spacing at the domain top. The 2-h-long truth simulations are initiated with an ellipsoidal thermal bubble of 2.5 K with 10-km radius in the horizontal direction and 1.4-km radius in the vertical direction that is placed at the center of the domain at *t* = 0 min. This thermal bubble develops into a convective cell within the first 30 min of the simulations; first echoes are seen by the radar emulator at approximately *t = 25* min. Over the next 30 min, the convective cell splits into two cells, one moving right toward the east and the other moving toward the northeast. During the second hour of the simulations, the right-moving cell tends to dominate the system with a few short-lived smaller cells developing in between the two main cells. The model grids are translated at *u* = 17 and *υ* = 7 m s^{−1} to keep the main storm near the center of the model domain.

*x*is the precipitation particle category,

*D*is the particle diameter (m),

*n*is the number concentration of particles per meter (m

_{x}^{−4}) with diameter

*D*,

*λ*is the slope parameter that defines the decrease in particle counts as diameter increases (m

^{−1}), and

*n*

_{0x}is the intercept parameter that defines the maximum number concentration of particles per unit volume at

*D =*0 size. The slope parameter varies with mixing ratio and is given by

*ρ*is the density of the particle,

_{x}*ρ*is the air density, and

_{a}*q*is the mixing ratio. From (8) and (9), it is obvious that the particle size distribution is strongly influenced by the selected values of

_{x}*n*

_{0x}and

*ρ*. The values of the density and the intercept parameters used for the two truth simulations are given in Table 1.

_{x}The intercept and the density parameters of the precipitation particles for the Truth_LFO and Truth_10ICE simulations.

While radar reflectivity and radial-velocity observations are generated from averaging radar pulses, a radar emulator is used here to construct synthetic reflectivity and radial-velocity values by averaging reflectivity and wind components from the three-dimensional, gridded truth simulations with a simplified version of a realistic volume-averaging technique (Wood et al. 2009). The mixing ratios of the hydrometers, air density, and the *u*, *υ*, and *w* wind components at model grid points within the radar beamwidth are scanned with the radar emulator to produce synthetic Weather Surveillance Radar-1988 Doppler (WSR-88D) radar reflectivity and radial-velocity observations. To reduce the heavy computational burden of observation assimilation, the synthetic reflectivity and radial-velocity observations are created along each radial at a coarser 1.0-km range sampling interval instead of the 0.25-km interval available from the operational WSR-88D radars. The antenna half-power beamwidths are assumed to be 0.89° with 1.0° azimuth interval and a 1.39° effective beamwidth to match the WSR-88D specifications. The synthetic radar observations are generated using volume coverage pattern (VCP) 11 precipitation mode scanning strategy consisting of 14 elevation angles. To assimilate the observations more realistically, the synthetic radar observations are generated for two–three sweeps every minute rather than assuming the entire volume is collected simultaneously. Out of the 14 elevation sweeps, the lower 12 sweeps of observations are generated 3 sweeps per minute for the first 4 min with the remaining upper 2 sweeps valid for the fifth minute of the volume scan. To account for the measurement and sampling errors for radial-velocity and reflectivity observations, random numbers are drawn from a Gaussian distribution of zero mean and standard deviations of 2 m s^{−1} and 2 dB*Z*, respectively, and added to the observations. The radar reflectivity observations are assimilated include nonprecipitating observations, while the radial-velocity observations are assimilated only where the observed reflectivity values are greater than 10 dB*Z*. Moreover, the radar location is stationary while the model domain moves with the simulated storm; all experiments in this study are conducted using a single radar location to observe the storms. Additional details of the radar emulator are discussed in Yussouf and Stensrud (2010). The number of observations assimilated during each 1-min assimilation period ranges from 630 to 25 360, depending on the location of the radar relative to the supercell thunderstorm, the height of radar scans, and storm intensity.

