Raindrop size distribution (DSD) is the fundamental property of precipitation, and its space–time variability depends on a variety of microphysical and dynamical processes inside and below clouds (Radhakrishna and Rao, 2009; Rao et al., 2009; Rosenfeld and Ulbrich, 2003). Such information is crucial even for numerical weather prediction models as these microphysical processes are fundamental blocks in microphysical schemes (Gao et al., 2011). Knowledge of DSD is not only required for fundamental understanding of microphysical processes, but also for a variety of operational applications in the fields of the hydrology, meteorology, agriculture, and road transportation sectors (Rosenfeld and Ulbrich, 2003; Serio et al., 2019; Uijlenhoet, 2001, and references therein). Disdrometers provide this crucial information continuously but only at the Earth's surface. Radars, on the other hand, provide DSD in both space and time and therefore play a major role in improving our understanding of microphysical processes in a variety of precipitating systems (Ryzhkov and Zrnic, 2019).

Remarkable progress has been made in polarimetric (dual-polarization) radar technology and its utilization for research and operational applications in the recent past (Bringi and Chandrasekar, 2001; Rauber and Nesbitt, 2018; Ryzhkov et al., 2022; Ryzhkov and Zrnic, 2019). Besides improving the rain rate estimation, the polarimetric radars offer unique information on microphysical properties of precipitation, like the DSD (Anagnostou et al., 2008a; Cao and Zhang, 2009; Gorgucci et al., 2001; Koffi et al., 2014; Maki et al., 2005; Moisseev and Chandrasekar, 2007; Penide et al., 2013; Seliga and Bringi, 1976; Zhang et al., 2001). They also provide information on the shape, orientation and phase state of hydrometeors by employing sophisticated hydrometeor classification algorithms like fuzzy logic and Bayesian classification (Liu and Chandrasekar, 2000; Marzano et al., 2007; Vivekanandan et al., 1999; Zrnic et al., 2001). Several earlier studies demonstrated that the DSD parameters can be used not only to understand the microphysics of precipitation and clouds, but also to improve rain rate estimation (Zhang et al., 2001; Gorgucci et al., 2001; Vivekanandan et al., 2003; Vulpiani et al., 2006; Brandes et al., 2004a; Cao et al., 2010, 2008; Gosset et al., 2010; Anagnostou et al., 2013; Koffi et al., 2014; Ryzhkov and Zrnic, 2019). They have shown that DSD-based rain rate estimation outperforms the fixed power law rainfall estimation from reflectivity fields and is equivalent to those derived with multi-parameter retrievals of rainfall with polarimetric radars (Anagnostou et al., 2010; Brandes et al., 2003; Vivekanandan et al., 2003).

Earlier studies followed various approaches to retrieve the DSD from polarimetric radars: statistical techniques and physics-based empirical relations between DSD model parameters and polarimetric products. Statistical methods, including neural networks (Vulpiani et al., 2006), Bayesian (Cao et al., 2010) and different variants of Bayesian, like variational methods (Cao et al., 2013; Yoshikawa et al., 2016), find the nonlinear relationships between DSD and polarimetric parameters by making use of mathematical techniques. These methods either train the chosen model or build an a priori database using existing information, which will then be used to retrieve DSD parameters. Physics-based methods assume that the DSD follows some functional form (exponential, gamma or normalized gamma) and derive a relation between DSD model parameters and polarimetric radar parameters empirically. Different methods have evolved over the years since: the Seliga and Bringi (1976) exponential method (Exp.), including constrained gamma (CG) (Zhang et al., 2001), Beta (β) (Gorgucci et al., 2000), normalized gamma (N-Gamma) (Bringi et al., 2002; Anagnostou et al., 2008a; Tokay et al., 2020a), the generalized gamma model (Thurai and Bringi, 2018), the double-moment model (Raupach and Berne, 2017), self-consistent with optical parameterization attenuation correction and microphysics estimation (SCOPE-ME) (Anagnostou et al., 2009), and the inverse model (Alcoba et al., 2022; Wen et al., 2018).

Among the above methods, the Exp., CG, N-Gamma and β methods are extensively used by researchers. The two-parameter exponential model assumes that the distribution of raindrops follows an exponential form, and its parameters can be retrieved from two polarimetric measurements, namely, the horizontal reflectivity factor (ZH) and differential reflectivity (ZDR) (Seliga and Bringi, 1976). The CG method assumes that the DSD follows a gamma distribution (Ulbrich, 1983), and the retrieval of the three gamma parameters is achieved using two independent polarimetric measurements and an empirically derived constrained relation between shape (μ) and slope (Λ) parameters of the gamma distribution (Brandes et al., 2004a; Zhang et al., 2001). The β method follows the normalized DSD concept, described in Willis (1984), Illingworth and Blackman (2002) and Testud et al. (2001). Here, the DSD is normalized with respect to liquid water content, which allows the study of variations in DSD shape by accounting for variations of water content. In addition, this method considers the raindrop shape–diameter relation to be a variable (Gorgucci et al., 2001) instead of a fixed relation for the equilibrium shape of a raindrop (Pruppacher and Beard, 1970). The ZH, ZDR and specific differential phase (KDP) are used to obtain the slope (β) of the above relation, which intrinsically considers changes in drop oblateness that increase with the size of a raindrop.

