## 1 Introduction  
The purpose of this paper is to develop and propose a semi-empirical method to convert 10 min rain rate time series into 1 min rain rate time series, the latter to be used afterward in this paper--as an example application--as input to the synthetic storm technique (SST) for simulating rain attenuation time series [1], which is a very reliable tool that can even reconstruct missing intervals in time series [2]. The rationale for the need for such a method lies in the fact that several meteorological institutes make the time series of the quantity of water accumulated every 10 min (therefore, providing 10 min rain rate time series) available compared to the past when, at most, the quantity of water collected referred to that in the period from 1 day to 1 h. Our previous theory on the de-integration of the accumulated rainfall--from 1 day to 1 h--refers only to the probability distributions of the rain rate [3, 4].  
To develop the semi-empirical method, we used a very large data bank of 1 min rain rate time series collected in Spino d'Adda (Table 1) from 1993 to 2002, representing 10 years of practically all rain events. First, we converted them into 10 min rain rate time series for obtaining the measured data bank allegedly provided by meteorological institutes, and then we developed a method that can simulate 1 min rain rate time series. A comparison between the experimental and simulated 1 min time series elucidated how good and precise the method is. The conversion method was then studied and developed for the other sites listed in Table 1 to assess whether it also works well in different climatic regions.  
The sites listed in Table 1 are useful study cases because (a) on-site 1 min rain rate time series were continuously recorded for several years; (b) important NASA and ESA satellite ground stations are located in these sites (Gera Lario, Fucino, Madrid, White Sands); (c) long-term radio propagation experiments were performed (Fucino, Gera Lario, Spino d'Adda, Madrid, Prague, White Sands); and (d) they are important cities (Madrid, Prague, Tampa, Vancouver).  
Afterward, we used the two sets of 1 min rain rate time series (i.e., that measured and simulated 10 min rain rate time series) as the input to the SST for simulating--as an example application--rain attenuation time series at 20.7 GHz in 35.5\({}^{\circ}\) slant paths to geostationary satellites at the sites in Table 1. The conclusion of our exercise is that the two rain attenuation probability distributions in an average year show no significant differences that can affect satellite system design at centimeter or millimeter frequencies; so, the method can be used very reliably for this purpose.  
The theory, however, can potentially be useful not only in satellite communications and radio propagation studies but also in other fields. In agriculture, for example, the measured daily rainfall needs to be disaggregated to predict runoff, erosion, and pollutant transport [5; 6; 7; 8]. In hydrology, kinetic energy determines the potential ability of the rainfall to detach soil, and it is used as a rain erosivity index [6]. Kinetic energy is strictly linked to the rain rate; therefore, data on the instantaneous rain rate (as 1 min rain rate times series can be considered) could improve the estimates in these fields.  
After this introductory section, in Section 2, we study the 1 min and 10 min rain rate data bank of Spino d'Adda; in Section 3, we develop a semi-empirical method for converting \(R_{10}(t)\) into 1 min rain rate time series, termed \(R(t)\), in Spino d'Adda; in Section 4, we develop and confirm the method for the other sites in Table 1; in Section 5, we recall the theory of the synthetic storm technique and report its application at the sites listed in Table 1; and in Section 6, we draw a conclusion.