The truth runs from both LFO and 10-ICE produce a qualitatively similar evolution of intense splitting supercell storms, but with some differences in the location, strength, and structure of the storms (Fig. 1). In particular, the LFO run produces splitting storms that have stronger cold pools (not shown) that lead to faster propagation speeds, as seen by the shift of the reflectivity maximum of the southern storm being ~5 km farther east in Truth_LFO compared to Truth_10ICE at the last output time (cf. Figs. 1e,f). The high-reflectivity cores of the cells from Truth_LFO are typically 5 dB*Z* more intense than the high-reflectivity cores of the cells from Truth_10ICE. The small cells in between the northern and southern supercells also are stronger in Truth_LFO compared to Truth_10ICE. Similar differences also are found for other variables at other vertical levels of the model domain and at other simulation times.

### b. The ensemble configuration and OSSE design

Each member of the 40-member ensemble uses the same WK sounding as in the truth runs to define the initial horizontally homogeneous environment. The sounding has a mixed-layer convective available potential energy of 2093 J kg^{−1}, with a hodograph favorable for intense supercell thunderstorms. The domain size and grid spacing for the ensemble members also are identical to the truth runs. To account for uncertainty in the ensemble initial condition, 3 thermal bubbles (1.5-K maximum ellipsoidal *θ* perturbations) are introduced at the initialization time (*t =* 0) into each ensemble member following the methodology of Dowell et al. (2004a). These bubbles have a 7.5 km (2.0 km) radius in the horizontal (vertical) direction and are placed at random horizontal locations within 10 km of the domain center and extend from 0.25 to 2.25 km in height. Thus, the bubbles in the ensemble members differ from the truth run in magnitude, size, and location. These thermal bubbles in the ensemble not only account for the uncertainty in the initial conditions, but also facilitate the development of storms and the region where the thermal perturbations are added includes the region where the truth runs produce radar echoes. Some of the cells in the ensembles are spurious in a sense that they are outside the domain of radar echoes in the truth run. However, the assimilation of nonprecipitating reflectivity observations (0 dB*Z*) suppresses the unwanted spurious convective cells. The ensemble members are then integrated forward in time for 25 min before the assimilation of the first radar observation sweep. During this time many of the thermal bubbles initiate convective cells, thereby producing the covariance information needed for the ensemble to successfully assimilate the radar data. This technique of using thermal bubbles to initialize the ensemble members has been shown to be successful by Aksoy et al. (2009, 2010), Dowell and Wicker (2009), and Yussouf and Stensrud (2010). The domains of the ensemble members also move at *u =* 17 and *υ =* 7 m s^{−1} following the truth runs to keep the storm inside the domain.

A 30-min-long assimilation period starts at *t* = 25 min and ends at *t* = 54 min. During this assimilation period, six volume scans of synthetic WSR-88D radar observations are assimilated. The radar is located at *x* = −3.6 km and *y* = −4.9 km from the southwest corner of the domain during the first volume scan. The observations valid within 1 min of the current time are assimilated followed by advancing the ensemble members 1 min to the next observation time. No additional localized perturbations (Dowell and Wicker 2009) or covariance inflation (Snyder and Zhang 2003; Dowell et al. 2004a; Tong and Xue 2005) are used to maintain the ensemble spread during the assimilation cycles. Results (not shown) indicate that total energy and total mixing ratio calculated over the model domain are conserved to better than 99.8% during the assimilation period as the model variables are adjusted by the EnSRF to match the synthetic radar reflectivity and velocity more closely. After the 30 min of data assimilation, all of the 40 ensemble members are used to produce a 1-h-long ensemble of forecasts.

#### 1) Perfect-model experiment

The ensemble members in the perfect-model experiment use the same microphysics scheme as in Truth_LFO while the synthetic WSR-88D reflectivity and radial-velocity observations assimilated are generated from the Truth_LFO run. Two experiments are conducted using identical background environments. The first ensemble is a single-parameter (SP) ensemble (Perfect_SP) that uses a single set of intercept and density parameters for the hydrometeor categories for all ensemble members as in the Truth_LFO (see Table 1). The ensemble members in the Perfect_SP experiment thus have the identical base environment and microphysics scheme as in the truth run, but differ from each other in the location and magnitude of the starting thermal bubbles. Thus, we are assuming that the model is perfect and the environmental condition is perfectly represented, thereby giving the ensemble data assimilation and forecast system the best chance to produce excellent results.

