### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Measured Attenuation Dynamics and Rain Rate Dynamics
- 3. Theoretical Expression for the Variance of Relative Fade Slope
- 4. Comparison With Measurements
- 5. Conclusions
- Appendix A:: Variance of Relative Fade Slope Related to Rain Rate
- Appendix B:: Autocorrelation of Attenuation
- Acknowledgments
- References
- Supporting Information

[1] A key parameter of the statistics of the rate of change of rain attenuation (“fade slope”) on satellite links is the variance of relative fade slope. This paper shows how this parameter can be derived from rain rate measurements without the use of satellite beacon measurements. Relating rain rate directly to attenuation would give unrealistic results. A theoretical model to estimate the variance of relative fade slope is derived, using the integrating effect of rain rate variations along the propagation path and using as inputs several meteorological parameters only. The theoretical values are compared to measured results from a link in the United Kingdom at 50 GHz. The agreement is good on average and also in their correlation with the type of rain and with the rain height. With the wind speed, the theoretical values increase more strongly than the measured results, which may be due to the limited spatial resolution of the meteorological data.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Measured Attenuation Dynamics and Rain Rate Dynamics
- 3. Theoretical Expression for the Variance of Relative Fade Slope
- 4. Comparison With Measurements
- 5. Conclusions
- Appendix A:: Variance of Relative Fade Slope Related to Rain Rate
- Appendix B:: Autocorrelation of Attenuation
- Acknowledgments
- References
- Supporting Information

[2] In frequency ranges at V band and above, rain attenuation on satellite links becomes so severe that static fade margins are not feasible anymore, and dynamic adaptive fade countermeasures are necessary, usually referred to as fade mitigation techniques (FMT) [*Sweeney and Bostian*, 1999; *European Cooperation in the Field of Science and Technology*, 2002]. For the design of FMTs, information on the dynamic behavior of rain attenuation is essential. Knowledge of the rate of variations of attenuation is useful to design a control loop of an FMT system that can follow signal variations, and also to allow a better short-term prediction of the propagation conditions. Information on the dynamic behavior of rain attenuation is also useful for the development of models of this dynamic behavior [e.g., *Maseng and Bakken*, 1981; *Sweeney and Bostian*, 1992].

[3] Commonly, the dynamic behavior of rain attenuation is studied in terms of the rate of change of attenuation, or “fade slope.” In order to provide information on rain fade slope, various experiments have been carried out to analyze its statistical properties [e.g., *Matricciani*, 1981; *Stutzman et al.*, 1995; *Feil et al.*, 1997; *Schnell and Fiebig*, 1997; *van de Kamp*, 2003a]. An empirical prediction model of fade slope, based on measurements from various sites [*van de Kamp*, 2003a] is currently recommended by *International Telecommunication Union* (*ITU*) [2005]. An overview of published measurements and models is given by *van de Kamp and Castanet* [2002].

[5] The fade slope *ζ* can be calculated from two attenuation sample values *A* (dB) and the time interval Δ*t* between them:

This definition is used in most (but not all) studies cited above. Values of Δ*t* used in practice vary between 2 s [*van de Kamp*, 2003a] and 10 s [*Feil et al.*, 1997]. The resulting fade slope depends also on Δ*t* [*van de Kamp and Clérivet*, 2004; *ITU*, 2005].

[6] Since the rain fade slope *ζ* is stochastic, measured results are studied statistically. The distribution of *ζ* is generally found to be symmetrical around a zero mean. In many studies the distribution of *ζ* is evaluated conditional to the coinciding attenuation value *A*. This conditional distribution is also symmetrical, and has, for all attenuation values above a certain minimum threshold *A*_{th}, the following properties [*van de Kamp*, 2003a; *ITU*, 2005]: a zero mean, a constant shape (independent of *A*), and a standard deviation approximately proportional to *A*.

[7] The minimum value *A*_{th} must be regarded because of the following: since the standard deviation of *ζ* is proportional to *A*, when *A* becomes small *ζ* also becomes small. So even after low-pass filtering, the fluctuations may be dominated by residual scintillation and noise, which show different dynamical characteristics.

