### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Experimental Power Spectrum of the Tropospheric Channel
- 3. Rain Attenuation and Scintillation Experimental Data
- 4. Thin Layer Model
- 5. Experimental Relationship Between Scintillation and Rain Attenuation
- 6. Conclusions
- Appendix A:: Signal-to-Noise Ratio in Scintillation Measurements
- Appendix B:: Amplitude Fluctuation as a Function of Scintillation and Rain Attenuation
- Acknowledgments
- References

[1] We present experimental results on the relationship between rain attenuation and simultaneous scintillation, obtained with measurements of attenuation time series at 18.7 GHz, collected in two slant paths to Italsat (13.2° E): at Spino d'Adda (Italy, a 37.7° slant path, times series of 50 samples/s and 1 sample/s) and Darmstadt (Germany, a 32.7° slant path, 1 sample/s). We have found that the average rain attenuation *A* (dB) and the average standard deviation *σ*_{m} (dB) in 1-min intervals (calculated from the 1 sample/s time series), of the scintillation standard deviation *σ* (dB), are linked by a power law *σ*_{m} = *C*_{3a}*A*, or by the more refined model *σ*_{m} = *C*_{3b}*A*, that can be due to an effective thin layer of turbulence aloft. The constants *C*_{3a,3b} can be estimated by applying the ITU−R formula for scintillation predictions. For conservative design, a best fit model of the conditional standard deviation of *σ* is also provided, as a function of *A*, as *σ*_{s} = *C*_{s}*A*^{d} (dB), where *C*_{s} and *d* are constants.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Experimental Power Spectrum of the Tropospheric Channel
- 3. Rain Attenuation and Scintillation Experimental Data
- 4. Thin Layer Model
- 5. Experimental Relationship Between Scintillation and Rain Attenuation
- 6. Conclusions
- Appendix A:: Signal-to-Noise Ratio in Scintillation Measurements
- Appendix B:: Amplitude Fluctuation as a Function of Scintillation and Rain Attenuation
- Acknowledgments
- References

[2] Any time signal fading in satellite communication is to be compensated by real time fade countermeasures, it is necessary to know or predict attenuation time series during rain. System designers may not have experimental attenuation time series available and, moreover, when available, they have been recorded only for specific sites, frequencies and elevation angles, and their scaling to other sites is not straightforward. Instead of using real data collected from expensive and long propagation experiments, the alternative is to generate synthetic rain attenuation and scintillation time series with mathematical synthesizers, which, of course, must be tested for their reliable use [*Lemorton et al.*, 2004].

[3] Both for modeling and gaining a deeper physical insight into a complex phenomenon, we present the results of a sound and meaningful analysis of the instantaneous relationship between rain attenuation and simultaneous scintillation obtained from experimental time series of fading during rain collected at 18.7 GHz in two slant paths to the geostationary satellite Italsat (13.2° E) at Spino d'Adda in Italy (45.4°N, 9.5°E, 84 m above sea level, offset parabolic antenna of 3.5 m diameter, 37.7° slant path) and Darmstadt in Germany (49.9°N, 8.6°E, 180 m asl, offset parabolic antenna of 3.7 m diameter, 32.7° slant path).

[4] At Spino d'Adda data were sampled both at 50 samples/s (“fast mode” acquisition) and at 1 sample/s (“conventional mode” acquisition); at Darmstadt data were sampled only at 1 sample/s. The unusually high sampling rate of the fast mode acquisition is useful because it allows to assess, in the power spectrum of the time series of fading, the full range of frequencies in which either rain attenuation or scintillation prevail (and also to set the system noise floor).

[5] We have found that rain attenuation and scintillation are, on the average, linked, for both localities, by the same power law, and that this power law can be theoretically derived by considering an effective turbulent thin layer aloft, a model already applied at 19.77 GHz [*Matricciani et al.*, 1996] and 12.5 GHz to satellite Olympus data [*Matricciani et al.*, 1997], and to a small preliminary data bank of Italsat at 49.5 GHz [*Matricciani et al.*, 1995], all collected at Spino d'Adda.

