# Validation of GPM DPR Rainfall and Drop Size Distributions Using Disdrometer Observations in the Western Mediterranean

Eric Peinó, Joan Bech, Francesc Polls, Mireia Udina, Marco Petracca, Elisa

Adirosi, Sergi Gonzalez, Brice Boudevillain

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Eric Peinó, Joan Bech, Francesc Polls, Mireia Udina, Marco Petracca, et al. Validation of GPM DPR Rainfall and Drop Size Distributions Using Disdrometer Observations in the Western Mediterranean. Remote Sensing, 2024, 16 (14), pp.2594. 10.3390/rs16142594. hal-04801608

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Article

# Validation of GPM DPR Rainfall and Drop Size Distributions Using Disdrometer Observations in the Western Mediterranean

Eric Peinó<fnref n="1" />, Joan Bech <fnref n="1" />,<fnref n="2" />,*<fnref n="1" />, Francesc Polls <fnref n="1" />, Mireia Udina <fnref n="1" />, Marco Petracca <fnref n="3" />, Elisa Adirosi <fnref n="3" />, Sergi Gonzalez <fnref n="4" /> and Brice Boudevillain <fnref n="5" />

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**Citation:** Peinó, E.; Bech, J.; Polls, F.; Udina, M.; Petracca, M.; Adirosi, E.; Gonzalez, S.; Boudevillain, B. Validation of GPM DPR Rainfall and Drop Size Distributions Using Disdrometer Observations in the Western Mediterranean. *Remote Sens.* **2024**, *16*, 2594. https://doi.org/10.3390/rs16142594

Academic Editor: Won-Ho Nam

Received: 7 June 2024

Revised: 10 July 2024

Accepted: 11 July 2024

Published: 16 July 2024

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**Copyright:** © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
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4.0/).

**Abstract:** Dual-frequency precipitation radar (DPR) on the Core GPM satellite provides spaceborne three-dimensional observations of precipitation fields and surface rainfall rate with quasi-global coverage. The present study evaluates the behavior of liquid precipitation intensity, radar reflectivity factor ($Z_{Ku}$ and $Z_{Ka}$) and drop size distribution (DSD) parameters (weighted mean diameter $D_m$ and intercept parameter $N_w$) of the GPM DPR-derived products, version 07, from 2014 to 2023. Observations from seven Parsivel disdrometers located in different topographic zones in the Western Mediterranean are taken as ground references. Four matching techniques between satellite estimates and ground level observations were tested, and the best results were found for the so-called optimal comparison approach. Overall, GPM DPR products captured the variability of the observed DSD well at different rainfall intensities. However, overestimation of the mean $D_m$ and underestimation of the mean $N_w$ were observed, being much more sensitive to errors in drop diameters larger than 1.5 mm. Moreover, the lowest errors were found for radar reflectivity factor and $D_m$, and the highest for $N_w$ and rainfall rate. In addition, the GPM DPR convective and stratiform classification was tested, and a substantial overestimation of stratiform cases compared to disdrometer observations were found.

**Keywords:** dual-frequency precipitation radar (DPR); GPM; disdrometer; ground validation; precipitation estimates; Western Mediterranean

## 1. Introduction

Satellite precipitation estimates are an essential input to provide a complete perspective of the hydrological cycle at the global scale, including the monitoring of extreme events and complementing traditional ground-based observation methods based on rain gauge and weather radar networks [1]. The Tropical Rainfall Measuring Mission (TRMM) of the National Aeronautics and Space Administration (NASA) and Japan Aerospace Exploration Agency (JAXA), launched in 1997, was the first satellite equipped with weather radar, operating at Ku-band (13.6 GHz), dedicated to measuring precipitation at latitudes between 35° S and 35° N [2]. In 2014, the same agencies launched the Core Observatory satellite (CO) on the Global Precipitation Measurement (GPM) mission [3] to provide precipitation estimates between 65° S and 65° N and become the basis for future long-term analyses [4]. To this end, GPM CO became the first spaceborne dual-frequency precipitation radar (DPR), operating at Ka- (35.5 GHz) and Ku-band (13.6 GHz) to offer three-dimensional measurements of the precipitation structure. Compared to TRMM precipitation radar, DPR is more sensitive to light rainfall rates, and because of simultaneous measurements from

<footnote n="1">¹ Department Applied Physics—Meteorology, Universitat de Barcelona, 08028 Barcelona, Spain; epeino@meteo.ub.edu (E.P.); fpolls@meteo.ub.edu (F.P.); mudina@meteo.ub.edu (M.U.)</footnote>

<footnote n="2">² Water Research Institute, Universitat de Barcelona, 08028 Barcelona, Spain</footnote>

<footnote n="3">³ National Research Council of Italy, Institute of Atmospheric Sciences and Climate (CNR-ISAC), 00133 Rome, Italy; m.petracca@isac.cnr.it (M.P.); elisa.adirosi@artov.isac.cnr.it (E.A.)</footnote>

<footnote n="4">⁴ WSL Institute for Snow and Avalanche Research SLF, CH-7260 Davos, Switzerland; sergi.gonzalez@slf.ch</footnote>

<footnote n="5">⁵ Institute for Geosciences and Environmental Research, University Grenoble Alpes, CNRS, INRAE, IRD, Grenoble INP, IGE, 38000 Grenoble, France; brice.boudevillain@univ-grenoble-alpes.fr</footnote>

<footnote>* Correspondence: joan.bech@ub.edu</footnote>
overlapping Ka/Ku bands, new information on the drop size distribution over moderate precipitation intensities can be obtained [5].

Because of the valuable information provided by the DPR and their multiple applications, validation exercises are essential. In fact, GPM precipitation recovery algorithms have been subject to frequent updates (seven versions in the first 10 years). Therefore, it is necessary to identify biases and improve future versions, with ground validation being an important component for evaluating and improving the performance of the DPR algorithm [6]. Some of the DPR-derived variables are estimated at ground level, so specific information about precipitation drop size distributions (DSDs) at that level is needed for their verification. For this reason, ground-based disdrometers, able to measure DSDs from which integral rainfall parameters, such as reflectivity, intensity and liquid water content can be computed, are a key instrument for the verification of DPR-derived products. A number of field campaigns promoted by the NASA Ground Validation program and other research groups have been carried out in recent years [7] deploying different disdrometer types (two-dimensional video disdrometer (2DVD) from Joanneum Research, Inc. in Graz, Austria; OTT Parsivel Model 2; and Joss-Waldvogel) [8]. Table 1 lists different GPM DPR validation studies using disdrometers and their region of study.

<table><caption>**Table 1.** Validation studies of GPM DPR products using disdrometers.</caption><thead><tr><th>DPR Version</th><th>Disdrometer Type</th><th>Variables Studied *</th><th>Region of Study</th><th>Reference</th></tr></thead><tbody><tr><td>-</td><td>OTT Parsivel <sup>2</sup></td><td>RR, N<sub>w</sub>, D<sub>m</sub>, Z, k</td><td>Iowa, USA</td><td>Liao et al., (2014) [9]</td></tr><tr><td>V03</td><td>RD-80</td><td>RR, N<sub>w</sub>, D<sub>m</sub>, Z, k</td><td>Gadanki, India</td><td>Radhakrishna et al., (2016) [10]</td></tr><tr><td>V05</td><td>2DVD</td><td>RR, DSD, Z</td><td>Italian Peninsula</td><td>D'Adderio et al., (2019) [11]</td></tr><tr><td>V06</td><td>OTT Parsivel <sup>2</sup></td><td>RR, N<sub>w</sub>, D<sub>m</sub>, Z, μ</td><td>Jianghuai, China</td><td>Wu et al., (2019) [12]</td></tr><tr><td>V06</td><td>2DVD</td><td>RR, D<sub>m</sub></td><td>Several international sites</td><td>Chase et al., (2020) [13]</td></tr><tr><td>V06</td><td>OTT Parsivel <sup>2</sup></td><td>RR, N<sub>w</sub>, D<sub>m</sub>, Z, k</td><td>Central Andes, Peru</td><td>Del Castillo-Velarde et al., (2021) [14]</td></tr><tr><td>V06</td><td>Thies, OTT Parsivel <sup>2</sup></td><td>RR, N<sub>w</sub>, D<sub>m</sub>, Z</td><td>Italian Peninsula</td><td>Adirosi et al., (2021) [15]</td></tr><tr><td>V07</td><td>Joss-Waldvogel</td><td>RR, N<sub>w</sub>, D<sub>m</sub>, Z</td><td>North Taiwan</td><td>Seela et al., (2023) [8]</td></tr></tbody></table>

A first study was performed simulating the DPR algorithm before the GPM CO launch with disdrometer data [9]. In a comparative DPR-disdrometer study over Gadanki, India [10], it was observed that the D_m values obtained from GPM DPR were severely underestimated at high rainfall rates (R > 8 mm/h) during the SW monsoon season. Meanwhile D'Adderio et al. [11] obtained statistical scores that did not differ significantly between land and sea [13,14] and found that the GPM DPR showed superior performance in estimating rainfall parameters in stratiform precipitation than in convective precipitation. Adirosi et al. [15] compared the precipitation and drop size distribution parameters of a large network of disdrometers in Italy with the DPR GPM. The sensitivity analysis revealed, regardless of the type of DPR algorithm (dual- or single-frequency algorithm), a superior agreement for the mass-weighted mean raindrop diameter (D_m) and a lower agreement for the normalized gamma DSD intercept parameter (N_w), similar to the results of [8,11,16]. Del Castillo-Velarde et al. [14] concluded that differences with respect to convective rainfall could be associated with the setting of the shape parameter (μ) in the DPR algorithm. They also suggested that in the central Andes, the estimation of DSD parameters in stratiform rainfall is strongly affected by the limitation of the dual-frequency (DF) algorithm in estimating D_m < 1 mm.