[8] The “relative fade slope” is defined as

From the abovementioned properties of the distribution of *ζ* conditional to *A*, it follows that the distribution of *ζ*_{r} conditional to *A* also has a zero mean and a constant shape, but also an approximately constant standard deviation. This means that the unconditional distribution of *ζ*_{r}, for all values of *A* > *A*_{th}, also has the same standard deviation. The square of this standard deviation, the “variance of the relative fade slope” *σ*_{ζr}^{2}, will be used as the characterizing parameter of fade slope in this paper. It is defined as

where *E* denotes an average. From the above follows that the standard deviation of the distribution of fade slope *ζ* conditional to attenuation *A* is (approximately) given by *A*. The relation between *σ*_{ζr}^{2} and the current ITU-R model of fade slope is given as

where *S* is a link-dependent parameter and *F*(*f*_{B}, Δ*t*) is a factor expressing the dependence on the low-pass filter bandwidth *f*_{B} and the time interval Δ*t* [*van de Kamp*, 2003a; *ITU*, 2005].

[9] The variance of relative fade slope *σ*_{ζr}^{2} is useful for more purposes than the prediction of fade slope statistics: with Δ*t* = 10 s it is also one of the parameters from which the input parameters of the “two-sample model” can be derived, as is shown by *van de Kamp* [2003b, 2005a]. The two-sample model [*van de Kamp*, 2002; *van de Kamp*, 2003c] predicts the probability distribution of rain attenuation a short time after a measured value, dependent on the values of two previous samples of rain attenuation. This model can be applied in the design of FMT systems, for the short-term prediction of rain attenuation. It can also be used to generate simulated rain attenuation time series, which can be used for testing FMT systems. *Sweeney and Bostian* [1999] showed that 10 s is within the range of useful time constants for the design of FMTs. It is therefore clear that *σ*_{ζr}^{2} is an invaluable parameter in the design of FMT systems.

[10] The variance of relative fade slope *σ*_{ζr}^{2} is a site- and climate-dependent parameter. There are indications that the rain fade slope depends on the elevation angle [*Feil et al.*, 1997], and on meteorological parameters such as the wind speed, the path length through rain, and the type of rain (widespread/convective) [*Buné et al.*, 1988]. Therefore the parameter *σ*_{ζr}^{2} should be determined for every separate link and meteorological condition for which FMT systems are to be designed.

[11] In this paper relations are studied between the dynamic properties of rain attenuation and those of the rain itself. The main purpose of this study is to predict *σ*_{ζr}^{2} for specific sites from rain gauge data, without having to use rain attenuation data.

### 2. Measured Attenuation Dynamics and Rain Rate Dynamics

- Top of page
- Abstract
- 1. Introduction
- 2. Measured Attenuation Dynamics and Rain Rate Dynamics
- 3. Theoretical Expression for the Variance of Relative Fade Slope
- 4. Comparison With Measurements
- 5. Conclusions
- Appendix A:: Variance of Relative Fade Slope Related to Rain Rate
- Appendix B:: Autocorrelation of Attenuation
- Acknowledgments
- References
- Supporting Information

[12] As a first step, the variance of rain fade slope *σ*_{ζr}^{2} derived from measured data will be compared to an equivalent parameter determined from measured rain rate data [*van de Kamp*, 2005b].

[13] *σ*_{ζr}^{2} has been determined from beacon data from the satellite Italsat, which had been measured by Rutherford Appleton Laboratory in Sparsholt, United Kingdom. The signals, at 18.69, 39.59 and 49.49 GHz were recorded during 43 months from April 1997 to January 2001. The elevation angle of the satellite was 29.9°. In this paper, only the measurements at 49.49 GHz are considered (hereafter referred to as 50 GHz).