Earlier studies derived/generated several empirical relations relating to polarimetric variables at different frequencies to obtain the DSD parameters. Some of these relations are obtained from simulations or parameterizations, and the others from observations (Adirosi et al., 2020; Anagnostou et al., 2008a, b; Brandes et al., 2004a, b; Gorgucci et al., 2001; Maki et al., 2005; Rao et al., 2006; Seliga and Bringi, 1976; Tang et al., 2014; Tokay et al., 2020b; Zhang et al., 2001, and references therein). Unfortunately, the above relations are found to be quite different at different locations due to large DSD variations (Brandes et al., 2004b; Chen et al., 2017; Chu and Su, 2008; Kim et al., 2020; Kumar et al., 2011; Rao et al., 2006; Seela et al., 2018; Tang et al., 2014; Zhang et al., 2001; Zheng et al., 2020). Not only between regions, the DSD and μ-Λ relations are also found to vary between different regimes (i.e., eye wall and rain bands) of a cyclone (Bao et al., 2020). These variations are caused primarily by different prevailing atmospheric conditions (in different geographical regions), in which the drop forms and the DSD evolves (Lee and Zawadzki, 2005). The above-reported relations are based on the data from America, Japan, Taiwan, Singapore, Italy and China and therefore are more appropriate for the above regions, while such relations do not exist for India (barring one study by Rao et al. (2006) using a limited dataset). The first objective of this paper is to derive suitable DSD retrieval relations at the X-band for monsoonal rainfall over the Indian region, where several X-band polarimetric radars are either installed or being installed. An X-band dual-polarization radar (DROP-X – Dual polarization Radar for Observing Precipitation at X-band), developed indigenously, recently became operational at Gadanki (13.5∘ N, 79.18∘ E) (Rao et al., 2023).

It is also known from earlier studies that the DSD varies not only with the climatic regime, but also with the season at the same location. For example, the DSD at a single station can be influenced by both the oceanic and continental systems, depending on the wind and circulation patterns (Kozu et al., 2006; Radhakrishna and Rao, 2009; Rao et al., 2009, 2001; Tokay et al., 2002; Lavanya et al., 2019). Recently, Rao et al. (2018) noted large differences in the coefficients of attenuation correction relations in different seasons. Given such large variability in DSD from one season to the other in southeastern peninsular India, one should also examine the impact of the observed seasonal variation on the DSD retrieval methods. This forms the second objective of this paper.

There have been differences of opinion about the validity of the retrieval of the above relations (μ–Λ relation and β method), the usage of the DSD models (exponential vs. gamma vs. normalized gamma) and the drop shape–size relations (linear and constant vs. linear but variable vs. polynomial). Earlier, a few studies compared different DSD retrieval techniques (Anagnostou et al., 2008b, a; Brandes et al., 2006, 2004a; Tokay et al., 2020b; Zhang et al., 2006). Such efforts were not made for monsoonal rain. Given the large seasonal variability in DSD, it is important to evaluate such schemes using observations from polarimetric radars. The present study, therefore, evaluates the retrieved mass-weighted mean diameter (Dm) and intercept parameter (N0) or normalized intercept parameter (NW) of DSD from DROP-X measurements and the derived relations.

The remainder of this paper is organized as follows. Section 2 describes the instruments, data and methodology (scattering simulations, deriving polarimetric products and DSD models) used in the present study. Relations between polarimetric products and exponential/gamma model parameters are empirically derived in Sect. 3. Seasonal dependence of the coefficients of the above relations and their variation with temperature are also discussed in Sect. 3. The retrieved DSD parameters from radar measurements are evaluated against the independent reference dataset in Sect. 4. Section 5 summarizes important findings from the present study.

The degree of agreement of radar-derived DSD parameters with disdrometer-derived parameters depends on several factors: (1) the differences in sampling volumes of the radar and disdrometer, (2) the vertical variability of DSD from the radar-measured volume to the surface (or disdrometer measurement height) and (3) the accuracy of the empirical relations between polarimetric parameters (ZDR, ZH and KDP) and DSD model parameters (Dm, N0, μ and Λ). The radar sampling volume depends on the range, beam width and pulse length. For the given radar beam width of 1∘, range resolution of 150 m and range of 450 m, the estimated sampling volume of the radar is 7264 m3. To match the radar temporal resolution, the disdrometer data are averaged over 6 min (360 s). The sampling volume of the disdrometer for a given surface area of 50 cm2 (for the JW disdrometer) and a characteristic drop size, represented by a Dm (or terminal velocity) of 2 mm (6.5 m s-1), is less than 12 m3. Thus, the sampling volumes differ by a factor greater than 600, which is much less than the similar comparisons made elsewhere, wherein the sampling volumes differ by a factor of 105 to 107 (Cao et al., 2008; Tokay et al., 2020a). This is mainly due to the fact that the comparisons were made at a longer range in earlier studies. Another advantage of using a shorter range for comparison studies, as is done in the present study, is the proximity of the radar-measuring volume to the surface. In the present study, the sampling volume is at a height of ∼ 20 m above the disdrometer location. This reduces the bias caused by the time–height ambiguity due to the vertical variability of DSD. The retrieval accuracy also depends on empirical relations between the radar and DSD parameters, as these relations vary with season (as shown in Sect. 3). However, appropriate relations have been used for comparison in the present study to reduce such ambiguity.