[6] Our present results confirm, with a very large data base at 18.7 GHz and two distant sites (Spino d'Adda and Darmstadt), that the thin layer modeling is a very good physical approach to establish the long-term relationship between rain attenuation and average standard deviation of the simultaneous scintillation, and that the two phenomena are statistically dependent, at least for the climate of the these two sites.

[7] In section 2 we first derive and discuss the power spectrum of attenuation time series measured during rain at Spino d'Adda in the fast mode (50 samples/s), then we derive and discuss the power spectra obtained for Spino d'Adda and Darmstadt in the conventional mode (1 sample/s). In section 3 we report the analysis conducted on simultaneous scintillation and rain attenuation. In section 4 we recall the main features of a turbulent thin layer aloft and far from a receiving antenna. In section 5 we establish the average experimental relationship between the average standard deviation of scintillation and rain attenuation measured in 1-min intervals, and in section 6 we draw some conclusions.

### 2. Experimental Power Spectrum of the Tropospheric Channel

- Top of page
- Abstract
- 1. Introduction
- 2. Experimental Power Spectrum of the Tropospheric Channel
- 3. Rain Attenuation and Scintillation Experimental Data
- 4. Thin Layer Model
- 5. Experimental Relationship Between Scintillation and Rain Attenuation
- 6. Conclusions
- Appendix A:: Signal-to-Noise Ratio in Scintillation Measurements
- Appendix B:: Amplitude Fluctuation as a Function of Scintillation and Rain Attenuation
- Acknowledgments
- References

[8] To distinguish between rain attenuation (signal fluctuations caused by absorption and scattering by hydrometeors) and scintillations (signal fluctuations caused by air turbulence), we must separate the two phenomena as much as possible by filtering the experimental attenuation time series [*Mousley and Vilar*, 1982; *Karasawa and Matsudo*, 1991; *Poiares Baptista and Davies*, 1994]. The effects of the two phenomena are additive when the received signal amplitude (or the attenuation) relative to clear sky is expressed in decibels, as is common practice and in the following analysis.

[9] For the fast mode analysis (done only for Spino d'Adda) the experimental data consist of *N* = 35 attenuation time series collected during rain with a postdetection sampling rate of 50 samples/s. The data bank spans the period May–August 1996, and refers to an overall attenuation time equal to 227 hours and 26 min.

[10] Before separating the two phenomena we have computed the power spectrum *S*_{i}(*f*), *i* = 1, 2, 3…, *N* for each time series, and then we have calculated the average power spectrum given by

[11] For each site, the spectra were calculated for each time series on the same number of sample, *M*, which is the first power of 2 greater than the number of sample in the longest event of the site data bank. Therefore, every event, obviously shorter than *M*, was then bordered with zeroes up to *M* (zero-padding). For Spino d'Adda we used *M* = 2^{17} samples for the 1 sample/s time series and *M* = 2^{21} samples for the 50 samples/s time series. For Darmstadt (1 sample/s time series) we used *M* = 2^{16} samples.

[12] The averaging of the logarithm of power spectra of the single rain attenuation time series done in equation (1) is useful because the power spectra of the rain attenuation and scintillation have different log-slopes, theoretically known, which remain distinct if the averaging of logarithm is performed on the experimental data. This allows us to compare the experimental results with theory.

