1 Introduction

The geographical location of the Philippines makes it susceptible to rainfall-inducing weather systems such as Tropical cyclones and Monsoons (Bagtasa 2019; Cayanan et al. 2011). Heavy rainfall is frequently experienced in Metro Manila during the Southwest Monsoon (SWM) period (Cruz et al. 2013). The SWM, locally known as Habagat, brings significant amounts of rainfall during the months of June to September in the western regions of the Philippines (Matsumoto et al. 2020). Asuncion and Jose (1980) reported that 43% of the average annual rainfall in the Philippines is derived from the SWM period. While rainfall is a valuable water resource, it remains a disaster threat during extreme rainfall events (Jamandre and Narisma 2013). It is essential, therefore, to have accurate rainfall estimates in the country, especially in the highly-urbanized area of Metro Manila. Measurements from rain gauges are usually considered to be the reference rainfall (Villarini et al. 2008). However, due to gaps in observation sites and time resolution of data, rain gauges are limited in providing accurate rainfall measurements for a wide range of areas. Satellite-derived rainfall measurements are also used to provide rain information at a global scale. However, satellite observations are not always available in real-time and are limited to lower spatial resolutions (Macuroy et al. 2021). High-quality rainfall measurements are important in numerical weather prediction models and hydrometeorological applications (Lee et al. 2019). Hence, it is necessary to have simultaneous rainfall observations with higher temporal and spatial resolution. Polarimetric weather radars are preferred over rain gauges and satellites in producing Quantitative Precipitation Estimates (QPE) because of their ability to cover a larger spatial range and provide real-time rainfall information (You et al. 2022). Weather radars estimate rainfall by measuring the resulting reflectivity (Z) scattered by raindrops within a scanning volume measured in decibels relative to Z (dBZ). One of the most common methods of retrieving rainfall from radar reflectivity is the use of Reflectivity-Rain rate (R(Z)) relations. The R(Z) relation is often expressed as a power law (Z = a*Rb), wherein the values of a and b vary for different seasons, locations, and weather systems. Globally, the most used R(Z) relations are the Marshall & Palmer relation (Z = 200*R1.6; Marshall & Palmer 1948), Rosenfeld tropical relation (Z = 250*R1.2, Rosenfeld et al. 1993), and the United States WSR-88D radar network relation (Z = 300*R1.4, Ulbrich & Lee 1999). However, using a single R(Z) relation may result in inaccurate rainfall estimates since Z is highly variable for different rain types and locations (Seela et al. 2017). Hence, it is highly recommended to calibrate the R(Z) relationship for a specific region in order to improve its performance in rainfall retrieval (Ji et al. 2019).

In addition to the conventional R(Z) relations, rainfall can also be estimated from dual-polarimetric variables (will be referred to as dual-pol variables from hereon). Dual-pol relations are known to have advantages over the conventional R(Z) relation (Zhang et al. 2019). Dual-pol variables such as differential reflectivity (ZDR) and specific differential phase (KDP) can be used to estimate rain rate (R) with greater accuracy because they can constrain environmental factors such as signal attenuation and partial beam blocking as compared to the single-polarization Z (Thompson et al. 2018). The radar parameters being used for rainfall retrieval are related to the microphysical characteristics of rainfall thru the raindrop size distribution (DSD), which is a fundamental property of rainfall defined as the number concentration of raindrops as a function of diameter (Tapiador et al. 2010). DSD variability reflects the relative importance of governing microphysical processes such as collision-coalescence, drop break-up, evaporation, and cloud ice-water interactions (Houze 2014). The DSD also varies with rainfall type (i.e., stratiform and convective), seasons, and topography (Thurai et al. 2016). Bringi et al. (2003) demonstrated that convective rainfall over tropical oceans is characterized by a higher number concentration of smaller raindrops (D < 1 mm) compared to continental locations. Moreover, Seela et al. (2018) and Zeng et al. (2019) reported relatively larger raindrops during the summer monsoon compared to the winter monsoon in Northern Taiwan and the South China Sea, respectively. Marzuki et al. (2013) and Seela et al. (2017) also reported terrain-induced convection resulting to drop size enhancements in Indonesia and Taiwan, respectively. More recently, Ibanez et al. (2023) reported larger raindrops in Clark, Pampanga compared to Metro Manila, which also demonstrates the effects of terrain-enhanced convections on the DSD.

In terms of radar applications, DSD measurements are of great importance in having accurate rainfall retrievals since Z is proportional to the sixth moment of the raindrop diameter (Hachani et al. 2017; Wu et al. 2018). Disdrometers are commonly paired with weather radars as they can explicitly measure the fall velocities and diameter of precipitation. (Tokay et al. 2013; Thompson et al. 2015). Integral rainfall parameters (IRPs) such as rain rate (R, mm h−1), total number concentration (Nt, m−3), liquid water content (LWC, g m−3), and reflectivity factor (Z, mm−6 m−3) can also be derived from disdrometer measurements (Angulo-Martinez et al. 2018). You et al. (2018) derived dual-pol parameters and relations for different rainfall events in a coastal area in Korea using an optical disdrometer. It was found that using a linear ensemble method composed of R(Z, ZDR) and R(KDP) provided more accurate QPE than the conventional R(Z) relation. The applicability of ZDR and KDP for tropical oceanic rain also was studied by Cifelli et al. (2011) (hereinafter C11) by creating a blended QPE algorithm based on continental convection in Colorado. Thompson et al. (2015, 2018) (hereinafter TH15 and TH18, respectively) hypothesized that smaller raindrops observed in the Tropical oceans resulted in lower values of ZDR and KDP for a given LWC. Hence, TH18 lowered the threshold values of C11 for ZDR in order to utilize it and explore precipitation in the Tropical Ocean. Previous radar QPE studies in the Philippines used pre-calculated values derived from other areas (Heistermann et al. 2013; Crisologo et al. 2014). The recent study of Macuroy et al. (2021) (will be referred to as MC21 from hereon) was the first study in the country to derive dual-pol parameters from DSD measurements using an optical disdrometer during the wet period in Southern Luzon. Results showed that although the R(Z) relation performed well in terms of correlation and root mean square error, the R(KDP) relation statistically outperformed other relations and provided more accurate QPE. However, the results of the study are only limited to a single radar wavelength (i.e., C-band) and do not necessarily reflect the optimal QPE relations and DSD properties for other regions in the Philippines. Notably, the DSD properties and their application in calibrating dual-pol rainfall relations are rarely explored for Metro Manila.

