The observation site (43.12°N, 86.82°E, 3 835 m a.s.l) is located in the source area of the Urumqi River in the southeastern part of the Tianshan Mountains in the central region of Asia. There are seven glaciers in the head watershed region, among which the Urumqi Glacier No. 1 (43°06′N, 86°49′E) is the largest, and it is the “reference glacier” of the World Glacier Monitoring Service (WGMS) in the arid regions of Central Asia and one of the ten major modern glaciers monitored globally (Li et al., 2010). Annual precipitation is about 468 mm on average, the summer season contributed approximately about 77% (Li et al., 2007) (Figure 1).

The DSD data used in this paper were obtained from PWS100 produced by Campbell Scientific (particle diameter, falling velocity and drops count of raindrops). The instrument was installed in the meteorological observation field on the moraine ridge at the terminal of the Urumqi Glacier No. 1, and equipped with Geonor T-200B series precipitation gauges, Young 05103 wind monitor, HC2-S3 temperature and humidity sensor, and other instruments (Figure 1B). Table 1 were meteorological parameters, comparison of precipitation and precipitation days between T-200B and PWS100, the correlation between the two instruments was 0.8. The PWS100 disdrometer observation is based on the laser Doppler measurement principle, which can automatically measure the amount of precipitation and visible meteorological factors. The instrument configuration defines an area of 40 cm² by overlap measurement regions of two detectors. Thus, the PWS100 is capable of classifying precipitation as rain, snow, hail, etc., and it delivers output messages about the type of particles detected. Using advanced detection technology and a fuzzy logic, the PWS100 disdrometer can measure the size and speed of raindrops with an accuracy of 5% (Ellis et al., 2006). The rainfall total resolution was 0.0001 mm, and the accuracy was better than 10% (Campbell Scientific, Inc., 2015). When connected to the air temperature and humidity sensor, making the measurement is more accurate (Ellis et al., 2006). The nominal detection size and velocity of individual particles range from 0.1 to 30 mm and 0.1–30 m s⁻¹, respectively. The outputs are arranged in 34 by 34 size and velocity bins.

The data selected in this paper were the observation results of the disdrometer in the summer of 2017–2019, with a time resolution of 1 min. When a particle is partially within the beam, it can be misclassified as a small particle that falls faster than other particles observed at that size (Yuter et al., 2006). Moreover, during heavy rains, strong winds and splashes on the surface of the instrument may produce unrealistically slow falling large particles (Friedrich et al., 2013). Therefore, in its built-in algorithm, PWS100 assumes that similar distortions will occur when raindrops fall, and its internal algorithm excludes precipitation particles with abnormal velocity and simultaneously outputs incorrect particle numbers and unknown particle numbers (Campbell Scientific, Inc., 2015). This may result in a low number of raindrops less than 0.4 mm (Montero-Martinez et al., 2009; Montero-Martinez and Eduardo, 2016). To minimize the effects of edge fall, wind, and splash, first, we removed data with minute raindrop counts below 10 and rainfall rates less than 0.1 mm h⁻¹. Secondly, precipitation particles with diameters below 0.45 mm and above 6 mm were excluded. Finally, the disdrometer data with precipitation duration of less than thirty minutes were excluded. After these steps, there must be no precipitation period of at least one hour between each rainfall event to reduce statistical error according to the previous studies (Atlas et al., 1973; Chen et al., 2013; Chen et al., 2016; Tang et al., 2014). The total number of samples available for this study was 40,214.

The number concentration of raindrops can be obtained by the following equation: N ( D i ) = ∑ j = 1 34 n i j υ j ⋅ S ⋅ t ⋅ Δ D i (1) where N(D _i ) (mm⁻¹m⁻³) is the number concentration of raindrops per unit volume per unit size interval for raindrop diameter D _i (mm); n _ij is the number of raindrops within the size bin i and velocity bin j; S(m²) and t(s) are the sampling area (set to 40 cm² in this study) and sampling time (set to 60 s in this study), respectively; and v _j (m s⁻¹) is the falling speed for velocity bin j. and ∆D _i (mm) is the interval between the diameter classes. In this study, the falling speed measurements from the PWS100 disdrometer are not used because of measurement error. Instead, the model-based velocity relation proposed by Atlas et al. (1973) as shown in below. υ ( D i ) = 9.65 − 10.3 ⁡ exp ( − 0.6 D i ) (2)