The number of verification studies using disdrometer data has been growing over the years, but it is still much lower than the number of studies with rain gauges, limited to comparing precipitation amounts. As disdrometers are not frequently deployed in operational

<footnote>* Variables considered are rain rate (RR), mass weighted mean drop diameter (D_m), intercept parameter (N_w), shape parameter (μ), radar reflectivity (Z), specific attenuation (k).</footnote>
networks, their observations are relatively scarce, so the difficulties of sampling satellite overpasses matching precipitation events is an important limitation of such studies. For instance, in the Mediterranean basin, an area vulnerable to climate change, hydrometeorological extremes and uncertainty of projections regarding precipitation [17], studies of this type have focused only on the area over Italy using DPR products V05 and V06A. Therefore, the limited number of geographical regions examined and the continuous upgrade of DPR product versions require more validation studies of this type. To contribute to fill this gap, the objective of this study is to evaluate the behavior of the precipitation intensity, radar reflectivity factor (ZKu and ZKa) and DSD parameters (Dm, Nw) of the GPM DPR level 2 version 07B, the latest available. For this purpose, data from seven disdrometers (OTT Parsivels 1, 2) covering the period 2015 to 2024 located in different topographic areas of Catalonia, Spain were used as references.

The remainder of this paper is organized as follows. Section 2 provides a description of the study area, datasets used (disdrometers and GPM DPR data), the comparison methodology adopted in this study and the evaluation metrics employed. Section 3 shows results using disdrometer data, DPR data and their matches, the latter validated considering four different approaches. The most significant results are discussed in Section 4, and a summary and conclusion are provided in Section 5.

## 2. Materials and Methods

### 2.1. Datasets

#### 2.1.1. GPM-DPR

The GPM CO operates in low Earth orbit, carrying two instruments to measure the Earth's precipitation and serving as a calibration standard for other members of the GPM satellite constellation [18]. The satellite was developed and tested in-house at NASA's Goddard Space Flight Center and launched from the Tanegashima Space Center, Japan, on 27 February 2014 [5]. The orbit height has been 442 km since November 2023, and the orbit inclination is 65°.

GPM-DPR Version 07B Level 2 products provide three main classes of precipitation products: (1) Ku-band frequency, derived over a 245 km-wide swath in so-called full scan (FS, low sensitivity) mode; (2) Ka-band frequency, which, as of May 2018, occupies a 125 km-wide swath in FS mode and the rest of the swath in high scan mode (HS, high sensitivity); and (3) dual-frequency-derived data in FS and HS modes. Finally, the swath structures can be categorized into single- and double-beam pixels based on the availability of radar reflectivity within the Ku and Ka bands [19].

The derivation of the DSD using the single-frequency (SF) and dual-frequency (DF) algorithms in the liquid phase intervals assumes a gamma-shaped droplet size distribution with three parameters: Nw, Dm and the shape parameter (μ). To reduce the number of unknown parameters from three to two, GPM DPR algorithms consider a constant value for μ, set to μ = 3 [6]. To determine Dm and Nw, relationships between Dm and k/Ze or DFR are used, where k is the specific attenuation in dB/km, Ze is the effective reflectivity factor, and DFR is the dual-frequency ratio. A brief summary of SF and DF algorithms is shown in Appendix A and further information is available in [19–21].

An important relationship is assumed between the precipitation rate R and Dm [15]. In the current version, V07B, the R-Dm relationship is given by

$$R = \varepsilon^{\tau} \alpha D_{m}^{\beta} \qquad (1)$$

where R is the precipitation rate in mm/h for temperatures between –50 °C and 50 °C and α, β and τ are constants of 0.392, 6.131 and 4.815, respectively. To reconcile possible inconsistencies arising from the use of different attenuation estimation techniques [22], the equation includes an adjustment factor ε. Different R-Dm relations were tested by varying ε from 0.2 to 5.0. Assuming a gamma DSD with a fixed shape parameter, it is possible
to establish a relationship between R and Dm for various effective reflectivity values [20]. Thus, given ε = 1.25, a pair (R, Dm) can be obtained.

In this study, DF (DPR) and SF (Ku-band and Ka-band) products were used in the FS mode of the GPM DPR version 07B from 2014 to 2023. Note that the FS format is available in version V07 for observations recorded both before and after the scan pattern change of the Ka-band in May 2018 [21]. The output variables selected to be evaluated were the precipitation intensity (precipRateNearSurface, mm/h) estimated in the clutter-free bin closest to the surface (binClutterFreeBottom, CFB), the reflectivity factor with attenuation correction at the CFB (zFactorFinalNearSurface, dBZ), the normalized gamma DSD parameters (paramDSD) and N<sub>w</sub> (dB) and D<sub>m</sub> (mm) evaluated at the CFB, as well as the precipitation type (TypePrecip) for the case of the DF product. It is worth mentioning that HS mode data results were not yet available for processing during this study.

2.1.2. Disdrometer Locations

Data from seven disdrometers deployed at different sites in the region of study were used. The Department of Applied Physics-Meteorology of the University of Barcelona manages six disdrometers. Three of them, plus a fourth one from the University of Grenoble-Alpes, were used during the Land Surface Interactions with the Atmosphere over the Iberian Semi-Arid Environment (LIAISE) field campaign in the Eastern Ebro subbasin [23]. The rest of the disdrometers were at Das Aerodrome (in the Eastern Pyrenees, during the Cerdanya-2017 [24] and the ARTEMIS field campaigns), the roof of the Faculty of Physics of the University of Barcelona and the Fabra Observatory of Royal Academy of Sciences and Arts of Barcelona also supporting the ARTEMIS campaign. Table 2 provides detailed information about each site, including temporal period covered and valid rainfall data for each site after quality control (for details, see next section).

<table><caption>**Table 2.** Information about the Parsivel disdrometers (model 1 and 2 as indicated by the superindex) used in the present study.</caption><thead><tr><th>Disdrometer Type</th><th>Disdrometer Site</th><th>Label (Subregions)</th><th>Lon (°E)</th><th>Lat (°N)</th><th>Height (m)</th><th>Start Date</th><th>End Date</th><th>Valid Data (min)</th></tr></thead><tbody><tr><td>Parsivel <sup>1</sup></td><td>Barcelona University</td><td>C01 (Coast)</td><td>2.11</td><td>41.38</td><td>98</td><td>1 January 2015</td><td>1 February 2024</td><td>51,679</td></tr><tr><td>Parsivel <sup>2</sup></td><td>Fabra Observatory</td><td>C02 (Coast)</td><td>2.12</td><td>41.42</td><td>411</td><td>26 July 2022</td><td>13 February 2024</td><td>12,537</td></tr><tr><td>Parsivel <sup>1,2</sup></td><td>Das</td><td>M01 (Mountain)</td><td>1.87</td><td>42.39</td><td>1097</td><td>9 December 2016</td><td>8 February 2024</td><td>59,388</td></tr><tr><td>Parsivel <sup>2</sup></td><td>Tarrega</td><td>P01 (Plain)</td><td>1.16</td><td>41.67</td><td>427</td><td>4 May 2021</td><td>14 June 2022</td><td>10,218</td></tr><tr><td>Parsivel <sup>2</sup></td><td>Mollerussa</td><td>P02 (Plain)</td><td>0.87</td><td>41.62</td><td>247</td><td>27 April 2021</td><td>5 December 2022</td><td>12,855</td></tr><tr><td>Parsivel <sup>2</sup></td><td>Tordera</td><td>P03 (Plain)</td><td>1.22</td><td>41.68</td><td>388</td><td>30 April 2021</td><td>14 June 2022</td><td>12,035</td></tr><tr><td>Parsivel <sup>2</sup></td><td>Cendrosa</td><td>P04 (Plain)</td><td>0.93</td><td>41.69</td><td>239</td><td>9 April 2021</td><td>12 October 2021</td><td>3616</td></tr></tbody></table>

The locations of the disdrometers are representative of three key areas with different climatic and orographic characteristics typical of Catalonia: mountain (in the Pyrenees mountains), plain (inland plain of the Segre River Valley) and coast. The disdrometers of Tordera, Mollerussa, Tarrega and Cendrosa are in the plain subregion, characterized by flat terrain with few orographic contrasts and an arid Köppen climate (Figure 1), conditioned by precipitation deficit. Disdrometers located at the Faculty of Physics and Fabra Observatory represent the coast subregion with a hot-summer Mediterranean Köppen climate, more exposed to Mediterranean heavy precipitation. The disdrometer at Das was in a valley at 1094 m a.s.l. in the mountain subregion and had a temperate Köppen climate.
<figure><img src="image_5.png" /><figcaption>Figure 1. Digital elevation model of the region of study and the three subregions considered (mountain, plain and coast) and disdrometer sites (black dots), showing range circles of 5 km (thin red line) and 10 km (black dotted line) around each site where GPM-DPR data were collected for the present study. The lower right corner shows a map with the Köppen climate classification of the region and the disdrometer locations.</figcaption></figure>

### 2.1.3. Disdrometer Data

The OTT Parsiveloptical disdrometer uses a 650 nm laser device with a power of 3 mW [25,26]. The laser emits a horizontal sheet of light 30 mm wide and 180 mm long. With a horizontal sampling area of 54 cm², particles passing through it cause a reduction in light intensity, resulting in the measurement of their size. The signal duration and particle size allow the estimation of particle velocity [27]. The size and fall speed of each particle is classified into 32 classes ranging from 0.05 to 20 m/s and 32 particle diameter classes ranging from 0.062 mm to 24.5 mm. Based on the recorded size and fall speed spectra, different variables are computed, including the present weather type (synop code 4677 [25]). Temporal resolution was set to 1 min aggregation periods for all disdrometers.