[13] Figure 1 shows the average power spectrum obtained from (1) for the fast mode data and also for the complete (1 sample/s) data bank of Spino d'Adda (i.e., for *N* = 2685 time series of fading, see section 3). We can notice the typical shape predicted for a turbulent atmosphere with a complex refractive index [*Ott*, 1978] (although Figure 1 is not explicitly related to clear sky - as in the analysis conducted in the work of *Ott* [1978] - but to rain), with two frequency intervals identified with slopes of different and definite magnitude, separated by a frequency interval with a more flat spectrum. Up to frequencies of a few hundredths of Hz there is the classical −20 dB/decade slope due only to rain attenuation [*Matricciani*, 1994]. The magnitude starts to change at about 0.02–0.03 Hz, confirming our previous results for slant paths of about the same elevation angle at the same site at 19.77 GHz [*Matricciani et al.*, 1996, 1997]. The more flat range spreads for about a decade so that for frequencies larger than about 0.2–0.3 Hz the classical −80/3 dB/decade slope due only to scintillation (i.e., −8/3 in natural units [*Tatarski*, 1961, 1967]) clearly appears. A distinct scintillation corner frequency is not clearly visible because many spectra were averaged. Finally, receiver noise, given by the flat portion above about 3 Hz, limits the range of the measurements.

[14] The “spikes” observed in Figure 1 are due to the satellite nutation, but they can be neglected because most of their power can be filtered out together with the receiver noise above 0.5 Hz.

[15] Figure 1 also shows the average power spectrum obtained for Darmstadt with 848 attenuation time series, postdetection sampling rate of 1 sample/s. The data bank spans the period October 1993 to December 1997 (50 months of observation), and refers to an overall attenuation time equal to 2401 hours and 40 min. We notice part of the typical shape predicted for a turbulent atmosphere with a complex refractive index and we can again observe the −20 dB/decade slope, due only to rain attenuation, up to about 0.02–0.03 Hz.

[16] In conclusion, for both sites we can separate the two phenomena, namely rain attenuation and scintillation in the conventional mode (1 sample/s data), by low pass filtering the time series with a cutoff frequency of the order of few hundredths of Hz, namely at 0.025 Hz, to obtain rain attenuation alone, and by band pass filtering them in the bandwidth 0.025–0.5 Hz to obtain scintillation alone.

### 3. Rain Attenuation and Scintillation Experimental Data

- Top of page
- Abstract
- 1. Introduction
- 2. Experimental Power Spectrum of the Tropospheric Channel
- 3. Rain Attenuation and Scintillation Experimental Data
- 4. Thin Layer Model
- 5. Experimental Relationship Between Scintillation and Rain Attenuation
- 6. Conclusions
- Appendix A:: Signal-to-Noise Ratio in Scintillation Measurements
- Appendix B:: Amplitude Fluctuation as a Function of Scintillation and Rain Attenuation
- Acknowledgments
- References

[17] Let us first consider Spino d'Adda. For the conventional mode (1 sample/s) the experimental data consist of 2685 attenuation time series (those shorter than 1 min were excluded) collected during rain in the years 1994–2000 (7 years of observation, see [*Matricciani and Riva*, 2005] for details on data preprocessing and data bank completeness, an issue less relevant for this study). The database amounts to an overall attenuation time equal to 2541 hours and 30 min.

[18] Let us first consider rain attenuation. We have filtered the experimental time series (expressed in dB) with a fifth order low-pass Butterworth filter, with cutoff frequency 0.025 Hz to get rid of most scintillation power and leave rain attenuation only. Then, we have averaged the attenuation of 60 samples of disjoint blocks of the low-pass filtered time series. In the end we have produced 1 sample/min time series with nominal highest frequency equal to 0.025 Hz, referred to as “rain attenuation” time series in the following. Because the highest frequency component is about 0.025 Hz, rain attenuation averaged over 1 min still retains its dynamical and statistical features. Notice that analysis below is insensitive on the “exact” value of cutoff frequency and it could be a little higher, [*Matricciani et al.*, 1995, 1996, 1997], without changing the findings and modeling. Figure 2 shows the long term complementary cumulative distribution functions (ccdfs) of rain attenuation only (scintillation, however, does not change it significantly) measured at Spino d'Adda and Darmstadt.

