# Assimilation of Polarimetric Radar Data in Simulation of a Supercell Storm with a Variational Approach and the WRF Model

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## Abstract

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## 1. Introduction

_{H}) and radial velocity (V

_{r}), are commonly used to improve the initial condition with advanced data assimilation (DA) methods, such as the three-dimensional variational (3DVAR) method [2,3,4,5,6,7], the four-dimensional variational (4DVAR) method [8,9,10,11,12], and the ensemble Kalman filter (EnKF) [13,14,15,16,17,18].

_{DR}), cross-correlation coefficient (ρ

_{hv}), and differential phase (ϕ

_{HV}), which is the range integral of the specific differential phase (K

_{DP}), benefits from polarimetric radar technology that has been widely concerned and developed rapidly after decades of effort. Nowadays, polarimetric radars are deployed and applied globally; in the U.S., for instance, the national radar network of WSR-88D has been upgraded with dual-polarization capability, and polarimetric radar data (PRD) are available nationally [19,20]. The PRD has been also successfully used in quantitative precipitation estimation (QPE) [21,22,23,24], hydrometeor classification (HC) [25,26,27], microphysics retrieval [28,29,30,31] and severe weather identification [32,33,34,35]. These mentioned advantages prompt us to further explore the possibility of assimilating PRD into NWP models to improve weather forecasts, especially for the prediction of convective-scale weather.

_{DR}arc, midlevel Z

_{DR}, and ρ

_{hv}ring [23,33] can only be reproduced by employing a DM MP scheme.

_{DR}[45] and K

_{DP}[46], respectively, and then assimilated them in the Weather Research and Forecasting (WRF) model using the WRF 3DVAR system. The consequences demonstrate the benefits of utilizing polarimetric variables to improve the initial condition and even short-term forecasts for mesoscale convective systems (MCSs). Then, Li et al. [47] developed an ice-phase operator embedded in the WRF 3DVAR system for assimilation of K

_{DP}, which can also additionally adjust solid water (cloud ice and snow) through DA. Consequently, a positive impact of extra K

_{DP}assimilation on analysis fields of rainwater in the lower troposphere and snow in the mid- to upper troposphere for an MCS was found. Aimed at applying J08 under the variational DA framework, Wang and Liu [48] rebuilt an operator named RadZIceVar (a package containing a reflectivity forward operator with its associated TL/AD operators together) and indicated its successful implement in DA use and significant improvement on short-term (2-5 h) precipitation forecast. However, the direct assimilation was applicable for Z

_{H}only, and assimilation of polarimetric variables was not supported.

_{DR}through the EnKF method for a real supercell case in the literature. Through a series of observing system simulation experiments (OSSEs) of an idealized supercell storm, Zhu et al. [18] investigated the impact of assimilating Z

_{DR}within an EnKF framework by using J10 coupled with a DM MP scheme, and also discussed the potential influence of updating hydrometeor number concentrations for DA effect. Their conclusions show that the assimilation of Z

_{DR}can improve the accuracy of analyzed hydrometeor fields in terms of pattern and intensity, and that updating the number concentrations with mixing ratios is important for deciding whether the benefit of assimilating Z

_{DR}can be preserved. However, they are all limited to the use of partial PRD for assimilation, and significant errors in the magnitude of analyzed PRD do exist, indicating that the operational application of PRD assimilation remains challenging.

_{DR}column signatures [33,52], which often occur in the strong updraft region within intense thunderstorms, to modify the temperature and moisture of the ARPS model. Encouraging results were obtained with improved analyses (more coherent updraft) and forecasts (more realistic reflectivity structures with better skill scores) compared to the original cloud analysis scheme without polarimetric information included. There are limits to the widespread use of this approach, however, due to the absence of Z

_{DR}columns in relatively weak precipitation systems and its high dependence on empirical relationships between model state variables and PRD.

## 2. Polarimetric Radar Observation Operator

#### 2.1. Microphysics Models and Parametrization

#### 2.2. Parameterized PRD Operators

_{H}, Z

_{DR}, K

_{DP}, ρ

_{hv}) are expressed by the integrals of DSD/PSD weighted by the scattering amplitudes [59,60]. The scattering amplitudes are normally calculated by the T-matrix method, and the integrals are calculated numerically for accurate results. This was done in J10. However, the complex integral form and resulting expensive computational cost make it not convenient for variational DA use, which usually needs the operators to be efficient and easy calculation of the derivative.