Quality control was applied, consisting of the following conditions: (1) to exclude non-liquid particles and errors associated with boundary effects [28], particle fall speeds did not differ more than ±50% from the empirical terminal fall speed V(D) [29]; (2) to further ensure liquid precipitation, the reported present weather (code 4677) was checked discarding all types containing solid particles [30]; (3) to compute DSD parameters consistently a minimum of 11 drops had to be present in each 1 min sample [15]. The DSD was computed according to the following expression:

$$N(D_i) = \frac{1}{A_{\text{eff}}(D_i) \times t \times \Delta D_i} \sum_{j=1}^{32} \frac{n_{ij}}{V_{D_i}} \quad (2)$$
where A<sub>eff</sub> is the effective sampling area (m²), t is the sampling time (60 s), ΔD<sub>i</sub> is the bin width (mm), n<sub>ij</sub> is the number of drops measured in the ith diameter class and jth drop velocity class, and V<sub>D<sub>i</sub></sub> is the drop velocity according to the theoretical diameter-drop velocity relationship [29]. In this case, as in [27,28], the edge effects mentioned above are considered: A<sub>eff</sub>(D<sub>i</sub>) = L(W - 0.5D<sub>i</sub>) where L = 180 mm and W = 30 mm (laser beam length and width, respectively). Finally, for each DSD, Z<sub>Ka,Ku</sub>, R, D<sub>m</sub>, N<sub>w</sub> and μ were calculated based on the nth-order moment (Mn) of the drop size distribution [8,31–33] using the following:

$$R(\text{mm/h}) = 3.6 \frac{\Pi}{6} 10^{-3} \sum_{i=1}^{32} N(D_i) V(D_i) D^3 \Delta D_i \quad (3)$$

$$Z_{K_a,K_u} (\text{mm}^6\text{m}^{-3}) = \frac{\lambda_{K_u,K_a}^4}{\Pi^5 |K_w|^2} \sum_{i=1}^{32} \sigma_{K_u,K_a}(D_i) N(D_i) \Delta D_i \quad (4)$$

$$M_n (\text{mm}^n \text{m}^{-3} \text{mm}^{-1}) = \sum_{i=1}^{32} D_i^n N(D_i) \Delta D_i \quad (5)$$

$$D_m (\text{mm}) = \frac{M_4}{M_3} \quad (6)$$

$$LWC (\text{gm}^{-3}) = \frac{\Pi 10^{-3} \rho_w}{6} \sum_{i=1}^{32} (D) D^3 \Delta D_i \quad (7)$$

$$N_w (\text{m}^{-3}\text{mm}^{-1}) = \frac{4^4 LWC}{\Pi D_m^4} \quad (8)$$

$$\mu = \frac{(7 - 11A) - \sqrt{(7 - 11A)^2 - 4(A - 1)(30A - 12)}}{2(A - 1)} \quad (9)$$

$$A = \frac{M_4^2}{M_2 M_6} \quad (10)$$

where LWC is the liquid water content (gm⁻³), λ is the wavelength (mm), K<sub>w</sub> is the complex dielectric constant of water, ρ<sub>w</sub> is the density of water (1 g/cm³), and σ<sub>Ku,Ka</sub> (mm²) is the backscatter radar cross-section for the Ku and Ka bands of a droplet of equivalent diameter D. For the calculation of the cross-sections, the T-matrix [34,35] estimation method was applied assuming (1) an ambient temperature of 20 °C; (2) the shape of hydrometeors according to the model proposed by Thurai et al. [36]; and (3) the distribution of hydrometeor canting angles modeled with a Gaussian distribution with mean 0° and standard deviation 10° [15]. These calculations were performed using the Python package pyTMatrix 0.3.3 [37].

Additionally, considering that the GPM assumes a normalized gamma-type DSD to estimate the DSD parameters (Equations (A1) and (A2), Appendix A) and based on Equations (A3)–(A8), the DSD measurements recorded by the disdrometers were used to compute k (specific attenuation), as well as the k/Z<sub>e</sub> and DFR (dB) ratios, by setting μ = 3 [10,14].

## 2.2. Methodology

As indicated above, the number of satellite overpasses coincident with rainfall events may be an important limiting factor when comparing satellite and disdrometer observations. To overcome or partly mitigate this problem, some previous studies considered not only satellite matches but also datasets of the area of study of both the disdrometers and the satellite without necessarily including satellite overpass matches [8,10]. Then these two relatively independent datasets (as opposed to the datasets with matches) can be compared, for example, by checking if biases are present, to better interpret the comparison of satellite matches with ground observations. According to this idea, below are described
the comparison of the so-called independent (non-matching) DPR and disdrometer datasets
and the matching approaches considered.

2.2.1. Comparison of Independent Datasets

To select DPR data comparable to the reference disdrometer data, precipitation DPR
observations within a 10 km radius around the location of the disdrometer sites were
considered. Duplicate information due to overlapping of the selected areas was eliminated
to avoid redundant data that could affect the characterization statistics (Figure 1). The
results of this analysis were stratified according to each disdrometer separately, into three
geographic areas with different climatic and orographic characteristics and considering all
the data together. For each subregion, it was ensured that only one data record existed at
any given time (in cases where more than one existed, only one was selected randomly).
The characterization of the entire domain was considered based on the union of all valid
data from the three subregions.

According to GPM documentation, the minimum detectable radar reflectivity and rainfall rate for the Ku- and Ka-bands are 13 dBZ, 17 dBZ and 0.5 and 0.2 mm/h, respectively. However, previous studies have observed improved detection of light precipitation using GPM DPR [38]. In addition, GPM DPR estimates over the study region showed minimum precipitation rates of 0.1 mm/h, therefore, in this work this threshold [8,15] was selected to fix precipitation events for both GPM DPR and disdrometer data.

2.2.2. Matching Approaches

Four different matching approaches were considered, based on similar previous stud-
ies [8,15]. An attempt was made to determine the most appropriate strategy considering
the performance of each GPM DPR scanning mode (Ka-FS, Ku-FS and DPR-FS). The four
methods are as follows:

a. Point: The disdrometer location was found within the footprint of the DPR (within the 5 km² pixel area) and so could be compared directly.

b. Mean 5 km: Disdrometer data were compared with the average of all DPR pixels within a 5 km radius of the disdrometer.

c. Mean 10 km: Disdrometer data were compared with the average of all DPR pixels within a radius of 10 km of the disdrometer.

d. Optimal: Disdrometer data were compared with the DPR pixel closest to the disdrometer within a 5 km radius and the nine DPR pixels containing the disdrometer. Finally, among these nine pixels, the pixel with closest radar reflectivity factor to that of the disdrometer was selected for comparison.

The methods proposed aim to reduce the spatiotemporal uncertainties that arise when comparing instantaneous measurements from space with ground-based measurements from disdrometers. Additionally, considering various methodologies allows us to understand the impact on the results and compare them with previous studies. As proposed by Adirosi et al. [15], due to advection processes, the significant DPR estimates determined in the CFB may not correspond to the corresponding pixel on the surface. To address this limitation, averaging methods are employed. The choice of a 5 km radius is based on the results of a sensitivity study and the physical considerations described by Adirosi et al. [15]. Similarly, in this work, a 10 km radius was used, which, while not considerably increasing the number of cases, yielded better results for some variables. A larger radius was not considered because, particularly in coastal areas, it would include parts of the sea, affecting the homogeneity of the selected terrain and potentially altering the microphysical characteristics of precipitation. Finally, the so-called optimal method, based on the work of Silvestro et al. [39], comparing ground-based weather radar observations with rain gauge data, primarily seeks to determine if the DPR can detect the characteristics of rain measured by the disdrometer, at least in its vicinity.

After selecting the GPM overpasses in rainy conditions, the 1-min DSD samples from
the disdrometers were averaged over a 10-min window to reduce the time and space
sampling problems between the GPM-DPR and the disdrometers. The results for the total number of precipitation exceedances and matches following these methodologies are shown in the results section. The comparison was made in terms of D<sub>m</sub> (mm), R (mm/h), Z<sub>Ku,Ka</sub> (mm<sup>6</sup>m<sup>-3</sup>) and N<sub>w</sub> (dB).

### 2.2.3. Verification Metrics

The comparison between the GPM DPR and disdrometer data was performed considering verification scores for both continuous variables and categorical events (Table 3).

<table><caption>**Table 3.** List of verification metrics used to evaluate DPR products.</caption><thead><tr><th>Name</th><th>Formula</th><th>Perfect Score</th></tr></thead><tbody><tr><td>Correlation Coefficient (CC)</td><td>CC = <sup>&Sigma;(O<sub>i</sub>-O&#773;<sub>i</sub>)(S<sub>i</sub>-S&#773;<sub>i</sub>)</sup>&frasl;<sub>&radic;&Sigma;(O<sub>i</sub>-O&#773;<sub>i</sub>)<sup>2</sup>&Sigma;(S<sub>i</sub>-S&#773;<sub>i</sub>)<sup>2</sup></sub></td><td>1</td></tr><tr><td>Normalized Mean Bias (NBias)</td><td>NBIAS = <sup><sup>1</sup>&frasl;<sub>n</sub>&sum;<sub>i=1</sub><sup>n</sup>(S<sub>i</sub>-O<sub>i</sub>)</sup>&frasl;<sub>O<sub>i</sub></sub> &times; 100</td><td>0</td></tr><tr><td>Normalized Mean Absolute Error (MAE)</td><td>NMAE = <sup>&sum;<sub>i=1</sub><sup>n</sup>|S<sub>i</sub>-O<sub>i</sub>|</sup>&frasl;<sub>n</sub></td><td>0</td></tr><tr><td>Normalized Root Mean Square Error (RMSE)</td><td>NRMSE = <sup>&radic;<><sup>1</sup>&frasl;<sub>n</sub>&sum;<sub>i=1</sub><sup>n</sup>(S<sub>i</sub>-O<sub>i</sub>)<sup>2</sup></sup>&frasl;<sub>O<sub>i</sub></sub></td><td>0</td></tr><tr><td>Accuracy</td><td>TP = <sup>TP</sup>&frasl;<sub>All classifications</sub></td><td>1</td></tr><tr><td>Precision</td><td>TP+FP = <sup>TP</sup>&frasl;<sub>TP+FN</sub></td><td>1</td></tr><tr><td>Recall</td><td>TP+FN = <sup>TP</sup>&frasl;<sub>All classifications</sub></td><td>1</td></tr></tbody></table>

S<sub>i</sub> is the value of satellite precipitation estimates for the ith event, O<sub>i</sub> is the value of disdrometer observation for the ith event, and n is the number of observed records. S&#773;<sub>i</sub> and O&#773;<sub>i</sub> are the mean of satellite and observations, respectively. The values of TP and FP are based on the confusion matrix (Table 4).

