1. Introduction
Heavy rainfall occurs frequently in South China and always leads to flooding, urban waterlogging, and mountain torrents. These heavy-precipitation processes seriously threaten the safety of people’s lives and property and often cause significantly economic losses. Accurate forecast of heavy rainfall in South China is essential for government disaster prevention and mitigation.
Numerical weather prediction (NWP) model is the main technical means and tool for modern weather forecast. As computing power continues to increase, the grid spacing of operational NWP models have reached the convective-scale resolution (≤4 km). At this resolution, the traditional cumulus parameterization scheme is no longer applicable, and the explicit processes of cloud and precipitation must use the microphysics schemes in NWP models. The microphysics processes have complex nonlinear interactions with dynamics and radiation process, and significantly affect the evolution of precipitation system (Johnson et al. 2018). As a result, the simulation accuracy of the microphysical scheme directly affects the accuracy of quantitative precipitation forecast (QPF) (Igel et al. 2015).
Bulk cloud microphysical parameterization schemes mainly include the one-moment (1M) scheme and the two-moment (2M) scheme. The 1M scheme only predicts the mass content Qx of the hydrometeor. The values of N0 and μ are fixed, and Λ is obtained by solving the Qx. The evolution of PSD is determined only by Λ, and it imposed great restrictions in comparison with observation results. Commonly used 1M schemes are the Lin (Lin et al. 1983), WSM6 (Hong and Lim 2006), Goddard (Tao et al. 1989), and State University of New York at Stony Brook bulk microphysical parameterization (BMP) scheme (SBU-YLIN; Lin and Colle 2011). Moreover, 2M schemes simultaneously predict the Qx and total number density Ntx of rainwater. The value of μ is fixed in most schemes (generally at zero), and the N0 and Λ can be solved by Qx and Ntx. For the 2M scheme, the description of particle spectrum is improved to some extent. The lack of physical constraints between the slope and intercept parameters often leads to the mismatch between them (Xu and Duan 1999). Commonly used 2M schemes are the Morrison (Morrison et al. 2005), WDM6 (Lim and Hong 2010), NSSL (Mansell et al. 2010), and Milbrandt (Milbrandt and Yau 2005a) schemes. Thompson (Thompson et al. 2008) is also a 2M scheme when simulating rain and ice.
All 1M and most of 2M schemes assume that μ is a constant (usually 0 for raindrops). However, extensive observational studies (e.g., Zhang et al. 2001; Bringi et al. 2002; Chen et al. 2013; Tang et al. 2014; Wen et al. 2016, 2017; Liu et al. 2018) have shown that the μ is not constant, but often varies from negative to positive values. This key technical bottleneck needs to be solved urgently.
Morrison et al. (2019) has proposed a general N-moment raindrop size distribution (RSD) normalization method. However, the uncertainty of estimating RSD is very large by using zeroth-moment (M0) and third-moment (M3) two-moment normalization (as shown in Fig. 5, described in more detail later) (Morrison et al. 2019). For the 2M microphysical scheme, after considering the air and rainwater density, M0 is equivalent to Ntr, and M3 is proportional to Qr. The accuracy of general RSD normalization method is not enough and needs to be further improved.
If we need to predict μ, then it is necessary to introduce a third equation to close the system. Milbrandt and Yau (2005b) introduced a prediction equation for the radar reflectivity factor Z to predict μ. However, the Z itself is not an independent forecast quantity but rather is a derived variable that depends on Qx and Ntx. Therefore, it cannot determine the value of μ (D. H. Wang et al. 2014). Zhang et al. (2016) used actual observations in East Asia to infer the value of μ (assuming that μ is between 0 and 6) from the model-predicted Λ value. This method simply inferred the value of μ through the μ–Λ relationship; however, there was no closure between the derived Γ RSD and the 2M model forecasts (Qr and Ntr), and no actual observation data were used to check whether the method is reasonable.