*n*

_{0r}, snow

*n*

_{0s}, hail/graupel

*n*

_{0h}, and the bulk densities of snow

*ρ*and hail/graupel

_{s}*ρ*. These values are varied within their typical range based on observational studies as reported in Table 1 of Tong and Xue (2008a). The upper and lower bounds of the parameters chosen for the MP ensemble are given in Table 2. The intercept and density parameters from the Truth_LFO are assigned to the ensemble member 1, while the upper and lower bound of the intercept parameters from the hydrometeor categories are assigned to two of the other ensemble members. The intercept parameters for the remaining ensemble members are then calculated from

_{h}*n*(=2, 3, …) is the ensemble member, and inc is the increment calculated from the upper

*P*

_{up}and lower

*P*

_{low}bounds of the intercept parameters using

*N*is the number of ensemble members. The density parameters are selected randomly within their upper and lower bounds. Using this approach, a set of candidate intercept and density parameter values from each of the hydrometeor categories is selected randomly for each ensemble member. However, it is possible that some combinations of these parameters will produce storms that differ too much in intensity from the truth runs. A reflectivity test is added to ensure that the synthetic reflectivity values from all the ensemble runs are similar in magnitude. The reflectivity factors

*Z*

_{er},

*Z*

_{es}, and

*Z*

_{eh}are calculated using (5), (6a), and (7a), respectively, using values from a grid point with a reflectivity value of 67 dB

*Z*in the Truth_LFO run (

*ρ*= 1.2 kg m

_{a}^{−3},

*q*= 0.004 06 kg kg

_{r}^{−1},

*q*= 0.002 63 kg kg

_{s}^{−1}, and

*q*= 0.004 75 kg kg

_{h}^{−1}). The total reflectivity value is calculated using

*Z*

_{e}= Z_{er}

*+ Z*

_{es}

*+ Z*

_{eh}, and is required to be within ±4 dB

*Z*of the 67-dB

*Z*reflectivity for the candidate set of intercept and density parameters to be used in the MP ensemble. Candidate parameter sets are tested until 40 separate sets are found that satisfy the reflectivity test and have mean values that are nearly identical to the Truth_LFO run values (see Table 3 and Fig. 2). Using the final set of microphysics parameters in Table 3, the minimum and maximum values of reflectivity calculated using (5)–(7) are 63.6 and 71 dB

*Z*, respectively. The resulting parameter distributions indicate that the parameters within the ensemble members span the observed range, but with smaller intercept values, larger hail densities, and smaller snow densities being more likely to pass the reflectivity and mean value tests (Fig. 2). A model run using the mean of the MP parameters reveals that the storm structure, intensity and location of the supercells are nearly identical to those produced by Truth_LFO (not shown). Thus, it is reasonable to label the MP ensemble as “Perfect_MP.”

The upper *P*_{up} and lower *P*_{low} bound of the intercept and density parameters of the precipitation particles for the multiparameter experiment.

List of ensemble members with the values of intercept parameters and densities of rain, hail/graupel, and snow particles from the LFO microphysics scheme.

Similar to Gilmore et al. (2004b) and Tong and Xue (2008a), results indicate that the variations in hail intercept and density parameters have more significant effects on the precipitation processes compared to equivalent changes in the snow intercept and density parameters (not shown). The use of a variety of density and intercept parameters across the ensemble members result in storm evolutions that are qualitatively very similar to each other, but differ in terms of the detailed storm structure, updraft and downdraft strength, and intensity as discussed in the following section. One may argue that these parameters should be perturbed based on their underlying probability distribution. However, as these parameters are not commonly observed, their underlying probability distribution is unknown. Hacker et al. (2011) uses a modified Latin Hypercube Sampling method for sampling the parameters from a parameter space. Similar methods or the assumption of an underlying probability distribution function for these parameters may provide better results and should be considered in future work. We recognize that this initial approach to perturb the parameters as discussed above may not be optimal and likely can be improved upon, but we believe this approach is sufficient to evaluate the potential value of an MP ensemble.