In this study, the DSD characteristics in Metro Manila during the SWM period were investigated using measurements from two optical disdrometers installed in Science Garden and La Mesa watershed, Quezon City. The impacts of DSD variability on dual-pol parameters were also investigated in order to develop dual-pol rainfall estimators for S-, C-, and X-band radars using the T-matrix method (Waterman 1971; Mishchenko et al. 1996). Considering the modernization program of the country’s weather bureau (i.e., Philippine Atmospheric, Geophysical, and Astronomical Services Administration or PAGASA), the DSD properties and rainfall estimators for different radar bands presented in this study can serve as a reference in optimizing the disdrometer and dual-pol radar network in different parts of the country. This study is organized as follows. Section 2 provides a brief discussion of the study site and data, which includes data cleaning and processing, and the calculation of IRPs and dual-pol radar parameters. The effects of DSD variability on the resulting radar parameters and rainfall estimators, as well as the utility of the dual-pol relations in different radar bands and rain types, are discussed in Sect. 3. Finally, Sect. 4 summarizes the results and provides the conclusion.

2 Data and methods

2.1 Instrumentation, data set, and study site

The DSD measurements during the wet period in Metro Manila (i.e., June–September) from 2020 to 2022 are collected from the 2nd-generation Particle Size Velocity Disdrometer (hereafter referred to as PARSIVEL2 disdrometer) installed in Science Garden, Quezon City (14.6° N, 121.04° E, 48 m.a.s.l.) and in La Mesa watershed, Quezon City (14.7° N, 121.07° E, 65 m.a.s.l.) (Fig. 1).

Fig. 1
figure 1

Digital elevation map showing the locations of the two disdrometer stations within the study site

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The PARSIVEL2 is an optical disdrometer that simultaneously measures the size and fall velocities of precipitation with a 1-min sampling interval. However, due to limitations in data transmission, the disdrometers used in this study were programmed to average the 1-min DSD measurements into 5-min samples. The measured raindrop diameter and fall velocities are stored in 32 × 32 diameter-velocity (D-V) bins with uneven intervals ranging from 0.062 to 24.5 mm and 0.05 to 20.8 m s−1, respectively. The first two bins that correspond to sizes less than 0.25 mm are left empty by the manufacturer because of the low signal-to-noise ratio (Loffler-Mang & Joss 2000). The PARSIVEL2 disdrometer is preferred over other disdrometer types and its first version model because of its better agreement with rain gauges and improved accuracy in measuring smaller raindrops (Tokay et al. 2014). To reduce sampling errors, the DSD measurements underwent data quality control (QC) procedures following the methods of previous studies (Seela et al. 2017; Angulo-Martinez et al. 2018). The QC procedure includes the removal of the following: (1) raindrops with diameters greater than 8 mm, (2) raindrops that have diameter and fall velocity values outside the 50% spread of the theoretical D-V curve of Beard (1976), and (3) DSD measurements corresponding to rain rates less than 0.1 mm h−1 and number concentration less than 10 m−3. 5-min DSD samples within the 1000 km effective radius of tropical cyclones (TCs) were also not included in the analysis as TC-induced rainfall is known to have different microphysical properties (Janapati et al. 2021). It was also reported by Ibanez et al. (2023) that there are no pronounced differences in the DSD properties observed between Science Garden and La Mesa watershed, hence the DSD measurements from the two disdrometer stations were combined. After the QC procedure, a total of 6,850 valid DSD samples were collected from the two stations.

2.2 DSD and integral rainfall parameters (IRPs)

The raindrop concentration per unit volume N(Di) can be calculated from the PARSIVEL2 disdrometer using the equation

$$N\left({D}_{i}\right)={\sum }_{i=1}^{32}\frac{{n}_{i}}{v\left({D}_{i}\right)At\Delta {D}_{i}}$$
(1)

where v(Di) is the raindrop fall velocity in m s−1, Di is the raindrop diameter in mm, A is the sampling area (A = 0.0054 m2), t is the sampling time (5 min = 600 s), and ΔDi is the width of the ith diameter bin. The terminal velocity v(Di) is approximated using the theoretical D-V curve equation of Beard K.V. (1976) given by:

$$v\left({D}_{i}\right)=9.58\left[1-exp\left(-{\left(\frac{{D}_{i}}{0.171}\right)}^{1.147}\right)\right]$$
(2)

The Integral rainfall parameters derived from the DSD, such as rain rate R (mm hr−1), liquid water content LWC (g m−3), total number concentration Nt (m−3), and reflectivity factor Z (mm6 mm−3) are calculated from N(D), Di, and v(Di) using the following equations:

$$R=6\pi \times {10}^{-4}{\sum }_{i=1}^{32}v\left({D}_{i}\right)N\left({D}_{i}\right){D}_{i}^{3}\Delta {D}_{i}$$
(3)
$$LWC=\frac{\pi }{6000}{\sum }_{i=1}^{32}N\left({D}_{i}\right){D}_{i}^{3}\Delta {D}_{i}$$
(4)
$${N}_{t}={\sum }_{i=1}^{32}N\left({D}_{i}\right)\Delta {D}_{i}$$
(5)
$$Z={\sum }_{i=1}^{32}N\left({D}_{i}\right){D}_{i}^{6}\Delta {D}_{i}$$
(6)

