Conversion of 10–min Rain–Rate Time Series into 1–min Time Series: Theory, Experimental Results and Application to Satellite Communications

1. Introduction

The purpose of this paper is to develop and propose a method to convert 10–min rain–rate time series into 1–min rain–rate time series, the latter to be afterwards used, in the paper, as input to the Synthetic Storm Technique (SST) for simulating rain–attenuation time series [1], a very reliable tool that can even reconstruct missing intervals in time series [2]. The rationale for the need of such a method lies in the fact that several meteorological institues make available time series of quantity of water accumulated every 10 minutes (therefore providing 10–min rain–rate time series), compared to the past when, at most, the quantity of water collected referred from 1 day to 1 hour. Our previous theory on the de–integration of the accumulated rainfall – from 1 day to 1 hour water – refers only to the probability distributions of rain–rate [3,4].

To develop the method, we use a very large data bank of 1–min rain–rate time series collected in Spino d’Adda (Table 1), from 1993 to 2002, 10 years of practically all rain events occurred in the period. First, we convert them to 10–min rain–rate time series for obtaining the measured data bank allegedly provided by some Meteorological Institutes, and then we develop the method that can simulate 1–min rain–rate time series. The comparison between the experimental and simulated 1–min time series will provide how good and precise the method is. The conversion method is then studied and developed for the other sites listed in Table 1, to assess whether it works well also 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) in these sites are located important NASA and ESA satellite ground stations (Gera Lario, Fucino, Madrid, White Sands); (c) long–term radio propagation experiments were performed (Fucino, Gera Lario, Spino d’Adda, Madrid, Prague, White Sands); (d) there are important cities (Madrid, Prague, Tampa, Vancouver).

Afterwards, we use the two sets of 1–min rain–rate time series (i.e., that measured and that simulated from 10–min rain–rate time series) as the input to the SST for simulating – as example of application – rain attenuation time series at 20.7 GHz in 35.5° slant paths to geostationary satellites at the sites of 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 systems design at centimeter or millimeter frequencies, so that the method can be used very reliably for this purpose.

The theory, however, can be potentially 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, the 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 of 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, our “laboratory” site; in Section 3 we develop the method to convert $R_{10}\left(t\right)$ into 1-min rain rate time series, termed $R\left(t\right)$ in Spino d’Adda; in Section 4 we develop and confirm the method for the other sites of Table 1; in Section 5 we recall the theory of the Synthetic Storm Technique and report its application to the sites of Table 1; in Section 6 we draw a conclusion.

2. Exploratory Data Analysis in Spino d’Adda

In this Section we use the 10–year 1–min rain rate data bank collected in Spino d’Adda, our “laboratoty” site, to study how the experimental 1–min rain–rate time series $R_1\left(t\right)$ (mm/h) are connected with the corresponding 10–min time series $R_{10}\left(t\right)$ (mm/h). The latter is generated by $R_1\left(t\right)$, according to the following relationhsip, applied to disjoint and contiguous 10–min intervals during the rainfall event:

$$R_{10}={\displaystyle \frac{1}{10}}{\sum }_{k=1}^{10}{R}_1\left(k\right)$$

(1)

Figure 1 shows an example of this linear tranformation. We notice, of course, that $R_{10}\left(t\right)$ is smoothed, therefore any application using $R_{10}\left(t\right)$ will underestimate the effect of the “instantaneous” largest rain rates whose intensity can be significantly reduced.

Since we transform $R_{10}\left(t\right)$ (mm/h) into an estimated/simulated $R_1\left(t\right)$, the first step is to study the distribution of $R_1$ within intervals of 10 min conditioned to $R_{10}$. These conditional distribution will be useful to simulate $R_1\left(t\right)$ from $R_{10}\left(t\right)$. This is possible because, as shown in the example of Figure 1, both $R_1\left(t\right)$, and $R_{10}\left(t\right)$ are available for Spino d’Adda and the other sites listed in Table 1, for several years. By considering the full data bank of similar results, we have therefore calculated the conditional histograms of $R_1$ within 10–min intervals, in the following ranges of $R_{10}$: 0–2, 2–4, 4–6, 6–8, 8–10, 10–15, 15–20, 20–30, 30–40, > 40 mm/h (samples with range maximum value are included in that range). Notice that the first range starts at 0.2 mm/h because of rain–gauge technology. Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6 show these histograms.

