*2.2. Drop Size Distribution Parameters*

Thies disdrometers are laser-based instruments that provide high temporal records of rain microstructure. When a precipitation particle passes between the transmitter and the receiver, the strength of the laser beam is reduced. Based on the magnitude and duration of this reduction, it is possible to estimate the size and velocity of the passing precipitation particle. The Thies disdrometers raw data output represents one-minute summaries of the number of particles in 22 non-linear size classes and 20 non-linear velocity classes. From the raw output, a number of parameters can be obtained. This study is focused particularly on rain intensity R, radar reflectivity Z, total number of drop concentration N, and median volume drop diameter D0.

Rain rate R (mm/h) is given by

$$R = \frac{6 \times 10^{-4} \times \pi}{\Delta T} \times \sum\_{i=1}^{i=22} \sum\_{j=1}^{j=20} \left( \mathbf{x}\_{i,j} \frac{D\_i^3}{A\_i} \right) \tag{1}$$

where

*xi*,*j*: Detected number of drops that fall in diameter range i and velocity range j,

Δ*T* (s): Temporal resolution (60 s in this case),

*Ai* (m2): Corrected detection area: *Ai* = <sup>228</sup> <sup>×</sup> -<sup>20</sup> <sup>−</sup> *Di* 2 /106,

*Di* (mm): Mean diameter of drops that fall in diameter range *i*.

The radar reflectivity Z (dBZ) is calculated with the following expression:

$$Z = 10 \* \log\_{10} \left( \sum\_{i=1}^{i=22} \sum\_{j=1}^{i=20} \left( \mathbf{x}\_{i,j} \frac{D\_i^6}{\left( A\_i \ V\_j \,\Delta T \right)} \right) \right) \tag{2}$$

where *Vj* (m/s) : Mean velocity of drops that fall in the velocity range *j*.

The total number of drops N (m<sup>−</sup>3) is computed according to

$$N = \sum\_{i=1}^{i=22} \sum\_{j=1}^{j=20} \left( \frac{x\_{i,j}}{A\_i \ V\_j \ W\_i \,\Delta T} \right) \tag{3}$$

where *Wi* (mm): the width of the diameter range *i*.

The rain microstructure is assumed to follow a gamma distribution [72]:

$$N(D) = N\_0 D^\mu \mathfrak{e}^{(-\Lambda D)}\tag{4}$$

where *N*(*D*) (mm<sup>−</sup>1m−3) is the number of drops for each diameter range per unit volume and unite size. The intercept *N*<sup>0</sup> (mm−1−<sup>μ</sup> m<sup>−</sup>3), the shape μ (-), and the slope Λ (mm<sup>−</sup>1) parameters are determined by the moments method [73]. The nth moment of the raindrop size distribution *Mn* (mm−1−<sup>μ</sup> m<sup>−</sup>3) is given by

$$M\_{\rm nl} = \int\_{D\_{\rm min}}^{D\_{\rm max}} D^{\rm n} N(D) dD \tag{5}$$

and the three gamma parameters are

$$N\_0 = \frac{\Lambda^{\mu+3} M\_2}{\Gamma(\mu+3)}\tag{6}$$

$$\mu = \frac{(7 - 11\eta) - \left[\left(7 - 11\eta\right)^2 - 4(\eta - 1)\left(30\eta - 12\right)\right]^{0.5}}{2(\eta - 1)}\tag{7}$$

$$
\Lambda = \left[\frac{(4+\mu)(3+\mu)M\_2}{M\_4}\right]^{0.5} \tag{8}
$$

where

$$
\eta = \frac{M\_4^2}{M\_2 M\_6} \tag{9}
$$

The mass weighted mean diameter Dm (mm), the median volume diameter D0 (mm) and the normalized intercept Nw (mm<sup>−</sup>1m−3) are calculated based on the parameters of gamma distribution:

$$D\_m = \frac{M\_4}{M\_3} \tag{10}$$

$$D\_0 = \frac{D\_m(\mu + 3.6\mathcal{T})}{\mu + 4} \tag{11}$$

$$N\_{\rm w} = \frac{4^4 \, M\_{\rm 3}}{6 \, D\_{\rm m}^{-4}} \tag{12}$$

Additionally, the classification of rain type into convective and stratiform requires the use of the following parameters: sd\_N\_10, sd\_D0\_10, and sd\_log10\_R\_10, where sd\_XX\_10 is the standard deviation of the values of XX (XX being N, D0 and R, respectively) over a time window of ten minutes.