The DSDs are parameterized using the widely used Gamma model (Ulbrich 1983) expressed as

$$N\left(D\right)={N}_{0}{D}^{\mu }{\text{exp}}\left(-\Lambda D\right)$$
(7)

where N0 is the number concentration parameter, µ is the shape parameter, and Λ (mm−1) is the slope parameter. The gamma parameters were calculated using the method of moments expressed as

$${M}_{n}={\int }_{{D}_{min}}^{{D}_{max}}{D}^{n}N\left(D\right)dD$$
(8)

where n stands for the nth moment of the DSD. A combination of 3.67th, 4th, and 6th moments based on MC21 was used to calculate the gamma parameters using the following equations:

$$\mu =\frac{11G-8+\sqrt{G\left(G+8\right)}}{2\left(1-G\right)},$$
(9)
$$\Lambda =\frac{\left(\mu +4\right){M}_{3.67}}{{M}_{4}},$$
(10)
$${N}_{0}=\frac{{\Lambda }^{\mu +4}{M}_{3.67}}{\Gamma \left(\mu +4\right)}$$
(11)

where:

$$G=\frac{{M}_{4}^{3}}{{M}_{3.67}^{2}{M}_{6}^{1}}$$
(12)

The mass-weighted mean diameter Dm (mm) is also computed using the 4th and 3rd DSD moments:

$${D}_{m}=\frac{{M}_{4}}{{M}_{3}}$$
(13)

The normalized intercept parameter Nw (m−3 mm−1), which represents the DSD when N(D) approaches the minimum value, is defined by Seela et al. (2017) as

$${N}_{w}=\frac{{4}^{4}}{\pi {\rho }_{w}}\left(\frac{{10}^{3}LWC}{{D}_{m}^{4}}\right)$$
(14)

where ρw is the density of water (1 × 103 kg m−3).

2.3 Derivation of dual-polarimetric variables

The dual-pol parameters were derived from the DSD using the openly available PyDSD python package (Hardin 2014). The PyDSD makes use of disdrometer data to retrieve dual-pol parameters (i.e., ZH, ZDR, and KDP) using the Mueller/T-matrix scattering method (Mishchenko et al. 1996). The process flow of implementing the T-matrix method using the PyDSD package is shown in Fig. 12 in the Appendices. To estimate the dual-polarization parameters using the T-matrix method, conditions such as axis ratio, canting angle distribution, raindrop temperature, diameter range, and corresponding radar frequency and elevation angle must be given. Using the proposed values in MC21, the raindrop temperature was set to be 20 °C, the diameter range was from 0.1 mm to 8 mm, and the elevation angle was set to 0.5°. The average canting angle distribution was assumed to follow a Gaussian distribution with a standard deviation about the mean up to 7.5° (TH15). The raindrop’s axis ratio used in this study is assumed to be oblate, which is based on the numerical simulations and wind tunnel tests of Brandes et al. (2002) and can be expressed as a fourth order polynomial equation:

$$\gamma =0.9951+0.0251D-0.03644{D}^{2}+0.005303{D}^{3}-0.0002492{D}^{4}$$
(15)

where γ is the axis ratio and D is the raindrop diameter. The dual-pol parameters were calculated for S, C, and X bands with frequencies 2.80Ghz, 5.61Ghz, and 9.67Ghz respectively.

The ZH and ZV, which correspond to the reflectivity factors in the horizontal and vertical polarization in dBZ, were calculated using the equation:

$${z}_{H,V}=10{\text{log}}\left(\frac{{\lambda }^{4}}{{\pi }^{5}{\left|{K}_{w}\right|}^{2}}\underset{{D}_{min}}{\overset{{D}_{max}}{\int }}{\sigma }_{H,V}\left(D\right) N\left(D\right)dD\right)$$
(16)

where λ is the radar wavelength in mm, \({\sigma }_{H,V}\left(D\right)\) is the backscattering cross section for horizontal or vertical polarization and \({K}_{w}\) is the dielectric constant of water at 20 °C. The quantities ZH and ZV are dependent on the drop diameter D6 and number concentration N (D) (see Eq. 6). The differential reflectivity (ZDR), which is the logarithmic ratio of ZH and ZV expressed in dB (Seliga & Bringi 1976), is expressed as

$${Z}_{DR}=10log\frac{{Z}_{H}}{{Z}_{V}}$$
(17)

The quantity ZDR is zero for spherical drops and increases as the raindrop become more oblate, which usually happens as D > 1 mm. The specific differential phase (KDP), expressed in ° km−1, can be calculated using the equation:

$${K}_{DP}=\frac{180}{\pi }\lambda {\int }_{Dmin}^{Dmax}Re \left[\left({f}_{vv}(D\right)-{f}_{hh}\left(D\right)\right]N(D)dD$$
(18)

where \({f}_{hh,vv}\) represents the real parts of the forward scattering amplitude for the horizontally and vertically polarized waves (Vivekanandan et al. 1991). KDP is directly proportional to the LWC and oblateness of the raindrop, and inversely proportional to the radar wavelength; hence KDP is higher at X-band than S-band. The dual-pol relations, R(ZH), R(KDP), R(ZH, ZDR), and R(KDP, ZDR) chosen for this study are expressed as

$$R\left({Z}_{H}\right)=a{Z}_{H}^{b},$$
(19)
$$R\left({K}_{DP}\right)=a{K}_{DP}^{b},$$
(20)
$$R\left({Z}_{H},{Z}_{DR}\right)=a{Z}_{H}^{b}{Z}_{DR}^{C},$$
(21)
$$R\left({K}_{DP},{Z}_{DR}\right)=a{K}_{DP}^{b}{Z}_{DR}^{C},$$
(22)

Where a, b, and c are the coefficient and exponents acquired by applying the least-mean-square fitting method to the polarimetric variables and rain rates calculated from the T-matrix and DSD measurements.