_{DR}and ρ

_{hv}. The shape and orientation of hydrometer particles follow the modeling and representation documented in J08. Similar to rain, for a given ice or mixed phase species x, including snow, hail, graupel, and their melting parts, PRD variables are calculated for a set of ${D}_{m}$ at a given ${\gamma}_{x}$, and then parameterized as functions of ${D}_{m}$ as follows:

## 3. The 3DVAR DA System

_{r}, Z

_{H}, Z

_{DR}, K

_{DP}, and ρ

_{hv}) is specified empirically as 2 m/s, 3 dBZ, 0.5 dB, 0.5 °/km, and 0.1 respectively. Finally, as the third term on the right-side, ${J}_{c}\left(\mathit{x}\right)$ represents the dynamic or equation constraints. This term has proved to be very important, particularly for this convective-scale 3DVAR system. For example, the mass continuity equation imposed as a weak constraint is helpful in producing more suitable wind analysis [2,3].

_{r}is regarded as a crucial variable in the DA procedure [3,4,10] because it provides important wind field information and dynamic characteristics of the precipitation event. In this 3DVAR DA system, the forward observation operator for radial velocity considering the effects of the Earth’s curvature is expressed as:

^{-14}, the R(a) values for all polarimetric variables (R_Z

_{H}, R_Z

_{DR}, R_K

_{DP}and R_ρ

_{hv}) are approaching 1 uniformly and steadily in the case of double precision. However, when a goes down to an extra small value, the R(a) starts to increase, which is caused by the rounding error of the calculator floating-point arithmetic operation. The above outcomes enable us to conclude that the assimilation system with the Z21 integrated passes the validity test and that the subsequent experimental research can be carried out reasonably.

## 4. Experimental Design

_{r}, Z

_{H}, Z

_{DR}, K

_{DP}, and ρ

_{hv}) can be simulated and used in the iterations of 3DVAR analysis to update the model state variables (${u}^{a}$, ${v}^{a}$, ${w}^{a}$, and ${q}_{x}^{a}$). After the analysis step, a 5 min forecast procedure is performed and the result is the background field (${u}^{b}$, ${v}^{b}$, ${w}^{b}$, and ${q}_{x}^{b}$) as the input of the DA analysis step in the next cycle. Considering the mixing ratios of precipitation are too small at the early stage of storm initiation, the idealized data are generated after the 30 min of model integration. Two assumed S-band radars are located in the southeast and southwest corner, respectively, and are performed on the VCP11 scan mode of WSR-88D radar (14 tilts from 0.5° to 19.5°). Idealized PRD is derived from the truth simulation output using the forward operators in Equations (7)–(11).

_{r}and Z

_{H}data, labeled as ExpVrZh, is considered as a reference for comparison purposes. Sensitivity experiments ExpVrZhZdr, ExpVrZhKdp, and ExpVrZhRhv are the same as the control run but with the additional assimilation of Z

_{DR}, K

_{DP}, and ρ

_{hv}, respectively. ExpVrZhPol is a special run with all PRD assimilated. These sensitivity experiments are performed to help understand the impact of PRD on the analysis of hydrometer variables. Therein, the number concentrations of hydrometeors will not be updated during the DA cycles on account of the complexity introduced by their wide dynamic range (from 0 to above 1E

^{12}m

^{-3}) and the problem of a significant increase of the degrees of freedom. The explanation of all abbreviations used in this paper can be found in Appendix A.

## 5. Results of 3DVAR Analysis

#### 5.1. The Root Mean Square Error Analysis

_{H}assimilation, the RMSEs of analysis and forecast displayed in Figure 2 are only calculated over precipitation grids with Z

_{H}exceeding 10 dBZ.

_{DP}is most sensitive to the presence of liquid water, the RMSE of ${q}_{r}$ (Figure 2f) of ExpVrZhKdp (blue line) has the most obvious difference. The better convergence through the whole DA cycle suggests that the satisfied rain water field much closer to the truth can be analyzed with additional K

_{DP}assimilation. In fact, as with ExpVrZhZdr, it also plays a positive role in obtaining a smaller RMSE in the last few cycles for ${q}_{i}$, ${q}_{h}$, and ${q}_{g}$. For ExpVrZhRhv (cyan line), the relatively distinct variations only occur in the first or last few cycles, but as a whole, its RMSEs almost follow that of ExpVrZh with no significant discrepancies, indicating the limited effect of ρ

_{hv}assimilation.