[19] Let us consider scintillation. To complement the previous analysis, we have filtered the original 1 sample/s time series with a Butterworth high-pass filter of the fifth order (in fact, the higher frequency cutoff at the Nyquist frequency 0.5 Hz was originally in the processing of the raw data prior to our analysis) with cutoff frequency 0.025 Hz to eliminate most rain attenuation power and leave scintillation alone, in the following referred to as “scintillation” time series.

[20] To correlate scintillation with the average rain attenuation in 1-min disjoint intervals, we have computed the standard deviation of the scintillation over 60 samples, and coupled it to the corresponding averaged rain attenuation. Notice that in 1-min intervals, scintillation is produced by stationary turbulence that can be modeled according to a statistical Gaussian process with zero average value [*Marzano and D'Auria*, 1998; *Vasseur*, 1999; *Otung and Savvaris*, 2003], if the amplitude is expressed in decibels, as in our case. As a consequence, scintillation is just characterized by its average power, i.e., its variance or its standard deviation, in decibels. In the end, we have a couple of simultaneous data every minute: rain attenuation *A* (dB) and standard deviation of scintillation *σ* (dB).

[21] Figure 3 shows the scatterplot of *A* and *σ* for Spino d'Adda. Couples with *σ* < 0.05dB have been excluded (40% of the sample size) because they are likely due to system noise because of the very low “signal-to-noise” ratio in this case, see section 5. Now, the total time of the data shown in Figure 3 amounts at 88593 min (1476.6 hours), in any case a large data bank.

[22] Let us consider Darmstadt. As mentioned in section 2 the experimental data bank consists of 848 attenuation time series for a total time of 2401 hours and 40 min. Figure 4 shows the scatterplot obtained after processing the time series with the same filtering applied to the Spino d'Adda data (again, couples with *σ* < 0.05 dB have been excluded, i.e., 31% of the sample size). Now the total time of the data shown in Figure 4 amounts at 98708 min (1645.1 hours).

[23] In both Figures 3 and 4, we can see that *σ* and *A* seem to be spread quite chaotically, and that a relationship may be hard to find. But this is not the case if we compute the average standard deviation for a given rain attenuation, as shown in the following, and compare these results to those predicted by a turbulent thin layer aloft, whose features are summarized in the next section.

### 4. Thin Layer Model

- Top of page
- Abstract
- 1. Introduction
- 2. Experimental Power Spectrum of the Tropospheric Channel
- 3. Rain Attenuation and Scintillation Experimental Data
- 4. Thin Layer Model
- 5. Experimental Relationship Between Scintillation and Rain Attenuation
- 6. Conclusions
- Appendix A:: Signal-to-Noise Ratio in Scintillation Measurements
- Appendix B:: Amplitude Fluctuation as a Function of Scintillation and Rain Attenuation
- Acknowledgments
- References

[24] We wish to assess if the thin layer model discussed and successfully applied in our previous work [*Matricciani et al.*, 1995, 1996, 1997] can also be applied to these Italsat data, so that we recall its fundamentals (see also [*Ho and Wheelon*, 2004] for a recent tutorial on scintillation due to troposphere turbulence).

[25] Let *L*_{R} (km) be the average rainy slant path length and *A* (dB) its rain attenuation. To a first approximation, we can write:

where *C*_{1a} is a constant.

[26] A refinement of the relationship (2a), discussed by [*Matricciani*, 2006] yields the non-linear relationship

where *C*_{1b} is a another constant and approximately *m* = 0.90 [*Matricciani*, 2006].

[27] Let us now consider scintillation. From Tatarski's [*Tatarski*, 1961, 1967] expression of variance of amplitude of scintillation at microwaves for a thin turbulent layer aloft of uniform intensity, that may simulate cloud turbulence in rainy conditions, we calculate the standard deviation as

where *k* = 2*π*/*λ* is the wave number, *λ* is the wavelength, *C*_{n} is the structure constant of the turbulence, *L*_{S} (km) is the slant distance of the turbulent layer from the receiving antenna, *D* (km) is the thickness of the turbulent layer, *C*_{2} (dB/km^{5/12}) is a constant.

