## 2 Materials and Methods
### Scatter in dB as a Measure of Dispersion of the Multiplicative Error from the Mean  
It is well known that quantitative precipitation estimates derived aloft by the spaceborne precipitation radar (e.g., [25; 43; 44]) and the ground-based weather radar (e.g., [7; 8; 9; 10; 11; 12; 13]) may suffer from a significant mean field bias (typically, underestimation) with respect to in situ point measurements at the ground. Since from a radar-meteorological point of view the multiplicative nature of the error is dominant relative to the additive one, the mean error is defined as the Log-transformed ratio between the spaceborne radar total and, for instance, the GAUGE total. Once the underestimation is compensated, the goal of our evaluation is to answer the following two questions: (1) how much does the Log-transformed ratio between the GPM precipitation product and the other three MeteoSwiss products vary from site to site in Switzerland? (2) For each given weighted contribution to the average annual precipitation amount, what is the range of variability of the Log-transformed ratio between GPM and the other products? (In the latter question we are trying to be more cautious and closer to the users' point of view).  
Let be \(x\) the Log-transformed ratio in dB of the precipitation product under test, \(P_{test}\), with respect to the reference product, \(P_{ref}\) in the formula:  
\[x=10\ Log\Bigg{(}\frac{P_{test}}{P_{ref}}\Bigg{)} \tag{1}\]  
A robust score for the dispersion of \(x\), which represents the variability in Switzerland of the multiplicative error, could simply be the median absolute deviation from the median (MADM), as defined in Equation (5) of [25]. This score adequately answers the first question. Furthermore, in cases where the cumulative distribution function (CDF) of \(x\) is symmetric, then the MADM is half of the widely used interquartile range. However, MADM is resilient along the abscissa but not over the ordinate (where the normalized CDF ranges from 0 to 1). The idea is that an ill-defined ratio that results from two small precipitation amounts should receive much less weight than an event characterized by much larger precipitation amounts. Bearing this in mind, in conceiving the score called scatter, each ratio data pair receives on the \(y\) axis a normalized contribution which is proportional to its contribution to the total amount [26; 38] (Section 4.1, pp. 53-54). The introduction of a weighted CDF is a crucial concept in the definition of the scatter. The scatter is defined as half the distance between the 16% and 84% percentiles of the weighted CDF. An example of a weighted CDF is shown in [45] (Figure 3). There, GAUGE is the reference, while RADAR is the product under test; this is relevant because an important aspect in the definition of scatter is that the weights, which affect the steps on the ordinate axis, are proportional to the amounts of the reference. Consequently, in relation to the present analysis, the scatter of GPM with respect to the that of the other three products is different from the scatter of each of the products with respect to that of GPM. Both the scatter and MADM are independent of any multiplicative factor that could affect the retrieved precipitation estimates. This means that both scatter and MADM are orthogonal to the bias. In addition, they are expressed in dB, like the bias.