#### 2) Imperfect-model experiment

Unlike the perfect-model experiment, the synthetic reflectivity and radial-velocity observations assimilated by the ensemble members in the imperfect-model experiment are generated from the Truth_10ICE run. The ensemble members in the SP ensemble (Imperfect_SP) use the LFO microphysics scheme with the same single set of constant precipitation particle parameters as in the Perfect_SP experiment (Table 1). The ensemble members in the MP ensemble (Imperfect_MP) also use the LFO microphysics scheme, but with the same variety in the intercept and density parameters as in Perfect_MP (Table 3). The initialization and other ensemble configuration details are identical to the previous experiment. The imperfect-model experiment explores the performance of the EnKF system for the same storm event in the presence of model error due to the use of a different microphysics scheme for the truth run.

The ultimate goal of storm-scale data assimilation is to obtain accurate short-term analyses and forecasts of severe storms events. To evaluate the accuracy of the ensemble forecasts from assimilating synthetic WSR-88D observations over a 30-min period, the 40 analyses from the last assimilation cycle are used as the initial conditions for each of the ensemble members and 1-h forecasts are produced.

## 3. Results

The accuracy of the analyses and forecasts for both perfect- and imperfect-model experiments, when using either SP or MP ensembles, are compared with the truth runs. The evaluation criteria include both statistical and graphical comparisons between the observations and the ensemble system. Statistical measures include observations-space diagnostics (Dowell and Wicker 2009; Aksoy et al. 2009; Dowell et al. 2011) of root-mean-square (rms) error and ensemble spread (ensemble standard deviation) for reflectivity and radial-velocity, model-space diagnostics of rms error and ensemble spread of the unobserved variables, and equitable threat scores (ETSs; Wilks 2006). The observation-space diagnostics are calculated for the 30-min assimilation period and only at locations where the synthetic reflectivity is greater than 15 dB*Z* to focus on the main convective cores (Dowell and Wicker 2009; Aksoy et al. 2009; Dowell et al. 2011). The model-space rms errors and ensemble spreads are calculated from averaging over only those model grid points where the sum of rain, snow, and hail mixing ratios are greater than 0.10 g kg^{−1} in the truth to provide a more accurate measure of the analyses and forecast quality where there is convection (Snyder and Zhang 2003; Tong and Xue 2005; Dowell and Wicker 2009). Various plots of ensemble maximum values are used to determine whether or not the ensembles capture the range of values found in the truth runs and to help determine if the MP approach is useful for predicting the extreme of an event occurring or not occurring (non event). This focus on extremes has not been done previously and represents a new evaluation framework for storm-scale ensembles.

To assess if the rms errors obtained are statistically significant, a bootstrap technique (Efron and Tibshirani 1993) is applied where resamples are randomly selected from the pool of gridpoint differences between the truth and the ensemble mean, and the error statistics for each of those resamples are generated. This resampling procedure is repeated 10 000 times to estimate the 95% bootstrap confidence interval of the error statistics. If the confidence intervals from two different methods do not overlap, then the differences are significant at more than the 95% level.

### a. Analyses

The observation-space diagnostics of the ensemble forecast and analysis during the 30-min assimilation period for the reflectivity and radial-velocity observations suggest that the ensemble data assimilation is working well (Fig. 3). While the observations are assimilated every 1 min, rms errors and ensemble spreads are calculated every 5 min so that each time interval encompasses one complete volume scan of radar observations. These choices produce the classic sawtooth pattern shown in Fig. 3. For all runs, the ensemble spread and the ensemble rms error are of similar magnitude, indicating that the ensembles are not underdispersive, and the ensemble spread and rms error decrease throughout the assimilation period as more observations are assimilated. The analysis rms errors for radial velocity decrease to near the assumed observational error of 2 m s^{−1} by the end of the assimilation period, suggesting that a good fit to the observations has been produced. The spread in the MP ensembles for reflectivity observations are typically 10%–20% larger than the spread in the SP ensembles as a result of the influence of the variations in microphysics parameters, consistent with other studies using multiple parameterizations in ensemble systems (Stensrud et al. 2000; Fujita et al. 2007; Meng and Zhang 2007; Snook et al. 2011).