[28] The scintillation that we have experimentally observed seems to be cloud scintillation, not clear-air scintillation, as treated by Tatarskii, and our results implicitly assume that the presence of water vapor and rain droplets within the cloud are conservative passive additives, i.e., the motion of the water vapor and droplets is affected by the turbulence within the atmosphere and not the other way around. Only in this case it is possible to assign a Kolmogorov-type fluctuation spectrum to the process, which is at the basis of Tatarskii's treatment of scintillation, and the model adopted in (3a).

[29] Now, if the turbulent layer is located aloft and far from the antenna, its exact position is unimportant to the modeling so long as it is “thin” i.e., *D* ≪ *L*_{S}, because to estimate *σ* we can always assume *L*_{S} ± D ≈ *L*_{S}, and this is a robust characteristic of the modeling. In fact, if we write *D* ≈ *rL*_{S}, with *r* ≪ 1, the ratio between the standard deviation *σ*_{±} estimated when the thin turbulent layer is at the extremes *L*_{S} ± *D* and the standard deviation given by (3a), is given by

For instance, if *r* = 0.1, equation (3b) yields a ratio equal to 0.96 or 1.04, i.e., only a ±4% change in the estimate of *σ*, and even less for smaller values, hence always a negligible quantity for our calculations and modeling.

[30] Now, if we simply assume that

by eliminating *L* from (2a), (2b), and (3a), *σ* turns out to be dependent only on *A*. Thus we get the theoretical relationships between *σ* and *A* for the linear (2a), or for the nonlinear (2b) relationships, and the thin layer model. i.e.:

where *C*_{3a,3b} are constants, numerically found by setting *A* = 1 dB in (5a) and (5b).

### 5. Experimental Relationship Between Scintillation and Rain Attenuation

- Top of page
- Abstract
- 1. Introduction
- 2. Experimental Power Spectrum of the Tropospheric Channel
- 3. Rain Attenuation and Scintillation Experimental Data
- 4. Thin Layer Model
- 5. Experimental Relationship Between Scintillation and Rain Attenuation
- 6. Conclusions
- Appendix A:: Signal-to-Noise Ratio in Scintillation Measurements
- Appendix B:: Amplitude Fluctuation as a Function of Scintillation and Rain Attenuation
- Acknowledgments
- References

[31] We wish to compare *σ* and *A* by calculating the average standard deviation *σ*_{m} (dB) and the standard deviation *σ*_{s} (dB) of *σ*, for a given rain attenuation. According to Appendix A we can measure a standard deviation *σ* (dB) of scintillation with signal-to-noise power ratio *ρ*_{s}, estimated from

where *ρ*_{o} (dB) is the signal-to-noise power ratio measured in clear sky conditions in 1-Hz bandwidth, *A* (dB) is the simultaneous rain attenuation and *σ* (dB) is the standard deviation of scintillation (the “signal”), *B* (Hz) is the postdetection bandwidth. Nominally *ρ*_{o} = 60.2 dB for the Italsat receiver at Spino d'Adda [*Clementi et al.*, 1990], and *ρ*_{o} = 64 dB for Darmstadt.

[32] Figure 5 shows values of *ρ*_{s} for the two receivers calculated from (6), as a function of rain attenuation. If we wish, for instance, to observe at Spino d'Adda scintillation (the “signal”) with standard deviation as low as 0.1 dB, according to (6) or to Figure 5, we need a signal-to-noise power ratio *ρ*_{0.1dB} = 60.2 − 1.3 + 3 − *A* − 38.7 = 23.2 − *A* dB. For instance, if *A* = 10 dB then *ρ*_{0.1dB} = 13.2 dB, so that the effective value of the scintillation (not in decibels) is about 4.6 times the noise effective amplitude.

