*2.4. Retrieving the Parameters of the Z–R Relation*

Weather radars usually provide the reflectivity Z which is transformed into rain intensity R using an exponential equation. In our case, R and Z are provided by the disdrometer; therefore, it is possible to get the values of A and b by fitting a linear model to the values of log10(R) and Z.

The radar reflectivity Z is assumed to be related to rain intensity R by the power law:

$$Z = A \times \mathbb{R}^b \tag{13}$$

In this equation, Z is expressed in mm<sup>6</sup> m<sup>−</sup>3. However, Z is usually expressed in the unit decibel relative to Z (dBZ):

$$Z\_{\text{[dBZ]}} = 10 \times \log\_{10} \left( Z\_{\text{[mm}^6 \text{ m}^{-3}]} \right) \tag{14}$$

By taking the log of Equation (13) and multiplying by 10:

$$10 \times \log\_{10}(Z) = 10 \times \log\_{10}(A) + 10 \times b \times \log\_{10}(R) \tag{15}$$

Moreover, based on Equation (14):

$$dBZ = 10 \times \log\_{10}(A) + 10 \times b \times \log\_{10}(R) \tag{16}$$

a simple linear model is fitted to the values of dBZ and log R which are calculated from the rain drop size distribution. This linear model has the equation:

$$\text{dBZ} = \text{intercept} + \text{slope} \times \log\_{10}(\mathcal{R}) \tag{17}$$

thus, by comparing Equations (16) and (17) the A and b parameters can be readily found:

$$b = \frac{slope}{10} \tag{18}$$

$$A = 10^{\frac{m \cdot {m \cdot {}\_{10}}}{10}} \tag{19}$$

Equations (13)–(19) represent the conventional way of retrieving A and b. An alternative method is to consider R as the dependent variable [75]. This method is more appropriate because the main purpose is to reduce errors in estimating R:

$$R = (1/A)^{1/b} \times Z^{1/b} \tag{20}$$

By taking the log10 of both sides of Equation (20):

$$
\log\_{10}(R) = \frac{1}{b} \times \log\_{10}(Z) - \frac{1}{b} \times \log\_{10}(A) \tag{21}
$$

$$
\log\_{10}(R) = \frac{dBZ}{10 \times b} - \frac{\log\_{10}(A)}{b} \tag{22}
$$

$$
\log\_{10}(R) = \text{intercept} + \text{slope} \times dBZ\tag{23}
$$

by comparing Equations (22) and (23):

$$b = \frac{1}{slope \times 10} \tag{24}$$

$$A = 10^{-b \times intercept} \tag{25}$$

Retrieval of A and b values was done for each event separately. Events with an accumulated rain amount of less than 1 mm were excluded to limit their influence on the fitting process. Additionally, the events were defined by a minimum interevent threshold of 15 min and a minimum duration of 15 min as in Jaffrain and Berne [75]. To ensure clear classification of the rain type on the event level, the fitting was restricted to events during which more than 60% of the event was convective, and events where all intervals were classified as stratiform. The remaining 2449 events contain 9914 h of rain (see Table A1).