The integral rainfall parameters including the liquid water content W (g m⁻³), rain rate R (mm h⁻¹), radar reflectivity factor Z (mm⁶ m⁻³) and total concentration of raindrops N _t (m⁻³) can be derived from N(D _i ) as follows. W = π 6 × 10 − 3 ∑ i = 1 34 D i 3 N ( D i ) Δ D i (3) R=6 π × 1 0 -4 ∑ i = 1 34 D i 3 V i N ( D i ) Δ D i (4) Z = ∑ i = 1 34 D i 6 N ( D i ) Δ D i (5) N t ∑ i = 1 34 N ( D i ) Δ D i (6)

The three-parameter gamma distribution is the most commonly accepted model for describing the measured raindrop spectra, and it is generally enough to describe the DSD fluctuations observed on small scale and has the capability of describing a broader variation in DSD (Ulbrich and Atlas., 1998), which is expressed as N ( D ) = N 0 D μ ⁡ exp ( − Λ D ) (7) where N ₀ is the intercept parameter in a unit of m⁻³ mm^−(1+µ), its related to the total number of rainfall particles; μ is the shape parameter (dimensionless), when μ is above zero, the curve is curved upward, and it is the opposite when the value of μ is below 0; Λ is the slope parameter in a unit of mm⁻¹, which represents the distribution characteristics of particles, and Λ decreases with the increase of rainfall rate, indicating that the number of large raindrops increases with the increase of rainfall rate; D (mm) is the diameter of the raindrop. Due to its ability to proportionally fit the moment of the above integral rainfall parameters, the method of moments (MoM) is considered in this study. In general, the nth moment of the DSD, M _n , is defined as: M n = ∫ 0 ∞ D n N ( D ) d D = ∑ i = 1 34 N ( D i ) D i n Δ D i (8)

Three moments are required to compute the Gamma distribution parameters. In this paper, the third, fourth, and sixth moments are used to calculate the Gamma parameter (Seela et al., 2018; Montero-Martinez et al., 2021; Wang et al., 2021). The 2/4/6 moment is also used to fit Gamma parameters in many studies (Chen et al., 2017; Wen et al., 2017; Zhang et al., 2019). However, Cao and Zhang (2009) showed that there was no significant difference between the 2/4/6 moment estimator and the 3/4/6 moment estimator. The formulas for the three parameters of the Gamma distribution are as follows (Kozu and Nakamura., 1991): μ = 11 G − 8 + G ( G + 8 ) 2 ( 1 − G ) (9) where, G = M 4 3 M 3 2 M 6 (10) Λ = ( μ + 4 ) M 3 M 4 (11) N 0 = Λ μ + 4 M 3 Γ ( μ + 4 ) (12) where, Г(x) is expressed as: Γ ( x ) = 2 π e − x x x − 1 2 (13)

The normalized intercept parameter N _w in the unit of mm⁻¹ m⁻³ and the mass-weighted mean diameter D _m (mm) were defined by Bringi et al. (2003) as: N w = 4 4 π ρ w ( 10 3 W D m 4 ) (14) where, ρ _ω (1.0 g cm⁻¹) is the density of the water. D m = M 4 M 3 (15)

Previous studies showed that the properties of DSD depend on rainfall rates and manifest as temporal and spatial variability (Tokay and Short., 1996; Bringi et al., 2003; Nzeukou et al., 2004). To examine the characteristics of the DSD for different rainfall regimes over the head watershed of the Urumqi River, the rain-rate was divided into five classes: class I (R < 0.1 mm h⁻¹), class II (0.1 < R < 1 mm h⁻¹), class III (1 < R < 5 mm h⁻¹), class Ⅳ (5 < R < 10 mm h⁻¹) and class V (R > 10 mm h⁻¹). The accumulated rainfall duration (977 h) and rain amount (1242 mm) for the five rainfall rate classes are shown in Figure 2. The longest precipitation duration is class II (0.1 < R < 1 mm h⁻¹) and class I (<0.1 mm h⁻¹) of precipitation, accounting for 69.1% of the total rain amount duration in the study period, but the rain amount only accounts for 13.8% of the overall. Class III (1 < R < 5 mm h⁻¹) had the largest accumulated rainfall amount, accounting for more than 45.8% of the total rainfall, but the precipitation duration only accounts for 25.1% of the total. In addition, although class V (R > 10 mm h⁻¹) had the shortest precipitation duration (14 h), the precipitation accounts for 21.4% of the total precipitation. The average rainfall rates of the five classes were 0.04, 0.41, 2.26, 6.67, and 18.50 mm h⁻¹, respectively.