This study also uses the blended optimization algorithm from TH18 and C11 which determines rain estimators used according to the following data quality thresholds.

\(R\left({Z}_{H}\right)\) if \({Z}_{DR}<0.5\) dB and \({K}_{DP}<0.{3}^{\circ }\) km−1

\(R\left({Z}_{H},{Z}_{DR}\right)\) if \({Z}_{DR}>0.5\) dB and \({K}_{DP}<0.{3}^{\circ }\) km−1

\(R\left({K}_{DP}\right)\) if \({Z}_{DR}<0.5\) dB and \({K}_{DP}>0.{3}^{\circ }\) km−1 and \({Z}_{H}>38\) dBZ.

\(R\left({K}_{DP},{Z}_{DR}\right)\) if \({Z}_{DR}>0.5\) dB and \({K}_{DP}>0.{3}^{\circ }\) km−1

Although these thresholds are optimized for S-band radars, they are designed to be wavelength independent (TH18). Hence, the algorithms can still be used for C- and X-band radars.

2.4 Statistical evaluation of the derived dual-pol relations

The rainfall values derived from the various relations (Rest) in Eqs. (18) to (19) were compared to the rainfall rate retrieved from the DSD measurements (RDSD) (i.e., considered as “ground truth”). In order to evaluate their QPE performance, four statistical validation variables were used in this study, namely: Pearson’s correlation coefficient (r), percent bias (pBias), mean error (ME), and root-mean-square error (RMSE).

$$Pearson{\prime}s\left(r\right)=\frac{{\sum }_{i=1}^{n}\left({R}_{DSD}-\overline{{R }_{DSD}}\right)-\left({R}_{est}-\overline{{R }_{est}}\right)}{\sqrt{{\sum }_{i=1}^{n}{\left({R}_{DSD}-\overline{{R }_{DSD}}\right)}^{2}{\sum }_{i=1}^{n}{\left({R}_{est}-\overline{{R }_{est}}\right)}^{2}}}$$
(23)
$$Mean \,\,error\left(ME\right)=\frac{{\sum }_{i=1}^{n}\left({R}_{est}-{R}_{DSD}\right)}{n}$$
(24)
$$Root \,mean \,square \,error\left(RMSE\right)=\sqrt{\frac{{\sum }_{i=1}^{n}{\left({R}_{est}-{R}_{DSD}\right)}^{2}}{n}}$$
(25)
$$Percent\, bias\left(pBIAS\right)=\frac{{\sum }_{i=1}^{n}\left({R}_{est}-{R}_{DSD}\right)}{{\sum }_{i=1}^{n}\left({R}_{DSD}\right)}*100\%$$
(26)

r and NSE are dimensionless, ME and RMSE are in mm h−1, and pBias is expressed as a percentage.

3 Results and discussions

3.1 Average DSD characteristics

The average and gamma-fitted DSD during the SWM season in Metro Manila are shown in Fig. 2. The number concentration (N(D)) in the y-axis is expressed in a logarithmic scale to account for large variations. The vertical dashed lines represent the raindrop size classification proposed by Krishna et al. (2016). Raindrops with diameters D < 1 mm are considered small, 1 ≤ D < 3 mm are midsize, and D > 3 mm are large. There is a good agreement between the observed and gamma-fitted DSD. Similar to the values reported by Ibanez et al. (2023), the average mass-weighted mean diameter (Dm) of the total rainfall in Metro Manila during the SWM period (Dm = 1.53 mm) is slightly higher than the value reported in MC21 in Southern Luzon (Dm = 1.45 mm) and relatively larger than the values reported by Seela et al. (2017) in Taiwan (1.24 mm) and in Palau (1.11 mm). To further investigate the DSD variability in Metro Manila, the DSD dataset was categorized into stratiform and convective rainfall type. Stratiform and convective rainfall are different in terms of cloud vertical structure and particle growth processes, hence, their DSD properties were also observed to be distinct (Tokay and Short 1996; Tao et al. 2010). This study implements a rain intensity (R) threshold of 10 mm h−1in classifying stratiform from convective rain types. DSD measurements corresponding to R < 10 mm h−1 were considered stratiform, while R ≥ 10 mm h−1 were considered convective (Banares et al. 2021). This 10 mm h−1 threshold is based on disdrometer, radar, and wind profiler measurements of tropical rainfall over the western pacific (Atlas et al. 1999; Tokay et al. 1999; Ulbrich and Atlas 2007; Sharma et al. 2009). The mean values of the integral rainfall parameters (IRPs) and the shape (μ) and slope (Λ) parameters for stratiform and convective rainfall are shown in Table 1. Results show that stratiform rains generally have lower values of Dm and higher values of Log10 Nw than convective rains. The higher standard deviation (SD) of Dm during convective rains (SD = 0.57) compared to stratiform (SD = 0.28) is a clear function of R, while the higher SD of Log10 Nw in stratiform (SD = 0.53) compared to convective rains (SD = 0.38) is due to different microphysical processes (Bringi et al. 2003; Houze 2014). Stratiform clouds with low concentrations of relatively large ice particles aloft result in DSD with relatively lower Log10 Nw and larger Dm. In radar observations, stratiform clouds exhibit a pronounced layer of high reflectivity called the bright band. The bright band is the layer where the downwards-settling ice particles start to melt (Yuter and Houze 1997). On the other hand, stratiform clouds with smaller ice particles aloft undergo complete melting (i.e., the bright band is not pronounced) before reaching the surface, resulting in DSD with high Log10 Nw and smaller Dm. Both stratiform cloud conditions are present during the SWM period and can be seen in most stratiform rain samples with mid-sized drops (1 mm < Dm ≤ 3 mm). Figure 3 also shows that the stratiform and convective rain samples during the SWM period in Metro Manila followed the c-s separation line proposed by Bringi et al. (2003). Meanwhile, convective rainfall types are observed to coincide with both maritime (MC) and continental (CC) clusters of Bringi et al. (2003, 2009). This suggests that the microphysical properties of convective rains in Metro Manila are related to both oceanic and continental convection. However, a higher percentage of convective samples fall within the CC cluster more than the MC cluster. This implies that convective rains in Metro Manila during the SWM period, particularly those with larger Dm values (Dm > 2 mm), are more continental in terms of origin.