<table><caption>**Table 4.** Confusion Matrix for multi-class classification (3 × 3).</caption><thead><tr><th rowspan="2">Predicted Class</th><th colspan="4">Observed Class</th><th rowspan="2">Total</th></tr><tr><th>A</th><th>B</th><th>C</th><th>Total</th></tr></thead><tbody><tr><th>A</th><td>TP<sub>A</sub></td><td>F<sub>BA</sub></td><td>F<sub>CA</sub></td><td>TP<sub>A</sub> + F<sub>BA</sub> + F<sub>CA</sub></td></tr><tr><th>B</th><td>F<sub>AB</sub></td><td>TP<sub>B</sub></td><td>F<sub>CB</sub></td><td>F<sub>AB</sub> + TP<sub>B</sub> + F<sub>CB</sub></td></tr><tr><th>C</th><td>F<sub>AC</sub></td><td>F<sub>BC</sub></td><td>TP<sub>C</sub></td><td>F<sub>AC</sub> + F<sub>BC</sub> + TP<sub>C</sub></td></tr><tr><th>Total</th><td>TP<sub>A</sub> + F<sub>AB</sub> + F<sub>AC</sub></td><td>F<sub>BA</sub> + TP<sub>B</sub> + F<sub>BC</sub></td><td>F<sub>CA</sub> + F<sub>CB</sub> + TP<sub>C</sub></td><td>All classifications</td></tr></tbody></table>

Note that the scores considered for verification of categorical forecasts are based on the so-called confusion matrix, also called the contingency table [40]. These scores are typically used in machine learning applications [41] but in this case are applied to multi-category events [42] considering a 3 × 3 confusion matrix (Table 4). As shown, TP<sub>A</sub>, TP<sub>B</sub> and TP<sub>C</sub> are the number of true positive samples in classes A, B and C, respectively. False negatives (FN) of any class, which are in a column, can be calculated by adding the errors in that class/column, whereas the false positives for any predicted class, which are in a row, represent the sum of all errors in that row. For example, the false positive rate in class A (FP<sub>A</sub>) is calculated as FP<sub>A</sub> = F<sub>BA</sub> + F<sub>CA</sub> and the false negative rate in the A class is FN<sub>A</sub> = F<sub>AB</sub> + F<sub>AC</sub> [43].

## 3. Results

### 3.1. GPM CO vs. Disdrometer-Derived Independent Estimates

Figure 2 shows the histograms of the probability of occurrence with respect to the following variables: reflectivity factor (Z<sub>Ka,Ku</sub>), precipitation intensity (R) and DSD parameters (D<sub>m</sub>, N<sub>w</sub>, μ) obtained for both GPM DPR and disdrometer independent datasets. The dashed lines represent the median of the distribution of each dataset, and the solid curve represents the kernel density estimation (KDE) curve associated with each distribution.
<figure><img src="image_6.png" /><figcaption>**Figure 2.** Histograms of (a) $Z_{Ku}$, (b) $Z_{Ka}$, (c) $\log_{10}(R)$, (d) $D_m$, (e) $10 \log_{10}(N_w)$ and (f) shape parameter ($\mu$) derived from all disdrometer and GPM DPR (DF) datasets.</figcaption></figure>

Despite the differences in the number of samples and period of record in the two independent datasets, important similarities can be observed in the distributions. However, the DPR distributions are slightly shifted to higher values compared to the disdrometer distributions in reflectivity ($Z_{Ka}$ and $Z_{Ku}$), precipitation intensity and $D_m$, which might indicate little skill in detecting the lower thresholds in these variables with respect to the data taken as reference. Specifically, in the case of $D_m$, while in the disdrometers, the highest probability of occurrence occurs for values less than 1 mm, in the DPR, this occurs between 1.0 and 1.5 mm. In contrast, the $N_w$ values obtained from the DSD of the disdrometers had a wider range, especially with a tendency to detect higher thresholds and a higher mean than that of the DPR. Figure 2f shows the discrepancies between the mean $\mu$ close to 10 in the case of the reference data, which is different from that set by the DPR algorithm ($\mu$ = 3). Several authors [8,9,28] discussed the limitations of setting this parameter.

Tables 5 and 6 show the number and median and maximum values of each variable analyzed for each dataset. In addition, to evaluate how the location of the disdrometers might affect the precipitation and DSD parameters, three zones with different orographic and climatic characteristics were analyzed. The plain, coastal and mountain regions were constructed by combining the data from the disdrometers that compose these homogeneous regions (Figure 1).

In general, there is little variability among the disdrometer statistics according to geographic location. The median reflectivity values range between 20 and 24 dBZ, precipitation intensities between 0.48 and 0.89 mm/h, $D_m$ close to 1 mm and the intercept parameter ($N_w$) around 35 dB. Similarly, in the analysis of the observations in regions with different climatologies, the behavior of the variables was similar. The coastal area shows slightly higher median values, and the maximum intensity is reported at the Fabra Observatory, which is consistent with previous rain gauge-based climatologies in this region reporting higher rainfall rates near or at the coast compared to inland areas [44]. This behavior of the variables can be compared with the results obtained by Adirosi et al. [15] in Italy, where median values of 23.4, 21.9, 0.73, 35.72 and 1.05 (see Table 5 last row) were obtained for the variables $Z_{Ka}$, $Z_{Ku}$, $R$, $N_w$ and $D_m$, respectively.
<table><caption>**Table 5.** Statistics of the different rainfall and DSD parameters for each single dataset, three similar orographic and climatic regions and all datasets together from the disdrometers.</caption><thead><tr><th rowspan="2">Dataset</th><th colspan="3">Z<sub>Ka</sub> (dBZ)</th><th colspan="3">Z<sub>Ku</sub> (dBZ)</th><th colspan="3">R (mm/h)</th><th colspan="3">N<sub>w</sub> (dB)</th><th colspan="3">D<sub>m</sub> (mm)</th></tr><tr><th>N</th><th>Median</th><th>Max</th><th>Median</th><th>Max</th><th>Median</th><th>Max</th><th>Median</th><th>Max</th><th>Median</th><th>Max</th><th>Median</th><th>Max</th><th>Median</th><th>Max</th></tr></thead><tbody><tr><td>C01</td><td>51,679</td><td>24.77</td><td>47.27</td><td>24.92</td><td>51.86</td><td>0.89</td><td>60.83</td><td>33.75</td><td>51.61</td><td>1.19</td><td>6.28</td><td></td><td></td><td></td><td></td></tr><tr><td>C02</td><td>12,537</td><td>22.79</td><td>52.40</td><td>22.68</td><td>55.83</td><td>0.77</td><td>183.27</td><td>35.23</td><td>53.06</td><td>1.04</td><td>4.40</td><td></td><td></td><td></td><td></td></tr><tr><td>M01</td><td>59,388</td><td>21.13</td><td>49.74</td><td>20.92</td><td>54.88</td><td>0.60</td><td>107.70</td><td>36.16</td><td>52.04</td><td>0.96</td><td>7.86</td><td></td><td></td><td></td><td></td></tr><tr><td>P01</td><td>10,218</td><td>20.30</td><td>47.83</td><td>20.04</td><td>52.47</td><td>0.48</td><td>74.61</td><td>34.77</td><td>50.17</td><td>0.98</td><td>3.85</td><td></td><td></td><td></td><td></td></tr><tr><td>P02</td><td>12,855</td><td>21.49</td><td>50.26</td><td>21.38</td><td>53.97</td><td>0.54</td><td>115.74</td><td>34.09</td><td>49.98</td><td>1.06</td><td>4.91</td><td></td><td></td><td></td><td></td></tr><tr><td>P03</td><td>12,035</td><td>21.04</td><td>50.16</td><td>20.79</td><td>53.76</td><td>0.56</td><td>114.86</td><td>35.27</td><td>50.00</td><td>1.00</td><td>7.59</td><td></td><td></td><td></td><td></td></tr><tr><td>P04</td><td>3616</td><td>21.56</td><td>50.71</td><td>21.44</td><td>55.20</td><td>0.54</td><td>132.65</td><td>34.75</td><td>50.88</td><td>1.04</td><td>4.75</td><td></td><td></td><td></td><td></td></tr><tr><td>Coast</td><td>60,570</td><td>24.26</td><td>52.40</td><td>24.34</td><td>55.83</td><td>0.85</td><td>183.27</td><td>34.04</td><td>53.06</td><td>1.16</td><td>6.28</td><td></td><td></td><td></td><td></td></tr><tr><td>Mountain</td><td>59,388</td><td>21.13</td><td>49.74</td><td>20.92</td><td>54.88</td><td>0.60</td><td>107.70</td><td>36.16</td><td>52.04</td><td>0.96</td><td>7.86</td><td></td><td></td><td></td><td></td></tr><tr><td>Plain</td><td>24,254</td><td>20.44</td><td>50.71</td><td>20.21</td><td>55.20</td><td>0.48</td><td>132.65</td><td>34.43</td><td>50.88</td><td>1.01</td><td>7.59</td><td></td><td></td><td></td><td></td></tr><tr><td>All</td><td>144,212</td><td>22.28</td><td>52.40</td><td>22.16</td><td>55.83</td><td>0.66</td><td>183.27</td><td>34.97</td><td>53.06</td><td>1.04</td><td>7.86</td><td></td><td></td><td></td><td></td></tr></tbody></table>