Extensive observation of RSD shows that the three parameters of the Γ function are not all completely independent (Zhang et al. 2001; Gorgucci et al. 2002; Brandes et al. 2004a,b; Chen et al. 2013; Wen et al. 2017; Liu et al. 2018; Wen et al. 2019). In particular, there is a positive correlation between μ and Λ of the RSD. All of the present studies describe the μ–Λ relationship with a parabolic function; these equations provide a one-variable sixth-order equation for μ. The one-variable sixth-order equation is difficult to solve analytically, and iterative solutions tend to oscillate back and forward without convergence. Solving this complex equation will consume a lot of computing resources. Furthermore, the equation can have as many as six solutions; therefore, it is difficult to eliminate invalid and wrong solutions and finally obtain a scientific and reasonable solution.
If the μ–Λ relationship can be reduced to a linear correlation, then one obtains a one-dimensional cubic equation that can be solved analytically. In comparison with the one-variable sixth-order equation, the solution speed of this method will be significantly faster. However, this raises two questions: First, does the fitting accuracy decrease significantly after the μ–Λ relationship is simplified to a linear relationship? Second, when there are two or three real solutions, how does one eliminate the invalid solutions and arrive at the unique solution that is scientifically valid?
To improve the understanding of precipitation microphysics characteristics and heavy-rainfall forecast skills during monsoon season in Southern China, unique program of the Southern China Monsoon Rainfall Experiment (SCMREX) project were conducted in Southern China from 2013 to 2018 (Luo et al. 2017). Several 2D video disdrometers (2DVDs) were collocated to observe the precipitation microphysics in South China in recent years, and the representative linear μ–Λ relationship of RSD was observed by 2DVD in this region. According to the linear μ–Λ relationship, a high-precision and rapid solution for Γ function based on the 0-moment and 3-moment has been established. The purpose of this study is to resolve the problem that current 2M microphysical schemes set μ to a constant and thereby improve the ability of the 2M schemes to simulate the RSD of heavy rainfall in South China.
2. Data and method
a. 2DVD and dataset
Aimed at the needs of rainstorms mechanism analyzing and development of NWP model in South China, the Guangzhou Institute of Tropical and Marine Meteorology of the China Meteorological Administration (ITMM/CMA) and the Chinese Academy of Meteorological Sciences (CAMS) have jointly set up the Longmen Cloud Physics Field Experiment Base, CMA, since 2014.
The 2DVDs are important instruments for measuring precipitation characteristic at the field experiment base. The locations and observation periods of these 2DVDs are shown in Fig. 1. The RSD samples observed by these 2DVDs from 2016 to 2019 were used in this study. The detailed information of the 2DVDs has been introduced by Liu et al. (2018). The data quality control and processing method for the observations of 2DVDs are similar to those of Tokay et al. (2013), Wen et al. (2016), and Liu et al. (2018). The 2DVD observations are processed at 1-min interval. For each 1-min 2DVD observation, if the total number is less than 10 or if rain rate is less than 0.1 mm h−1 then these 1-min data are considered as noise and are disregarded.
The geographical locations and observation time periods of 2DVDs at the Longmen Cloud Physics Field Experiment Base, CMA. Terrain is shaded (m).
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
The geographical locations and observation time periods of 2DVDs at the Longmen Cloud Physics Field Experiment Base, CMA. Terrain is shaded (m).
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
The geographical locations and observation time periods of 2DVDs at the Longmen Cloud Physics Field Experiment Base, CMA. Terrain is shaded (m).
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
b. Method of solving Γ raindrop size distribution function
Based on the μ–Λ relationship Zhang et al. (2001) proposed the constrained-gamma (C-G) method to retrieval RSD from polarimetric radar observations. According to the μ–Λ relations, the independent parameters of Γ distribution function will reduce from 3 to 2. Previous studies generally used parabolic functions to describe the μ–Λ relationships. In this study, linear functions were used to describe these relations and is referred to as the linear C-G method.
The RSD samples collected by 2DVDs from 2016 to 2018 in South China were used for the statistics of the linear μ–Λ relationship. Three typical heavy-rainfall processes occurred in 2019 and all RSD samples observed during 2019 in South China, as shown in Table 1, were mainly used to test the accuracy of linear (C-G) solution method.