[33] Figures 6 and 7show, for the two sites, the average standard deviation *σ*_{m} as a function of *A*, calculated from Figures 3 and 4 respectively. There are, clearly, two ranges with different slopes. In the following we show that: (a) the first range (low attenuation) is due to a turbulent thin layer aloft, thus confirming our previous results; (b) the second range (high attenuation) is due to system noise.

[34] Let us calculate the regression line in the first range, namely in the range 0.5 ≤ *A* ≤ 10 dB, where the signal-to-noise ratio is high enough (see Figure 5) to allow us to observe even low values of scintillation due to turbulence, instead of that due to the receiver noise.

[35] For Spino d'Adda (Figure 6), a regression line analysis in log-log scales yields a slope equal to 0.45 and a constant *C*_{3} = 0.12. The slope is practically equal to the slope predicted by (5a) and (5b), i.e., 5/12 = 0.42 for the linear relationship (2a), or (5/12)/0.9 = 0.46 for the non-linear relationship (2b). The fact that the experimental value is 0.45, i.e. very close to the predicted 0.46 of the thin layer modeling coupled to the non-linear relationship (2b) is another indication, besides the discussion by [*Matricciani*, 2006] that (2b) is a refinement of (2a) for satellite typical slant paths.

[36] On the other hand, if we impose the 5/12 slope, i.e. equation (5a), we get the same constant *C*_{3} = 0.12 (with two significant decimal digits). Hence, there is no real difference between the best fit and the theoretical thin layer model (5a) or (5b) in which *σ* = *σ*_{m}. Moreover, equation (5b) seems to be a better physical modeling.

[37] For Darmstadt (Figure 7), a regression line analysis in log-log scales, again in the range 0.5 ≤ *A* ≤ 10 dB, yields a slope equal to 0.44 (very close to the values predicted by (5a) or (5b)) and the same constant *C*_{3} = 0.12 obtained for Spino d'Adda. If we impose the 5/12 slope, we get the same constant *C*_{3} = 0.12 and again there is no real difference between the best fit and the theoretical thin layer model (5a) or (5b). The Darmstadt results seem to indicate that equation (2b), derived explicitly for Spino d'Adda [*Matricciani*, 2006], is also valid for this locality.

[38] Notice that if we consider all values of standard deviation, i.e., include also the couples with *σ* < 0.05 dB, for the 5/12 slope we get *C*_{3} = 0.11 for both sites.

[39] In conclusions, the relationship between the averages of *σ* and *A* is effectively predicted by a turbulent thin layer aloft, equation (5a) or (5b), independently of site, and agrees with a mathematical relationship that the higher the rain fade the higher the scintillation intensity [*Mertens and Vanhoenacker-Janvier*, 2001].

[40] In all cases, notice that the average standard deviation predicted by the thin layer models (5a) and (5b) becomes zero when rain attenuation becomes zero. In this case, however, there must be a “residual” turbulence due to clouds, always present before and after a rainstorm. The residual turbulence is estimated to be about 0.06 dB for the Italsat radio link and antenna diameter of Spino d'Adda, and 0.07 dB for Darmstadt, according to the *ITU-R* [2003]. These values are of the same order of magnitude of the constant *C*_{3}. In conclusion, as a first attempt, we could conservatively estimate the values of *C*_{3} by doubling the values predicted by the ITU-R and then apply (5). This modeling takes also care of frequency scaling of scintillation, an issue previously discussed by [*Matricciani et al.*, 1997].

[41] Now let us consider the second range and show that it is due to receiver noise. As Appendix B shows, the effective signal amplitude Δ*v* (dB) reported on the ordinate scale in Figures 6 and 7 can be estimated from

[42] Values calculated from (7) are also reported in Figures 6 and 7. It is clear that the first addend in (7) describes the trend at low attenuation, i.e., the turbulence produced by the thin layer, while the second addend describes the trend at higher attenuations produced by the receiver noise, not by the medium. This is confirmed by the reduction in the signal-to-noise power ratio *ρ*_{s}, which now, from (6) and (5), becomes

whose values are traced in Figure 8 for the two sites. The maximum is set at about *A* = 3.6 dB, independently of *ρ*_{o} and *B*.