<table><caption>**Table 6.** Same as Table 4, but with dual-frequency DPR data. Note that the number of Ka-band reflectivity data is different from other variables because only the Ka inner swath was available.</caption><thead><tr><th rowspan="2">Dataset</th><th colspan="3">Z<sub>Ka</sub> (dBZ)</th><th colspan="3">Z<sub>Ku</sub> (dBZ)</th><th colspan="3">R (mm/h)</th><th colspan="3">N<sub>w</sub> (dB)</th><th colspan="3">D<sub>m</sub> (mm)</th></tr><tr><th>N</th><th>Median</th><th>Max</th><th>N</th><th>Median</th><th>Max</th><th>Median</th><th>Max</th><th>Median</th><th>Max</th><th>Median</th><th>Max</th><th>Median</th><th>Max</th><th>Median</th><th>Max</th></tr></thead><tbody><tr><td>C01</td><td>291</td><td>25.10</td><td>45.70</td><td>351</td><td>24.46</td><td>51.13</td><td>0.96</td><td>57.07</td><td>33.19</td><td>51.29</td><td>1.19</td><td>3.00</td><td></td><td></td><td></td><td></td></tr><tr><td>C02</td><td>312</td><td>25.00</td><td>42.34</td><td>376</td><td>24.24</td><td>51.13</td><td>0.90</td><td>36.33</td><td>33.24</td><td>51.29</td><td>1.18</td><td>4.45</td><td></td><td></td><td></td><td></td></tr><tr><td>M01</td><td>360</td><td>23.47</td><td>37.54</td><td>423</td><td>23.25</td><td>46.34</td><td>0.78</td><td>18.49</td><td>33.04</td><td>50.65</td><td>1.16</td><td>3.00</td><td></td><td></td><td></td><td></td></tr><tr><td>P01</td><td>262</td><td>23.16</td><td>41.37</td><td>304</td><td>22.38</td><td>51.05</td><td>0.68</td><td>27.75</td><td>33.11</td><td>43.93</td><td>1.12</td><td>5.00</td><td></td><td></td><td></td><td></td></tr><tr><td>P02</td><td>203</td><td>24.23</td><td>38.67</td><td>232</td><td>23.18</td><td>44.61</td><td>0.75</td><td>11.71</td><td>33.34</td><td>41.34</td><td>1.12</td><td>3.00</td><td></td><td></td><td></td><td></td></tr><tr><td>P03</td><td>269</td><td>23.63</td><td>41.37</td><td>304</td><td>22.20</td><td>51.05</td><td>0.67</td><td>27.75</td><td>33.13</td><td>43.93</td><td>1.11</td><td>4.99</td><td></td><td></td><td></td><td></td></tr><tr><td>P04</td><td>234</td><td>23.42</td><td>37.44</td><td>273</td><td>22.38</td><td>47.42</td><td>0.68</td><td>13.18</td><td>33.34</td><td>41.51</td><td>1.11</td><td>3.56</td><td></td><td></td><td></td><td></td></tr><tr><td>Coast</td><td>603</td><td>25.05</td><td>45.70</td><td>727</td><td>24.40</td><td>51.13</td><td>0.92</td><td>57.07</td><td>33.21</td><td>51.29</td><td>1.18</td><td>4.45</td><td></td><td></td><td></td><td></td></tr><tr><td>Mountain</td><td>360</td><td>23.47</td><td>37.54</td><td>423</td><td>23.25</td><td>46.34</td><td>0.78</td><td>18.49</td><td>33.04</td><td>50.65</td><td>1.16</td><td>3.00</td><td></td><td></td><td></td><td></td></tr><tr><td>Plain</td><td>968</td><td>23.55</td><td>41.37</td><td>1113</td><td>22.59</td><td>51.05</td><td>0.69</td><td>27.75</td><td>33.21</td><td>43.93</td><td>1.11</td><td>5.00</td><td></td><td></td><td></td><td></td></tr><tr><td>All</td><td>1931</td><td>23.89</td><td>45.70</td><td>2263</td><td>23.19</td><td>51.12</td><td>0.77</td><td>57.07</td><td>33.17</td><td>51.29</td><td>1.14</td><td>5.00</td><td></td><td></td><td></td><td></td></tr></tbody></table>

The statistics obtained from DPR data showed higher median values for all variables, except N<sub>w</sub>. However, DPR data can capture the variability between different zones, exhibiting the highest median values in the coastal zones. Similarly, the maximum values observed by disdrometers were much higher in all datasets than those recorded by the DPR DF. Although other studies have commented on the limitations in the detection of extreme values by remote sensing products [45], in this case, we cannot draw any conclusions because such values are subject to the availability of DPR data at the time of the occurrence of this type of extreme event.

### 3.1.1. Rain Rate Effects

The DSD-derived precipitation characteristics were stratified into six rain rate intensity classes. For this purpose, the disdrometer and DPR DF records were considered together and grouped according to the three subregions (plain, mountain and coast) mentioned above. Figure 3 (top panel) shows the normalized density distributions for the datasets. A necessary condition to obtain the mean of these variables in each precipitation intensity interval was that they had at least 10 records. It is evident that all datasets behave similarly, with a high representation of data -as expected-, at lower precipitation intensities and much lower for moderate and high intensities. Despite the lack of temporal concurrence between the disdrometers and the DPR, both sources provide relatively similar results with similar qualitative behavior.
<figure><img src="image_7.png" /><figcaption>**Figure 3.** Comparison of normalized frequency distribution showing the ratio of data at each rain rate interval for disdrometer (in blue) and DPR (in red) data (top panel) and, similarly, comparison of DSD parameters D<sub>m</sub> (middle panel) and N<sub>w</sub> (bottom panel). All panels show values for the subregions plain (dashed line), coast (dotted line), mountain (semi-dashed line) and the whole region (all, thick line), for both disdrometer (DIS_) and DPR (DPR_) data.</figcaption></figure>

Figure 3 (middle and bottom panels) illustrates the variation in the mean D<sub>m</sub> and N<sub>w</sub> observed and estimated using the DF DPR algorithm as a function of precipitation intensity. The DSD parameters were similar in all regions for intensities below 4 mm/h. At thresholds higher than 16 mm/h, the D<sub>m</sub> for example differs by more than 0.5 mm between coastal and mountain areas according to the disdrometer data. The intercept parameter begins to differentiate in the regions from intensities between 8–16 mm/h with a difference of more than 5 dB between coastal and inland areas according to the DPR and close to 6 dB between coastal and mountain areas according to disdrometer data.

From these figures, the DSD parameters obtained by the DF algorithms capture the variability observed at different intensities, although with overestimates of the mean D<sub>m</sub> and underestimates of the mean N<sub>w</sub>, showing the greatest differences at moderate and high intensities and being much more sensitive to errors in drops greater than 1.5 mm. These results are similar to those found by Del Castillo-Velarde [14], in which it is stated that the DF algorithm is susceptible to the uncertainty of μ fixation, which causes an underestimation of N<sub>w</sub>.

Comparisons of the DSD parameters show an overestimation of D<sub>m</sub> of about 0.1 mm at low and moderate precipitation rates (0.1–1, 1–2, 2–4 mm/h) and of 0.4 mm at precipitation rates higher than 4 mm/h by the DF algorithm with respect to the disdrometer. In contrast, the behavior of N<sub>w</sub> was underestimated by the DPR, with a maximum value close to 6 dB at moderate precipitation rates (4–8 mm/h). Compared to the studies of [10], the magnitudes of underestimation and overestimation of D<sub>m</sub> and N<sub>w</sub> are very similar. However, here the behavior of the DSD parameters occurs in reverse; that is, the mean value of D<sub>m</sub> is
overestimated and $N_w$ is underestimated. This sensitivity analysis shows that the results presented here are consistent and that differences with other studies may be due to spatial and temporal sampling factors and differences arising from the use of other types of disdrometers [46,47].

### 3.1.2. Stratiform vs. Convective Regimes

In this section, we analyze the stratiform and convective regimes associated with the DPR and disdrometer data, as well as different related microphysical processes. For this purpose, we consider the classification for a given rainfall DSD proposed by Dolan et al. [48] based on the clustering of $D_0$ and $N_w$ values obtained from global disdrometric records. From this perspective, six groups with independent characteristics were defined: Group 1, Group 3, Group 5 and Group 6 (Figure 4b) are characterized by convective precipitation processes, while the Group 2 and Group 4 are stratiform precipitation processes, with increasing $D_0$ and decreasing $N_w$. Complementarily, we also considered the DPR algorithm classification, in which each pixel is assigned a so-called precipitation type label: stratiform, convective or other. Figure 4 plots all disdrometer records in the $D_0$-log ($N_w$) space overlaid with the diagram proposed by Dolan et al. [48].