To evaluate how well the storms are captured by the ensemble system during the 30-min assimilation period, the rms errors of several unobserved variables are examined, including the *w* wind component, temperature *T*, and total precipitation (sum of the rain, snow, and hail/graupel) mixing ratios from the ensemble analyses for both perfect- and imperfect-model experiments (Fig. 4). The slight increases and decreases in the error curve from assimilating observations every minute for each 5-min-long volume scan is due to the variations in both the number of radar observations and the quality of the ensemble analyses with height. The rms errors from all experiments are seen to decrease rapidly for all variables as more observations are assimilated. The rms errors for wind and temperature variables from both MP and SP experiments are very similar to each other with overlapping confidence bounds and approach 2 m s^{−1} and 1 K, respectively, at the end of the assimilation period, indicating that both variables are retrieved successfully. The rms errors from the Perfect_MP are smaller than those of the Perfect_SP during the last 15 min of the 30-min assimilation period for all three variables, but these differences are not significant at the 95% level (Figs. 4a,c,e). With the perfect-model assumption, the EnKF only has to correct the initial condition errors and so one would expect both SP and MP ensembles to yield good results. For the imperfect-model experiment, the rms errors of *w* and temperature from Imperfect_MP are similar to those from Imperfect_SP (Figs. 4b,d); however, the rms errors of total precipitation mixing ratio from Imperfect_MP are notably smaller than those from Imperfect_SP during the last 15 min of the assimilation period (Fig. 4f) and these differences are significant at the 95% level. Thus, in the presence of model error, the variation of the precipitation particle parameters in the Imperfect_MP helps to assimilate the synthetic observations more accurately and, hence, produce smaller rms errors in the analyses. This conclusion is consistent with multiparameterization ensemble studies on the mesoscale (Stensrud et al. 2000; Fujita et al. 2007; Meng and Zhang 2007).

The corresponding ensemble spread plots during the 30-min data assimilation period are shown in Fig. 5. The MP ensemble provides slightly larger spread compared to the SP experiments, particularly for the total precipitation mixing ratios (Figs. 5e,f). This conclusion agrees with Tong and Xue (2008a) that show that model precipitation fields are more sensitive to microphysical parameters than the wind fields. The ensemble spread (Fig. 5) and the ensemble mean rms errors (Fig. 4) are of similar magnitude for *w* and temperature, indicating that the filter is performing well even for unobserved variables. However, for the imperfect experiment the ensemble mean rms error for total precipitation mixing ratio is larger than the ensemble spread (cf. Figs. 4f and 5f), with the MP ensemble producing values that are closer to each other.

A more qualitative comparison of the SP and MP ensembles (not shown) indicates that all the ensembles produce intense, splitting supercell storms with similar structures and characteristics. The MP ensemble members have slightly more variability than the SP ensemble members in their maximum reflectivity values, but the variability in the general pattern of the splitting supercells at the end of the assimilation period is very similar between the two ensemble systems.

### b. Forecasts

The rms errors of the ensemble forecast mean fields during the subsequent 1-h forecast period for perfect and imperfect-model experiments indicate that the quality of the ensemble forecasts deteriorates with time as errors grow (Fig. 6) and the errors from the imperfect-model experiment are larger than those of the perfect-model experiment as expected. In the perfect-model experiment, the results are mixed with both SP and MP ensembles producing the lower rms errors at 95% level for some model fields. In addition the differences in rms errors between the perfect-model ensembles are less than 10%. However, in the imperfect-model experiment, the rms errors from Imperfect_MP yield notably smaller rms errors during the last 40 min of the forecasts for all three variables when compared to Imperfect_SP (Figs. 6b,d,f) and these differences in the rms errors are significant at the 95% level. In addition, these differences in rms errors exceed 20% at the end of the forecast time. Therefore, the variations in the microphysical parameters clearly help to reduce the mean forecast error in the presence of model error.

The corresponding ensemble spread plot (Fig. 7) during the 1-h forecast period indicates that the MP ensemble provides larger ensemble spread compared to the SP ensemble. For the imperfect experiment, the values of ensemble mean rms error and ensemble spread have similar magnitude in the MP ensemble, whereas the ensemble mean rms errors are at least 30% larger than the ensemble spread for the SP ensemble at the end of the forecast period (cf. Figs. 6 and 7). Therefore, the MP ensemble can be used as a tool to alleviate the underdispersion problem in storm-scale ensemble data assimilation and forecast without using any artificial methods (Anderson 2007, 2009; Dowell and Wicker 2009) to maintain spread. Snook et al. (2011) also show that using a mixed-microphysics scheme in storm-scale EnKF system increases the ensemble spread compared to a scheme that uses a single-microphysics scheme for all ensemble members.