[14] For this analysis, the measured rain attenuation data were divided in the periods: 0300–0900, 0900–1500, 1500–2100 and 2100–0300 of each day. For each of these 6-hour periods, the data were low-pass filtered to reduce the effect of scintillation, using a bandwidth *f*_{B} of 0.02 Hz, as recommended by *Poiares Baptista and Davies* [1994, p. 91]. The relative fade slope *ζ*_{r} was calculated as in equation (2) with Δ*t* = 10 s, and from this, the variance *σ*_{ζr}^{2} of relative fade slope was calculated as

where the integral is evaluated over the measured data in the 6-hour period for which *A* > *A*_{th} as mentioned in section 1, and *T* is the accumulated time of these data. The optimum value for *A*_{th} depends on frequency, since both scintillation and attenuation increase with frequency. It was found that *σ*_{ζr}^{2} shows stable results (i.e., becoming constant with respect to *A*_{th}) for *A*_{th} = 5 dB at 50 GHz. The resulting total amount of data used was 2,018,068 data points (seconds), 2.09% of the total measured time.

[15] It was checked that the results were free from residual scintillation, by calculating *σ*_{ζr}^{2} from selections of data with different values of scintillation measured in a higher frequency band (0.1–0.5 Hz) [*van de Kamp*, 2005b]. No scintillation dependence was found from this analysis; the values of *σ*_{ζr}^{2} calculated as described above can therefore be trusted to be due to rain attenuation only. The effect of the low-pass filter with *f*_{B} = 0.02 Hz was compensated by scaling the results to an infinite filter bandwidth by the factor *F*(*f*_{B}, Δ*t*) (see equation (4)) [*van de Kamp and Clérivet*, 2004; *ITU*, 2005].

[16] From the values of *σ*_{ζr}^{2} and *T* for each 6-hour period, the long-term values of *σ*_{ζr}^{2} were calculated as

where *σ*^{2}*ζ*_{ri} and *T*_{i} indicate the values of *σ*_{ζr}^{2} and *T* for 6-hour period *i*.

[17] These results were analyzed concurrent with meteorological data, which was obtained from the European Centre for Medium-Range Weather Forecasts (ECMWF) (http://www.ecmwf.int/) and the British Atmospheric Data Centre (http://badc.nerc.ac.uk/, from their Operational Data database, which is a continuation of their ERA-40 project). Various meteorological parameters are available recorded at reference times 0000, 0600, 1200 and 1800 (UTC) of each day, and for grid points around the globe 2.5° apart in longitude and latitude.

[18] The main meteorological parameter of interest for the characteristics of fade slope is the wind speed [*van de Kamp*, 2003a]. From ECMWF, the wind speed in two orthogonal directions is available on various height levels, expressed as pressure levels. The relation of *σ*_{ζr}^{2} with wind speed was studied using the velocity of wind in eastward direction *v*_{u} (m/s) and in northward direction *v*_{v} (m/s) at the pressure level 850 mbar, which is approximately 1.5 km high. This being the approximate height at which rain originates, it is expected that the wind speed at this height will show some correlation with the dynamics of rain attenuation. These data were downloaded for the period 1997 to 2001, and two-dimensionally linearly interpolated between the grid points to obtain the values in Sparsholt (latitude 51.07°; longitude –1.43°). This grid square is about 175 km (E-W) × 250 km (N-S), and contains much of southern England including its south coast, but no mountains to form orographic barriers. Thus, although the interpolated values cannot give exactly the momentary values on one spot, on average they are expected to be representative for the meteorological conditions at Sparsholt.

[19] From *v*_{u} and *v*_{v}, the absolute value of wind speed *v*_{w} was calculated as

[20] Next, the concurrent analyses was performed as follows: the values of *v*_{w} were classified in bins with bin size 2 m/s, and for every bin, the long-term values of *σ*_{ζr}^{2} were calculated over all values in the 6-hour periods coincident with *v*_{w} being in the respective bin. The results represent *σ*_{ζr}^{2} as a function of wind speed. The result is shown in Figure 1 (solid line).

[21] Another possible meteorological parameter which is likely to influence the statistics of fade slope, is the type of rain. To study this, the data were analyzed concurrent with the vertical velocity of the air *v*_{ver}, also obtained from ECMWF. Because convective rain is caused by vertical updraft of air, the vertical velocity can be an indicator for the degree of convectivity and thus for the type of rain. The unit of *v*_{ver} is Pa/s, so it is expressed as the change in pressure that an air package encounters in an updraft (negative values indicating upward movement). The vertical velocity *v*_{ver} was also obtained for the 850 mbar pressure level.