<figure><img src="image_8.png" /><figcaption>**Figure 4.** (a) Scatter density plot of raindrop size distribution measurements from all disdrometers in the $D_0$-log($N_w$) space overlapped by the stratiform region (limited by turquoise dashed line) and convective region (limited by red dashed line) defined by Dolan et al. [48]. Disdrometer data density increase from dark to white dots, and DPR DF convective and stratiform types are indicated by red and turquoise dots, respectively. (b) Convective, stratiform and microphysical dominant process regions in the $D_0$-log($N_w$) space according to Dolan et al. [48] overlapped by disdrometer (grey dots) and DF DPR (cyan dots) data.</figcaption></figure>

Note that in Figure 4a, most of the measurements were in the stratiform part of the plot, which is consistent with the values of the disdrometer parameters discussed above (low liquid water content and small mean droplet diameters). In the same figure, the values classified as stratiform (turquoise dots) and convective (red dots) by the DPR DF algorithm are plotted. Although the highest percentage of data are in the stratiform domain, there is a large scatter of data that does not fit this classification, especially for events classified as convective rain (Table 7).
<table><caption>**Table 7.** Percentage (%) of data according to the rainfall regime considering stratiform, convective, ambiguous (between stratiform and convective) and outlier (out of the stratiform and convective classification domain) according to Dolan et al. [48].</caption><thead><tr><th></th><th>Stratiform</th><th>Convective</th><th>Ambiguous</th><th>Outlier</th></tr></thead><tbody><tr><th>Disdrometer</th><td>53</td><td>31</td><td>13</td><td>2</td></tr><tr><th>GPM DPR (DF)</th><td>73</td><td>18</td><td>8</td><td>1</td></tr></tbody></table>

Figure 4b provides information on the dominant precipitation mechanisms in the disdrometer and DPR DF records following Dolan et al. [48]. According to this classification, the disdrometer stratiform rainfall was dominant (Table 7) and strongly influenced by vapor deposition followed by riming processes (Table 8). Convective events, on the other hand, are not associated with a single well-defined microphysical mechanism. Moreover, an important part of the events (13%) are classified as ambiguous (neither convective nor stratiform) and are associated with different microphysical mechanisms. DPR records have a large percentage of data (46%) in areas that fall outside the classification range, although the influence of stratiform precipitation processes by riming can be appreciated (Table 8).

<table><caption>**Table 8.** Percentage (%) of data associated with microphysical precipitation mechanism groups proposed by Doan et al. [48] (see Figure 4b).</caption><thead><tr><th rowspan="2"></th><th rowspan="2">Group 1</th><th colspan="4">Group 2</th><th colspan="3">Group 6</th></tr><tr><th>Vapor<br>Deposition</th><th>Weak<br>Convection</th><th>Aggregation/<br>Riming</th><th>Collision-<br>Coalescence</th><th>Ice-Based</th><th>Ambiguous</th><th>Outlier</th></tr></thead><tbody><tr><th>Disdrometer</th><td>4</td><td>29</td><td>4</td><td>14</td><td>0</td><td>1</td><td>12</td><td>32</td></tr><tr><th>GPM DPR (DF)</th><td>4</td><td>15</td><td>0</td><td>22</td><td>0</td><td>2</td><td>11</td><td>46</td></tr></tbody></table>

### 3.2. Analysis of Satellite Overpass Events Coincident with Disdrometer Data

This section evaluates the performance of the GPM-DPR estimates for the different matching methods with disdrometer data described in Section 2.2. Table 9 shows the number of overpasses of the GPM-DPR (2014–2023) with precipitation, and the number of matches from the different proposed methods. A comparison was carried out for the two algorithms: DPR DF and SF in the FS mode. Analyses of the variables Z, R, D<sub>m</sub> and N<sub>w</sub> were considered.

<table><caption>**Table 9.** Summary of overpasses over disdrometer sites without (Group A) and with (Group B) concurrent disdrometer rainfall data for DPR, Ka and Ku FS modes. Matching methods between GPM observations and disdrometer sites are point, 5 km, 10 km, 9 pixels (Group A) and optimal (Group B).</caption><thead><tr><th rowspan="2">GPM Product</th><th rowspan="2">Matching Method</th><th colspan="4">Group A: GPM CO Overpasses with Rain without Necessarily Matching Disdrometer Data</th><th colspan="4">Group B: GPM CO Overpasses with Rain Matching Disdrometer Data</th></tr><tr><th>Point</th><th>5 km</th><th>10 km</th><th>9 pixels</th><th>Point</th><th>Mean 5 km</th><th>Mean 10 km</th><th>Optimal</th></tr></thead><tbody><tr><td></td><td>DPR-FS</td><td>142</td><td>272</td><td>460</td><td>567</td><td>19</td><td>33</td><td>39</td><td>40</td></tr><tr><td></td><td>Ka-FS</td><td>69</td><td>157</td><td>289</td><td>328</td><td>12</td><td>27</td><td>33</td><td>34</td></tr><tr><td></td><td>Ku-FS</td><td>142</td><td>270</td><td>463</td><td>569</td><td>20</td><td>34</td><td>41</td><td>41</td></tr></tbody></table>

Between March 2014 and November 2023, the GPM CO passed (at least one footprint) over the region of Catalonia 2089 times and, of them, 1126 had at least one footprint with rainfall (hereafter rain overpasses). It is important to mention that the Ka-band presents a lower number of cases compared to the others (Table 9) because only inner swath data are available due to the DPR scan pattern change in May 2018. After that change, the data were reprocessed, leaving the payoffs in HS scan mode in the outer swath, for which no data are
available as of the date of this study. The differences in the number of overpasses between the DPR and Ku-band products in both Group A and Group B, although insignificant in this case, are due to differences in the precipitation estimation algorithms around the 0.1 mm/h threshold.

The main statistics for quantifying the error between the DPR estimates at the CFB and disdrometer ground-based values are shown in Table 10. Values marked in red represent the worst scores and those in green the best scores for each group of variables and error statistics analyzed.

<table><caption>**Table 10.** Statistics of the comparison between the GPM DF and SF products and disdrometer data for different variables and matching methods. Statistical significance of CC is indicated with an asterisk(*) and was tested with the t-test using a significance level of 0.05. The best and worst statistics obtained for each product, method and variable are marked in green and red, respectively. Note that all variables listed in the table are dimensionless.</caption><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="4">Point</th><th colspan="4">Mean 5 km</th><th colspan="4">Mean 10 km</th><th colspan="4">Optimal</th></tr><tr><th>NBIAS</th><th>NMAE</th><th>NRMSE</th><th>CC</th><th>NBIAS</th><th>NMAE</th><th>NRMSE</th><th>CC</th><th>NBIAS</th><th>NMAE</th><th>NRMSE</th><th>CC</th><th>NBIAS</th><th>NMAE</th><th>NRMSE</th><th>CC</th></tr></thead><tbody><tr><td rowspan="2">R</td><td>DF</td><td>-46.12</td><td>0.70</td><td>1.20</td><td>0.48*</td><td>-39.21</td><td>0.66</td><td>1.02</td><td>0.31</td><td>-16.88</td><td>0.59</td><td>0.90</td><td>0.70*</td><td>0.26</td><td>0.61</td><td>1.60</td><td>0.77*</td></tr><tr><td>SF</td><td>-49.84</td><td>0.69</td><td>1.18</td><td>0.37*</td><td>-46.93</td><td>0.64</td><td>1.01</td><td>0.30*</td><td>-31.63</td><td>0.66</td><td>0.99</td><td>0.44*</td><td>-35.35</td><td>0.43</td><td>0.78</td><td>0.78*</td></tr><tr><td rowspan="2">Z<sub>Ka</sub></td><td>DF</td><td>-6.07</td><td>0.19</td><td>0.27</td><td>0.61</td><td>-7.55</td><td>0.17</td><td>0.22</td><td>0.61*</td><td>-4.68</td><td>0.18</td><td>0.23</td><td>0.66*</td><td>-2.58</td><td>0.09</td><td>0.14</td><td>0.88*</td></tr><tr><td>SF</td><td>-13.55</td><td>0.16</td><td>0.20</td><td>0.27</td><td>-11.58</td><td>0.16</td><td>0.19</td><td>0.42*</td><td>-7.37</td><td>0.17</td><td>0.21</td><td>0.48*</td><td>-2.81</td><td>0.10</td><td>0.16</td><td>0.77*</td></tr><tr><td rowspan="2">Z<sub>Ku</sub></td><td>DF</td><td>-10.23</td><td>0.20</td><td>0.29</td><td>0.63*</td><td>-10.75</td><td>0.18</td><td>0.25</td><td>0.63*</td><td>-8.27</td><td>0.20</td><td>0.26</td><td>0.66*</td><td>-6.04</td><td>0.12</td><td>0.16</td><td>0.88*</td></tr><tr><td>SF</td><td>-9.23</td><td>0.20</td><td>0.29</td><td>0.63*</td><td>-11.53</td><td>0.18</td><td>0.27</td><td>0.64*</td><td>-8.12</td><td>0.20</td><td>0.25</td><td>0.65*</td><td>-5.37</td><td>0.10</td><td>0.16</td><td>0.88*</td></tr><tr><td rowspan="2">D<sub>m</sub></td><td>DF</td><td>-1.08</td><td>0.24</td><td>0.28</td><td>0.65*</td><td>2.05</td><td>0.21</td><td>0.27</td><td>0.56*</td><td>1.82</td><td>0.22</td><td>0.28</td><td>0.51*</td><td>0.96</td><td>0.14</td><td>0.18</td><td>0.83*</td></tr><tr><td>SF</td><td>-1.68</td><td>0.23</td><td>0.27</td><td>0.67*</td><td>2.82</td><td>0.23</td><td>0.32</td><td>0.38*</td><td>5.30</td><td>0.23</td><td>0.34</td><td>0.33*</td><td>2.99</td><td>0.14</td><td>0.19</td><td>0.83*</td></tr><tr><td rowspan="2">N<sub>w</sub></td><td>DF</td><td>-7.01</td><td>0.12</td><td>0.16</td><td>0.34</td><td>-7.59</td><td>0.11</td><td>0.14</td><td>0.32</td><td>-5.83</td><td>0.09</td><td>0.12</td><td>0.35*</td><td>-5.12</td><td>0.11</td><td>0.13</td><td>0.39*</td></tr><tr><td>SF</td><td>-6.77</td><td>0.12</td><td>0.15</td><td>0.34</td><td>-8.61</td><td>0.12</td><td>0.15</td><td>0.16</td><td>-8.36</td><td>0.12</td><td>0.15</td><td>-0.01</td><td>0.29</td><td>0.12</td><td>0.14</td><td>0.19</td></tr></tbody></table>

Figure 5 shows the scatterplots of rainfall rate R (mm/h), Z<sub>Ka</sub> (dBZ), Z<sub>Ku</sub> (dBZ), mass-weighted mean drop diameter (D<sub>m</sub>, mm) and N<sub>w</sub> (dB) of the GPM DPR and disdrometers. Regarding rainfall intensity, the point and mean methods show a certain dispersion of the data in general around the 1:1 line (dashed line). Although it is less evident in the optimal methods, for intensity values higher than 4 mm/h, it is again marked in all methodologies. In fact, the correlation values are generally higher than 0.7 in the optimal method, and in the point and mean methods most cases are lower than 0.5, showing a worse performance in the SF algorithm.