Fig. 2
figure 2

Average DSD (solid black line) and the fitted DSD using the gamma distribution (blue dashed line) for Metro Manila during the SWM period from 2020 to 2022. The vertical dashed lines represent the raindrop size classification

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Table 1 Mean of IRPs and gamma parameters during the SWM period in Metro Manila from 2020 to 2022
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Fig.3
figure 3

Scatterplot of the Dm vs. Log10 Nw values for stratiform (gray circles) and convective (black circles) rains in Metro Manila during the SWM periods of 2020 to 2022. The black solid line represents the convective-stratiform (c-s) separation line proposed by Bringi et al. (2003) while the blue and red boxes denote the maritime convective (MC) and continental convective (CC) clusters respectively

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3.2 Characteristics of DSD-derived dual-pol variables

The ZH derived from the disdrometer and the ZH simulated using the T-matrix method in different radar bands and rain types are compared in Fig. 4. Results show that the disdrometer-derived ZH shows good agreement with those derived by T-matrix in all radar bands, with r above 0.9. This shows that the T-matrix method is an effective tool for retrieving dual-pol radar parameters from DSD measurements. Figure 5 shows the frequency distribution of simulated ZH, ZDR, and KDP for different radar bands. The frequency distribution of dual-pol parameters in Fig. 5a shows that the simulated ZH values did not exceed 60 dBZ in all radar bands. Although S-band moderately had more points at ZH ≥ 25 dBZ, all radar bands’ mean values are notably close to ~ 29 dBZ. The ZDR peaks at ~ 0.4 dB in all radar bands but is ~ 2–3% higher at ZDR > 1.4 dB for X-band (Fig. 5b). The vertical broken lines in Figs. 5b and c depict the threshold values for ZDR and KDP adopted from the study of C11. In the study of TH18, the \({Z}_{DR}\) threshold was lowered from 0.5 dB to 0.25 dB as they observed that conditions needed to exceed the 0.5 dB threshold were rare for tropical oceanic rains. However, this is not the case in this study since ~ 55% of the simulated ZDR values in all radar bands exceed 0.25 dB. Hence, this study retained the 0.50 dB thresholds for ZDR and 0.3° km−1 for KDP. A lower ZDR threshold of 0.5 dB would also increase the utility of ZDR for rainfall estimation while remaining above the accepted noise level (ZDR > 0.1 dB; Ryzhkov et al. 2005). ~ 67% of KDP values are found at KDP < 0.1° km−1 in all radar bands while higher frequencies are found for X-band at KDP > 0.1° km−1(Fig. 5c). Although a 0.3° km−1 threshold for KDP seems restrictive, lowering it is no longer practical for most radar QPE applications because of phase instability (TH18). The 2D histogram plots of simulated dual-pol parameters in Fig. 6 also help visualize the difference between the dual-pol relations and the frequency of when they are utilized for different radar bands. In general, an increase in the use of dual-pol parameters (i.e., ZDR and KDP) can be observed with the increase in radar frequency. The bulk of the data points is found in the lower left quadrant of the 2D histogram for all radar bands. By following the blended algorithm of C11, this scenario suggests that R(ZH) is the most suitable QPE relation for S-band radars. Increased frequency of data points in the upper right quadrant is found for C- and X-band radars (Fig. 6b and c), which suggests the option for R(KDP) and R(KDP, ZDR) for QPE. To further elaborate on the effect of DSD variation on the utility of dual-pol relations, the average values of dual-pol variables in different radar bands and rainfall types are shown in Table 2. The average values of ZH are similar for stratiform (26.4 dBZ) and convective rainfall (46.4 dBZ) in all radar bands except for the X-band which is found to be a little higher during convective rains (48 dBZ).

Fig. 4
figure 4

Comparison between ZH products of the disdrometer and T-matrix for stratiform and convective rainfall types in Metro Manila

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Fig. 5
figure 5

Frequency distribution of simulated Dual-pol variables using the T-matrix method: (a) Zh, (b) ZDR, and (c) KDP for Metro Manila during the SWM period. The broken lines in Fig. 4b and c represent ZDR and KDP threshold values proposed by TH18 and C11

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Fig. 6
figure 6

2D Histogram plot of simulated ZDR and KDP for (a) S-band, (b) C-band, and (c) X-band radar. The red horizontal and vertical broken lines represent the 0.5 dB and 0.3° km−1thresholds for Z­DR and KDP, respectively

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Table 2 Average values of dual-pol parameters in different radar bands for stratiform, convective, and total rainfall in Metro Manila
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The average ZDR values for convective rainfall are also higher than stratiform rainfall in all bands. Compared to stratiform rainfall, convective types have higher ZH, ZDR, and KDP. This demonstrates that raindrops during convective rainfall are relatively larger in size than those of stratiform rainfall, hence the greater difference between ZH and ZV which results in larger diameter and more shape deformation. This is also consistent with the larger average Dm of convective rainfall in Table 1. Since KDP is directly related to the liquid water content (LWC) and total number concentration (Nt) (Tang et al. 2014), the KDP of convective rainfall is also higher compared to stratiform rainfall in all radar bands. This observation is also consistent with the higher LWC and Nt of convective rainfall in Table 1.