To quantify the forecast accuracy from the ensemble forecasts, the ETS is calculated by comparing the ensemble mean forecast with the truth for reflectivity values exceeding 45 dB*Z* and for total precipitation mixing ratios exceeding 1.0 g kg^{−1}. Results indicate that for the perfect-model experiment, the ETS from SP is similar to MP ensemble for reflectivity, but higher than MP for total precipitation mixing ratios (Figs. 8a,c). In contrast, for the imperfect-model experiment, the Imperfect_MP yields a 20% higher ETS throughout the 1-h forecast period compared to that of the Imperfect_SP for both reflectivity and mixing ratio (Figs. 8b,d). The differences in ETS values from the Imperfect_MP and Imperfect_SP ensembles grow larger with time and highlight the value associated with the MP approach.

The ability of the EnKF to forecast the important storm characteristics is illustrated by comparing the forecast time series of the maximum vertical vorticity at 5000 m above ground level (AGL; Fig. 9) and the minimum cold pool temperature at the lowest model level (100 m AGL; Fig. 10) from each ensemble member for both perfect- and imperfect-model experiments. All runs produce strong midlevel rotation (Fig. 9), indicating that the dominant characteristic of a rotating storm is reproduced in all the members. This is due to the use of identical environmental conditions that are very conducive to developing midlevel rotation in the updraft region. However, near the ground the ensemble members from the SP runs for both the perfect- and imperfect-model experiment provide insufficient ensemble spread, with the coldest truth run temperatures (indicative of cold pool strength) falling outside the ensemble envelope for some forecast periods (Figs. 10). This result suggests that methods to increase the spread are needed. In contrast, the MP experiments increase the ensemble spread while also capturing the truth run minimum temperatures well within the envelope of the ensemble members. In particular, a few of the Imperfect_MP ensemble members have warmer cold pools and an associated weaker low-level rotation, which agree better with the Truth_10ICE run. The differences in cold pool strength are due to differences in precipitation evaporation below the cloud base, which is strongly influenced by the microphysics scheme. Runs with small graupel (i.e., smaller density and larger intercept values for hail) generally produce weaker, more expansive cold pools than runs with large hail, since small graupel leaves the storm at higher levels and is slower to reach the ground (Gilmore et al. 2004b). The intercept parameter for rain also plays a role in determining the evaporation rate, with evaporation increasing as the intercept parameter increases (all else being equal; Stensrud 2007).

Another critical forecast variable is storm total rainfall. The ground-relative storm total rainfall valid at the end of 1-h forecast period from the imperfect experiment shows that the total rainfall from the Imperfect_MP ensemble mean forecast more closely resembles the truth run than the total rainfall from the Imperfect_SP experiment (Fig. 11). The Imperfect_SP produces higher rainfall amounts from the northern and the southern storms cells when compared to the Truth_10ICE run. Rainfall totals from the perfect-model experiments are nearly identical and so are not shown. The sensitivity of total rainfall to the microphysics parameters is also seen in Gilmore et al. (2004b).

The maximum vertical vorticity at 300 m AGL (Fig. 12), maximum rainwater mixing ratio (Fig. 13), maximum hail mixing ratio (Fig. 14), and the maximum mean hail diameter (Fig. 15) at the lowest model level from anywhere in the model domain during the 1-h forecast period for the perfect- and imperfect-model experiments are used to explore the ability of the ensemble to predict the extremes of an event occurring or not occurring. Extremes are important to the prediction of severe weather, but are not an indication of how well the ensembles capture the overall characteristics of the storms. The rms errors, ETSs, midlevel rotation, and storm rainfall totals described earlier are much better indicators of overall ensemble analysis and forecast accuracy. The extreme values are important, however, as they indicate whether or not the ensembles have sufficient spread to capture these more unlikely events that have important societal impacts. Results from the Perfect_SP ensembles show that the truth run extremes often lay on the edge of or outside the ensemble envelope. One interpretation of this result is that the SP ensembles are not adequately sampling the microphysical characteristics that influence the production of these extremes as shown by Snook and Xue (2008) for tornadogenesis. In contrast, the Perfect_MP ensembles capture the truth run extremes within the ensemble members.