[22] The concurrent analyses with *v*_{w} and *v*_{ver} together was performed by calculating the long-term values of *σ*_{ζr}^{2} over all values in the 6-hour periods coincident with *v*_{ver} < −0.15 Pa/s and for every value of *v*_{w}; the same was done using the condition of *v*_{ver} > −0.05 Pa/s. These threshold values were arbitrarily chosen, to cut off approximately the upper and lower one-third part of the distribution. The results of this are included in Figure 1 (dotted lines).

[23] Figure 1 shows that *σ*_{ζr}^{2} increases with wind speed, for all data, as well as for both categories of *v*_{ver}. Furthermore, the values for *v*_{ver} > −0.05 Pa/s (supposedly corresponding to stratiform rain) are higher than those for *v*_{ver} < −0.15 Pa/s (convective rain). All curves have a relatively small offset.

[24] The results for *σ*_{ζr}^{2} increasing with wind speed, with a small offset, suggests that the major part of variations in rain attenuation observed are influenced by wind, and are most likely caused by horizontal variations in the structure of the rain storm, which is moved across the propagation path. The gradient of the variance as a function of the wind speed is likely to be an indication of the variance of spatial horizontal variations of the rain intensity in the structure of the rainstorm. The observation that this gradient is larger during stratiform rain suggests that although convective rainstorms are smaller in horizontal size, stratiform rain contains more spatial variations.

[25] A relation of this parameter with rain rate dynamics may be searched as follows. According to the ITU-R recommendation P.618-8 [*ITU*, 2003a], rain attenuation is related to the rainfall intensity, or “rain rate” *R* (mm/hr) as

where *L*_{E} is the effective path length through rain, and *a* and *b* are frequency-, elevation- and polarization-dependent coefficients; *b* takes values around 1 in the centimeter wave region. Although *a* and *b* can be expected to depend on the drop size distribution, which may vary between rainstorms and even within a rainstorm, this model with static values of *a* and *b* is considered adequate to predict attenuation statistics [*ITU*, 2003b]. It may therefore be expected that a similar dynamic behavior as for the attenuation would be found for the rain rate.

[26] In order to study this, 94 months of rain rate measurements in Sparsholt and Chilbolton (8 km from Sparsholt) have been analyzed. The rain measurements were performed in three different measurement campaigns by Rutherford Appleton Laboratory, using drop counter rain gauges. These rain gauges release the collected rainwater as drops of a constant size, which are then counted. More details are given in Table 1. Even though these measurements are not completely concurrent with the rain attenuation data from Sparsholt, they come from (nearly) the same site, and the relation between the dynamics of rain and climatic characteristics can be expected to be the same for both data sets.

Table 1. Details About the Measurements of Rain Rate UsedLocation | Period | Length, months | Integration Time, s | Quantization Level, mm/hr |
---|

Sparsholt | Apr 1997 to Dec 1999 and Jan 2001 | 31 | 10 | 1.44 |

Sparsholt | Jan 2003 to May 2004 and Jan to Dec 2005 | 29 | 10 | 1.44 |

Chilbolton | Jan 2003 to Oct 2004 and Jan to Dec 2005 | 34 | 10 | 1.44 |

[27] The rain rate data were analyzed in the same way as the attenuation data, and the variance of relative rain rate slope was calculated similarly as equations (5) and (2), where *A*(*t*) is replaced by the rain rate *R*(*t*), and Δ*t* = 10s. The integral of equation (5) was evaluated only over samples in each 6-hour period which exceed 8 mm/hr, in order to avoid effects of the quantization level of 1.44 mm/hr (Table 1). The resulting total amount of rain data used was 34,637 data points (= 346,370 s), 0.146% of the total measured time. This relative portion of data is smaller than for the attenuation data; this is because a rain rate of 8 mm/hr causes at 50 GHz about 22 dB of attenuation, so this threshold is effectively higher than *A*_{th}, the threshold used for the attenuation.