_{DR}(Figure 2j) and Z

_{H}(Figure 2i). The above conclusions agree well with those of Jung et al. [40], except for benefits shown earlier in DA cycles here. This is largely the same as the result of Zhu et al. [18]. Apparently, ExpVrZhKdp produces consistently better K

_{DP}(Figure 2k) in both analysis and forecast cycles, as expected. Moreover, there are some subtle advancements of Z

_{H}in the last few cycles, but no contributions to the simulation of Z

_{DR}and ρ

_{hv}(Figure 2l). As confirmed by the variation of the RMSEs, the assimilation of ρ

_{hv}cannot bring benefits in observation space as well. Although both Z

_{H}and ρ

_{hv}can converge reasonably, the influence of ρ

_{hv}assimilation is too small to be noticed, which may be caused by the relatively small variation range of itself (on the order of 0.01). Noted that the reverse changing problem (the RMSE decreases in the forecast cycle, but increases in the corresponding analysis cycle) occurring in Z

_{DR}and K

_{DP}can be resolved by assimilating the corresponding PRD variable.

_{DR}and ρ

_{hv}. Actually, with respect to most variables, the sharpest downward variation range occurring in the first cycle of ExpVrZhPol indirectly reveals the greatest potential capability of revising the model analysis field.

#### 5.2. Evaluation of PRD Assimilation

_{H}(Figure 3a), all OSSEs represent the hook echo signature and the strong Z

_{H}core at the leading edge of the storm. Among them, ExpVrZh (Figure 3b) and ExpVrZhRhv (Figure 3e) have very similar results with obviously high Z

_{H}(greater than 15 dBZ overestimated) in the rear stratiform precipitation area (hereafter SPA). However, the Z

_{H}intensity described by ExpVrZhZdr (Figure 3c) is superior to them, both in the front convective precipitation area (CPA) and the rear SPA. However, the weak echo (<20 dBZ) on the northern edge is over-adjusted to deviate from the truth. On the contrary, although the weak echo is well analyzed, a much smaller Z

_{H}exists in the SPA of ExpVrZhKdp (Figure 3d). In terms of intensity and structure, the overall result of ExpVrZhPol (Figure 3f) is much closer to the truth, particularly in the echo edge and rear SPA, except for a certain overestimation in the CPA. There is a small Z

_{DR}patch (<1 dB) in the southeast corner of the main storm in ExpVrZh (Figure 4b), and also the same in ExpVrZhRhv (Figure 4e). Beyond that, the Z

_{DR}in the SPA is significantly smaller than the truth (Figure 4a), with the largest difference over 1.5 dB. ExpVrZhKdp (Figure 4d) has even worse simulation with much smaller Z

_{DR}. However, the intensity of Z

_{DR}in ExpVrZhZdr (Figure 4c) is much enhanced in the fore-mentioned two underestimated areas, particularly in the SPA. However, to some extent, overestimation persists in the weak echo at the edge of storm. ExpVrZhPol (Figure 4f) equally gives the best analysis among all OSSEs, except for the underestimation that occurred in the south-east of the storm. Although the dominant distribution of K

_{DP}is well exhibited in ExpVrZh (Figure 5b), ExpVrZhZdr (Figure 5c) and ExpVrZhRhv (Figure 5e) without K

_{DP}assimilation, the central strength of the leading edge and the hook echo are clearly weaker. The assimilation of K

_{DP}is beneficial that the simulated K

_{DP}shown in Figure 5d is perfectly matched with the truth (Figure 5a) in both intensity and structure. Undoubtedly, as a result of additional assimilation of K

_{DP}, ExpVrZhPol (Figure 5f) also has a similar good performance to ExpVrZhKdp. Based on the simulated result of ExpVrZh (Figure 6b) which is pretty close to the true ρ

_{hv}(Figure 6a), ExpVrZhRhv (Figure 6e) mainly brings improvement in details, whereas ExpVrZhZdr (Figure 6c) and ExpVrZhKdp (Figure 6d) make further overestimation of ρ

_{hv}in certain areas. Affected by Z

_{DR}and K

_{DP}, ExpVrZhPol (Figure 6f) does not exhibit better ρ

_{hv}distribution, but instead has an underestimated area (<0.9).

_{H}, the whole structure and extended height of the strong echo are both close to the truth (Figure 7a), with the exceptions of the ring-shape core (>55 dBZ) in ExpVrZh (Figure 7b), ExpVrZhZdr (Figure 7c), and ExpVrZhRhv (Figure 7e). Additionally, Z

_{H}in these OSSEs are overestimated markedly in the middle level of the forward flank (5 km horizontally, the same hereafter), which is reduced properly in ExpVrZhKdp (Figure 7d). Additionally, the overestimation in the low level of the rear SPA (30~40 km) has been alleviated by assimilating Z