The parameterization used in the Truth_LFO run for the perfect-model experiment yields a very intense storm in low levels, while the parameterization in the Truth_10ICE run for the imperfect-model experiments produces a warmer cold pool, less surface precipitation, and weaker low-level rotation. For the Imperfect_SP runs in Figs. 12–15, the truth is below the envelope of ensemble solutions, indicating that the SP ensemble is overpredicting the likelihood of producing extremes. In contrast, some members from the Imperfect_MP run produce values less than the truth and the truth is contained within the envelope of solution. Therefore the MP ensemble may help in reducing false alarms. The MP results show how variations in microphysical parameters alter the prediction of mixing ratios and hail size in agreement with Gilmore et al. (2004b) and low-level rotation as seen in Snook and Xue (2008). Note that while the maximum mixing ratios evolve more slowly (Figs. 13 and 14), the maximum hail sizes change more rapidly owing to the important role played by the slope and intercept parameters in diagnosing the maximum hail size (Fig. 15).

One may question whether or not these variations in mixing ratio and hail size are reasonable. The large differences between the SP and MP ensembles for maximum hail size would seem to indicate differences in storm structure. Yet all the MP storms are splitting supercells and have reflectivity values within 8 dB*Z* of the truth runs. Instead, these results highlight the variety of hydrometeor combinations that can produce a given value of radar reflectivity. The varied microphysics parameters in the MP ensemble cause the precipitation to be partitioned in different ways between snow, rain, and graupel/hail, which are then adjusted during the radar reflectivity assimilation. The different microphysics parameters also then influence the evaporation of rain, leading to a wider variety of cold pool minimum temperatures that encompass the minimum temperature of the truth run (Fig. 10). It may be that the microphysics parameters can be estimated during the data assimilation process, as done by Tong and Xue (2008a,b), but there is no guarantee that these estimated parameters will produce an accurate storm forecast as the microphysics parameter values can change throughout the storm lifetime.

To determine whether or not the large hail sizes found in some members of the MP ensemble are reasonable, the results of Renick and Maxwell (1977) are used to estimate maximum hail size. Using their method, the maximum estimated hail size from the MP ensemble members exceeds 40 mm. In addition, a tool frequently used by operational forecasters to discriminate between significant hail (diameter ≥2 in. or 50 mm) and nonsignificant hail (diameter <2 in. or 50 mm) is the significant hail parameter (SHiP; more information is available online at http://www.spc.noaa.gov/exper/soundings/help/ship.html). SHiP is determined using five parameters calculated from an environmental sounding. The value of SHiP calculating from the WK sounding used to initialize the base environment of all the ensemble members is 1.87, which indicates that this environment is favorable for producing hail larger than 50 mm. These calculations indicate that the presence of large hail in some of the MP ensemble members is reasonable. The large spread in maximum hail diameter from the MP ensembles is supported by results from the Severe Hazards Analysis and Verification Experiment (SHAVE; Ortega et al. 2009). This experiment collects high temporal and spatial resolution reports to document hail sizes, wind damage, and flash flooding produced by severe thunderstorms. Results from the project reveal that supercell storms can produce a wide spectrum of hail size within a single storm, while results from previous studies and SHAVE suggest that radar reflectivity values above 55 or even 65 dB*Z* are not always associated with hail at the ground (Mather et al. 1976; Dye and Martner 1978; K. Ortega 2010, personal communication). Thus, the larger variations in maximum hail mixing ratio and size from the MP ensemble appear reasonable. The real question is whether or not the probabilities of large hail (or other extremes) produced by an MP ensemble are reliable. This question, unfortunately, is outside the purview of this initial study and will require real-data experiments to answer conclusively. These results highlight the potential importance of using an MP ensemble in the presence of model error, especially when examining the extreme values of the model fields that would be most helpful in determining and identifying potential hazards.