Detailed information for four heavy-rainfall events in 2019.
If the RSD [N(D); mm−1 m−3] is obtained, the corresponding radar reflectivity factor (Z; mm6 mm−3) and rain rate (R; mm h−1), rainwater content (W; g m−3), and raindrop total number concentration (Nt; m−3) can be calculated following the methods as employed in Wen et al. (2016) and Liu et al. (2018). As can be seen from Eq. (2), Z and M6 are equivalent, Nt is equal to M0, and W can be easily calculated from M3.
Joint normalized PDFs of 2DVD observed and Γ fit for (a) M0, (b) M3, (c) M6, and (d) M9 of all RSD samples from 2016 to 2018.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
Joint normalized PDFs of 2DVD observed and Γ fit for (a) M0, (b) M3, (c) M6, and (d) M9 of all RSD samples from 2016 to 2018.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
Joint normalized PDFs of 2DVD observed and Γ fit for (a) M0, (b) M3, (c) M6, and (d) M9 of all RSD samples from 2016 to 2018.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
We attempt to fit the μ–Λ relationship with a high-precision linear function and simplify Eq. (5) into a one-variable cubic function. If achieved, the equation f(μ) = 0 can be easily solved analytically.
c. Assessing the accuracy of calculated results
As the RSD of two-moment bulk microphysical scheme is continuous, the accuracy of the linear C-G method was verified by using the Γ fit from 2DVD observations to reduce truncation error. Since the existing two-moment bulk microphysical schemes use exponential function to describe RSD, the accuracy of the exponential function method was also verified in the same way.
To assess the accuracy of the calculated results using the above method, we regard the R, M2, M6, and M9 values calculated from Γ-fitted RSDs of 2DVD observations as the true values. The R, M2, M6, and M9 values calculated by RSDs from linear C-G method solutions and exponential method solutions are compared with these true values.
3. A high-precision rapid solution method for Γ RSD suitable for 2M microphysical schemes
a. Linear μ–Λ relationship
During 2016–18, about 2 × 105 RSD samples have been collected by all of the 2DVDs at the Longmen Cloud Physics Field Experiment Base, CMA, as shown in Fig. 1. These mass RSD samples include different rainfall conditions in different seasons, which can well represent the precipitation characteristics in South China.
(a) Histogram of rainwater content of RSD samples from 2016 to 2018. (b) PDF of M0/M3 values. (c) Joint normalized PDFs of M0/M3 and M3 with units of decibels. (d) A scatterplot of μ–Λ with DSD sorting. The gray crosses indicate relationships of all DSD samples from 2016 to 2018; the heavy red dotted line indicates fitted curves of all DSD samples. In addition, the relationships from Zhang et al. (2003), Cao et al. (2008), Chen et al. (2013), Jin et al. (2015), and Liu et al. (2018) are provided.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
(a) Histogram of rainwater content of RSD samples from 2016 to 2018. (b) PDF of M0/M3 values. (c) Joint normalized PDFs of M0/M3 and M3 with units of decibels. (d) A scatterplot of μ–Λ with DSD sorting. The gray crosses indicate relationships of all DSD samples from 2016 to 2018; the heavy red dotted line indicates fitted curves of all DSD samples. In addition, the relationships from Zhang et al. (2003), Cao et al. (2008), Chen et al. (2013), Jin et al. (2015), and Liu et al. (2018) are provided.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
(a) Histogram of rainwater content of RSD samples from 2016 to 2018. (b) PDF of M0/M3 values. (c) Joint normalized PDFs of M0/M3 and M3 with units of decibels. (d) A scatterplot of μ–Λ with DSD sorting. The gray crosses indicate relationships of all DSD samples from 2016 to 2018; the heavy red dotted line indicates fitted curves of all DSD samples. In addition, the relationships from Zhang et al. (2003), Cao et al. (2008), Chen et al. (2013), Jin et al. (2015), and Liu et al. (2018) are provided.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
As the accuracy of the equation solution of μ is closely related to the μ–Λ relationship, it is very important to increase the fitting accuracy of μ–Λ relationship. If there are too few samples used in fitting the μ–Λ relationship, the error would propagate to the fitted ones. To obtain high precision and reliability for the μ–Λ linear correlation relationship, the pair of moments M0/M3 and M3 are discretize into 1 dB × 1 dB bins by following the method proposed by Kumjian et al. (2019). Within each bin that has more than 50 RSD samples, the μ–Λ linear relationships are obtained. The RSD sample numbers in most bins are more than 200. Figs. 4a–c show the statistical results and linear fit results in three typical bins. The bin shown in Fig. 4a has most RSD samples. The RSD samples in the bin shown in Fig. 4b are heavy-rainfall cases. The RSD samples in the bin shown in Fig. 4c are weak rainfall cases. The scatterplot of Dm–δm (δm: standard deviation of the mass-weighted diameter distribution) calculated from RSDs samples agrees well with the corresponding μ–Λ linear relationships, which indicate they are not artifacts. Meanwhile, following the procedure documented in Zhang et al. (2003), the numerical simulation results by adding independent random errors to the moments are given in Figs. 4g–i. The results shown that high linear correlations between