_{DR}, such that the intensity is much closer to the truth and the echo goes from ungrounded to grounded. ExpVrZhRhv shows no enhancement compared with ExpVrZh, or even worse results (overestimation in some areas, Figure 7e vs. Figure 7a and Figure 7b). More than that, the ahead-developing convective cell (~6 km) and inner strong echo of the main storm are both better described in ExpVrZhKdp and ExpVrZhPol (Figure 7f). In comparison with the truth (Figure 8a), the signature of the Z

_{DR}column (~5 km) related to the updraft within the storm is more evidently discerned in ExpVrZdr (Figure 8c) and ExpVrKdp (Figure 8d), whereas the sagged high Z

_{DR}area (~25 km) attributed to the existence of a small number of large raindrops is well retained in ExpVrZh (Figure 8b), ExpVrZdr, and ExpVrRhv (Figure 8e). Modifications of amplitude and shape of Z

_{DR}can be noticed in the tail (~35 km) in ExpVrZhZdr and ExpVrZhKdp. With respect to above features, the best analysis result in terms of intensity and structure is given in ExpVrZhPol (Figure 8f). ExpVrZhKdp reveals an extremely high degree of similarity with the truth (Figure 9d vs. Figure 9a), including the K

_{DP}core associated with high liquid water content (LWC) within the storm, an overshooting structure linked to the strong updraft rushing through the melting layer, and even the small K

_{DP}area hanging ahead. The effect of K

_{DP}assimilation is maintained in ExpVrZhPol (Figure 9f), showing almost identical distribution characteristics to that of ExpVrZhKdp. Comparatively, other OSSEs (Figure 9b,c,e) nearly fail to reproduce these mentioned signatures. The melting layer associated with the melting hydrometeors is precisely described by a ρ

_{hv}drop in these OSSEs, but with a wider melting band and a few underestimated areas above. Other than a small positive contribution from the ρ

_{hv}assimilation (Figure 10e) in the low-value area located in the middle level of the forward edge (~5 km) of the storm, there are no notable improvements in the distribution of analyzed ρ

_{hv}for different OSSEs.

#### 5.3. Evaluation of Hydrometeor Analysis

_{DR}or K

_{DP}assimilation are also retained in ExpVrZhPol. In addition, there are many other improvements such as the preferable descriptions of relatively small ${q}_{i}$ (<0.2 g/kg), ${q}_{h}$ (<0.6 g/kg), and ${q}_{g}$ (<4.5 g/kg) in the upper left corner f, Figure 13f and Figure 14f), the superior ${q}_{r}$ analysis much closer to the truth (Figure 12f vs. Figure 12a), and so on. However, apart from minor differences, all analyzed mixing ratios in ExpVrZhRhv are largely the same as the corresponding ones in ExpVrZh (Figure 11e, Figure 12e, Figure 13e and Figure 14e vs. Figure 11b, Figure 12b, Figure 13b and Figure 14b, respectively). Additionally, analyses of cloud water ${q}_{c}$ and snow ${q}_{s}$ yield similar results as other hydrometeor variables, that is, ${q}_{c}$ is similar to ${q}_{r}$, and ${q}_{s}$ is similar to other ice species (${q}_{i}$, ${q}_{g}$, and ${q}_{h}$), which are not detailed here.

_{DR}assimilation on the analysis of solid hydrometeor particles, more benefits of K

_{DP}assimilation in improving the simulation of rain water rather than ice hydrometeors, and the limited usefulness of ρ

_{hv}in terms of enhancing model hydrometeor analysis. Attributing to the combined positive impacts from PRD assimilations, more comprehensive hydrometeor analysis results in both intensity and pattern can be obtained. The conclusions are also in accordance with the findings of RMSE analysis.

_{DR}increases as the raindrop size increases or ice particles melt. Therefore, assimilation of Z

_{DR}yields improved analysis of all hydrometeors. K

_{DP}depends on both the raindrop size and the number concentration, hence, its assimilation improves the analysis of ${q}_{r}$. Since ρ

_{hv}reduction occurs only in the melting/mixing regions which have a very small percentage of the storm and it has a small dynamic range with relatively large errors, the improvement with assimilation of ρ

_{hv}is minimal.

#### 5.4. Evaluation of Forecast

_{DR}assimilation in increasing the prediction accuracy of ice species. Further, due to the high sensitivity of K

_{DP}to liquid water, ExpVrZhKdp (blue line) illustrates more superiorities in the forecasting of water vapor ${q}_{v}$ (Figure 15f), ${q}_{c}$, and ${q}_{r}$. Comparatively speaking, ExpVrZhRhv (cyan line) has fully comparable behaviors with ExpVrZh (black line), demonstrating the limited benefits of ρ

_{hv}assimilation.

