Raindrop Size Distribution Characteristics of the Western Paciﬁc Tropical Cyclones Measured in the Palau Islands

: Due to the severe threat of tropical cyclones to human life, recent years have witnessed an increase in the investigations on raindrop size distributions of tropical cyclones to improve their quantitative precipitation estimation algorithms and modeling simulations. So far, the raindrop size distributions of tropical cyclones using disdrometer measurements have been conducted at coastal and inland stations, but such studies are still missing for oceanic locations. To the authors’ knowledge, the current study examines—for the ﬁrst time—the raindrop size distributions of fourteen tropical cyclones observed (during 2003–2007) at an oceanic station, Aimeliik, located in the Palau islands in the Western Paciﬁc. The raindrop size distributions of Western Paciﬁc tropical cyclones measured in the Palau islands showed unlike characteristics between stratiform and convective clusters, with a larger mass-weighted mean diameter and smaller normalized intercept parameter in the convective type. The contribution of the drop diameters to the total number concentration showed a gradual decrease with the increase in drop diameter size. Raindrop size distributions of Western Paciﬁc tropical cyclones measured in the Palau islands differed slightly from Taiwan and Japan. The helpfulness of empirical relations in raindrop size distribution in rainfall estimation algorithms of ground-based ( Z – R , µ – Λ , D m –R , and N w –R ) and remote-sensing ( σ m –D m , µ o –D m , D m –Z ku , and D m –Z ka ) radars are evaluated. Furthermore, the present study also related the rainfall kinetic energy of fourteen tropical cyclones with rainfall rate and mass-weighted mean diameter ( KE time –R , KE mm –R , and KE mm –D m ). The raindrop size distribution empirical relations appraised in this study offer a chance to: (1) enhance the rain retrieval algorithms of ground-based, remote sensing radars; and (2) improve rainfall kinetic energy estimations using disdrometers and GPM DPR in rainfall erosivity studies.


Introduction
Raindrop size distribution (RSD) is one of the essential parameters of precipitation, offering investigation into cloud and rain microphysics [1]. RSD information can improve

The Joss-Waldvogel Disdrometer (JWD) Data and Methods
The Joss-Waldvogel disdrometer (JWD) can measure 0.3-5.3 mm diameter raindrops [38]. Once the raindrops hit the JWD, it can infer the size information of the raindrops from the voltage produced by its Styrofoam cone (cross-sectional area 50 cm 2 ), and this information is stored in 20 size intervals. As mentioned in the literature [9], to reduce the sampling errors, the current study also discarded the 1-min RSD samples if the raw drops count was less than ten and the rainfall rate was <0.1 mm h −1 . Using the number of raindrops from the JWD, the N(D) and other integral rain parameters, such as R (rainfall rate, mm h −1 ), Z (radar reflectivity, dBZ), Nt (total number concentration, m −3 ), and LWC (liquid water content, gm −3 ) are estimated: where, ni, A(m 2 ), Δt (s), Di (mm), ΔDi (mm)), V(Di) (m s −1 ) are the number of raindrops counted in size bin i, sampling area, sampling time, drop diameter for the size bin i, diameter interval for the drop size bin i, and terminal velocity of the raindrops in the ith channel, respectively. The terminal velocity of raindrops at each channel is estimated using the below equation [39].

The Joss-Waldvogel Disdrometer (JWD) Data and Methods
The Joss-Waldvogel disdrometer (JWD) can measure 0.3-5.3 mm diameter raindrops [38]. Once the raindrops hit the JWD, it can infer the size information of the raindrops from the voltage produced by its Styrofoam cone (cross-sectional area 50 cm 2 ), and this information is stored in 20 size intervals. As mentioned in the literature [9], to reduce the sampling errors, the current study also discarded the 1-min RSD samples if the raw drops count was less than ten and the rainfall rate was <0.1 mm h −1 . Using the number of raindrops from the JWD, the N(D) and other integral rain parameters, such as R (rainfall rate, mm h −1 ), Z (radar reflectivity, dBZ), N t (total number concentration, m −3 ), and LWC (liquid water content, gm −3 ) are estimated: where, n i , A(m 2 ), ∆t (s), D i (mm), ∆D i (mm)), V(D i ) (m s −1 ) are the number of raindrops counted in size bin i, sampling area, sampling time, drop diameter for the size bin i, diameter interval for the drop size bin i, and terminal velocity of the raindrops in the ith channel, respectively. The terminal velocity of raindrops at each channel is estimated using the below equation [39].
Z (dBZ) = 10 × log 10 LWC g m −3 = π 6 10 −3 ρ w The nth-order moment (M n ) of raindrop size distribution can be expressed as: The mass-weighted mean diameter, D m , can be expressed as: The one-min RSD samples are fitted to gamma function as given below [40]: where D (mm) is the drop diameter, N(D) (m −3 mm −1 ) is the number of drops per unit volume per unit size interval, N 0 (m −3 mm −1 ) is the number concentration parameter, µ (-) is the shape parameter, and Λ (mm −1 ) is the slope parameter, The normalized intercept parameter, N w , can be expressed as [10]: The σ m (mass spectrum standard deviation, mm) can be estimated using [41,42]: The JWD measurements of fourteen WP TCs are also used to estimate the rainfall kinetic energy (KE), which can be expressed as KE flux (KE time , in J m −2 h −1 ) and KE content (KE mm , J m −2 mm −1 ), and the formulations for KE mm and KE time can be found in [43][44][45].