[43] Now, if we subtract from the average measured values reported in Figures 6 and 7 the second addend of (7), that depends on the system noise, we can get a more reliable estimate of *σ*_{m} (Figures 6 and 7). However, if on these “noiseless” data we calculate the new values of *C*_{3}, they do not practically change.

[44] So far, we have discussed the average relationship obtained from the thin-layer modeling. For a conservative system design, however, we must consider the large variations likely to occur in the instantaneous *σ*(A) relationship, as shown by the scatterplots in Figures 3 and 4.

[45] Specifically, we have computed the distributions of the standard deviation of *σ* as a function of *A*, in the more stable attenuation range shown in Figures 6 and 7. Figure 9 shows these conditional distributions for the two sites, and Figure 10 shows the conditional standard deviation *σ*_{s} of *σ* as a function of *A*. Figures 6 and 7 have already shown the +1*σ*_{s} bounds. Again, the two sites are similar. For Spino d'Adda, *σ*_{s} can be modeled as

For Darmstadt

Equations (9) and (10) then complete our statistical modeling and provide more conservative estimates for the two sites. For example, the linear model (5a) can be completed as

with *n* = 1, 2, 3 for more and more conservative design, and *C*_{s} = 0.07, *d* = 0.28 for Spino d'Adda, *C*_{s} = 0.08, *d* = 0.26 for Darmstadt. Since the three constants in equation (11) take almost the same numerical value for both sites, we suggest to consider the most conservative relationship (Darmstadt) and apply it to other similar sites.

### 6. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Experimental Power Spectrum of the Tropospheric Channel
- 3. Rain Attenuation and Scintillation Experimental Data
- 4. Thin Layer Model
- 5. Experimental Relationship Between Scintillation and Rain Attenuation
- 6. Conclusions
- Appendix A:: Signal-to-Noise Ratio in Scintillation Measurements
- Appendix B:: Amplitude Fluctuation as a Function of Scintillation and Rain Attenuation
- Acknowledgments
- References

[46] Our results confirm that the power law *σ* = *C*_{3a}*A*^{5/12} (dB), a relationship predicted by a thin layer turbulence aloft and a linear relationship between rain attenuation and rainy path length, or the more refined formula *σ* = *C*_{3b}*A*^{(5/12)/0.9} (dB), which assumes a non-linear relationship between rain attenuation and rainy path length, can be used to estimate the average value of standard deviation of the simultaneous scintillation, for a given average rain attenuation, both calculated in intervals of 1 min, for Spino d'Adda and Darmstadt.

[47] The power law presented cannot be used in its inverse form, i.e. to predict average rain attenuation using the average of observed scintillation intensity because, obviously, scintillation is most often unaccompanied by rain and can itself impact system design and performance. In general, scintillation must be taken into account with a statistical prediction model, much like that used for rain attenuation. Our results model only a part of a much wider range of phenomena, i.e. only rain attenuation and simultaneous scintillation.

[48] For a particular radio link, the value of *C*_{3} can be estimated from the ITU−R formulas that give the standard deviation of clear-sky scintillation. In other words, for other sites with weather similar to that of Spino d'Adda and Darmstadt, we could first estimate the standard deviation predicted for no rain conditions (i.e., an estimate of the constant *C*_{3}) and then apply (5a) or the more refined model (5b) to get the full model. The +*σ*_{s} bounds shown in Figures 6 and 7, or the mathematical modeling (9)(10), can be used to calculate more conservative values, e.g. +2*σ*_{s}, +3*σ*_{s} according to equation (11). Since our measurements refer to a particular set of radio electrical and geometrical parameters, the values of the “constants” *C*_{3} and *C*_{s} must be scaled [e.g., see *ITU-R*, 2003] according to frequency, elevation angle and antenna size.