From these figures we assume that a log–normal probability density function (PDF) can adequately model the central part of the experimental histograms, therefore we have calculated mean value and standard deviation of $\mathrm{l}\mathrm{o}\mathrm{g}\left(R_1\right)$ – natural logarithm – and assumed a log–normal probability PDF chacterized by the values reported in Table 2, together with the correlation coefficient between two successive $\mathrm{l}\mathrm{o}\mathrm{g}\left(R_1\right)$ samples within the same 10–min interval. These parameters are fundamental because they will be used in Section 3 to simulate $R_1\left(t\right)$ from $R_{10}\left(t\right)$.

Table 1. Mean value, standard deviation of $\mathrm{l}\mathrm{o}\mathrm{g}\left(R_1\right)$ and correlation coefficient between two successive $\mathrm{l}\mathrm{o}\mathrm{g}\left(R_1\right)$ samples within the same 10–min interval, for the indicated $R_{10}$ ranges. The first range starts at 0.2 mm/h.

$R_{10}$(mm/h)	Mean	Standard Deviation	Correlation Coefficient
0–2	–0.60	0.75	0.94
2–4	0.94	0.37	0.76
4–6	1.51	0.41	0.70
6–8	1.83	0.49	0.72
8–10	2.07	0.52	0.68
10–15	2.35	0.61	0.72
15–20	2.62	0.76	0.71
20–30	3.03	0.68	0.70
30–40	3.32	0.77	0.75
> 40	3.95	0.72	0.76

Table 1. Mean value, standard deviation of $\mathrm{l}\mathrm{o}\mathrm{g}\left(R_1\right)$ and correlation coefficient between two successive $\mathrm{l}\mathrm{o}\mathrm{g}\left(R_1\right)$ samples within the same 10–min interval, for the indicated $R_{10}$ ranges. The first range starts at 0.2 mm/h.

$R_{10}$(mm/h)	Mean	Standard Deviation	Correlation Coefficient
0–2	–0.60	0.75	0.94
2–4	0.94	0.37	0.76
4–6	1.51	0.41	0.70
6–8	1.83	0.49	0.72
8–10	2.07	0.52	0.68
10–15	2.35	0.61	0.72
15–20	2.62	0.76	0.71
20–30	3.03	0.68	0.70
30–40	3.32	0.77	0.75
> 40	3.95	0.72	0.76

3. From ${\mathit{R}}_{\mathbf{10}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ to $\mathit{R}\mathbf{\left(}\mathit{t}\mathbf{\right)}$

In this section we describe how to convert $R_{10}\left(t\right)$ into the final 1-min rain rate series to $R\left(t\right)$. In Appendix A we report the list of mathematical symbols. First, we show how to get the first version of $R\left(t\right)$, termed simulated $R_{1,sim}\left(t\right)$. First, we estimate the sample $R_{1,sim}(k+1)$ from the previous sample $R_{1,sim}\left(k\right)$ within a 10–min interval, then we filter $R_{1,sim}\left(t\right)$ to reduce the high–frequency noise introduced by the simulation. In all cases, the experimental quantity of water collected in each 10–min interval in conserved by using a suitable scaling factor.

3.1. Simulation Steps

Since any conditional PDF is modelled as log–normal within each 10–min interval, we model the bivariate probability density function between two succesive values $R_{1,sim}\left(k\right)$ and $R_{1,sim}(k+1)$, in the 10–min interval, as log–normal – we think this modelling can adequately describe at least the central part of the bivariate probability density – therefore $R_{1,sim}(k+1)$ can be estimated from $R_{1,sim}\left(k\right)$ by considering the conditional log–normal PDF, as follows.

Let $x={\mathrm{l}\mathrm{o}\mathrm{g}(R}_{1,sim}\left(k\right))$ and $y={\mathrm{l}\mathrm{o}\mathrm{g}(R}_{1,sim}(k+1))$, then the conditional PDF has mean value and standard deviation given by [9,10,11]:

$$m_{y/x}=m_y+r{\displaystyle \frac{s_y}{s_x}}(x-m_x)$$

(2)

$$s_{y/x}=s_y\sqrt{1-r^2}$$

(3)

Since within the same 10–min rain rate range (see Table 2) $m_x=m_y=m$ and $s_x=s_y=s$, Eqs.(2)(3) become:

$$m_{y/x}=(1-r)m+rx$$

(4)

$$s_{y/x}=s\sqrt{1-r^2}$$

(5)