JWD Data Validation
Before using the RSD information of JWD for further analysis, the daily accumulated rainfall amounts of WP TCs measured with JWD were correlated with the collocated rain gauge, as shown in Figure 2. The comparison clearly shows a good correlation between JWD and the rain gauge measurements, indicating that the JWD measurements are worth enough for further analysis.  Figure 3 illustrates the distributions of Do [Do = (3.67+ μ)/Λ] and log10Nw in different rainfall rates (<5, 5-10, 10-30, 30-50, and >50 mm h −1 ) and radar reflectivity (<10, 10-20, 20-30, 30-40, and >40 dBZ) classes for the WP TCs measured in the Palau islands. The stratiform and convective classification lines of [46,47] are denoted with slant solid and horizontal dotted lines. The distributions of Do and log10Nw narrowed with the increase in rainfall rates and reflectivity classes. The Do and log10Nw data points for rainfall rates less (higher) than 10 mm h −1 are distributed in the stratiform (convective) regions of [46], as seen in the precipitation classification line (inclined solid line in Figure 3). Likewise, the Do and log10Nw data   10-20, 20-30, 30-40, and >40 dBZ) classes for the WP TCs measured in the Palau islands. The stratiform and convective classification lines of [46,47] Figure 3 illustrates the distributions of Do [Do = (3.67+ μ)/Λ] and log10Nw in different rainfall rates (<5, 5-10, 10-30, 30-50, and >50 mm h −1 ) and radar reflectivity (<10, 10-20, 20-30, 30-40, and >40 dBZ) classes for the WP TCs measured in the Palau islands. The stratiform and convective classification lines of [46,47] are denoted with slant solid and horizontal dotted lines. The distributions of Do and log10Nw narrowed with the increase in rainfall rates and reflectivity classes. The Do and log10Nw data points for rainfall rates less (higher) than 10 mm h −1 are distributed in the stratiform (convective) regions of [46], as seen in the precipitation classification line (inclined solid line in Figure 3). Likewise, the Do and log10Nw data The distributions of D o and log 10 N w narrowed with the increase in rainfall rates and reflectivity classes. The D o and log 10 N w data points for rainfall rates less (higher) than 10 mm h −1 are distributed in the stratiform (convective) regions of [46], as seen in the precipitation classification line (inclined solid line in Figure 3). Likewise, the D o and Remote Sens. 2022, 14, 470 6 of 22 log 10 N w data points corresponding to radar reflectivity values higher (less) than 40 dBZ lie above and below the stratiform and convective precipitation line of [46]. The mean values of D o and log 10 N w in different rainfall rates (R, mm h -1 ) and radar reflectivity (Z, dBZ) classes are depicted in Figure 3c,d. Mean D o values are increased with the increase in R and Z classes. Moreover, for higher rainfall rates classes (R > 10 mm h −1 ), mean D o, and log 10 N w values were distributed in the convective region of [46] (Figure 3). Furthermore, the WP TCs mean log 10 N w values were higher than the [47] rainfall classification line for rainfall rates > 10 mm h −1 (Figure 3).

RSD in Stratiform and Convective Precipitation
The RSDs and their corresponding microphysical properties exhibit profound disparities between stratiform and convective precipitations [9]. Using disdrometer measurements, previous researchers have adopted different methods to segregate the rainfall into two categories (stratiform and convective types) [9,10,14]. The current study separated the WP TCs rainfall measured in the Palau islands into convective and stratiform types using the modified form of [10]. Particularly, if the mean rainfall rate of ten successive 1-min RSD samples was greater than 5 mm h −1 and the standard was greater than 1.5 mm h −1 , those samples were considered as convective types, and if this condition was not satisfied, then they were considered as stratiform type. With this classification criteria, around 80% (20%) of RSDs were the stratiform (convective) type, and they contributed to rainfall accumulations of 33% (67%).
The stratiform and convective precipitation mean raindrop size distributions of WP TCs are portrayed in Figure 4a. Except for the first drop size bin, the mean RSDs show a higher concentration in convective than stratiform (Figure 4a). It is apparent from Figure 4a that the stratiform precipitation shows a closely exponential distribution, whereas the convective rainfall exhibited a broader distribution, which could be due to the enhanced collision-breakup processes in the convective compared with the stratiform type [48]. Figure 4b illustrates the scatter plot of D m and log 10 N w values for the stratiform and convective precipitations and the corresponding mean values. From the figure, the WP TCs' convective and stratiform precipitation mean log 10 N w and D m values align above and below the stratiform-convective segregation line of [10] (inclined black dotted line). The rectangular gray boxes in Figure 4b denote the [10] maritime and continental clusters of RSD measurements from different geographic locations. The comparison of WP TCs' D m and log 10 N w values shows that the stratiform mean D m and log 10 N w have lower values than the maritime and continental clusters; however, the convective rainfall average D m is in alignment with maritime convective clusters. points corresponding to radar reflectivity values higher (less) than 40 dBZ lie above and below the stratiform and convective precipitation line of [46]. The mean values of Do and log10Nw in different rainfall rates (R, mm h -1 ) and radar reflectivity (Z, dBZ) classes are depicted in Figure 3c,d. Mean Do values are increased with the increase in R and Z classes. Moreover, for higher rainfall rates classes (R > 10 mm h −1 ), mean Do, and log10Nw values were distributed in the convective region of [46] (Figure 3). Furthermore, the WP TCs mean log10Nw values were higher than the [47] rainfall classification line for rainfall rates > 10 mm h −1 (Figure 3).