In the analysis of the intensity of precipitation, the optimal and mean 10 km methods, generally, show the lowest values of NBIAS, NMAE and NRMSE, as well as higher values of correlation, displaying also a substantial improvement of the DF algorithm over the SF over a mean of 10 km. For the rest of the methods, the behavior of the SF and DF is similar. The NBIAS shows a marked difference in the DF algorithm between the point method (−46%) and optimal (0.26%); however, this could be due to error compensation, a disadvantage associated with this statistic.

The errors associated with the reflectivity in both Ka and Ku-bands are generally below 10%. Again, the worst results are observed with the point method and are better in the optimal method, with errors not exceeding 6%. It should be noted that the optimal method precisely optimizes the comparison with respect to reflectivity. The improvements in SF and DF behave similarly, although a slight improvement is observed in the SF returns associated with the Ka-band, which may be associated with a smaller number of records in the selected sample. A higher dispersion of the Ka-band reflectivity can also be observed, which can be verified with slightly lower correlation values and errors in the Ku-band. In terms of the MAE and RMSE, there were hardly any differences between the point and mean methods. However, these values were almost halved, with values barely exceeding 0.15 in the optimal method.
<figure><img src="image_9.png" /><figcaption>**Figure 5.** Scatterplots between disdrometer (x-axis) and GPM products (y-axis) considering rainfall rate (first row), $Z_{Ka}$ (second row), $Z_{Ku}$ (third row), $D_m$ (fourth row) and $dBN_w$ (fifth row) and four matching methods (point, mean 5 km, mean 10 km and optimal).</figcaption></figure>

Agreement with respect to the $D_m$ values depends on the method applied to the selection of cases. In the point method, unlike the others, the $D_m$ values tended to be
underestimated by approximately 1 mm and 1.7 mm under the SF and DF algorithms, respectively. However, the mean methods show an overestimation that reaches a maximum value of 5.3 mm in the SF (mean 10 km). In these methods, the lowest correlation values were also obtained, whereas the best values were recorded in the optimal method with 0.83, and the highest point was 0.65. There is a clear improvement in the DF algorithms with respect to SF, especially in the mean methods, where values of 1 and 2 mm are overestimated, and values higher than 2 mm are underestimated.

As in the results of Adirosi et al. [15], the concordance in terms of N<sub>w</sub> was not satisfactory. Although the correlations this time turned out to be better, they lack statistical significance: the NMAE values were very similar, close to 12% in all cases, while the NBIAS was higher, similar to the work of Seela et al. [8], increasing the underestimation in our cases. Although earlier versions of the DPR were used in those works, it is shown that the deficiencies in N<sub>w</sub> estimates remain. This may support the idea that the discrepancies between satellite- and disdrometer-based N<sub>w</sub> may be due to the parameterization used by the GPM to model DSD.

3.2.1. Single- vs. Dual-Frequency-derived Estimates

Figure 6 shows the behavior of N<sub>w</sub> versus D<sub>m</sub> comparing all disdrometer data and two overpass matching methods (9-pixels and optimal) with GPM CO single- and dual-frequency estimates. The GPM data follow the typical D<sub>m</sub>-10log<sub>10</sub> N<sub>w</sub> behavior reported by Adirosi et al. [15], although they are concentrated, mainly SF, at approximately 30–35 dBN<sub>w</sub>. As illustrated in Figure 6, comparing single- vs. dual-frequency estimates, it is apparent that the dual-frequency pattern in the D<sub>m</sub>-10log<sub>10</sub> N<sub>w</sub> space is closer to disdrometer data than the single-frequency pattern, which is consistent with the improved scores obtained by DF-derived estimates seen in Table 10.

<figure><img src="image_10.png" /><figcaption>**Figure 6.** D<sub>m</sub> vs. N<sub>w</sub> for all available disdrometer data (grey dots), GPM data of nine pixels around the disdrometers (cyan dots) and GPM data coincident with disdrometers (violet dots) under optimal method showing GPM single-frequency (a)- and dual-frequency (b)-derived estimates. The black and red dots with the error bars represent the averages and standard deviations of the disdrometer dataset and GPM 9 pixels method.</figcaption></figure>

Further insight about differences between single- and dual-frequency-derived esti-
mates can be seen in Figure 7, which shows a Taylor Diagram that displays the STD-
normalized CC and RMSE with the data obtained by the point method and the optimal
method for the five variables of analysis. The benchmark represents the standard deviation
and the correlation coefficient equal to unity. We observe an improvement using the optimal method, especially in the estimates of $Z_{Ka,Ku}$ and $D_m$. However, considering the analysis of precipitation intensity using the DF algorithm, the optimal yielded worse results. The differences between the methods may be related to the variability in precipitation in the pixels around the disdrometer.

<figure><img src="image_11.png" /><figcaption>**Figure 7.** Taylor diagram with the data obtained by the point and the optimal methods for R, $Z_{Ka}$, $Z_{Ku}$, $D_m$ and $N_w$ for single-frequency (a) and dual-frequency (b) GPM-derived estimates.</figcaption></figure>

### 3.2.2. Stratiform vs. Convective Regimes

The precipitation type classification (including stratiform, convective and other regimes) provided by the DPR (variable named TypePrecip) was compared with the classification proposed by Dolan et al. [48] applied to disdrometer data. Table 11 shows the confusion matrix obtained by comparing the two datasets and considering the matches with the optimal method. Note that an “ambiguous” class appears to classify records that do not belong to stratiform or convective regimes using either method.

<table><caption>**Table 11.** Confusion matrix between ambiguous, stratiform and convective regimes classified by DPR DF and disdrometer data matched with the optimal approach, listing totals for each regime.</caption><thead><tr><th rowspan="2"></th><th rowspan="2"></th><th colspan="4">Disdrometer</th></tr><tr><th>Ambiguous</th><th>Convective</th><th>Stratiform</th><th>Total</th></tr></thead><tbody><tr><th rowspan="4">DPR DF</th><th>Ambiguous</th><td>1</td><td>0</td><td>1</td><td>2</td></tr><tr><th>Convective</th><td>2</td><td>0</td><td>0</td><td>2</td></tr><tr><th>Stratiform</th><td>5</td><td>15</td><td>17</td><td>37</td></tr><tr><th>Total</th><td>8</td><td>15</td><td>18</td><td>41</td></tr></tbody></table>

According to Table 11, disdrometer data presents a similar proportion of convective (37%) and stratiform cases (44%) and a smaller ratio of ambiguous cases. However, this is not the case for the DPR data where stratiform cases are clearly predominant (90%) and convective and ambiguous cases are marginal (5% each). The overall DPR classification is rather limited according to the value of the accuracy (below 0.50), as only 46% (precision of 0.46) of the predominant predicted regime (stratiform) is correctly done, despite 94% of
cases identified as stratiform by the disdrometers where correctly predicted. Worse scores
are obtained for convective cases.

4. Discussion

As discussed in previous research, discrepancies observed in DSD parameter values between disdrometer data and GPM-DPR estimates may be associated with spatial and/or temporal sampling problems [49,50]. The spatial sampling uncertainties are due to the difference in the observation area of the two instruments, as the disdrometer has a sampling area of about 50 cm² and the GPM-DPR footprint circular radius is 5 km at nadir [10]. Another aspect to consider is the effect of updrafts and downdrafts present from the lowest GPM-DPR measurement to the ground, which can actually modify the estimated DSD at ground level [51]. As mentioned above there is also a problem associated with the limited GPM-DPR overpasses over a given region which implies a low probability of coincidence with observing precipitation over the disdrometer sites.

Similarly to other investigations [14], results found here indicate the DF algorithm overestimated the mean Dm values and underestimated the intercept parameter Nw. Such a problem has been primarily associated with the DPR assumption of a constant shape parameter. In Section 3.1, Figure 2f, we compared the μ distribution observed by disdrometers with the fixed value set by DPR (μ = 3) finding that it corresponded to the mode value but differed from the median (μ = 7). To better understand the limitation of fixing the shape parameter μ in the calculation of DSD-derived variables, we examine the k/Ze ratio vs. the Dm, and the DFR vs. Dm with the disdrometer DSD (Figure 8). The k/Ze ratio was calculated assuming the SF algorithm applied to ZKu and the DFR was calculated considering the DF algorithm.