_{H}(Figure 15m), and mainly in the late forecast stage (after 30 min) of Z

_{DR}(Figure 15n), K

_{DP}(Figure 15o) and ρ

_{hv}(Figure 15p), which is of great significance for quantitative applications of PRD (precipitation estimation, hydrometeor classification, and so on). By contrast, ExpVrZhKdp is the second choice, having a suboptimal effect comprehensively. It is noteworthy that the RMSEs of ExpVrZhRhv nearly coincide with that of ExpVrZh for all PRD variables, further indicating the minor positive impact of ρ

_{hv}assimilation on the forecasts.

## 6. Summary and Conclusions

_{DR}, specific differential phase K

_{DP}, cross-correlation coefficient ρ

_{hv}, as well as horizontal reflectivity Z

_{H}and radial velocity V

_{r}, cycled 3DVAR DA and forecast schemes are performed. The truth is generated with a 3 h simulation of an idealized supercell storm using the WRF model. The PRD is constructed from the idealized truth simulations via Z21 forward operators. Then, these observations are assimilated into the WRF model at a 5 min interval with the initial DA beginning at 30 min of model integration.

_{H}and V

_{r}as the benchmark run, OSSEs with additional assimilation of Z

_{DR}, K

_{DP}, and ρ

_{hv}, respectively, are performed to examine the impact of each polarimetric variable on DA results. Additionally, an extra OSSE which assimilates all PRD is conducted to see whether a more balanced result can be achieved. The results suggest that the Z

_{DR}assimilation adjusts all model state variables to varying degrees, and helps reduce RMSEs at almost each DA cycle. Moreover, the assimilation process makes some polarimetric signatures, such as hook echo, Z

_{DR}column and melting layer, more obvious, and improves the distributions of hydrometeors and observations much closer to the truth as well. High sensitivity to liquid water of K

_{DP}is believed to be responsible for the more improved analysis of rainwater, and the intensity and pattern of PRD are also enhanced with additional assimilation of K

_{DP}, especially for Z

_{H}and K

_{DP}itself. The small discrepancy between OSSEs with and without ρ

_{hv}assimilation indicates its limited ability in adjusting model state variables and improving the analysis accuracy. In contrast, due to the accumulation of favorable DA effects of different polarimetric variables, assimilating all PRD obtains more balanced analysis results, as expected. The above polarimetric signatures are also well presented in observation space, and many other improvements have been made, including the optimization of rain water analysis which is perfectly matched with the reference truth, and better descriptions of some small value areas in mixing ratios of cloud water, snow, and graupel. These different assimilation results are attributed to the different information contents and error effects from PRD variables. It is noted that these characteristics of assimilation PRD can change depending on the type of storm and model microphysics as well as observation errors, which needs further study. A series of prediction experiments, which initiate at the 120 min of different OSSEs and run over a 1 h period at a 15 min interval, are conducted to further study the impact of improved initial conditions on ensuing forecasts. The RMSEs of forecasts are calculated and the results show that the Z

_{DR}assimilation can bring obvious benefits for forecasts of ice hydrometeor species including cloud ice, hail, and graupel and even for the vertical wind field and perturbation pressure in the late stage of prediction. Assimilating K

_{DP}is found to improve the prediction of water vapor, cloud water, and rainwater, which is consistent with its performance in the liquid water analysis. There is no visible promoting effect on the prediction of most variables, suggesting that ρ

_{hv}assimilation is not very helpful for the improvement of NWP. Similar to analysis results, the OSSE assimilating all PRD shows a more comprehensive performance in terms of RMSEs, having the most accurate (smallest RMSE) results of horizontal wind field, perturbation potential temperature, cloud water, rain water, snow, hail, and graupel in the end of forecast period. In addition, the forecasts in observation space are also better, with higher accuracy in the second half of period.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Abbreviation/Acronym | Explanation |
---|---|