RSD in Stratiform and Convective Precipitation
The RSDs and their corresponding microphysical properties exhibit profound disparities between stratiform and convective precipitations [9]. Using disdrometer measurements, previous researchers have adopted different methods to segregate the rainfall into two categories (stratiform and convective types) [9,10,14]. The current study separated the WP TCs rainfall measured in the Palau islands into convective and stratiform types using the modified form of [10]. Particularly, if the mean rainfall rate of ten successive 1-min RSD samples was greater than 5 mm h −1 and the standard was greater than 1.5 mm h −1 , those samples were considered as convective types, and if this condition was not satisfied, then they were considered as stratiform type. With this classification criteria, around 80% (20%) of RSDs were the stratiform (convective) type, and they contributed to rainfall accumulations of 33% (67%).
The stratiform and convective precipitation mean raindrop size distributions of WP TCs are portrayed in Figure 4a. Except for the first drop size bin, the mean RSDs show a higher concentration in convective than stratiform (Figure 4a). It is apparent from Figure  4a that the stratiform precipitation shows a closely exponential distribution, whereas the convective rainfall exhibited a broader distribution, which could be due to the enhanced collision-breakup processes in the convective compared with the stratiform type [48].     The WP TCs in the Palau islands showed average Dm values of 1.23 mm, 1.15 mm, 1.52 mm, and average log10Nw values of 3.59, 3.52, and 3.89, for total, stratiform and convective rainfall, respectively. The RSDs of WP TCs measured over Japan (using JWD) indicated that the mean values of mass-weighted mean diameter (Dm = 1.25 mm) and number concentration (log10Nw = 3.74) were larger than the meiyu-baiu front rainfall (Dm = 1.15 mm, log10Nw = 3.59) [8]. On the other hand, summer season rainfall segregated to typhoon and non-typhoon weather conditions in Taiwan showed lower mean Dm (1.25 mm) and higher mean log10Nw (3.63) values in the typhoon than the non-typhoon rainfall (Dm = 1.29 mm, log10Nw = 3.41) [3]. Despite distinctions in the seasonal rainfall RSDs between Palau and Taiwan [35], there was a likeness in the RSDs of WP TCs measured between Palau and the Taiwan islands. Furthermore, the WP TCs measured in Palau also showed nearly The WP TCs in the Palau islands showed average D m values of 1.23 mm, 1.15 mm, 1.52 mm, and average log 10 N w values of 3.59, 3.52, and 3.89, for total, stratiform and convective rainfall, respectively. The RSDs of WP TCs measured over Japan (using JWD) indicated that the mean values of mass-weighted mean diameter (D m = 1.25 mm) and number concentration (log 10 N w = 3.74) were larger than the meiyu-baiu front rainfall (D m = 1.15 mm, log 10 N w = 3.59) [8]. On the other hand, summer season rainfall segregated to typhoon and non-typhoon weather conditions in Taiwan showed lower mean D m (1.25 mm) and higher mean log 10 N w (3.63) values in the typhoon than the non-typhoon rainfall (D m = 1.29 mm, log 10 N w = 3.41) [3]. Despite distinctions in the seasonal rainfall RSDs between Palau and Taiwan [35], there was a likeness in the RSDs of WP TCs measured between Palau and the Taiwan islands. Furthermore, the WP TCs measured in Palau also showed nearly identical RSDs to Japan. The WP TCs measurements with a similar type of disdrometer (JWD) inter- estingly showed nearly identical D m values among these three islands (mean D m values in Palau, Taiwan, and Japan are 1.23 mm, 1.25 mm, 1.25, respectively), which signifies that the TCs of WP show no much variation in RSDs with the geographical location. Figure 6 elucidates the WP TCs' raindrops contribution to N t (total number concentration) and R (rainfall rate). From the figure, we can notice that with the increase in raindrop sizes, the number concentration decreases, and rainfall rates increase and then decrease for total, stratiform and convective rainfall (Figure 6a,b). Similar characteristics were also reported for the WP TCs measured in Japan and Taiwan [3,8]. identical RSDs to Japan. The WP TCs measurements with a similar type of disdrometer (JWD) interestingly showed nearly identical Dm values among these three islands (mean Dm values in Palau, Taiwan, and Japan are 1.23 mm, 1.25 mm, 1.25, respectively), which signifies that the TCs of WP show no much variation in RSDs with the geographical location. Figure 6 elucidates the WP TCs' raindrops contribution to Nt (total number concentration) and R (rainfall rate). From the figure, we can notice that with the increase in raindrop sizes, the number concentration decreases, and rainfall rates increase and then decrease for total, stratiform and convective rainfall (Figure 6a,b). Similar characteristics were also reported for the WP TCs measured in Japan and Taiwan [3,8]. Small-size drops (<1 mm) largely contributed to Nt for the total, stratiform, and convective rainfall (Figure 6a). The percentage contributions of smaller drops (<1 mm) to the number concentration were predominantly higher in stratiform precipitation than the convective precipitations; conversely, the percentage contribution of raindrops >1 mm diameter were higher in convective than the stratiform rainfall ( Figure 6c). Raindrops of diameter 1-2 mm contributed more to the rainfall rate than the raindrops of other diameters (Figure 6b). On the other hand, smaller drops' (<1 mm) contribution percentage to the rainfall rate was higher in stratiform precipitation than the convective precipitations, and the contribution of raindrops above 1 mm diameter to rainfall rate was predominantly higher in convective than stratiform rainfall (Figure 6d). It is apparent from Figure 6a,b that the raindrops up to 2 mm diameter contribute primarily to the total number concentration and rainfall rate.

Radar Reflectivity-Rainfall Rate Relations
The empirical relationship between radar reflectivity and rainfall rate (also called Z-R relation), which can be expressed in the form of a power law, i.e., Z = AR b , with R in mm Small-size drops (<1 mm) largely contributed to N t for the total, stratiform, and convective rainfall (Figure 6a). The percentage contributions of smaller drops (<1 mm) to the number concentration were predominantly higher in stratiform precipitation than the convective precipitations; conversely, the percentage contribution of raindrops >1 mm diameter were higher in convective than the stratiform rainfall ( Figure 6c). Raindrops of diameter 1-2 mm contributed more to the rainfall rate than the raindrops of other diameters (Figure 6b). On the other hand, smaller drops' (<1 mm) contribution percentage to the rainfall rate was higher in stratiform precipitation than the convective precipitations, and the contribution of raindrops above 1 mm diameter to rainfall rate was predominantly higher in convective than stratiform rainfall (Figure 6d). It is apparent from Figure 6a,b that the raindrops up to 2 mm diameter contribute primarily to the total number concentration and rainfall rate.