Now, we can detail the simulation steps:

• Sample 1 of $R_{1,sim}$(t). According to the sample 1 of $R_{10}\left(t\right)$ – in a real application this would be the first $R_{10}$ value of the $R_{10}\left(t\right)$ provided by Meteorological Institutes – the mean value and the standard deviation of the conditional PDF and the correlation coefficient are selected from Table 1.
• A standard Gaussian random number $X$ is generated and denormalized according to the values of step 1, according to the following relationships:

  $$X={\displaystyle \frac{x-m}{s}}$$

  (6)

  $$x=sX+m=\mathrm{l}\mathrm{o}\mathrm{g}\left(R_{1,sim}\right)$$

  (7)

  $$R_{1,sim}=\mathrm{e}\mathrm{x}\mathrm{p}(sX+m)$$

  (8)
  For example, let $R_{10}=12$ mm/h and $X=-0.40$ (randomly generated), then (from Table 2) $m=2.35$, $s=0.61$, $r=0.72$, therefore $R_{1,sim}=\mathrm{e}\mathrm{x}\mathrm{p}(0.61\times (-0.4)+2.35)=8.21$ mm/h.
• Samples 2 to 10. A standard Gaussian random number is generated for each sample as in step 2, but now it is denormalized according to Eqs (4)(5). Continuing the example of step 2: $R_{1,sim}\left(k\right)=8.21$ mm/h; let $X=0.62$ (randomly generated), then $m_{y/x}=(1-0.72)\times 2.35+0.72\times \mathrm{l}\mathrm{o}\mathrm{g}\left(8.21\right)=2.17$ and $s_{y/x}=0.61\times \sqrt{1-{0.72}^2}=0.42$, therefore $R_{1,sim}(k+1)=\mathrm{e}\mathrm{x}\mathrm{p}(0.42\times 0.62+2.17)=11.36$ mm/h. Since the next sample depends only the previous one, the simulation process is a first–order Markov process [12].
• The 1–min rain–rate sample $R_{1,sim}\left(k\right)$is scaled to conserve the quantity of water $Q_T$ of the original $R_{10}$ sample, the latter given by:

  $$Q_T={\sum }_{k=1}^{10}{R}_1\left(k\right)=10\times R_{10}$$

  (9)
  Now, the quantity of water in the simulation is given by:

  $$Q_{T,sim}={\sum }_{k=1}^{10}{R}_{1,sim}\left(k\right)$$

  (10)
  Since in general, $Q_{T,sim}\ne Q_T$, to conserve the quantity of water each 1–min sample must be scaled to obtain the rain–rate time series $R_{sim}\left(t\right)$:

  $$R_{sim}(t)={\displaystyle \frac{Q_T}{Q_{T,sim}}}{R}_{1,sim}(t)$$

  (11)
• The simulation process is repeated by considering the successve value $R_{10}$, till the last sample of the rain event. The passage from sample 10 simulated from $R_{10}\left(h\right)$ to the sample 1 of the next sample $R_{10}(h+1)$ is memoryless, therefore step 1 is repeated without recalling sample 10 of the previous interval. Moreover, notice that we neglect the “border” distortion due to the last 10–min sample of $R_{10}\left(t\right)$ in a rain event because this interval may not be completely rainy (an information lost, of course, also in real experimental data).

The 1–sample memory steps 1–4 and the memoryless step 5 simplify the simulation but introduce high frequency noise as shown in Figure 7. This noise can be reduced by digital filtering, as we show in the next sub–section.

3.2. Optimum Filter

To reduce the noise produced by the sumulation steps, $R_{sim}\left(t\right)$ is filtered with a low–pass Butterworth filter of order 10 [13,14] therefore producing $R_{filt}\left(t\right)$. Also $R_{filt}\left(t\right)$ must be scaled to conserve the experimental quantity of water of the 10–min interval. In conclusion, by filtering and scaling $R_{sim}\left(t\right)$ we get the final 1–min rain–rate time series $R\left(t\right)$, according to:

$$R={\displaystyle \frac{Q_T}{Q_{T,R_{filt}}}}{R}_{filt}$$

(12)

With:

$$Q_{T,R_{filt}}={\sum }_{k=1}^{10}{R}_{filt}\left(k\right)$$

(13)

The only parameter which defines the transfer function of the filter is its cut–off frequency $f_c\le f_N$ ${(\mathrm{m}\mathrm{i}\mathrm{n}}^{-1})$, where $f_N=0.5$ ${\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$is the Nyquist frequency [13,14].