Scatterplots of (a)–(c) μ–Λ, (d)–(f) Dm–δm, (g)–(i)
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
Scatterplots of (a)–(c) μ–Λ, (d)–(f) Dm–δm, (g)–(i)
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
Scatterplots of (a)–(c) μ–Λ, (d)–(f) Dm–δm, (g)–(i)
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
As shown in Fig. 5, in comparison with the fixed linear relation of all RSD samples given by Eq. (11), the CCs of the original (direct linear correlation between μ and Λ in each 1 dB × 1 dB bin) and improved μ–Λ linear relationships are greatly increased. The CCs of original μ–Λ linear relationships are higher than 0.99 in more than 91% of the bins, and the CCs are about 0.999 in 61% of the bins. The CCs of improved μ–Λ linear relationships (corrected by the linear correlation between ΔΛ and M0/M3) are higher than 0.95 in all of the bins. The CC values are higher than 0.99 in 98% of the bins, and more than 85% of them are higher than 0.9995. These indicate that the linear μ–Λ correlation relationships obtained by the improved method have higher accuracy. The improved method effectively overcome the defect of linear correlation by dividing all RSD samples into 1 dB × 1 dB bins and can well represent the internal connection between the Γ RSD parameters μ and Λ in South China. Since the accuracy of the Γ function solution result is closely related to the accuracy of the linear μ–Λ correlation relationships, the step of improvement is very important.
PDF of CC values for the original and improved μ–Λ linear relationships in all (M0/M3, M3) 1 dB × 1 dB bins. In comparing with the original μ–Λ linear relationships, it is seen that the improved μ–Λ linear relationships are further corrected by the ΔΛ–M0/M3 relations. The red dotted line shows the CC value for μ–Λ linear relationship of all DSD samples from 2016 to 2018.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
PDF of CC values for the original and improved μ–Λ linear relationships in all (M0/M3, M3) 1 dB × 1 dB bins. In comparing with the original μ–Λ linear relationships, it is seen that the improved μ–Λ linear relationships are further corrected by the ΔΛ–M0/M3 relations. The red dotted line shows the CC value for μ–Λ linear relationship of all DSD samples from 2016 to 2018.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
PDF of CC values for the original and improved μ–Λ linear relationships in all (M0/M3, M3) 1 dB × 1 dB bins. In comparing with the original μ–Λ linear relationships, it is seen that the improved μ–Λ linear relationships are further corrected by the ΔΛ–M0/M3 relations. The red dotted line shows the CC value for μ–Λ linear relationship of all DSD samples from 2016 to 2018.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
b. Three-parameter solution method based on the linear constrained gamma function
Three typical bins are chosen to show the variations of Eq. (6) for different values of M0/M3, and the results are given in Fig. 6. When f(μ) = 0, there may be one, two, or three real number solutions. If there is only a unique real solution, then μ directly adopts the solution. If there are two or three real solutions, one of the real solutions is about −2 or even smaller in most cases. Since μ is about −2 or even smaller, the corresponding Λ value obtained from the linear μ–Λ relationship is less than 0. For an actual RSD, the value of Λ must be greater than 0; that is, the larger the raindrop size is, the lower is the number density. Otherwise, the rain intensity will be infinite. Therefore, this solution μ may be rejected since its corresponding Λ does not meet the actual atmospheric observation results. If there are only two real solutions, the remaining μ is the final solution. If there are three real solutions, one of the real solutions can be removed by the negative value of Λ. Then, only two real solutions of μ1 and μ2 remain, and the two values of Λ1 and Λ2 are obtained. Since the Z accumulates according to D6, a high density of large sized raindrops can increase the value of Z. If the μ is obviously different with observations, the obtained Z is also significant deviate from normal Z–R relationship. At this point, the Z–R relationship can be used to eliminate the unreasonable solutions and arrive at the only valid μ solution.