Radar Reflectivity-Rainfall Rate Relations
The empirical relationship between radar reflectivity and rainfall rate (also called Z-R relation), which can be expressed in the form of a power law, i.e., Z = AR b , with R in mm h −1 , Z in mm 6 m −3 , can offer the operation radars to estimate the rainfall rate from the observed radar reflectivity. This empirical relationship can be derived by fitting the straight line to logarithmic reflectivity versus logarithmic rainfall rate plot. The radar-Remote Sens. 2022, 14, 470 9 of 22 derived rainfall information from the Z-R relations has tremendous applications in hydrometeorological models. These relations showed substantial variations with precipitation types, geographical locations, and intensely rely on RSD features [1,35]. The coefficient (A) and exponent (b) values of Z-R relations (Z = AR b ) can infer the microphysics of given precipitation. The bigger the raindrops, the higher the coefficient A will be. If the exponent is > 1, the size-controlled (collision-coalescence) process is the dominant characteristic of the precipitation. If exponent = 1, then the number-controlled (collision, coalescence, and breakup) processes are related to the given rainfall [49][50][51]. Previous studies have demonstrated that implementing the region and precipitation-specific Z-R relations could reduce the rainfall estimation uncertainties [12,52].
The radar reflectivity and rainfall rate scatter plots for total, stratiform and convective precipitation of WP TCs and the corresponding linear regression lines are depicted in Figure 7, which clearly shows that the Z-R relations differ substantially between stratiform and convective precipitation with a higher coefficient and exponent values in stratiform than the convective. Using a similar kind of disdrometer (JWD), the Z-R relations of WP TCs were documented as Z = 189 R 1.38 in Japan [8] and Z = 217.02 R 1.35 in Taiwan [3]. If we compare the Z-R relationship of the WP TCs evaluated in the present study (Z = 221.51 R 1.35 ) with the WP TCs measured over Taiwan, both the locations showed identical exponent values (b = 1.35), which can hint that the WP TCs exhibited similar microphysical processes at these two islands. h −1 , Z in mm 6 m −3 , can offer the operation radars to estimate the rainfall rate from the observed radar reflectivity. This empirical relationship can be derived by fitting the straight line to logarithmic reflectivity versus logarithmic rainfall rate plot. The radar-derived rainfall information from the Z-R relations has tremendous applications in hydro-meteorological models. These relations showed substantial variations with precipitation types, geographical locations, and intensely rely on RSD features [1,35]. The coefficient (A) and exponent (b) values of Z-R relations (Z = AR b ) can infer the microphysics of given precipitation. The bigger the raindrops, the higher the coefficient A will be. If the exponent is > 1, the size-controlled (collision-coalescence) process is the dominant characteristic of the precipitation. If exponent = 1, then the number-controlled (collision, coalescence, and breakup) processes are related to the given rainfall [49][50][51]. Previous studies have demonstrated that implementing the region and precipitation-specific Z-R relations could reduce the rainfall estimation uncertainties [12,52]. The radar reflectivity and rainfall rate scatter plots for total, stratiform and convective precipitation of WP TCs and the corresponding linear regression lines are depicted in Figure 7, which clearly shows that the Z-R relations differ substantially between stratiform and convective precipitation with a higher coefficient and exponent values in stratiform than the convective. Using a similar kind of disdrometer (JWD), the Z-R relations of WP TCs were documented as Z = 189 R 1.38 in Japan [8] and Z = 217.02 R 1.35 in Taiwan [3]. If we compare the Z-R relationship of the WP TCs evaluated in the present study (Z = 221.51 R 1.35 ) with the WP TCs measured over Taiwan, both the locations showed identical exponent values (b = 1.35), which can hint that the WP TCs exhibited similar microphysical processes at these two islands.

The Shape-Slope Relationship
The three-parameter Gamma distribution is widely used in bulk microphysics schemes and rain retrieval algorithms of ground-based radars, and it provides better characterization of rain RSDs than two-parameter exponential distribution [40,[53][54][55]. Converting three-parameter gamma distribution to two-parameter gamma distribution (also called constrained gamma distribution) using empirical relations between the slope and shape parameters (μ-Λ relations) can reduce the errors in the polarimetric radar rainfall estimators. Initially, [56] argued that, due to the statistical errors in the RSD moment estimation, the empirical relationship between the slope and shape parameters (μ-Λ relations) could not represent the microphysics of precipitation. However, the subsequent study demonstrated that the μ-Λ relationship captured the RSDs' physical nature, and was least influenced by the errors in the assessment of μ and Λ from the RSD moments