<figure><img src="image_12.png" /><figcaption>**Figure 8.** (a) Attenuation (k) to reflectivity (Z<sub>h</sub>) ratio as a function of D<sub>m</sub> at Ku-band frequency obtained from disdrometer measurements without and with fixed shape parameter (μ = 3). (b) As panel (a) but for DFR estimated with the dual-frequency algorithm.</figcaption></figure>

Each panel of Figure 8 shows the variable considered in two ways: fixing the shape parameter (grey dots) and not fixing it (blue dots). When μ is set to 3 in both single- and dual-frequency algorithms (Figure 8a,b), a much higher variability of the data is observed. Radhakrishna et al. [10] posited that the high scatter in the values of (k/Z<sub>h</sub>) and DFR is caused by the high variability in the DSD of convective rainfall. When μ is fixed, part of this variability is lost, increasing the uncertainty in the estimates. These results and the strong correlation between μ and N<sub>w</sub> [33,52] are among the factors that generate the differences
in the DSD parameter values between the disdrometers and the DPR DF. In addition, Del Castillo-Velarde [14] and Radhakrishna et al. [10] showed that the DF algorithm has limitations when attempting to estimate Dm for DFR values less than 0 dB, where there are two possible Dm values for a DFR measurement as the DF algorithm selects the drop with the highest Dm if two solutions exist [19]. Here we also observe that, for DFR values below 0 dB, Dm grows when DFR decreases for Dm below 0.5 mm as reported [10,14].

As seen in Equation (1), the GPM DPR rainfall rate computation is based on an adjustable R-Dm relationship linked to ε. In version 6 of the DPR algorithm, ε was assumed to be invariant, which imposed constraints on rainfall retrieval and caused the natural DSD variations along the rainfall column to be missed. In version 7, a two-scale model of ε was introduced, allowing it to vary with range [6]. To assess the consistency of this approach with our disdrometer observations, the Dm vs R observations were plotted overlaid with the corresponding GPM DPR curves computed for values of ε equal to 1.25 and 0.2 and 5.0 thus covering the possible range of values (Figure 9).

<figure><img src="image_13.png" /><figcaption>**Figure 9.** Scatter density plot of log₁₀ (R) vs. Dm observed by disdrometers overlayed with the relation used in the GPM DPR algorithm for ε equal to 0.2 (upper dashed line), 1.25 (solid line) and 5.0 (lower dashed line).</figcaption></figure>

The disdrometer data are clearly contained in the region limited by the GPM DPR lines and higher observation densities correspond well to the ε = 1.25 for Dm < 1.5 mm, but for larger diameters, the GPM DPR relationships tend to underestimate R. As posited by [8,15], this may be due to the use of predefined constants (α, β and τ) in the relationships between the precipitation rate and mass-weighted mean diameter (Equation (1)) that may not be adequate for the rainfall characteristics of the region of study. In addition, factors such as the attenuation effect, multiple scattering, non-uniform beam filling, and terrain interference directly affect the accuracy of the GPM DPR parameter estimation [6,53].

The results of the analysis of the matches between GPM DPR overpasses with disdrometers were consistent with those of similar studies [8,15]. The superiority of the optimal matching approach and the lower errors associated with the radar reflectivity factor and the mass-weighted mean diameter, as well as the poorer agreement between the intercept parameter and the rainfall rate, are indications of limitations in the DPR algorithm. In addition, it is worth mentioning that there is no clear trend of improvement of the DF algorithms over SF in version 07, which agrees with what was observed with version 6 by Adirosi et al. [15]. Finally, the limited ability to detect the convective precipitation type,
documented in studies analyzing previous GPM DPR previous versions [11,13,14], is also
found here for version 07 as reported by Seela et al. [8].

**5. Conclusions**

In this study, more than nine years of data ranging from March 2014 to November 2023 recorded by the GPM DPR over different geographical areas of the northeastern Iberian Peninsula were analyzed. Based on information from seven disdrometers, the first part of the study focuses on the characterization and comparison of DSD-derived parameters obtained from both datasets independently considering 162,328 and 2263 min of precipitation records observed by disdrometers and the GPM DPR, respectively. Results were stratified by orographic and climatic characteristics and several rainfall rate intensity thresholds. The second part of the study focused on validating four spatial matching methods between DPR overpasses and disdrometers. The main results are as follows:

1. The behavior of DSD-derived variables among the plain, mountain and coastal sub-regions showed some differences according to the disdrometer data, which were captured by the DPR DF algorithm. However, the GPM DSD parameters show an overestimation of Dm by about 0.1 mm at low and moderate precipitation rates (0.1–1, 1–2, 2–4 mm/h) and by 0.4 mm at precipitation rates greater than 4 mm/h by the DF algorithm with respect to the disdrometer. In contrast, the behavior of Nw was underestimated by the DPR, with a maximum value close to 6 dBNw at moderate precipitation rates (4–8 mm/h).

2. Disdrometer data indicated that the shape parameter mode over the area of study corresponds to the DPR fixed value (μ = 3), but the median was higher (μ = 7). Moreover, μ presents a distribution with a substantial natural variability which implies an increase in the uncertainty of DSD estimates based on the constant value assumption.

3. The superiority of the optimal matching approach was observed when validating the GPM DPR rainfall parameters with disdrometers. The GPM DPR estimates showed better verification statistics for the radar reflectivity factor in both Ku and Ka bands and the mass-weighted mean diameter, while worse results were found for the rainfall rate and the shape parameter N<sub>w</sub>.

4. According to the available sample of overpass matches (41 cases) the DPR DF rainfall classification algorithm showed little ability to detect events identified as convective by the disdrometers.

To the authors' knowledge, this validation study is the first of its kind covering the Iberian Peninsula and one of the few carried out in areas with a Mediterranean climate. Moreover, this is one of the first analyses in which recent updates incorporated in version 7 are validated. Results reported here may contribute to enhance our understanding of potential applications and limitations of satellite precipitation observations and can be considered in the development of future satellite precipitation retrievals.

**Author Contributions:** Conceptualization, E.P. and J.B.; methodology, E.P. and J.B.; formal analysis, E.P.; investigation, E.P. and J.B.; data curation, E.P., F.P., J.B., M.U., S.G. and B.B.; writing—original draft preparation, E.P. and J.B.; writing—review and editing, E.P., J.B., F.P., M.U., M.P., E.A., S.G. and B.B.; visualization, E.P.; supervision, J.B. and M.U. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partly funded by the projects WISE-PreP (RTI2018-098693-B-C32), ARTEMIS (PID2021-124253OB-I00 MINECO/FEDER) and the Water Research Institute (IdRA) of the University of Barcelona.

**Data Availability Statement:** Data are contained within the article.

**Acknowledgments:** Satellite and gauge data for this study were provided by NASA/JAXA. The authors wish to acknowledge the Fabra Observatory of the Royal Academy of Sciences and Arts of Barcelona and the support from the Secretariat for Universities and Research of the Ministry of Business and Knowledge of the Government of Catalonia and the European Social Fund.
**Conflicts of Interest:** The authors declare no conflicts of interest.

## Appendix A

The GPM algorithm assumes a gamma-type DSD (normalized version) to estimate the DSD parameters of the form:

$$N(D) = N_{w}f(D; \mu; D_{m}) \qquad (A1)$$

$$f(D; \mu; D_{m}) = \frac{6(\mu + 4)^{(\mu + 4)}}{4^{4}\Gamma(\mu + 4)} \left(\frac{D}{D_{m}}\right)^{\mu} \exp\left[-\frac{(\mu + 4)D}{D_{m}}\right] \qquad (A2)$$

where N(D) is the drop size distribution (in mm⁻¹m⁻³), D is the diameter of the raindrop (mm), Dm (mm) is the mass-weighted mean diameter which represents a mean particle size of the distribution. Nw is the normalized scaling parameter for concentration (mm⁻¹m⁻³), μ the shape parameter for gamma distribution and Γ denotes the gamma function. Letting σb (in mm²) and σe (in mm²) be the backscattering cross section and the extinction cross section of raindrops at a given temperature, respectively, K as a constant defined as a function of complex refractive index and λ the radar wavelength (in mm), the equivalent reflectivity Z (in mm⁶/m³) and specific attenuation k (in dB/km) are expressed as follows:

$$Z_{e} = N_{w}F(\lambda; \mu; D_{m}) \qquad (A3)$$

$$F(\lambda; \mu; D_{m}) = \frac{\lambda^{4}}{\Pi^{5}|K|^{2}} \int f(D; \mu; D_{m}) \sigma_{b} \Delta D \qquad (A4)$$

$$k = N_{w}G(D; \mu; D_{m}) \qquad (A5)$$

$$G(D; \mu; D_{m}) = 4.343 \times 10^{-3} \int f(D; \mu; D_{m}) \sigma_{e} \Delta D \qquad (A6)$$

Equations (A3) and (A5) are used by the SF and DF algorithms to determine the DSD parameters. The terms F(λ; μ; Dm) and G(D; μ; Dm) refer to normalized radar reflectivity and specific attenuation (in dB/km) and are the same as F(Dm) and G(Dm) in [19] and Ib (Dm, μ, λ) and Ie (Dm, μ, λ) in [9]. To determine parameters Dm and Nw, the GPM-DPR uses the SF and DF algorithms defined by Equations (A7) and (A8), respectively.

$$\frac{k}{Z_{e}} = \frac{G(D; \mu; D_{m})}{F(\lambda; \mu; D_{m})} \qquad (A7)$$

$$DFR = 10 \log_{10} \left( \frac{F(\lambda_{1}; \mu; D_{m})}{F(\lambda_{2}; \mu; D_{m})} \right) \qquad (A8)$$

The aim of these algorithms is to define a monotonic function that only depends on Dm [10,14,19]. It means that from measurements of k/Ze or DFR we will obtain a value of Dm and then, Nw can be calculated by replacing Dm in (A3) or (A5).

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