The sudden peak of ZDR at ZH > ~ 38 dBZ in stratiform rainfall (Fig. 7a) could be a result of relatively larger raindrops and can also be a suggestive signal of the 38 dBz threshold for stratiform-convective separation regime (Gamache and Houze 1981). For convective rainfall, C-band has the largest ZDR values while a higher percentage of simulated ZH exceeding 55 dBZ is found for X-band (Fig. 5b). The distribution of the simulated polarimetric variables in Figs. 7 and 8 are also observed to be more dispersed in the C-band, especially for convective rains. The possible reasons for this lie in the DSD properties of convective rains and dependency of the T-matrix simulation on the implemented initial conditions. Zrnić et al. (2000) reported that ZDR and KDP values for C-band radars are highly dependent on the DSD and raindrop temperature, both of which can be extremely variable for convective rains. Additionally, Teschl et al., (2008) showed that resonance effect occurs for raindrop sizes larger than about 5 mm at 5-cm wavelength. Since C-band operates between 4 and 8 cm, the distribution of simulated polarimetric variables, especially KDP, are expected to be noisier. The values of simulated ZDR and ZH for convective rainfall are found to be more continental in nature, hence the higher magnitude compared to the dominantly oceanic DSD properties in TH18 (Fig. 7b). The differences between maritime and continental DSDs in the tropics can be explained by using the observed differences in the ZDR vs. ZH distributions. Compared to maritime convection, continental convection has stronger updrafts and more dominant ice microphysical processes, resulting in the formation of graupel and hail that can melt and reach the surface as larger raindrops (Marzuki et al. 2013). Large DSDs with lower Nw would lead to larger ZH and ZDR (TH18). Moreover, the continental convective cluster of DSDs in the tropics, as defined by Bringi et al. (2003), is more prone to evaporation below the cloud base which can reduce small raindrops and increase the ZDR.

Fig. 7
figure 7

ZDR—ZH relations with fitted curves for (a) stratiform and (b) convective rainfall during the 2020–2022 SWM period in Metro Manila

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Fig. 8
figure 8

KDP—ZDR relations with fitted curves for (a) stratiform and (b) convective rainfall during the 2020–2022 SWM period in Metro Manila

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The distribution of ZDR and KDP in Fig. 8 shows that a considerable percentage of both stratiform and convective samples met the 0.50 dB threshold for ZDR. This motivates the option to use R(ZH, ZDR) for QPE. However, most stratiform DSD samples did not meet the 0.3° km−1 KDP threshold, especially for the S-band (Fig. 8a). Furthermore, Fig. 8 also illustrates that DSD samples with KDP > 0.3° km−1 are always associated with ZDR > 0.5 dB in all radar bands and rain types. Similar observations were reported in TH18, but for a lower threshold of ZDR > 0.25 dB. The distribution of simulated dual-pol variables and the ZDR and KDP thresholds suggest that R(ZH) and R(ZH, ZDR) relations are for stratiform rain types (R < 10 mm h−1), while R(KDP) or R(KDP, ZDR) can be utilized for convective rain types (R ≥ 10 mm h−1).

3.3 Evaluation of derived dual-polarimetric relations

Results discussed in Sect. 3.2 clearly demonstrated the applicability of dual-pol parameters on QPE differs for different DSD properties and radar bands. Table 3 presents the derived dual-pol relations for different radar bands and rain types during the SWM period in Metro Manila. It can be observed that coefficient a in R(KDP) and R(KDP, ZDR) are larger compared to R(ZH) and R(ZH, ZDR) in all radar frequencies and rainfall types. The coefficient a in R(KDP) and R(KDP, ZDR) derived from the total rainfall decreases as the radar frequency increases from S-band to X-band, while the coefficient c has a negative value for all rainfall types in order to constrain the positive correlation of ZH and KDP to R (TH18). It can also be noticed that there were no derived R(ZH) and R(ZH, ZDR) relations for convective rainfall in X-band. This is due to the implementation of KDP and ZDR thresholds of C11 and TH18 as discussed in Sect. 3.2. In comparison with MC21, the R(KDP) obtained in this study have similar values of a but slightly higher values of b compared to MC21 (a = 21.18, b = 0.71). In terms of R(ZH, ZDR), MC21 reported a relatively lower value of a, and higher values of b and c (a = 0.0025, b = 0.9340, c = − 0.86). Finally, the R(KDP, ZDR) found in this study also has similar b but different a and c values compared to MC21 (a = 31.27, b = 0.95, c = − 0.70). The differences in the obtained dual-pol relations in Metro Manila and Southern Luzon show distinct DSD properties between the two regions despite being affected by a similar synoptic system during the SWM period. These observations also show the need to implement localized QPE relations for Metro Manila.