To determine the optimum value $f_c=f_o$, we consider the error $\epsilon$(mm/h) between the rain rate simulated$R$ and the rain rate measured $R_1$ at equal probabiltiy exceeded:

$$\epsilon =R\left(P\right)-R_1\left(P\right)$$

(14)

Figure 8 shows mean value and standard deviation of $\epsilon$ versus $f_c$. It can be noted that: (a) the mean error is very small in the entire range ($\epsilon \lesssim 0.6$ mm/h); (b) the standard deviation is minimum at about $f_c=1/3$ ${\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$, therefore, in the following we fix $f_o=1/3$ ${\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$ for all simulations.

Figure 9 show the same time series of Figure 7 after the full simulation process just described. Finally, Figure 10 shows the overall result of this exercise by drawing the probability distrbution (PD) that the 1–min rain rate in abscissa is exceeded in the experimental data, i.e. $P\left(R_1\right)$, and in the simulated 1–min data, $P\left(R\right)$. The two curves are almost indistinguishable, in agreement with the error and standard deviation reported in Figure 8 at $f_c=1/3$ ${\mathrm{m}\mathrm{i}\mathrm{n}}^{-1}$.

4. From ${\mathit{R}}_{\mathbf{10}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ to $\mathit{R}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ in Other Sites

In this section we apply the method obtained in our “laboratory” of Spino d’Adda at the sites reported in Table 1, for which 1–min rate times series are available for many years. First, in Figure 11, we show the scatterplots between mean values, standard deviations and correlation cofficients of Spino d’Adda (the values of Table 2) versus those of the othe sites of Table 1 (Appendix B reports the numerical values). We can see a very tight relationship between the mean values. This means that the rain rate process, although in sites with different weather and rainfall intensity, can be modelled with log–normal PDFs with the same mean value. Differences arise in the standard deviation and correlation coefficient, although these differences do not impact significantly on the simulation predictions as we show next.

Figure 11. Scatterplots of mean values (left panel), standard deviations (central panel) and correlation coefficients (right panel) between the values of the sites of Table 1 and Spino d’Adda. Gera Lario: green; Fucino: blue; Madrid: cyan; Prague: yellow; Tampa: red; White Sands: magenta; Vancouver: black.

Figure 11. Scatterplots of mean values (left panel), standard deviations (central panel) and correlation coefficients (right panel) between the values of the sites of Table 1 and Spino d’Adda. Gera Lario: green; Fucino: blue; Madrid: cyan; Prague: yellow; Tampa: red; White Sands: magenta; Vancouver: black.

Figure 12. Gera Lario. Probability distrbution that the 1–min rain rate in abscissa is exceeded in the experimental data, $P\left(R_1\right)$, blue line, and in the simulated 1–min data, $P\left(R\right)$, black line. Left panel: $P\left(R\right)$ is obtained by using local values of the conditional PDFs; Right panel: $P\left(R\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 12. Gera Lario. Probability distrbution that the 1–min rain rate in abscissa is exceeded in the experimental data, $P\left(R_1\right)$, blue line, and in the simulated 1–min data, $P\left(R\right)$, black line. Left panel: $P\left(R\right)$ is obtained by using local values of the conditional PDFs; Right panel: $P\left(R\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 13. Fucino. Probability distrbution that the 1–min rain rate in abscissa is exceeded in the experimental data, $P\left(R_1\right)$, blue line, and in the simulated 1–min data, $P\left(R\right)$, black line. Left panel: $P\left(R\right)$ is obtained by using local values of the conditional PDFs; Right panel: $P\left(R\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 13. Fucino. Probability distrbution that the 1–min rain rate in abscissa is exceeded in the experimental data, $P\left(R_1\right)$, blue line, and in the simulated 1–min data, $P\left(R\right)$, black line. Left panel: $P\left(R\right)$ is obtained by using local values of the conditional PDFs; Right panel: $P\left(R\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