Variation of f(μ) as a function of μ for different values of M0/M3 in three typical bins.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
Variation of f(μ) as a function of μ for different values of M0/M3 in three typical bins.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
Variation of f(μ) as a function of μ for different values of M0/M3 in three typical bins.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
To illustrate the above problems, three typical heavy-rainfall processes and all RSD samples observed by 2DVDs during 2019 in South China are selected to perform cases test. Table 1 gives the detailed information of these three rainfall processes and RSD samples in 2019. Meanwhile, Fig. 7 shows the solution results based on the above method in different precipitation intensities. It can be seen that there are only one or two valid solutions of μ. If there is only one valid solution, the Γ RSD obtained by the linear constrained-gamma method is very close to the original fitted Γ RSD. When there are two valid solutions, one of the solutions (μ1) will be very close to the original Γ function–fitted μ0. Another one (μ2) will be an obviously higher or lower value in comparison with the μ0. In this situation, the R calculated by the Γ RSD of μ2 is close to the 2DVD observations, but the Z is significantly higher or lower than observation results. This makes its Z–R relationship significantly deviate from the average Z–R relationship in this bin (not shown). By comparing the degree of deviation from Z–R relationship, one of the solutions (here, μ2) can be excluded.
Raindrop size distribution (black dots) observed by 2DVDs for different precipitation intensities in three typical rainfall processes. The blue dashed line represents the curve fitted to the original gamma function, the red solid line represents the gamma function curve obtained by solving for μ1 of the linear C-G method, the deep-green dashed line represents the gamma function curve obtained by solving for μ2 of the linear C-G method, and the deep pink dashed line represents the fitted curve of the exponential function. This figure also shows the rain rate and M6 observed by the 2DVDs, as well as the rain rate and M6 obtained by the RSD fitting methods.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
Raindrop size distribution (black dots) observed by 2DVDs for different precipitation intensities in three typical rainfall processes. The blue dashed line represents the curve fitted to the original gamma function, the red solid line represents the gamma function curve obtained by solving for μ1 of the linear C-G method, the deep-green dashed line represents the gamma function curve obtained by solving for μ2 of the linear C-G method, and the deep pink dashed line represents the fitted curve of the exponential function. This figure also shows the rain rate and M6 observed by the 2DVDs, as well as the rain rate and M6 obtained by the RSD fitting methods.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
Raindrop size distribution (black dots) observed by 2DVDs for different precipitation intensities in three typical rainfall processes. The blue dashed line represents the curve fitted to the original gamma function, the red solid line represents the gamma function curve obtained by solving for μ1 of the linear C-G method, the deep-green dashed line represents the gamma function curve obtained by solving for μ2 of the linear C-G method, and the deep pink dashed line represents the fitted curve of the exponential function. This figure also shows the rain rate and M6 observed by the 2DVDs, as well as the rain rate and M6 obtained by the RSD fitting methods.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
Three lookup tables are established in each (M0/M3, M3) 1 dB × 1 dB bin based on the RSD samples observed by 2DVDs from 2016 to 2018. For each (M0/M3, M3) 1 dB × 1 dB bin, we collect the μ and Λ values of all RSD samples in this bin. Then we get the linear μ–Λ relationship and linear M0/M3–ΔΛ relation. The first lookup table in each 1 dB × 1 dB bin is the linear μ–Λ relationship (Λ = aμ + b). The second one is the linear M0/M3–ΔΛ relation [ΔΛ = c(M0/M3) + d]. The third one is the Z–R relationship (Z = eRf).