The Shape-Slope Relationship
The three-parameter Gamma distribution is widely used in bulk microphysics schemes and rain retrieval algorithms of ground-based radars, and it provides better characterization of rain RSDs than two-parameter exponential distribution [40,[53][54][55]. Converting three-parameter gamma distribution to two-parameter gamma distribution (also called constrained gamma distribution) using empirical relations between the slope and shape parameters (µ-Λ relations) can reduce the errors in the polarimetric radar rainfall estimators. Initially, [56] argued that, due to the statistical errors in the RSD moment estimation, the empirical relationship between the slope and shape parameters (µ-Λ relations) could not represent the microphysics of precipitation. However, the subsequent study demonstrated that the µ-Λ relationship captured the RSDs' physical nature, and was least influenced by the errors in the assessment of µ and Λ from the RSD moments [57]. Hence, the µ-Λ relationship is widely used to understand RSD variability, and estimate the rainfall from remote-sensing and ground-based radars [54,[57][58][59][60]. It has been demonstrated that µ-Λ relations differ by region and rain type, and it is always essential to customize the regionor precipitation-specific µ-Λ relationship.
The empirical relationships between µ and Λ are derived by adopting the quality control procedure of [60] to the total, stratiform and convective rainfall of WP TCs. As the µ-Λ relations estimated for the light rain/drizzle can lead to statistical errors, the µ and Λ values corresponding to rainfall rate < 5 mm h −1 were removed in the current study [58]. Moreover, values of µ and Λ, higher than 20 and 20 mm −1 , respectively, were also discarded [57]. A polynomial least-square fit to the total, stratiform, and convective precipitations of the WP TCs are represented with solid black lines in Figure 8a-c, and their corresponding equations are also depicted in the respective figure panels. The inclined dashed grey lines in Figure 8a-c are computed for different D m values (D m = 1, 1.5, 2, and 3 mm) from the relationship ΛD m = 4 + µ [61]. Along with the present WP TCs' µ-Λ relations, previously reported TCs' µ-Λ relations [16,29,30] are also given in Figure 8. We can notice that the WP TCs measured in Taiwan and the coastal location of China showed nearly identical fit lines to that of the WP TCs in the Palau islands. However, there is an apparent disparity between continental convective/WP TCs rainfall fit lines and those of the WP TCs in the Palau islands.
Remote Sens. 2022, 14, x FOR PEER REVIEW 10 of 22 [57]. Hence, the μ-Λ relationship is widely used to understand RSD variability, and estimate the rainfall from remote-sensing and ground-based radars [54,[57][58][59][60]. It has been demonstrated that μ-Λ relations differ by region and rain type, and it is always essential to customize the region-or precipitation-specific μ-Λ relationship. The empirical relationships between μ and Λ are derived by adopting the quality control procedure of [60] to the total, stratiform and convective rainfall of WP TCs. As the μ-Λ relations estimated for the light rain/drizzle can lead to statistical errors, the μ and Λ values corresponding to rainfall rate < 5 mm h −1 were removed in the current study [58]. Moreover, values of μ and Λ, higher than 20 and 20 mm −1 , respectively, were also discarded [57]. A polynomial least-square fit to the total, stratiform, and convective precipitations of the WP TCs are represented with solid black lines in Figure 8a-c, and their corresponding equations are also depicted in the respective figure panels. The inclined dashed grey lines in Figures 8a-c are computed for different Dm values (Dm = 1, 1.5, 2, and 3 mm) from the relationship ΛDm = 4 + μ [61]. Along with the present WP TCs' μ-Λ relations, previously reported TCs' μ-Λ relations [16,29,30] are also given in Figure 8. We can notice that the WP TCs measured in Taiwan and the coastal location of China showed nearly identical fit lines to that of the WP TCs in the Palau islands. However, there is an apparent disparity between continental convective/WP TCs rainfall fit lines and those of the WP TCs in the Palau islands.

Relationship of Rainfall Rates with Dm and Nw
The Dm and log10Nw can be used to inspect the microphysics of given precipitation, and these two parameters exhibit profound variations with weather systems and precipitation types [1,19]. To understand the variability of Dm and log10Nw with rainfall rate, Figure 9 displays the distributions of Dm and log10Nw with rainfall rate for the total, stratiform, and convective precipitations of WP TCs. The Dm and log10Nw values show broader distribution at lower rainfall rates, and its spread is reduced with the increase in rainfall rate. The reduction in the spread of Dm values at higher rainfall rates is related to the equilibrium conditions attained by the raindrops through collision-coalescence and breakup process, and further increases in rainfall rate under the equilibrium condition infer further raindrop concentration increases [10,48,62]. On the other hand, for rainfall rates > 25 mm

Relationship of Rainfall Rates with D m and N w
The D m and log 10 N w can be used to inspect the microphysics of given precipitation, and these two parameters exhibit profound variations with weather systems and precipitation types [1,19]. To understand the variability of D m and log 10 N w with rainfall rate, Figure 9 displays the distributions of D m and log 10 N w with rainfall rate for the total, stratiform, and convective precipitations of WP TCs. The D m and log 10 N w values show broader distribution at lower rainfall rates, and its spread is reduced with the increase in rainfall rate. The reduction in the spread of D m values at higher rainfall rates is related to the equilibrium conditions attained by the raindrops through collision-coalescence and breakup process, and further increases in rainfall rate under the equilibrium condition infer further raindrop concentration increases [10,48,62]. On the other hand, for rainfall rates > 25 mm h −1 , the spread in D m values is relatively more in convective precipitation than in the stratiform type. However, the spread in log 10 N w values is more in stratiform precipitation than the convective for lower rainfall rates (<10 mm h −1 ). For a given rainfall rate, comparison of present TCs relationships (D m =1.143 R 0.145 , log 10 N w = 3.553 R 0.033 ) with TCs measured in Taiwan (D m = 1.133 R 0.153 , log 10 N w = 3.572 R 0.031 ) showed a slight difference in D m /log 10 N w values between these two islands [3]. Even though there were reports on the seasonal differences in the RSDs between Taiwan and the Palau islands [35], comparison of TCs' D m -R and log 10 N w -R relations between these two islands revealed slight variation. h −1 , the spread in Dm values is relatively more in convective precipitation than in the stratiform type. However, the spread in log10Nw values is more in stratiform precipitation than the convective for lower rainfall rates (<10 mm h −1 ). For a given rainfall rate, comparison of present TCs relationships (Dm =1.143 R 0.145 , log10Nw = 3.553 R 0.033 ) with TCs measured in Taiwan (Dm = 1.133 R 0.153 , log10Nw = 3.572 R 0.031 ) showed a slight difference in Dm/log10Nw values between these two islands [3]. Even though there were reports on the seasonal differences in the RSDs between Taiwan and the Palau islands [35], comparison of TCs' Dm-R and log10Nw-R relations between these two islands revealed slight variation.