From these figures we notice that the simulation with the local conditional PDFs (Appendix B) gives better results than that with the parameters of Spino d’Adda (Table 2), as expected. However, notice that in the simulations with the data of Spino d’Adda, the largest errors mostly occur at the lowest probabilities. In real applications, as the one we show in the next sections, these probabilities correspond to few minutes. For example, in Madrid – Figure 14, the worst site for this comparison –, in the arithmetic average year of the 9–year period here considered, $R_1\gtrsim 0$ for about 2.2% of the time, i.e. about $365\times 24\times 60\times 0.022=11563$ min. Now, Figure 14 shows that the error is less than $4\sim 5$ mm/h for probabilities smaller than $4\times {10}^{-3}$ therefore only for $4\times {10}^{-3}\times 11563=46$minutes the errror is larger than $4\sim 5$ minutes. In other words, for almost all the time the error is negligible.

Figure 14. Madrid. Probability distrbution that the 1–min rain rate in abscissa is exceeded in the experimental data, $P\left(R_1\right)$, blue line, and in the simulated 1–min data, $P\left(R\right)$, black line. Left panel: $P\left(R\right)$ is obtained by using local values of the conditional PDFs; Right panel: $P\left(R\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 14. Madrid. Probability distrbution that the 1–min rain rate in abscissa is exceeded in the experimental data, $P\left(R_1\right)$, blue line, and in the simulated 1–min data, $P\left(R\right)$, black line. Left panel: $P\left(R\right)$ is obtained by using local values of the conditional PDFs; Right panel: $P\left(R\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 15. Prague. Probability distrbution that the 1–min rain rate in abscissa is exceeded in the experimental data, $P\left(R_1\right)$, blue line, and in the simulated 1–min data, $P\left(R\right)$, black line. Left panel: $P\left(R\right)$ is obtained by using local values of the conditional PDFs; Right panel: $P\left(R\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 15. Prague. Probability distrbution that the 1–min rain rate in abscissa is exceeded in the experimental data, $P\left(R_1\right)$, blue line, and in the simulated 1–min data, $P\left(R\right)$, black line. Left panel: $P\left(R\right)$ is obtained by using local values of the conditional PDFs; Right panel: $P\left(R\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 16. Tampa. Probability distrbution that the 1–min rain rate in abscissa is exceeded in the experimental data, $P\left(R_1\right)$, blue line, and in the simulated 1–min data, $P\left(R\right)$, black line. Left panel: $P\left(R\right)$ is obtained by using local values of the conditional PDFs; Right panel: $P\left(R\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 16. Tampa. Probability distrbution that the 1–min rain rate in abscissa is exceeded in the experimental data, $P\left(R_1\right)$, blue line, and in the simulated 1–min data, $P\left(R\right)$, black line. Left panel: $P\left(R\right)$ is obtained by using local values of the conditional PDFs; Right panel: $P\left(R\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 16. White Sands. Probability distrbution that the 1–min rain rate in abscissa is exceeded in the experimental data, $P\left(R_1\right)$, blue line, and in the simulated 1–min data, $P\left(R\right)$, black line. Left panel: $P\left(R\right)$ is obtained by using local values of the conditional PDFs; Right panel: $P\left(R\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 16. White Sands. Probability distrbution that the 1–min rain rate in abscissa is exceeded in the experimental data, $P\left(R_1\right)$, blue line, and in the simulated 1–min data, $P\left(R\right)$, black line. Left panel: $P\left(R\right)$ is obtained by using local values of the conditional PDFs; Right panel: $P\left(R\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 17. Vancouver. Probability distrbution that the 1–min rain rate in abscissa is exceeded in the experimental data, $P\left(R_1\right)$, blue line, and in the simulated 1–min data, $P\left(R\right)$, black line. Left panel: $P\left(R\right)$ is obtained by using local values of the conditional PDFs; Right panel: $P\left(R\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 17. Vancouver. Probability distrbution that the 1–min rain rate in abscissa is exceeded in the experimental data, $P\left(R_1\right)$, blue line, and in the simulated 1–min data, $P\left(R\right)$, black line. Left panel: $P\left(R\right)$ is obtained by using local values of the conditional PDFs; Right panel: $P\left(R\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

In the next section, as an example of the possible applications, we apply the theory to the important case of estimating the rain attenuation in slant paths to satellites in the Geostationary orbit with a powerful tool, the Synthetic Storm Technique.