To discern the characteristics of DSD among different rainfall rates, we calculated the averaged size spectra of each rain-rate class and put them on the same figure (Figure 3). The corresponding integral rainfall parameters computed from the average spectra are shown in Table 2. We can know that the spectral widths of the five rainfall rates don’t widen with the increase of rainfall rates compared to Motuo and Naqu (Chen et al., 2017; Wang et al., 2021), but first, widen and then narrow, and at class III reaches the maximum value. With the increase in rainfall rates, the concentration of raindrops showed an increasing-decreasing-increasing distribution shape when the particle diameter is below 1.22 mm. However, the raindrop concentration is different from the trend proposed by Wang et al. (2021) and Chen et al. (2017). The raindrop concentration increased with the rainfall rate and the largest was at class V, of which the mass-weighted mean diameter is above 1.74 mm. In this paper, with the enhancement of rainfall rate, although the raindrop concentration also showed an increasing trend, the maximum raindrop concentration was at class Ⅳ. The liquid water content W, radar reflectivity factor Z, and the mass-weighted mean diameter D _m reach the maximum value at class Ⅳ, while the total concentration of raindrops N _t and the normalized intercept parameter N _w increased with the increase in rainfall rate.

The relation between terminal fall velocity and drop size as that in Atlas et al. (1973) was used in our research, whereas the actual drop fall velocity was relative to the ambient air flows in real atmosphere. Updrafts could decrease the fall velocity of raindrops, resulting in an underestimation of the drop size. Furthermore, the liquid water content and rainfall rate were proportional to the third DSD moments in Eq. 3 and the third velocity-weighted DSD moments in Eq. 4, respectively. Therefore, these two bulk properties were also underestimated in case of updrafts. This effect would also be due to enhanced collision process aloft or strong attenuation along the path in the convective rainfall.

To further understand the distribution of DSD and its relative contribution to the total rainfall, we divided the droplet diameter of the observed samples at the headwaters of the Urumqi River into six classes, and the corresponding diameter values were shown in Figure 4. The raindrop diameter < 1 mm contributes the most to the precipitation, accounting for 83% of the total concentration of raindrops and 20% of the rainfall rate. In addition, particles with a diameter class of 1–2 mm contributed 15% to the total raindrop concentration but contributed 39% to the rainfall rate.

Previous studies showed a clear distinction between convective and stratiform precipitation (Tokay and Short, 1996; Maki et al., 2001; Ulbrich and Atlas, 2007). DSD characteristics of different rain types over a wide range of climatic regimes were studied by Bringi et al. (2003), and they found distinct variations between stratiform and convective regimes of the maritime and continental clusters. As the microphysical characteristics of raindrop spectra vary from stratiform to convective precipitation, the total dataset was further divided into convective rain type and stratiform rain type to analyze DSD characteristics according to the time series of rainfall rates. Many precipitation classification schemes have been proposed by previous researchers through different types of instruments (e.g., Disdrometer, profiler, and radar) (Tokay and Short, 1996; Bringi et al., 2003; Campos et al., 2006; Das et al., 2017). In this study, the classification method of Bringi et al. (2003) was used to classify precipitation into convective and stratiform. Specifically, the precipitation was classified as stratiform if the R values are between 0.5 and 5 mm h⁻¹ and the standard deviation (σR) is < 1.5 mm h⁻¹ over at least ten consecutive 1 min DSD samples. It was classified as convective rain when the R values were higher than 5 mm h⁻¹ and the standard deviation (σR) was more than 1.5 mm h⁻¹ over the same period. The results displayed after classification that the stratiform precipitation lasted 19228 min, accounting for 47% of the total amount of stratiform precipitation duration. The precipitation amount was 506 mm (40%), and the average rainfall rate was 1.57 mm h⁻¹. The duration of convective precipitation was 390 min, the cumulative precipitation was 38 mm (accounting for 3%), and the average rainfall rate was 5.87 mm h⁻¹.

The average raindrop spectra for the stratiform and convective precipitation types in the head watershed of the Urumqi River were shown in Figure 5. Stratiform precipitation exhibits broader spectra than Convective precipitation. The concentration of convective precipitation drops decreased with the increase of median diameter, while the concentration of stratiform precipitation drops increased first (the largest diameter was 1.6 mm) and then decreased. Moreover, when the diameter was smaller than 1.5 mm, the raindrop concentration of stratiform precipitation was greater than that of convective precipitation. However, when the diameter was above 1.5 mm, the raindrop concentration of convective precipitation was always higher than that of stratiform precipitation with the increase in diameter. The integral rain parameters obtained from the average spectra were listed in Table 3. The total concentration of raindrops N _t (m⁻³), liquid water content W (g m⁻³), and radar reflectivity factor Z (dBZ) for stratiform precipitation were 150.0, 0.216, and 27.5, respectively; and for convective precipitation were 309.7, 1.124, and 42.4, respectively.