RSD Implications for Satellite Rainfall Retrievals
The scarcity of statistical individuality among three parameters of the normalized gamma distribution can lead to bias in the GPM satellite rainfall rate retrievals. To reduce this bias, [42] proposed a new framework based on Dm (the mass-weighted mean diameter) and the standard deviation of the mass spectrum (σm). They showed that the rainfall rates estimated from Dm-μ constraint relations-using two independent physical attributes (Dm and normalized mass spectrum, σ'm)-produced smaller biases than assuming a constant μ [63]. Moreover, the Dm-μ constraint relations derived for ice-phase particles also showed an enhanced improvement in retrieving reflectivity than the currently used algorithms in ice-phase precipitation [64]. The σm-Dm relations ( = ) differ with microphysical mechanisms, and recent studies have demonstrated that these relations vary from geographical location and rain regimes, and also infer rain RSD features [12,41,60,[65][66][67][68]. Figure 10 demonstrates the distribution of Dm and σm (standard deviation of the mass spectrum) for the total, stratiform, and convective precipitations of WP TCs. The scatter points in Figure 10 are the RSD samples eligible for the quality-controlled method of [42]. The 1-min RSDs should have a minimum number of 50 raindrops for a minimum of three different diameter bins, the Z (radar reflectivity) should be >10 dBZ, and the rainfall rate should be >0.1 mm h −1 . Moreover, the σm values related to Dm < 0.5 mm are discarded. A total of 8185, 6449, and 1736 1-min RSDs in total, stratiform, and convective precipitations were qualified for the above-mentioned conditions.

RSD Implications for Satellite Rainfall Retrievals
The scarcity of statistical individuality among three parameters of the normalized gamma distribution can lead to bias in the GPM satellite rainfall rate retrievals. To reduce this bias, [42] proposed a new framework based on D m (the mass-weighted mean diameter) and the standard deviation of the mass spectrum (σ m ). They showed that the rainfall rates estimated from D m -µ constraint relations-using two independent physical attributes (D m and normalized mass spectrum, σ m )-produced smaller biases than assuming a constant µ [63]. Moreover, the D m -µ constraint relations derived for ice-phase particles also showed an enhanced improvement in retrieving reflectivity than the currently used algorithms in ice-phase precipitation [64]. The σ m -D m relations (D m = a m σ b m m ) differ with microphysical mechanisms, and recent studies have demonstrated that these relations vary from geographical location and rain regimes, and also infer rain RSD features [12,41,60,[65][66][67][68]. Figure 10 demonstrates the distribution of D m and σ m (standard deviation of the mass spectrum) for the total, stratiform, and convective precipitations of WP TCs. The scatter points in Figure 10 are the RSD samples eligible for the quality-controlled method of [42]. The 1-min RSDs should have a minimum number of 50 raindrops for a minimum of three different diameter bins, the Z (radar reflectivity) should be >10 dBZ, and the rainfall rate should be >0.1 mm h −1 . Moreover, the σ m values related to D m < 0.5 mm are discarded. A total of 8185, 6449, and 1736 1-min RSDs in total, stratiform, and convective precipitations were qualified for the above-mentioned conditions.
Comparable to the findings of previous studies, a higher correlation between σm and Dm can be seen for the total, stratiform, and convective precipitations of the WP TCs [42,68]. Using eight years of in-situ shipboard global ocean RSD, ref. [68] evaluated the σm-Dm relations for seven latitude bands, and computed σm-Dm relation for the Northern Tropics (0°N to 22.5°N) is σm = 0.313 Dm 1.438 . Figures 11-13 illustrate the mapping of (Dm, μo) space from (σm, Dm) space for total, stratiform, and convective precipitations of WP TCs, respectively. The occurrence frequency of σm with Dm for total, stratiform, and convective precipitations are depicted, in Figure 11a, Figure 12a and Figure13a, respectively, and the normalized PDF of the mass spectrum (σm) are given in Figure 11b, Figure 12b and Figure 13b. The normalized mass spectrum, σ'm, a statistically independent attribute from Dm [42,67], is computed for total (σ'm = σm/Dm 1.122 ), stratiform (σ'm = σm/Dm 1.223 ), and convective (σ'm = σm/Dm 1.018 ) precipitations, and are plotted in Figure 11c, Figure 12c and Figure 13c with the two-dimensional frequency of occurrence. The normalized mass spectrum mean ( ) and standard deviation (std(σ'm)) values were 0.2912 and 0.0653 for the total precipitation of the WP TCs. The mean and standard deviations of the normalized mass spectrum were 0.2902 (0.2918) and 0.0649 (0.0605) for stratiform (convective) precipitation. The solid black line in Figure 11a, Figure 12a and Figure 13a, represents the total, stratiform, and convective σm-Dm relations (Equations (15)-(17)), respectively, and the solid red and blue lines denote the upper (Equations (21), (23), and (25)) and lower (Equations (22), (24), and (26)) σ'm bounds. The σ'm mean value for total, stratiform, and convective precipitation is represented with a black dash-dotted line, in Figure 11c,d, Figure 12c,d and Figure 13c,d, respectively.
Comparable to the findings of previous studies, a higher correlation between σ m and D m can be seen for the total, stratiform, and convective precipitations of the WP TCs [42,68].  Figures 11-13 illustrate the mapping of (D m , µ o ) space from (σ m , D m ) space for total, stratiform, and convective precipitations of WP TCs, respectively. The occurrence frequency of σ m with D m for total, stratiform, and convective precipitations are depicted, in Figures 11a, 12a and 13a, respectively, and the normalized PDF of the mass spectrum (σ m ) are given in Figures 11b, 12b and 13b. The normalized mass spectrum, σ m , a statistically independent attribute from D m [42,67], is computed for total (σ m = σ m /D m 1.122 ), stratiform (σ m = σ m /D m 1.223 ), and convective (σ m = σ m /D m 1.018 ) precipitations, and are plotted in Figures 11c, 12c and 13c with the two-dimensional frequency of occurrence. The normalized mass spectrum mean (σ m ) and standard deviation (std(σ m )) values were 0.2912 and 0.0653 for the total precipitation of the WP TCs. The mean and standard deviations of the normalized mass spectrum were 0.2902 (0.2918) and 0.0649 (0.0605) for stratiform (convective) precipitation. The solid black line in Figures 11a, 12a and 13a, represents the total, stratiform, and convective σ m -D m relations (Equations (15)-(17)), respectively, and the solid red and blue lines denote the upper (Equations (21), (23), and (25)) and lower (Equations (22), (24), and (26)) σ m bounds. The σ m mean value for total, stratiform, and convective precipitation is represented with a black dash-dotted line, in Figure 11c     The upper ( + std(σ'm) = 0.3565, 0.3551, and 0.3523, for total, stratiform, and convective rainfall, respectively) and lower bounds ( − std(σ'm)= 0.2258, 0.2254, and 0.2313, for total, stratiform, and convective rainfall, respectively) of σ'm are depicted with red and blue dash-dotted lines in Figure 11c,d, Figure 12c,d and Figure13c,d, for total, stratiform, and convective rainfall, respectively. These bounds cover 73.19%, 73.87%, and 74.25% of σ'm data points, for total, stratiform, and convective rainfall, respectively.
The expected value of σm for total, stratiform, and convective rainfall are: The upper and lower bounds of σm for total, stratiform and convective rainfall are: The upper (σ m + std(σ m ) = 0.3565, 0.3551, and 0.3523, for total, stratiform, and convective rainfall, respectively) and lower bounds (σ m − std(σ m ) = 0.2258, 0.2254, and 0.2313, for total, stratiform, and convective rainfall, respectively) of σ m are depicted with red and blue dash-dotted lines in Figure 11c,d, Figure 12c,d and Figure 13c,d, for total, stratiform, and convective rainfall, respectively. These bounds cover 73.19%, 73.87%, and 74.25% of σ m data points, for total, stratiform, and convective rainfall, respectively.
The expected value of σ m for total, stratiform, and convective rainfall are: The upper and lower bounds of σ m for total, stratiform and convective rainfall are: The σ m and D m assessments of WP TCs were transformed into µ o estimates using below-mentioned equations, and are depicted in Figures 11e, 12e and 13e. The µ o PDF distributions and the normalized Gaussian curve with its mean (black dash-dotted line), mean plus standard deviation (red dash-dotted line), and mean minus standard deviation (blue dash-dotted line) values are illustrated in Figures 11f, 12f and 13f, for total, stratiform and convective rainfall, respectively.
The expected value of µ o with the lower and upper bounds for total, stratiform, and convective rainfall was evaluated with the above equation and is shown in Figures 11e, 12e and 13e with black, blue, and red solid lines, respectively.
In the GPM DPR rain-retrieval algorithms, the radar reflectivity at Ka-and Ku-bands (Z Ka and Z Ku ) and the difference between these two radar reflectivities, known as differential frequency ratio (DFR = 10log 10 Z Ku -10log 10 Z Ka , in dB), are used to retrieve the RSD parameters (D m and log 10 N w ). While estimating D m values from DFR, previous studies noticed two D m values for a given negative DFR value, called a dual-value problem, which arises due to the predominance of Rayleigh scattering in light rain at Ku-and Ka-band reflectivities [69,70]. Consistent with the previous studies, we also noticed a double solution problem (figure not shown); hence, rather than relating the D m values to the DFR, empirical relations between D m and Z Ku /Z Ka are derived. The T-matrix simulations with 25 • C temperature were applied to the disdrometer data to obtain the radar reflectivity at Kuand Ka-bands for the WP TCs [71]. Figure 14 depicts the scatter plots of D m versus Z Ka /Z Ku for total, stratiform, and convective rainfall of WP TCs. With the increase in reflectivity at Ku/Ka-band frequency, the D m values increased in WP TC rainfall. The second-degree polynomial fit lines and the corresponding D m -Z ku and D m -Z ka relations are also depicted in the figure. Along with the present study polynomial fit lines, D m -Z ku /Z ka relations derived for the Southwestern Pacific summer season rainfall from [13] are also depicted in the figure. For a given radar reflectivity, especially for reflectivity values greater than 20 dBZ, WP TCs measured in the present study showed lower D m values than the oceanic summer season rainfall.