## 4. Discussion

The goal of this study is to evaluate the potential value of using a range of intercept and density parameters within the same microphysics scheme to produce more accurate storm analyses and forecasts using an EnKF approach for radar data assimilation. Two reference simulations of a splitting supercell storm are generated using LFO and 10ICE microphysics schemes in an identical storm environment. Two sets of OSSEs are then conducted from both a perfect- and an imperfect-model framework using an EnKF data assimilation technique with 1) constant intercept and density parameters for the hydrometeors in all ensemble members and 2) a range of different values of the intercept and density parameters for the hydrometeors in the different ensemble members. Synthetic WSR-88D reflectivity and radial-velocity observations are created from the truth runs using a realistic volume-averaging technique and these observations are assimilated into the two ensemble systems over a 30-min period. The 40 ensemble analyses valid at last assimilation cycle are then used to make 1-h forecasts.

Results show that the EnKF system performs well under both the perfect- and imperfect-model assumptions and that both ensembles yield similar results under the perfect-model assumption. However, under the imperfect-model assumption a multiparameter ensemble (Imperfect_MP) generates more accurate ensemble mean precipitation mixing ratio analyses and more accurate forecasts of ensemble mean winds, temperature, precipitation mixing ratios, and accumulated rainfall compared to that of the single-parameter model ensemble (Imperfect_SP). The 1-h forecasts for minimum cold pool temperature, maximum hail, and rainwater mixing ratios at 100 m AGL and the maximum vertical vorticity at 300 m AGL indicates that the truth run values almost always lie within the envelope of ensemble members for the MP ensemble, whereas the truth run values more often lie on the edge or outside the ensemble envelope for the SP ensemble. The MP ensemble also yields larger ensemble spread than the SP ensemble. Underdispersion is a common problem in storm-scale data assimilation and is an active area of research. The results from this study show that the MP ensemble can provide increased ensemble spread without applying any artificial method to increase spread. The results from this study support the idea that some amount of microphysical parameter diversity across the ensemble members may be beneficial to a storm-scale ensemble forecasting system. These conclusions regarding the ability of an MP ensemble to produce more accurate analyses and forecasts are consistent with results obtained using mesoscale model ensembles (Stensrud et al. 2000; Fujita et al. 2007; Meng and Zhang 2007, 2008).

The results obtained in this study are based on an idealized modeling framework where the lower model boundary is flat, and no surface fluxes or radiative effects are included. In addition, only the microphysics and turbulent mixing coefficient are parameterized to represent the cloud particle formation, growth, dissipation, and the energy associated with the subgrid-scale eddies, respectively. While the main source of model error for this study originates from differences in the microphysics parameterization scheme, the error from turbulent mixing scheme and the model numeric also plays a role in storm-scale data assimilation and forecasts. Furthermore, the results obtained in this study are based on an OSSE, whereas in a real-observation assimilation, the model error can potentially be larger than that considered in this study. We further emphasize that the selection of density and intercept parameters for the MP ensemble (Table 3 and Fig. 2) likely is far from optimal and other perturbation approaches, such as described by Hacker et al. (2011), but may yield better results. The MP ensemble system clearly needs to be tested over a broader range of experiments using real radar observations of severe weather events. Because of our limited understanding of microphysical processes, it is likely that the use of even more sophisticated microphysics parameterizations will face challenges in some storm environments. However, our results suggest that by using a variety of observed intercept and density parameters, an ensemble system is more likely to span the observations and provide improved short-range forecasts for a wide range of storm systems.

## Acknowledgments

The authors are thankful to two anonymous reviewers for their valuable comments that led to substantial improvements in the manuscript. The authors also thank Ted Mansell and Louis Wicker for their valuable suggestions. Local computer assistance provided by Brett Morrow, Steven Fletcher, Brad Swagowitz, and Karen Cooper are greatly appreciated. Partial funding for this research was provided by the NOAA/Office of Oceanic and Atmospheric Research under NOAA–University of Oklahoma Cooperative Agreement NA17RJ1227 (U.S. Department of Commerce).

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