Evaluation of DROP-X derived DSD parameters, using retrieved coefficients in the different DSD formulations discussed above, is carried out by comparing them with those derived with disdrometer observations. For this purpose, the disdrometric dataset during 2019–2020, which has not been used for the retrieval of coefficients, is used for comparison. Long-duration events (longer than 2 h) are selected for the evaluation of DSD retrieval techniques. A total of six events each from SWM and NEM are selected for this purpose (Table 5). These events include a variety of precipitating systems, including thunderstorms and mesoscale convective systems.

Five years of disdrometric measurements and 2 years of DROP-X measurements have been used, for the first time, to (i) obtain relations for the retrieval of DSD parameters appropriate for monsoonal rain and to study their dependency on temperature and drop size–shape relations, (ii) understand the seasonal variation of coefficients and (iii) assess the DROP-X-derived DSD by various DSD retrieval methods. Using 3 years of disdrometer-measured DSD, various polarimetric parameters have been computed using T-matrix simulations. Coefficients of four commonly used DSD relations are retrieved empirically from simulated data. Important results coming from the study are summarized as follows.

The coefficients for obtaining DSD parameters by the exponential, CG, N-Gamma and β methods for monsoonal rain are found to be different from other regions, indicating that they are region-dependent. The mean value of β estimated at Gadanki is closer to the default value (0.062) given by Pruppacher and Beard (1970) during the NEM, whereas the values obtained for PRE and SWM are much smaller, indicating that the slope of the drop-shape-sized relation is season-dependent and 0.062 is more applicable for the colder season. To understand the dependency of the coefficients of these relations on temperature and drop shape models, the coefficients of the Exp. method are retrieved for different temperatures and drop shape models. It is found that the variation in Dm or N0, for a given ZDR and ZH, due to temperature variation is within 5 % in any season and is much less in NEM (< 2 %). However, the dependency of coefficients in the Dm–ZDR equation on the drop shape model is high (7 %–15 % in the prefactor and 28 %–28 % in the exponent) and in fact is higher than on seasons. The dependency of coefficients on drop shape models is found to be different in different geographical regions. While the dependency is found to be high at Gadanki and in Africa, it is found to be weak along the western coast of the United States of America.

The present study corroborates some of the earlier studies that showed that the μ–Λ relation is region-dependent. It clearly shows that this relation is also season- and temperature-dependent, as we see a gradual change in coefficients from the warmest PRE to the coldest NEM. Also, the warmest seasons of PRE and SWM have higher slope and curvature values compared to those in NEM. This means that μ will be higher during PRE and SWM than in NEM for the same Λ for the majority of data (i.e., when Λ and μ values are less than 8). A comparison of μ–Λ relations obtained in different seasons at Gadanki with those available in the literature elsewhere clearly reveals that warm seasons/regions typically have larger curvature and slope values than cold seasons/regions.

The disdrometer data clearly show large seasonal variation with a preponderance of smaller drops during NEM compared to the warm seasons, corroborating earlier findings (Rao et al., 2001, 2009; Radhakrishna et al., 2009). As a result, the obtained coefficients also show large seasonal variation. From the retrieved coefficients it is clear that the Dm values will be larger for the same ZDR during PRE and SEM than in NEM. Though the prefactor is nearly equal in all seasons, the variation in the exponent makes a difference of ∼ 20 %–30 % in the N0 value between the seasons for the same ZH/N0 and Dm. Among seasons, the variation in coefficients of DSD relations with temperature is larger in hot seasons than in the cold season (i.e., NEM) by a factor of 2 to 6. However, the impact of seasonal variations of coefficients on derived DSD parameters is relatively larger and is up to 20 %. Therefore, appropriate coefficients need to be used while retrieving DSD from polarimetric measurements.

The four commonly used radar retrieval methods of DSD are evaluated with the help of two case studies (one each from SWM and NEM) and data from 12 events. All the methods retrieve Dm reasonably well and produce a high correlation and small RMSE against the reference. The β method alone produced a wide range of Dm values similar to that of the disdrometer. However, the scatter is large, particularly in convection, mainly due to the fact that the comparison is made close to the radar site, where the differential phase is often contaminated by a differential backscattering phase. As a result, the RMSE exhibited by the β method is also found to be large. Comparison of retrievals of N0/NW with those of the disdrometer shows the superiority of the N-Gamma method over other methods. All other methods compare poorly with disdrometer-derived N0/NW, with small r2 and large RMSE values. Considering all the factors (Table 4), the N-Gamma method is found to be better in retrieving the DSD parameters. However, such assessment studies are also planned at longer ranges (10 and 35 km) with DROP-X to understand the strengths and limitations of the above methods in retrieving DSD accurately.