RSD Implications for Rainfall Kinetic Energy Retrievals
The energy with which the raindrops from the cloud base reach the ground surface is called the kinetic energy (KE) of rain or rainfall KE. The rainfall KE plays a crucial role in estimating the rainfall erosivity factor of the universal soil loss equation, a physical parameter that describes surface erosion caused by rainfall [2,[72][73][74]. Due to the expensive experimental setup required for the direct measurement of rainfall KE, indirect measurements such as the utilization of RSD information from the ground-based disdrometers have been adopted globally [2,26,[75][76][77][78].
The rainfall KE can be estimated using raindrop size and velocity information. Relating the rainfall KE with the rainfall rate can provide the opportunity to estimate the KE at places with rain gauges and lack of disdrometer measurements. Figure 15 displays the distribution of rainfall KE with the rainfall rate for total, stratiform, and convective rainfall. From the figure, it is apparent that the KE time increases with the increase in rainfall rate, whereas the KE mm showed a steep increase for the rainfall rate less than 20 mm h −1 , and above 20 mm h −1 , a flat increase can be seen. The linear, power, logarithmic, and exponential forms of rainfall KE are provided in Figure 15, and their statistical values are given in Table 1. Remote Sens. 2022, 14, x FOR PEER REVIEW 16 of 22 Figure 14. Scatter plots of mass-weighted mean diameter radar reflectivity at Ku-band (a-c) and Kaband (d-f) for total, stratiform, and convective rainfall of WP TCs.