PRD | Polarimetric radar data |

NWP | Numerical weather prediction |

DA | Data assimilation |

TL | Tangent linear model |

AD | Adjoint model |

3DVAR | Three-dimensional variational system/method |

4DVAR | Four-dimensional variational system/method |

EnKF | Ensemble Kalman filter |

EnSRF | Ensemble square-root Kalman filter |

V_{r} | Radial velocity |

Z_{H} | Horizontal reflectivity |

Z_{DR} | Differential reflectivity |

ϕ_{HV} | Differential phase |

K_{DP} | Specific differential phase |

ρ_{hv} | Cross-correlation coefficient |

QPE | Quantitative precipitation estimation |

HC | Hydrometeor classification |

MP | Microphysical parameterization scheme of model |

SM | Single-moment scheme |

DM | Double-moment scheme |

WRF | Weather research and forecasting model |

WRF-ARW | Advanced research weather research and forecasting model |

MCS(s) | Mesoscale convective system(s) |

OSSE(s) | Observing system simulation experiment(s) |

DSD | Drop size distribution |

PSD | Particle size distribution |

$q$ | Mixing ratio |

${N}_{t}$ | Number concentration |

$W$ | Water content |

${\rho}_{a}$ | Air density |

${\rho}_{w}$ | Water density |

${D}_{m}$ | Mass-weighted diameter |

${\gamma}_{x}$ | Percentage of melting |

APRS | Advanced Regional Prediction System |

CAPS | Center for Analysis and Prediction of Storms |

NSSL | National Severe Storms Laboratory |

J08 | Represents the article of Jung et al., (2008) |

J10 | Represents the article of Jung et al., (2010) |

Z21 | Represents the article of Zhang et al., (2021) |

${\mathit{x}}^{b}$ | Background vector in the cost function |

$\mathit{x}$ | Analysis vector in the cost function |

${\mathit{y}}^{o}$ | Observation vector in the cost function |

$\mathit{B}$ | Background error covariance matrix of model |

$\mathit{R}$ | Observation error covariance matrix of model |

$H\left(\mathit{x}\right)$ | Forward operator |

${J}_{c}\left(\mathit{x}\right)$ | Constraints in the cost function |

RUC | Rapid Update Cycle |

AGL | Above ground level |

$u$ | Horizontal wind in u-direction |

$v$ | Horizontal wind in v-direction |

$w$ | Vertical velocity |

$\theta $ | Perturbation potential temperature |

${q}_{c}$ | Mixing ratio of cloud water |

${q}_{i}$ | Mixing ratio of cloud ice |

${q}_{r}$ | Mixing ratio of rain water |

${q}_{s}$ | Mixing ratio of snow |

${q}_{h}$ | Mixing ratio of hail |

${q}_{g}$ | Mixing ratio of graupel |

${N}_{c}$ | Number concentration of cloud water |

${N}_{i}$ | Number concentration of cloud ice |

${N}_{r}$ | Number concentration of rain water |

${N}_{s}$ | Number concentration of snow |

${N}_{h}$ | Number concentration of hail |

${N}_{g}$ | Number concentration of graupel |

VCP | Volume coverage pattern |

WSR-88D | Weather surveillance radar—1988 Doppler |

RMSE(s) | Root mean square error(s) |

ExpVrZh | Experiment on assimilation of V_{r} and Z_{H} |

ExpVrZhZdr | Experiment on assimilation of V_{r}, Z_{H}, and Z_{DR} |

ExpVrZhKdp | Experiment on assimilation of V_{r}, Z_{H}, and K_{DP} |

ExpVrZhRhv | Experiment on assimilation of V_{r}, Z_{H}, and ρ_{hv} |

ExpVrZhPol | Experiment on assimilation of V_{r}, Z_{H}, Z_{DR}, K_{DP}, and ρ_{hv} |

CPA | Convective precipitation area |

SPA | Stratiform precipitation area |

LWC | Liquid water content |

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**Figure 1.**Illustration of DA and forecast cycles used for the idealized case study. Observations represent the assimilated PRD including radar radial velocity, reflectivity, and all polarimetric variables simulated from the truth simulation output of the WRF run. The observations are assimilated by the 3DVAR system based on the background field of previous 5 min forecast result.

**Figure 2.**The mean RMSEs of the forecasts and analyses throughout the 5 min 3DVAR DA cycles for ExpVrZh (black line), ExpVrZhZdr (red line), ExpVrZhKdp (blue line), ExpVrZhRhv (cyan line), and ExpVrZhPol (purple line), respectively, averaged over grid points where the observed (simulated) Z

_{H}is greater than 10 dBZ. (

**a**) $u$, (

**b**) $v$, (

**c**) $w$, mixing ratios of (

**d**) cloud water ${q}_{c}$, (

**e**) cloud ice ${q}_{i}$, (

**f**) rain water ${q}_{r}$, (

**g**) snow ${q}_{s}$, (

**h**) hail ${q}_{h}$, and graupel ${q}_{g}$, and the same calculating procedure as truth simulations for polarimetric radar variables (

**i**) Z

_{H}, (

**j**) Z

_{DR}, (

**k**) K

_{DP}, and (

**l**) ρ

_{hv}via Z21 forward operators.