The linear C-G solution method is established from these lookup tables. This method has the following four steps:
Calculate the M3 and M0 values of the rainwater. Then, convert M0/M3 and M3 to dB values.
Based on the three lookup tables, the improved linear μ–Λ relationship and Z–R relationship are obtained through the dB values of M0/M3 and M3. If the M0/M3 or M3 decibel values are out of the lookup table, the fixed μ–Λ relationship of Eq. (11) is suggested for use in this step.
Equation (6) is solved by using Shengjin’s formula. Through the positive or negative values of μ, as well as the degree of deviation from the Z–R relationship, the unreasonable solutions of μ are eliminated, and only the reasonable solution of μ is left.
Calculate Λ according to the μ and the μ–Λ relationship. Calculate N0 according to μ, Λ, and M3. Then, the high-precision Γ RSD is obtained.
4. Verification using real data
To check the fitting accuracy of the Γ function parameter solving by the linear C-G method developed in this study, we selected the three typical heavy-rainfall processes, as well as all RSD samples observed by 2DVD during 2019 in South China as shown in Table 1. The weather conditions of these three processes include cold air, monsoon, and typhoon, which are typical conditions of heavy rainfall in South China during the flood season. The RSD samples of these three processes are all more than 1000, and the heaviest rain rates are all larger than 100 mm h−1.
The values of M0 and M3 were calculated using the 1-min interval Γ-fit RSD measured by the 2DVDs. Using the linear C-G method, the three parameters (N0, μ, and Λ) of the Γ function were solved from the M0/M3 and M3 values. In addition, the two parameters (N0 and Λ) of the exponential function were solved from the M0 and M3 values. The R, M2, M6, and M9 values calculated from the Γ RSD parameters (N0, μ, Λ) derived from 2DVD measurements were setting as the reference standard. We conducted a quantitative assessment of the fitting accuracy of the RSD function described above using Eqs. (7)–(10). Comparisons of R values calculated from the linear C-G Γ function and exponential function with the Γ-fit RSD from 2DVD observations are shown in Fig. 8. The comparison results for the M2, M6, and M9 values are also given in Figs. 9–11, respectively.
(a)–(d) Comparison of the rain rate calculated from the linear C-G method solution with Γ-fitted RSD from 2DVD observations in 2019 and three typical heavy-rainfall processes. (e)–(h) Comparison of rain rate of the exponential function RSD with Γ-fitted RSD from 2DVD observations in 2019 and three typical rainfall processes.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
(a)–(d) Comparison of the rain rate calculated from the linear C-G method solution with Γ-fitted RSD from 2DVD observations in 2019 and three typical heavy-rainfall processes. (e)–(h) Comparison of rain rate of the exponential function RSD with Γ-fitted RSD from 2DVD observations in 2019 and three typical rainfall processes.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
(a)–(d) Comparison of the rain rate calculated from the linear C-G method solution with Γ-fitted RSD from 2DVD observations in 2019 and three typical heavy-rainfall processes. (e)–(h) Comparison of rain rate of the exponential function RSD with Γ-fitted RSD from 2DVD observations in 2019 and three typical rainfall processes.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
As in Fig. 8, but for the values of M2.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
As in Fig. 8, but for the values of M2.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
As in Fig. 8, but for the values of M2.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
As in Fig. 8, but for the values of M6.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
As in Fig. 8, but for the values of M6.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
As in Fig. 8, but for the values of M6.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
As in Fig. 8, but for the values of M9.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
As in Fig. 8, but for the values of M9.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
As in Fig. 8, but for the values of M9.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
As shown in Fig. 8, the R values calculated from linear C-G Γ method are generally consistent with the Γ-fit RSD from 2DVD observations. The CCs of the linear C-G method are all up to 0.997 in these three processes and all samples in 2019. Meanwhile, all of the RMSE values are less than 1.3 mm h−1, the NAE values are less than 0.071, and the NRE values approach 0. The R values calculated from the exponential method are higher than the Γ-fit RSD from 2DVD observations when rain rates are less than 20 mm h−1 but lower when rain rates are heavier than 50 mm h−1. The errors of the exponential method are obviously larger than that of linear C-G method. The CCs are all less than 0.99. Meanwhile, the RMSE and NAE values of exponential method are more than 2 times that of linear C-G method.