Table 3 Dual-pol relations with their corresponding a, b, and c values for different rainfall types and radar bands using the Developed dual-pol relations at S-, C-, and X-bands derived from the DSD measurements for the 2020–2022 SWM period in Metro Manila
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Rainfall data from the Science Gardena and La Mesa watershed disdrometer stations were used to evaluate the performance of the relationships. For this section and the succeeding discussions, the dual-pol relations will have the subscripts TOT, STR, and CNV which correspond to the derived relationships for the total, stratiform, and convective rainfall, respectively. The scatterplots of the observed rain rates with those derived from the dual-pol relationships for the C-band radar are shown in Fig. 9. A significant improvement in the statistics was observed when the relationship was change from the classic R(ZH) to R(ZDR) and R(KDP) or a combination of KDP and ZDR. The same improvements were observed for the S- and X-band but were not shown here. R(KDP)TOT and R(KDP, ZDR)TOT significantly reduced the ME and RMSE when compared to R(ZH)TOT which suggests that the relationship between R and KDP is more linear in nature. Furthermore, R(KDP, ZDR)TOT statistically outperformed the other dual-pol relation and shows that a multiparameter relation can significantly lower the errors and biases in the rainfall estimates. To evaluate the performance of the derived dual-pol relations in generating QPEs, two continuous rain events in Metro Manila during the study period were chosen as test cases. For future operational purposes, only the dual-pol relations derived for the C-band Radar will be evaluated in the next sub-sections since the nearest dual-pol Radar in Metro Manila operates within the C-band. The performance of each dual-pol relation is discussed in the succeeding sub-sections.

Fig. 9
figure 9

Scatterplots of rain rate estimates from the C-band relations for the total rainfall during the SWM period in Metro Manila. The correlation coefficient (r), root mean square error (RMSR), mean error (ME), and percent bias (pBias) are also included

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3.3.1 Event 1: 24 June 2021 heavy rainfall

Event 1 was recorded by the Science Garden disdrometer station and lasted for ~ 2 h with an average R of 8.58 mm h−1. The highest R were recorded between 12:05–12:30 UTC and 13:20–13:45 UTC with maximum values of 33.6 mm h−1 and 42 mm h−1, respectively. The average mass-weighted mean diameter (Dm) recorded during the entire event was 1.83 mm. Figure 10 shows the time series and scatter plots of R derived from the Science Garden disdrometer station and from the dual-pol relations. The standard Marshall & Palmer (R(ZMP)) relation (Z = 200R1.6) was also used for comparison. The time series shows similar troughs and peaks throughout the rain event (Fig. 10a). However, large discrepancies were observed during high rain rate periods between 12:00–12:30 UTC and 13:10–13:50 UTC. R(ZMP) and R(ZH)Tot generally overestimate rainfall with a pBias of (+)29% and (+)39%, respectively. Meanwhile, R(ZH, ZDR)TOT is observed to underestimate rainfall by (−)27%. Among the relationships, R(KDP)TOT and R(KDP, ZDR)TOT performed relatively better compared with other dual-pol relations, with r values of 0.96 and 0.99, respectively (Fig. 10 b). R(KDP)TOT and R(KDP, ZDR)TOT also statistically outperform all other ZH-based QPEs in terms of RMSE [2.63 mm h−1 and 1.48 mm h−1, respectively], ME [0.49 mm h−1 and 0.58 mm h−1, respectively], and pBias [(+)5.43% and (+)6.32%, respectively]. Since Event 1 is a heavy rainfall event, the QPE products of dual-pol relations for convective rain are also evaluated in Fig. 10c and d. Results show that both R(ZH)C and R(ZH, ZDR)C generally underestimated the rainfall, while R(KDP)CNV and R(KDP, ZDR)CNV outperformed all dual-pol QPEs. In fact, R(KDP)CNV and R(KDP, ZDR)CNV performed better than R(KDP)TOT and R(KDP, ZDR)TOT in terms of all the statistical validation parameters. This can be easily observed by comparing the fitted lines of R(KDP)T and R(KDP, ZDR)T in Fig. 10b to the linear regression fit of R(KDP)C and R(KDP, ZDR)C in Fig. 11b. R(KDP)CNV and R(KDP, ZDR)CNV also significantly reduced the RMSE [1.9 mm h−1 and 1.05 mm h−1, respectively], ME [-0.097 mm h−1 and 0.059 mm h−1, respectively], and pBias [(−)1.14% and (+)0.68%, respectively] compared to R(KDP)TOT and R(KDP, ZDR)TOT.

Fig. 10
figure 10

Comparison between the time series and scatter plots of R derived using the Marshall & Palmer relation (Z = 200R1.6) and the C-band dual-pol relations. a and b show the time series and scatterplot of derived R using the dual-pol relations for the total rainfall, while c and d show the derived R using the dual-pol relations for convective rainfall

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Fig. 11
figure 11

Comparison between the time series and scatter plots of R derived using the Marshall & Palmer relation (Z = 200R1.6) and the C-band dual-pol relations. a and b show the time series and scatterplot of derived R using the dual-pol relations for the total rainfall, while c and d show the derived R using the dual-pol relations for stratiform rainfall

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3.3.2 Event 2: 19 July 2021 stratiform rain