5. Rain Attenuation in Slant Paths to Geostationary Satellites: The Synthetic Storm Technique

In this section, as an example of the possible applications of the conversion of $R_{10}\left(t\right)$ into $R\left(t\right)$, we use the theory to estimate the rain attenuation in slant paths to geostationary satellites by using the a very reliable tool available today, namely the Synthetic Storm Technique (SST), which need as input 1–min rain–rate time series [1]. We first recall briefly the SST theory and then we appply it to the sites of Table 1, by simulating, at each site, reliable rain attenuation time series in 35.5° slant paths to a geostationary satellite with a 20.7 GHz carrier, circular polarization of the electromagnetic wave.

5.1. Synthetic Storm Technique

Satellite communication links at centimeter and millimeter wave frequencies are faded by rainfall. For a reliable link‒budget design, we need to know, at the very least, the annual probability distribution function $P\left(A\right)$– i.e. the fraction of time in an average year – of rain attenuation $A$ (dB) measured/predicted in the up‒ or down‒links to a satellite. Instead of long and expensive measurements of beacon attenuation, prediction models are used for estimating $P\left(A\right)$ from locally, measured or estimated, annual probability distributions, $P\left(R\right),$ of 1‒min rain rate $R$. The Synthetic Storm Technique [1] is a powerful and accurate tool, as is now recognized, that can produce all the necessary statistics of rain attenuation, not only $P\left(A\right)$, but also fade durations and rate of change of attenuation because it provides reliable rain attenuation time series $A\left(t\right)$ and their power spectra [15,16]. From knowing the rain‒rate time series $R\left(t\right)$ (mm/h), recorded at a site, the SST can generate rain attenuation time series $A\left(t\right)$, with time resolution of the order of one second – at any frequency and polarization, and for any slant path above about 10°. Because it reproduces reliable $A\left(t\right)$, it has been used in the last 30 years for many purposes, by several researchers [17,18,19,20,21,22,23,24]. In this paper we consider only $P\left(A\right)$; fade durations and rate–of–change will be studied in a future work.

5.2. Application of the SST to All Sites

We apply the SST twice: first by using as input the experimental 1–min rain–rate time series, namely $R_1\left(t\right)$, secondly by using the simulated $R\left(t\right)$. Figure 10 shows the annual average rain–attenuation probability distributions obtained with $R_1\left(t\right)$ and with $R\left(t\right)$. There are no appreciable differences that can impact on satellite system design.

Figure 18. Average annual probability distrbution $P\left(A\right)$ – namely, the fraction of time of a year – that the rain attenuation $A$ (dB) in abscissa is exceeded, estimated with the SST; cyan line: experimental $R_1\left(t\right)$, magenta line: simulated $R\left(t\right)$.

Figure 18. Average annual probability distrbution $P\left(A\right)$ – namely, the fraction of time of a year – that the rain attenuation $A$ (dB) in abscissa is exceeded, estimated with the SST; cyan line: experimental $R_1\left(t\right)$, magenta line: simulated $R\left(t\right)$.