The M2 values calculated from the linear C-G Γ method are also consistent with the Γ-fit RSD from 2DVD observations (sea Fig. 9). The CC values in these three heavy-rainfall processes and all samples in 2019 are all larger than 0.993. The RMSE values are all smaller than 0.59 dB. Meanwhile, the NAE values are less than 0.019, and the NRE values are nearly 0. The M2 values calculated from the exponential method are relatively smaller than the Γ-fit RSD from 2DVD observations when M2 are less than 30 dB but are higher when M6 is larger than 35 dB. The errors of the exponential method are obviously larger than that of the linear C-G method. The RMSE and NAE values of the exponential method are more than 3 times that of the linear C-G method.
The M6 values calculated from the linear C-G Γ method are roughly the same as the Γ-fit RSD from 2DVD observations (see Fig. 10). The CCs of linear C-G method are all up to 0.94 in these three cases and all samples in 2019. The RMSE values are smaller than 3.3 dB, the NAE values are less than 0.1, and the NRE values are close to 0. The M6 values calculated from the exponential method are obviously higher than the Γ-fit RSD from 2DVD observations when M6 are less than 40 dB but are lower when M6 is higher than 45 dB. The errors of the exponential method are much larger than that of the linear C-G method. The RMSE and NAE values of the exponential method are about 3 times that of the linear C-G method.
The M9 values calculated from the linear C-G Γ method are basically consistent with the Γ-fit RSD from 2DVD observations as shown in Fig. 11. The RMSE values are smaller than 8.0 dB, the NAE values are less than 0.20, and the NRE values are also close to 0. The M9 values calculated from the exponential method are obviously higher than the Γ-fit RSD from 2DVD observations when M9 are less than 50 dB but are lower when M9 is higher than 60 dB. The RMSE and NAE values of the exponential method are about 2.5 times that of the linear C-G method.
To analyze in depth the performance of the linear C-G Γ method and exponential method under different rainfall intensities, rainfall samples are divided into six categories based on the rain-rate PDF in South China. They are weak rain (0.1–1.0 mm h−1), light rain (1.0–5.0 mm h−1), moderate rain (5.0–10.0 mm h−1), heavy rain (10.0–20.0 mm h−1), rainstorm (20.0–50.0 mm h−1), and downpour (heavier than 50.0 mm h−1). This classification method is based on the local characteristics of precipitation and climate and is different from the daily precipitation classification method.
The RMSE values of both the linear C-G Γ method and exponential method in every R range for three heavy-rainfall processes and all samples in 2019 are shown in Fig. 12. Meanwhile, the NRE values are shown in Fig. 13. The RMSE values of R, M2, M6 and M9 of the linear C-G Γ method are all obviously smaller than the exponential method in every R ranges. The NRE values of R, M2, M6, and M9 of the linear C-G Γ method are all very close to 0. However, the NRE values of R, M6 and M9 for the exponential method are significantly higher than 0 when R ranges from weak rain to heavy rain and are obviously smaller than 0 when R is in the downpour range. The NRE values of M2 for the exponential method are evidently smaller than 0 when R ranges from weak rain to heavy rain and are larger than 0 when R is in the downpour range. The results show that, no matter what the precipitation-class situation is, the errors of the linear C-G Γ method are obviously smaller than those of the exponential method. Thus, the linear C-G Γ method can significantly improve the accuracies of simulated RSD for 2M bulk microphysical schemes.