**Figure 3.**The reflectivity Z

_{H}(unit: dBZ) at 1 km altitude at 80 min for (

**a**) truth, and analysis fields of (

**b**) ExpVrZh, (

**c**) ExpVrZhZdr, (

**d**) ExpVrZhKdp, (

**e**) ExpVrZhRhv, and (

**f**) ExpVrZhPol, respectively. The two assumed radars are located at the bottom-left and top-right corners as shown in Figure 3a.

**Figure 4.**The same as Figure 3, but for the differential reflectivity Z

_{DR}(unit: dB) of (

**a**) truth, and analysis fields of (

**b**) ExpVrZh, (

**c**) ExpVrZhZdr, (

**d**) ExpVrZhKdp, (

**e**) ExpVrZhRhv, and (

**f**) ExpVrZhPol, respectively.

**Figure 5.**The same as Figure 3, but for the specific differential phase K

_{DP}(unit: °/km) of (

**a**) truth, and analysis fields of (

**b**) ExpVrZh, (

**c**) ExpVrZhZdr, (

**d**) ExpVrZhKdp, (

**e**) ExpVrZhRhv, and (

**f**) ExpVrZhPol, respectively.

**Figure 6.**The same as Figure 3, but for the cross-correlation coefficient ρ

_{hv}(dimensionless) of (

**a**) truth, and analysis fields of (

**b**) ExpVrZh, (

**c**) ExpVrZhZdr, (

**d**) ExpVrZhKdp, (

**e**) ExpVrZhRhv, and (

**f**) ExpVrZhPol, respectively.

**Figure 7.**Vertical cross-sections of reflectivity Z

_{H}along the black dashed line in Figure 3a for (

**a**) truth, and analysis fields of (

**b**) ExpVrZh, (

**c**) ExpVrZhZdr, (

**d**) ExpVrZhKdp, (

**e**) ExpVrZhRhv, and (

**f**) ExpVrZhPol, respectively.

**Figure 8.**The same as Figure 7, but for the differential reflectivity Z

_{DR}of (

**a**) truth, and analysis fields of (

**b**) ExpVrZh, (

**c**) ExpVrZhZdr, (

**d**) ExpVrZhKdp, (

**e**) ExpVrZhRhv, and (

**f**) ExpVrZhPol, respectively.

**Figure 9.**The same as Figure 7, but for the specific differential phase K

_{DP}of (

**a**) truth, and analysis fields of (

**b**) ExpVrZh, (

**c**) ExpVrZhZdr, (

**d**) ExpVrZhKdp, (

**e**) ExpVrZhRhv, and (

**f**) ExpVrZhPol, respectively.

**Figure 10.**The same as Figure 7, but for the cross-correlation coefficient ρ

_{hv}of (

**a**) truth, and analysis fields of (

**b**) ExpVrZh, (

**c**) ExpVrZhZdr, (

**d**) ExpVrZhKdp, (

**e**) ExpVrZhRhv, and (

**f**) ExpVrZhPol, respectively.

**Figure 11.**The same as Figure 7, but for the mixing ratio of cloud ice ${q}_{i}$ (unit: g/kg) of (

**a**) truth, and analysis fields of (

**b**) ExpVrZh, (

**c**) ExpVrZhZdr, (

**d**) ExpVrZhKdp, (

**e**) ExpVrZhRhv, and (

**f**) ExpVrZhPol, respectively.

**Figure 12.**The same as Figure 7, but for the mixing ratio of rain water ${q}_{r}$ (unit: g/kg) of (

**a**) truth, and analysis fields of (

**b**) ExpVrZh, (

**c**) ExpVrZhZdr, (

**d**) ExpVrZhKdp, (

**e**) ExpVrZhRhv, and (

**f**) ExpVrZhPol, respectively.

**Figure 13.**The same as Figure 7, but for the mixing ratio of hail ${q}_{h}$ (unit: g/kg) of (

**a**) truth, and analysis fields of (

**b**) ExpVrZh, (

**c**) ExpVrZhZdr, (

**d**) ExpVrZhKdp, (

**e**) ExpVrZhRhv, and (

**f**) ExpVrZhPol, respectively.

**Figure 14.**The same as Figure 7, but for the mixing ratio of graupel ${q}_{g}$ (unit: g/kg) of (

**a**) truth, and analysis fields of (

**b**) ExpVrZh, (

**c**) ExpVrZhZdr, (

**d**) ExpVrZhKdp, (

**e**) ExpVrZhRhv, and (

**f**) ExpVrZhPol, respectively.

**Figure 15.**The mean RMSEs of 1 h forecasts at a 15 min intervals initiated at 120 min for ExpVrZh (black line), ExpVrZhZdr (red line), ExpVrZhKdp (blue line), ExpVrZhRhv (cyan line), and ExpVrZhPol (purple line), respectively, averaged over grid points with observed (simulated) Z