The mean values of log ₁₀ N _w and D _m for the convective and stratiform precipitation types in the head watershed of the Urumqi River were compared (Figure 6). The red and blue symbols denote the results from the stratiform and convective precipitation data, respectively. Two outlined dark grey rectangles, corresponding to the maritime-like (D _m :1.5–1.75 mm; log ₁₀ N _w :4–4.5 m⁻³mm⁻¹) and continental-like (D _m : 2–2.75 mm; log ₁₀ N _w : 3–3.5 m⁻³mm⁻¹) convective clusters defined by Bringi et al. (2003), were superimposed upon the scatterplot. The D _m value of convective precipitation in the headwaters was larger than that of stratiform precipitation. In terms of stratiform precipitation, D _m and N _w in the headwaters are different from those in other parts of China. The stratiform precipitation in headwaters had a large D _m and a small N _w value, indicating that the precipitation was formed by the melting of large and dry snowflakes. The larger liquid water content causes the raindrops to collide and coalesce during the falling process, which increases the mass-weighted mean diameter of raindrops and reduced the raindrops’ concentration to a certain extent, leading to the smaller N _w . Based on the present observations and comparison to the study of Bringi et al. (2003), the convective precipitation in the head watershed of the Urumqi River could be identified as continental-like precipitation.

Figure 7 showed the relative frequency histogram of D _m and log ₁₀ N _w for the whole data set and convective and stratiform subsets calculated from the disdrometer. In addition, three key parameters, such as mean, standard deviation, and skewness were used. As shown in Figures 7A,B, for the whole dataset, the mean value of D _m and N _w is 1.54 mm and 2.85, respectively. The D _m was larger and the log ₁₀ N _w was smaller than the value proposed by Zhang et al. (2019) in South China (1.46 and 3.86). The histogram of D _m had a highly positive skewness, and on the contrary, the log ₁₀ N _w was slightly negative skewness. The standard deviation of D _m and log ₁₀ N _w were large (0.83 mm and 0.44, respectively), indicating high variability in D _m and log ₁₀ N _w . The convective histogram and stratiform histogram of D _m were both positively skewed (0.19 and 1.24); however, the log ₁₀ N _w skewness of convective and stratiform precipitation was different, with convective precipitation being positively skewed (0.51) and stratiform precipitation being negatively skewed (−0.11). Meanwhile, the D _m value of the three datasets are all bimodal.

Figure 8 is a scatter plot of the D _m -R and log ₁₀ N _w -R relationships of convective and stratiform precipitation obtained by the PWS100 disdrometer. The red solid line was the corresponding power-law relationship fitted by the least-squares method. In terms of the log ₁₀ N _w -R relationship, which are more scattered, especially for convective precipitation. The coefficient of convective precipitation was larger than that of stratiform precipitation but the index was smaller. Notably, the exponent of the log ₁₀ N _w -R relationship was negative, indicating that log ₁₀ N _w -R was lower at higher rainfall rates, which were related to collision and coalescence mechanisms during precipitation. As far as the D _m -R relationship was concerned, the scatter was not as spread out as the log ₁₀ N _w -R relationship. The index of the power-law relationship was positive, indicating that large particles contribute the most to the rainfall rate. The D _m value of convective precipitation increased with the increase in rainfall rates, while the D _m value of stratiform precipitation tended to be stable (1–2 mm) at higher rainfall rates. This indicated that the DSD tended to be in an equilibrium state.

To study the DSD difference between daytime and nighttime rainfall in the source area of the Urumqi River, the convective precipitation and stratiform precipitation were further divided into two types: daytime and nighttime. The separation of day and night was based on the local mean sunrise and sunset time in summer, with the beginning times at approximately 0700 and 2100 BST (UTC + 8), respectively. A rain event was considered a daytime event (nighttime event) if all the spectra were from daytime hours (nighttime hours). The classification results were shown in Table 4. The duration of daytime precipitation is 25,400 min, the cumulative precipitation is 825.1 mm, and the average rainfall rate is 1.95 mm h⁻¹, accounting for 62% and 65% of the precipitation in headwater, respectively. The night precipitation lasted 15,260 min, the accumulated precipitation is 439.1 mm, and the average rainfall rate is 1.72 mm h⁻¹, accounting for 38% and 35% of the precipitation in headwater, respectively.