RMSE histograms of (a),(b) rain rate, (c),(d) M2, (e),(f) M6, and (g),(h) M9 for (left) the linear C-G method and (right) the exponential fit method in different rain-rate ranges.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
RMSE histograms of (a),(b) rain rate, (c),(d) M2, (e),(f) M6, and (g),(h) M9 for (left) the linear C-G method and (right) the exponential fit method in different rain-rate ranges.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
RMSE histograms of (a),(b) rain rate, (c),(d) M2, (e),(f) M6, and (g),(h) M9 for (left) the linear C-G method and (right) the exponential fit method in different rain-rate ranges.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
As in Fig. 12, but for the values of NRE.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
As in Fig. 12, but for the values of NRE.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
As in Fig. 12, but for the values of NRE.
Citation: Journal of Applied Meteorology and Climatology 60, 10; 10.1175/JAMC-D-21-0043.1
5. Conclusions and discussion
Based on the linear μ–Λ correlation relationship acquired by several 2DVDs in South China, we constructed a high-precision fast Γ function solution using a linear C-G method. In this solution, the three parameters (N0, μ, and Λ) of Γ RSD function are calculated from M0 and M3, which can be easily obtained from the mass content Qr and total number density Ntr of the rainwater that is simulated by the 2M microphysical scheme. The obtained RSDs are obviously closer to the observations in comparison with the exponential function solution and avoid the problem of setting the shape parameter to a constant (usually 0) in the existing 2M microphysical scheme.
Based on about 2 × 105 RSD samples observed by several 2DVDs from 2016 to 2018 at the Longmen Cloud Physics Field Experiment Base, CMA, the statistical linear μ–Λ correlation relationships in each (M0/M3, M3) 1 dB × 1 dB bin were obtained. The CC values in 98% of the bins are higher than 0.99, and the values are higher than 0.9995 in more than 85% of the bins. These μ–Λ linear relationships reflect characteristics of actual RSDs and have good regional representations in South China. Based on these linear μ–Λ correlation relationships, we obtained a one-dimensional cubic equation for solving μ that is dependent only on the value of M0/M3. The value of M0/M3 can be easily calculated from Qr and Ntr. Furthermore, analytical solutions are obtained using Shengjin’s formula. When multiple solutions existed, the invalid solutions are excluded using the positive–negative relationship of Λ as well as the deviation from the Z–R relationship. Only one reasonable analytical solution of μ is obtained. Then, the values of N0 and Λ can be calculated.
Three typical heavy-rainfall processes and all RSD samples observed by 2DVDs during 2019 were selected to verify the accuracy of the linear C-G method. Analysis results show that, relative to the exponential method usually employed in the 2M microphysical scheme, the R, M2, M6, and M9 values obtained by the linear C-G method are significantly in better agreement with the Γ-fit RSD from 2DVD observations. The CC values of the linear C-G method are higher than those of the exponential method. Meanwhile, the RMSE, NAE, and NRE values are obviously lower.
In summary, a high-precision and fast linear C-G solution method based on M0 and M3 has been established in this study. The proposed method has effectively solved the problem that the shape parameter in the 2M microphysical scheme is set to a constant, and the obtained Γ RSD are much closer to the observations. The simulated radar reflectivities agree well with observations. This method has great potential to be applied to the 2M microphysical scheme to improve the simulation of heavy precipitation in South China, and we will further analyze and apply this method in future studies.
Nevertheless, this work only conducted systematic studies of RSD characteristics in South China. Because the μ–Λ relationship changes depending on climatology, geographical location, and rain type (Liu et al. 2018), the lookup table values cannot be simply applied to other areas. Future in-depth research is needed, particularly for different climate regions, to ensure that this method is broadly applicable and impactful.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (41975138, 41905047, 41705020, and 41705120), Guangdong Province Science and Technology Project (2015B020217001 and 2017B020244002), and Natural Science Foundation of Guangdong Province (2019A1515010814).
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