_{H}exceeding 10 dBZ. (

**a**) $u$, (

**b**) $v$, (

**c**) $w$, (

**d**) perturbation potential temperature $\theta $, (

**e**) perturbation pressure $p$, mixing ratios of (

**f**) water vapor ${q}_{v}$, (

**g**) cloud water ${q}_{c}$, (

**h**) cloud ice ${q}_{i}$, (

**i**) rain water ${q}_{r}$, (

**j**) snow ${q}_{s}$, (

**k**) hail ${q}_{h}$, and (

**l**) graupel ${q}_{g}$, and simulated PRD in observation space: (

**m**) Z

_{H}, (

**n**) Z

_{DR}, (

**o**) K

_{DP}, and (

**p**) ρ

_{hv}.

α | R_Z_{H} | R_Z_{DR} | R_K_{DP} | R_ρ_{hv} |
---|---|---|---|---|

1E^{0} | 0.699045165073124 | 0.483074981957829 | 1.02528819132801 | 0.490697018550403 |

1E^{-1} | 0.954886643047696 | 0.884597784411837 | 0.995278460334951 | 0.931147252389009 |

1E^{-2} | 0.995227804958050 | 0.987614013476037 | 0.999521333803539 | 0.992331436494260 |

1E^{-3} | 0.999519983964757 | 0.998752196825316 | 0.999952065828401 | 0.999224559666156 |

1E^{-4} | 0.999951970230481 | 0.999875126965556 | 0.999995205904266 | 0.999922370046164 |

1E^{-5} | 0.999995196913338 | 0.999987513140014 | 0.999999520577862 | 0.999992241080020 |

1E^{-6} | 0.999999520221590 | 0.999998772473737 | 0.999999952019359 | 0.999999175462863 |

1E^{-7} | 0.999999959777296 | 1.00000006087191 | 0.999999997600995 | 0.999999828029847 |

1E^{-8} | 1.00000001030094 | 1.00000292308563 | 1.00000002346292 | 1.00001425319475 |

1E^{-9} | 0.999999336652349 | 1.00002350529665 | 1.00000031440953 | 0.999973038437884 |

1E^{-10} | 1.00001449374567 | 1.00003817816193 | 0.999999667861502 | 0.999114397669863 |

1E^{-11} | 1.00019974710850 | 1.00046831421264 | 1.00001583156215 | 0.999457853977071 |

1E^{-12} | 1.00036815925652 | 0.995141863117389 | 1.00004815896344 | 0.996023290904985 |

1E^{-13} | 1.01047288813790 | 1.08940999759569 | 1.00214944004741 | 0.686912614417231 |

1E^{-14} | 1.01047288813790 | 1.38688726631185 | 1.01831314069334 | 3.43456307208615 |

Experiments | Observations | Description |
---|---|---|

ExpVrZh | V_{r}+Z_{H} | V_{r} and Z_{H} assimilated |

ExpVrZhZdr | V_{r}+Z_{H}+Z_{DR} | As ExpVrZh with additional Z_{DR} assimilated |

ExpVrZhKdp | V_{r}+Z_{H}+K_{DP} | As ExpVrZh with additional K_{DP} assimilated |

ExpVrZhRhv | V_{r}+Z_{H}+ρ_{hv} | As ExpVrZh with additional ρ_{hv} assimilated |

ExpVrZhPol | V_{r}+Z_{H}+Z_{DR}+K_{DP}+ρ_{hv} | As ExpVrZh with all PRD assimilated |

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**MDPI and ACS Style**

Du, M.; Gao, J.; Zhang, G.; Wang, Y.; Heiselman, P.L.; Cui, C.
Assimilation of Polarimetric Radar Data in Simulation of a Supercell Storm with a Variational Approach and the WRF Model. *Remote Sens.* **2021**, *13*, 3060.
https://doi.org/10.3390/rs13163060

**AMA Style**

Du M, Gao J, Zhang G, Wang Y, Heiselman PL, Cui C.
Assimilation of Polarimetric Radar Data in Simulation of a Supercell Storm with a Variational Approach and the WRF Model. *Remote Sensing*. 2021; 13(16):3060.
https://doi.org/10.3390/rs13163060

**Chicago/Turabian Style**

Du, Muyun, Jidong Gao, Guifu Zhang, Yunheng Wang, Pamela L. Heiselman, and Chunguang Cui.
2021. "Assimilation of Polarimetric Radar Data in Simulation of a Supercell Storm with a Variational Approach and the WRF Model" *Remote Sensing* 13, no. 16: 3060.
https://doi.org/10.3390/rs13163060