*2.3. Integral Rainfall Parameters*

Based on the DSD data, the number concentration of raindrops per unit volume per unit diameter interval for the *i*th size bin, *N*(*Di*) (m−<sup>3</sup> mm<sup>−</sup>1), can be calculated using Equation (2):

$$N(D\_i) = \sum\_{j=1}^{32} \frac{n\_{ij}}{A\_i \cdot \Delta t \cdot V\_j \cdot \Delta D\_i} \,\tag{2}$$

where *nij* is the number of raindrops at the *i*th size bin and the *j*th velocity class; *Ai* (m2) and Δ*Di* (mm) are the effective sampling area and width of the diameter interval at size *Di*; *Vj* (m s<sup>−</sup>1) is the fall speed for the *j*th velocity class; and Δ*t* is the sampling time interval, which was set to 60 s in this study.

To further understand the characteristics of rainfall, the integral parameters of total number concentration *NT* (m<sup>−</sup>3), rainwater content *W* (g m<sup>−</sup>3), rain rate *R* (mm h<sup>−</sup>1), median volume diameter *D*<sup>0</sup> (mm), mass-weighted mean diameter *Dm* (mm), normalized intercept parameter *Nw* (m−<sup>3</sup> mm<sup>−</sup>1), and mass spectrum standard deviation σ*<sup>m</sup>* (mm), were also calculated as follows:

$$N\_T = \sum\_{i=1}^{32} \sum\_{j=1}^{32} \frac{n\_{ij}}{A\_i \cdot \Delta t \cdot V\_j},\tag{3}$$

$$\mathcal{W} = \frac{\pi}{6} \times 10^{-3} \cdot \rho\_w \cdot \sum\_{i=1}^{32} \sum\_{j=1}^{32} D\_i^3 \frac{n\_{ij}}{A\_i \cdot \Delta t \cdot V\_j} \tag{4}$$

$$R = 6\pi \times 10^{-4} \cdot \sum\_{i=1}^{32} \sum\_{j=1}^{32} D\_i^3 \frac{n\_{ij}}{A\_i \cdot \Delta t} \,. \tag{5}$$

$$\frac{1}{2}\mathcal{W} = \frac{\pi}{6}\rho\_{w^\*} \int\_0^{D\_0} D^3 \mathcal{N}(D) dD\_\prime \tag{6}$$

$$N\_{\rm w} = \frac{3.67^4}{\pi \rho\_{\rm w}} \left(\frac{10^3 \,\mathrm{W}}{D\_0^4}\right) \tag{7}$$

$$D\_{\rm mf} = \frac{\sum\_{i=1}^{32} N(D\_i) \cdot D\_i^4 \cdot \Delta D\_i}{\sum\_{i=1}^{32} N(D\_i) \cdot D\_i^3 \cdot \Delta D\_i} \,\tag{8}$$

$$\sigma\_{\rm m} = \frac{\sum\_{i=3}^{32} (D\_i - D\_m)^2 \mathcal{N}(D\_i) \cdot D\_i^3 \cdot \Delta D\_i}{\sum\_{i=1}^{32} \mathcal{N}(D\_i) \cdot D\_i^3 \cdot \Delta D\_i},\tag{9}$$

where ρ*<sup>w</sup>* is the water density (1.0 g cm<sup>−</sup>3).

Considering the emerging development of X-band dual-polarization weather radar for urban hydrometeorological applications [42,43], a set of dual-polarization radar variables, including radar reflectivity in the horizontal (vertical) polarization *Zh* (*Zv*) (mm<sup>6</sup> m<sup>−</sup>3), differential reflectivity *Z*DR (dB) and specific differential phase *K*DP ( ◦ km−1), are derived from DSDs using the *T*-matrix scattering technique [44]:

$$Z\_{h, \upsilon} = \frac{4\lambda^4}{\pi^4 |K\_{\upsilon}|^2} \sum\_{i=1}^{32} \left| f\_{\text{hh,vv}}(D\_i) \right|^2 N(D\_i) \Delta D\_{i\prime} \tag{10}$$

$$Z\_{\rm DR} = 10 \log\_{10} \left( \frac{Z\_h}{Z\_v} \right) \tag{11}$$

$$K\_{\rm DP} = \frac{180\lambda}{\pi} \sum\_{i=1}^{32} \mathcal{Re} [f\_{\rm hh}(0\_i D\_i) - f\_{\rm Vv}(0\_i D\_i)] \mathcal{N}(D\_i) \Delta D\_{i\nu} \tag{12}$$

where *f*hh,vv(*Di*) is the backscattering amplitude of a droplet with horizontal and vertical polarization; *f*hh(0, *Di*) and *f*vv(0, *Di*) are the standard forward scattering amplitudes, which is related to the depolarization factor and relative permittivity of water dielectric [45]; *Kw* is the dielectric factor of water (0.9639); and λ (mm) is the radar wavelength (3 cm). Note that *Zh* (*Zv*) in the unit of mm6 m−<sup>3</sup> is replaced by *ZH* (*ZV*) in the unit of dBZ wherever required in this paper, and *ZH*,*<sup>V</sup>* = 10 × log10 *Zh*,*v*.