The average raindrop spectra for daytime and nighttime precipitation types in the head watershed of the Urumqi River are shown in Figure 9. The integral rain parameters obtained from the average spectra are listed in Table 4. For stratiform precipitation, daytime and nighttime precipitation had the same spectral width and curve trend, and the raindrop concentration decreases with the increase of particle diameter. For convective precipitation, the shape of the spectral curve was very different during daytime than at nighttime. When the diameter of raindrops is less than 1.5 mm, the concentration of raindrops in the daytime is much higher than that in the nighttime, and when the diameter of raindrops is about 2 mm, the concentration of raindrops in nighttime is higher than that in the daytime. Second, there are higher N _t and W and smaller Z values during the daytime than at night. The daytime/nighttime D _m -log ₁₀ N _w relationship of stratiform precipitation is 1.48–2.857/1.57–2.823, and the day/night D _m -log ₁₀ N _w relationship of convective precipitation is 2.62–3.493/2.58–3.407. In addition, regardless of daytime or nighttime precipitation, convective precipitation has larger N _t , W, Z, and D _m -log ₁₀ N _w values than stratiform precipitation.

The coefficients (A) and indices (b) of the radar reflectivity-rain rate (Z = A*R ^b ) power-law relation are strongly affected by many factors, such as geographic location, atmospheric conditions, and raindrop spectrometer type (Uijlenho, 2001; Chandrasekar et al., 2003; Rosenfeld and Ulbrich 2003; Tokay et al., 2008), which is also strongly dependent on the variability of DSD (Steiner et al., 2004; Lee and Zawadzki., 2005). The power-law relationship between Z-R has been widely applied in radar QPE algorithms. For example, the relationship of Z = 300R ^1.4 proposed by Fulton et al. (1998) was utilized by the Next Generation Weather Radar (NEXRAD) in midlatitudes for convective rain, and the relationship of Z = 200R ^1.6 was recommended in midlatitude areas for stratiform rain (Marshall and Palmer., 1948). Wu and Liu (2017) proposed that the Z-R relationship of stratiform and convective precipitation were Z = 170R ^1.3 and Z = 70R ^1.8 , respectively, through the study of raindrop spectrum data in Naqu, Qinghai-Tibet Plateau; while Wang et al. (2021) reported the Z-R relationships of stratiform and convective precipitation in Motuo over the TP are Z = 114R ^1.8 and Z = 53.7R ^1.7 , respectively. Therefore, the uncertainty of rainfall rate estimated by radar can be minimized by establishing the optimal reflectivity factor-rainfall rate (Z-R) relationship in specific locations, seasons, and precipitation types. Due to the complex topography of the alpine mountains and relatively limited observations, it is of great significance to research the Z-R relationship of precipitation in the head watershed of the Urumqi River basin based on the DSD observation data.

Scatterplots of the rain rate R and radar reflectivity factor Z, as well as the fitted power-law relationships using the least-squares method for the stratiform and convective rain types in the head watershed of the Urumqi River, are shown in Figure 10. The Z-R relationships of Naqu reported by Wu and Liu (2017) and the Z-R relationships of Motuo reported by Wang et al. (2021) are also imposed upon Figure 9 as a solid blue line and solid purple line. Compared with other study areas, the stratiform precipitation in the source area of the Urumqi River has a larger coefficient and index (A: 698.8, b: 2.0), while the convective precipitation has a smaller coefficient and a larger index (A: 47.1, b: 2.0). Compared with other regions, the D _m -N _w value in the source region of the Urumqi River is larger and the N _w value is smaller, which corresponds to the D _m -N _w relationship in Figure 5, which further indicates that the diversity of the Z-R relationship is mainly caused by the variability of DSD. For stratiform precipitation, the empirical relationship Z = 200R ^1.6 underestimates precipitation at a rainfall rate below 0.1 mm h⁻¹, while it overestimates precipitation at a rainfall rate above 0.1 mm h⁻¹, it is contrary to Wang et al. (2021). In terms of convective precipitation, compared with Naqu and Motuo, the Heyuan region has a smaller coefficient and a larger exponent, which may be related to the abundance of large raindrops in the Heyuan region. This indicates that under the same rainfall rates, the Heyuan area has higher radar reflectivity than the above two areas. The empirical relationship Z = 300R ^1.4 underestimates the convective precipitation in the head watershed of the Urumqi River.