Figure 18. Gera Lario. Average annual probability distrbution $P\left(A\right)$ that the rain attenuation $A$ (dB) in abscissa is exceeded, estimated with the SST; blue line: experimental $R_1\left(t\right)$, black line: simulated $R\left(t\right)$. Left panel: $P\left(A\right)$ is obtained by using local values of the conditional rain rate PDFs; Right panel: $P\left(A\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 18. Gera Lario. Average annual probability distrbution $P\left(A\right)$ that the rain attenuation $A$ (dB) in abscissa is exceeded, estimated with the SST; blue line: experimental $R_1\left(t\right)$, black line: simulated $R\left(t\right)$. Left panel: $P\left(A\right)$ is obtained by using local values of the conditional rain rate PDFs; Right panel: $P\left(A\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 19. Fucino. Average annual probability distrbution $P\left(A\right)$ that the rain attenuation $A$ (dB) in abscissa is exceeded, estimated with the SST; blue line: experimental $R_1\left(t\right)$, black line: simulated $R\left(t\right)$. Left panel: $P\left(A\right)$ is obtained by using local values of the conditional rain rate PDFs; Right panel: $P\left(A\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 19. Fucino. Average annual probability distrbution $P\left(A\right)$ that the rain attenuation $A$ (dB) in abscissa is exceeded, estimated with the SST; blue line: experimental $R_1\left(t\right)$, black line: simulated $R\left(t\right)$. Left panel: $P\left(A\right)$ is obtained by using local values of the conditional rain rate PDFs; Right panel: $P\left(A\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 20. Madrid. Average annual probability distrbution $P\left(A\right)$ that the rain attenuation $A$ (dB) in abscissa is exceeded, estimated with the SST; blue line: experimental $R_1\left(t\right)$, black line: simulated $R\left(t\right)$. Left panel: $P\left(A\right)$ is obtained by using local values of the conditional rain rate PDFs; Right panel: $P\left(A\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 20. Madrid. Average annual probability distrbution $P\left(A\right)$ that the rain attenuation $A$ (dB) in abscissa is exceeded, estimated with the SST; blue line: experimental $R_1\left(t\right)$, black line: simulated $R\left(t\right)$. Left panel: $P\left(A\right)$ is obtained by using local values of the conditional rain rate PDFs; Right panel: $P\left(A\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 21. Prague. Average annual probability distrbution $P\left(A\right)$ that the rain attenuation $A$ (dB) in abscissa is exceeded, estimated with the SST; blue line: experimental $R_1\left(t\right)$, black line: simulated $R\left(t\right)$. Left panel: $P\left(A\right)$ is obtained by using local values of the conditional rain rate PDFs; Right panel: $P\left(A\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 21. Prague. Average annual probability distrbution $P\left(A\right)$ that the rain attenuation $A$ (dB) in abscissa is exceeded, estimated with the SST; blue line: experimental $R_1\left(t\right)$, black line: simulated $R\left(t\right)$. Left panel: $P\left(A\right)$ is obtained by using local values of the conditional rain rate PDFs; Right panel: $P\left(A\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 22. Tampa. Average annual probability distrbution $P\left(A\right)$ that the rain attenuation $A$ (dB) in abscissa is exceeded, estimated with the SST; blue line: experimental $R_1\left(t\right)$, black line: simulated $R\left(t\right)$. Left panel: $P\left(A\right)$ is obtained by using local values of the conditional rain rate PDFs; Right panel: $P\left(A\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 22. Tampa. Average annual probability distrbution $P\left(A\right)$ that the rain attenuation $A$ (dB) in abscissa is exceeded, estimated with the SST; blue line: experimental $R_1\left(t\right)$, black line: simulated $R\left(t\right)$. Left panel: $P\left(A\right)$ is obtained by using local values of the conditional rain rate PDFs; Right panel: $P\left(A\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 23. White Sands. Average annual probability distrbution $P\left(A\right)$ that the rain attenuation $A$ (dB) in abscissa is exceeded, estimated with the SST; blue line: experimental $R_1\left(t\right)$, black line: simulated $R\left(t\right)$. Left panel: $P\left(A\right)$ is obtained by using local values of the conditional rain rate PDFs; Right panel: $P\left(A\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 23. White Sands. Average annual probability distrbution $P\left(A\right)$ that the rain attenuation $A$ (dB) in abscissa is exceeded, estimated with the SST; blue line: experimental $R_1\left(t\right)$, black line: simulated $R\left(t\right)$. Left panel: $P\left(A\right)$ is obtained by using local values of the conditional rain rate PDFs; Right panel: $P\left(A\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 24. Vancouver. Average annual probability distrbution $P\left(A\right)$ that the rain attenuation $A$ (dB) in abscissa is exceeded, estimated with the SST; blue line: experimental $R_1\left(t\right)$, black line: simulated $R\left(t\right)$. Left panel: $P\left(A\right)$ is obtained by using local values of the conditional rain rate PDFs; Right panel: $P\left(A\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Figure 24. Vancouver. Average annual probability distrbution $P\left(A\right)$ that the rain attenuation $A$ (dB) in abscissa is exceeded, estimated with the SST; blue line: experimental $R_1\left(t\right)$, black line: simulated $R\left(t\right)$. Left panel: $P\left(A\right)$ is obtained by using local values of the conditional rain rate PDFs; Right panel: $P\left(A\right)$ is obtained by using Spino d’Adda conditional PDFs (Table 2).

Now, since for satellite communications the tolerated outage probabilty due to rain attenuation, i.e. $P\left(A\right)$, is in the range ${10}^{-1}-{10}^{-2}$ (%) (i.e. outage time from 525 to 52.5 minutes, respectively), from these figures it turns out that for most probability range there are no signficant differences between rain attenuation $A$ estimated with local or with Spino d’Adda parameters.