RSD Implications for Rainfall Kinetic Energy Retrievals
The energy with which the raindrops from the cloud base reach the ground surface is called the kinetic energy (KE) of rain or rainfall KE. The rainfall KE plays a crucial role in estimating the rainfall erosivity factor of the universal soil loss equation, a physical parameter that describes surface erosion caused by rainfall [2,[72][73][74]. Due to the expensive experimental setup required for the direct measurement of rainfall KE, indirect measurements such as the utilization of RSD information from the ground-based disdrometers have been adopted globally [2,26,[75][76][77][78].
The rainfall KE can be estimated using raindrop size and velocity information. Relating the rainfall KE with the rainfall rate can provide the opportunity to estimate the KE at places with rain gauges and lack of disdrometer measurements. Figure 15 displays the distribution of rainfall KE with the rainfall rate for total, stratiform, and convective rainfall. From the figure, it is apparent that the KEtime increases with the increase in rainfall rate, whereas the KEmm showed a steep increase for the rainfall rate less than 20 mm h −1 , and above 20 mm h −1 , a flat increase can be seen. The linear, power, logarithmic, and exponential forms of rainfall KE are provided in Figure 15, and their statistical values are given in Table 1. The GPM DPR offers the RSD parameters (Dm and log10Nw) of given precipitation globally with the dual-polarization capability. The empirical relationship between the rainfall KE and RSD parameters (Dm and log10Nw) can aid in evaluating the rainfall KE using GPM DPR at the places where disdrometers and rain gauges are scarce. Previous  The GPM DPR offers the RSD parameters (D m and log 10 N w ) of given precipitation globally with the dual-polarization capability. The empirical relationship between the rainfall KE and RSD parameters (D m and log 10 N w ) can aid in evaluating the rainfall KE using GPM DPR at the places where disdrometers and rain gauges are scarce. Previous researchers reported a reasonable agreement between the rainfall/RSD parameters of GPM DPR and the ground-based disdrometer. For instance, [66] compared the seasonal RSD of PARSIVEL disdrometer with the GPM DPR data products over eastern China, and they reported better agreement for the winter rainfall. A reasonable agreement was reported for the indirect comparison of GPM DPR D m and log 10 N w values with worldwide disdrometer measurements [79].
Recently, a good agreement between RSD parameters (R and D m ) of ground-based disdrometers in Italy and the GPM DPR was reported by [80]. The studies mentioned above have evidently proven the reliability of GPM DPR rain/RSD parameters for hydrometeorological applications; hence, we can use the D m values from GPM DPR to estimate the rainfall KE using the empirical relation between D m and KE. Figure 16 illustrates the distributions of srainfall KE with the mass-weighted mean diameter values for total, stratiform, and convective precipitation. Recently, a good agreement between RSD parameters (R and Dm) of ground-based disdrometers in Italy and the GPM DPR was reported by [80]. The studies mentioned above have evidently proven the reliability of GPM DPR rain/RSD parameters for hydrometeorological applications; hence, we can use the Dm values from GPM DPR to estimate the rainfall KE using the empirical relation between Dm and KE. Figure 16 illustrates the distributions of srainfall KE with the mass-weighted mean diameter values for total, stratiform, and convective precipitation. The figure shows that the KEmm is highly correlated with the Dm for the WP TCs rainfall with little spread from the respective fit lines. Second-degree polynomial relations computed for KEmm and Dm are depicted in the respective panels, and their statistical values are given in Table 1. Along with the present TCs fit lines, the KEmm-Dm relation of WP TCs measured in Taiwan by [3] is portrayed in Figure 16. It is evident that the KEmm-Dm relations of the WP TCs measured in Palau and Taiwan islands showed minor variations, which can be attributed to no variation in the RSDs of WP TCs measured at these two islands, as pointed out in Section 3.2. The figure shows that the KE mm is highly correlated with the D m for the WP TCs rainfall with little spread from the respective fit lines. Second-degree polynomial relations computed for KE mm and D m are depicted in the respective panels, and their statistical values are given in Table 1. Along with the present TCs fit lines, the KE mm -D m relation of WP TCs measured in Taiwan by [3] is portrayed in Figure 16. It is evident that the KE mm -D m relations of the WP TCs measured in Palau and Taiwan islands showed minor variations, which can be attributed to no variation in the RSDs of WP TCs measured at these two islands, as pointed out in Section 3.2.

Summary and Conclusions
In this study, the raindrop size distribution (RSD) statistical characteristics of fourteen tropical cyclones (TCs) were investigated using Joss-Waldvogel disdrometer (JWD) measurements conducted at an oceanic site in the Palau islands in the Western Pacific (WP). The WP TCs D o and log 10 N w distribution diagram displayed that the mean D o and log 10 N w values were located below (above) the [46] rain classification line for rainfall rates of less than (greater than) 10 mm h −1 . The WP TCs RSDs revealed distinct segregation between stratiform and convective types, and the convective RSDs of the considered TCs presented similar features to the maritime convective clusters. The contribution of small-size drops (<1 mm) to number concentration (N t )/rainfall rate (R) was predominant in the stratiform precipitation than the convective type, and an opposite characteristic is recognized for the raindrops of diameter greater than 1 mm.
For a given type of disdrometer (JWD), average D m values of WP TCs measured at Palau islands showed minor variations with Japan and Taiwan. Regardless of measurements from different disdrometer types, the WP TCs in Palau inferred nearly identical µ-Λ relations with the WP TCs measured in Taiwan and coastal China; however, these relations were different from the WP TCs measured over inland. In addition, an identical exponent value of the Z-R relations (b = 1.35 in Z = AR b ) observed in Taiwan and the Palau islands inferred that the WP TCs of these two islands were related with identical microphysical processes.
The Z-R, µ-Λ, D m -R, N w -R, σ m -D m , and µ o -D m relations of the WP TCs measured in the Palau islands showed an inequality between stratiform and convective precipitation. The present study offered unique RSD characteristics of WP TCs measured at an oceanic site in the Palau islands, and these results can deliver possible implications for rain retrieval algorithms. For instance, evaluated RSD relations (σ m -D m , µ o -D m , D m -Z ku , and D m -Z ka ) can aid in optimizing the constraints associated with the global precipitation measurement (GPM) dual-precipitation radar (DPR) rain retrieval algorithms. In addition to the implications of the rain retrieval algorithms, the proposed rainfall kinetic relations (KE time -R, KE mm -R, and KE mm -D m ) can be used to estimate the rainfall kinetic energy of WP TCs using rain gauge and remote sensing (GPM DPR) measurements, and the obtained rainfall KE can offer a better appraisal of rainfall erosivity, an essential parameter used in the soil erosion modeling.