Event 2 was recorded by the La Mesa watershed disdrometer station. The rainfall event lasted for ~ 3 h and 30 min. with an average R of 1.5 mm h−1. The maximum R = 7.97 mm h−1 was observed at the beginning of the rain event around 17:15 UTC. The average mass-weighted mean diameter (Dm) recorded during the entire event was 1.23 mm. Compared to Fig. 10a, R(ZH)TOT performed relatively better in stratiform than convective rainfall events. Although R(ZH)TOT has a slight overestimation, it still has lower pBias [( +)16%] and ME (0.29 mm h−1) compared to the R(ZMP) [pBias = (+)21%, ME = 0.4 mm h−1)]. On the other hand, R(ZH, ZDR)TOT performed relatively better in Event 1 than here in Event 2 as it generally overestimated R having an RMSE = 1.62 mm h−1 and pBias = (+)46.7%. R(KDP)TOT also performed relatively poorer here in Event 2 and underestimated R (Fig. 11b) having a pBias = (−)42% and ME = -0.44 mm h−1. R(KDP, ZDR)TOT statistically outperformed the other dual-pol relations having the lowest RMSE = 0.16 mm h−1, ME = -0.1 mm h−1 and pBias = − 7.4%. R(KDP, ZDR)TOT was able to capture the rainfall peaks better compared to the other dual-pol relations. R(ZH)STR provided the best statistics in Fig. 11c and d in terms of the stratiform dual-pol relations. Similar to Fig. 11a, R(KDP)STR, R(ZH, ZDR)STR, and R(KDP, ZDR)STR failed to capture most of the rainfall peaks and overestimated R. R(ZH)STR also outperformed R(ZMP) in terms of lower RMSE, ME, and pBias. The results presented in Events 1 and 2 show that KDP and ZDR can provide a more accurate QPE under heavy rain conditions compared to ZH, while ZH can still be considered a better estimator for light rains compared to R(ZMP). All in all, R(KDP, ZDR) has the best performance in both convective and stratiform rain events. These findings agree with other dual-pol studies that R(KDP) and R(KDP, ZDR) result in better rainfall estimates compared to conventional single-parameter relations (Chen et al. 2017; Voormansik et al. 2020) and further prove the effectivity of the threshold-based utilization of KDP and ZDR in C11 and TH18.

4 Summary and conclusion

In this study, the three-year worth of DSD data collected from the Science Garden and La Mesa watershed disdrometer stations during the Southwest monsoon (SWM) period were used to investigate the microphysical characteristics of rainfall in Metro Manila and develop QPE relations for S-, C-, and X-band dual-polarimetric radars. The DSD characteristics during the SWM period are discussed and the performance of the QPE relations is also evaluated. The major conclusions are as follows.

  1. 1

    The observed DSD characteristics in Metro Manila show higher variability in terms of raindrop sizes compared to neighboring countries and regions (Seela et al. 2017). The smaller values of μ and Λ parameters in Metro Manila during the SWM period also indicate that despite the similarities in Dm and Nw values in Southern Luzon (Macuroy et al. 2021), Metro Manila DSD is still more distributed to larger raindrops. A clear distinction between the DSD properties of stratiform and convective rainfall was also observed. The stratiform and convective DSD samples during the SWM period follow the convective-stratiform separation line of Bringi et al. (2003) and suggest that the microphysical processes of convective rainfall in Metro Manila during the SWM period are influenced by both continental and maritime convection.

  2. 2

    The derived ZH values using the T-matrix scattering method have good agreement with the DSD-derived ZH values, thus showing that the T-matrix is an effective method in simulating dual-pol parameters using disdrometer measurements. In all radar bands, the simulated ZH values for the total rainfall in Metro Manila during the SWM period did not exceed 60 dB. Moreover, 55% of simulated ZDR were also found to be less than 0.25 dB, and 67% of KDP values were less than 0.1° km−1. Meanwhile, ZDR > 1.4 dB and KDP > 0.1° km−1 are found to have higher frequencies in X-band. In terms of rainfall type, the average value of ZH of convective rains is found to be the same for S- and C-band (46.4 dBZ) but slightly higher for X-band (48 dBZ).

  3. 3

    The distribution of the dual-pol parameters among different radar bands and rain types shows that there is a need to implement certain data quality thresholds to determine the usability of a certain dual-pol relation. The 0.5 dB and 0.3° km−1 thresholds for ZDR and KDP based on the blended algorithm of C11 and TH18 show that dual-pol relations involving ZDR and KDP are recommended to be used especially for C- and X-band. Localized dual-pol estimators such as R(ZH), R(KDP), R(ZH, ZDR), and R(ZDR, KDP) were also developed by applying the thresholds to the simulated dual-pol parameters. In general, the localized dual-pol relations can decrease the RMSE and ME by at least 7.43% and 30.25%, respectively relative to the conventional R(ZMP). Evaluation of the QPEs from the dual-pol relations for the C-band radar shows that R(ZH) is most sensitive to DSD variations hence its poor performance, especially during convective rains. Moreover, according to MC21, R(ZH) and R(ZH, ZDR) relations are more sensitive to the number of small raindrops than the proportion of large raindrops. Hence, these two rainfall estimators are not recommended for convective rain types since they contain higher concentrations of large raindrops compared to stratiform rain types. On the other hand, the relatively good performance of R(KDP) and R(KDP, ZDR) can be attributed to their lesser sensitivity to DSD variation compared to ZH (Zhang et al. 2019) and to the immunity of KDP to radar attenuation and calibration (MC21).

The comprehensive analysis of DSD properties is an important step in developing localized QPE relations since variation in the DSD is one of the major sources of error in radar QPE products. Hence, this study investigated the DSD characteristics of rainfall in Metro Manila during the SWM period using DSD measurements from two PARSIVEL2 disdrometer stations. The study also introduced an effective method of developing dual-pol relations for S-, C-, and X-band radars using DSD measurements. Since this study is focused on the performance of the QPE products in C-band radar only, other dual-pol relations mentioned in this study can be further evaluated for S- and X-band. The DSD properties observed in this study, together with the derived localized QPE relations do not necessarily reflect the DSD characteristics and dual-pol relations of other monsoon seasons and locations in the Philippines. Nevertheless, the results presented in this study, especially the derived dual-pol relations, can provide possible improvements in the general rainfall retrieval operations of the country’s dual-pol and single-pol radar networks.