[28] The measured variance of relative rain rate slope was compensated for the filtering effect caused by the 10 s integration time of the rain gauge, using the same procedure as the attenuation data. The data were analyzed concurrent with the wind speed *v*_{w} and vertical velocity *v*_{ver} as obtained from ECMWF, in the way described above for the attenuation data. The results are shown in Figure 2.

[29] On the basis of the almost linear relation between rain rate and attenuation as given in equation (8), it would be expected that the variance of relative rain rate slope will be similar to that of relative fade slope. Figure 2 shows however a quite different picture. The variance of relative rain rate slope also increases with wind speed, and, although slightly less pronouncedly, with the vertical velocity. However, note the different vertical axis scales: the variance of relative rain rate slope is much larger than that of fade slope. For instance, at *v*_{w} = 10 m/s the variance of relative fade slope is 8 × 10^{–5} s^{–2}, while that of rain rate slope is 1.4 × 10^{–3} s^{–2}, over 17 times larger!

[30] Such a difference must mean that the dynamic relation between rain rate and attenuation is less straightforward than suggested by equation (8). If rain rate measurements are to be used for the prediction of the variance of relative fade slope, a more elaborate relation between these two will have to be used. This will be addressed in the next section.

### 4. Comparison With Measurements

- Top of page
- Abstract
- 1. Introduction
- 2. Measured Attenuation Dynamics and Rain Rate Dynamics
- 3. Theoretical Expression for the Variance of Relative Fade Slope
- 4. Comparison With Measurements
- 5. Conclusions
- Appendix A:: Variance of Relative Fade Slope Related to Rain Rate
- Appendix B:: Autocorrelation of Attenuation
- Acknowledgments
- References
- Supporting Information

[60] The theoretical expression for the variance of relative fade slope *σ*_{ζr}^{2}, composed of equations (10), (11) and (21) will now be tested by comparing to measurements. For this purpose, the measured results of rain attenuation from Sparsholt at 50 GHz (see section 2) are again analyzed. A more extensive comparison can be found elsewhere [*van de Kamp*, 2005b]. The theoretical expression depends on the following parameters (with the values of the system parameters of the link in Sparsholt given): the magnitude of wind speed *v*_{w}, the azimuth angle of the wind direction θ_{w}, the azimuth angle of the link θ_{l} = 161.7°, the rain height *h*_{r}, the elevation angle of the link ɛ = 29.9°, the parameter indicating spatial variability of rain *B*_{x}, the long-term standard deviation of the log rain rate *σ*_{lnR}, the parameter *b* of the ITU-R rain attenuation model ≈0.87 for 50 GHz, and the time length Δ*t* = 10 s.

[61] To obtain all necessary input parameters for the theoretical expression, the following meteorological parameters were obtained from ECMWF, for a 6-hour time resolution and 2.5° horizontal resolution: the wind speeds *v*_{u} and *v*_{v} in (respectively) eastward and northward direction, at the 850 bar pressure level (around 1.5 km height), the vertical velocity *v*_{ver} at the 850 bar pressure level, and the temperature profiles up to 16 km height.

[62] From these, the magnitude of wind speed *v*_{w} (see equation (7)) was calculated, as well as the azimuth angle of wind direction:

and the rain height [*ITU*, 2001]:

where *h*_{0C} is the height at 0°C temperature, linearly interpolated from the temperature profile. All parameters were linearly interpolated to obtain the values in Sparsholt.

[63] In the comparison of the theoretical *σ*_{ζr}^{2} with measured values dependent on all these parameters, the dependence on wind speed and direction will here be represented as a dependence on effective wind speed, in order to increase the amount of data used in each bin and therefore the statistical significance. The usefulness of this is shown in the following.

[64] Figure 10 shows the theoretical *σ*_{ζr}^{2} calculated according to the procedure derived in section 3 (solid lines) as a function of wind azimuth angle θ_{w}, for three values of *v*_{w}, and specified values of the other parameters. Figure 10 shows that *σ*_{ζr}^{2} depends strongly on both *v*_{w} and θ_{w}. This also demonstrates that the assumption θ_{w} = θ_{l}, as used in the synthetic storm technique [*Matricciani*, 1996], would introduce large errors in the calculation of the variance of relative fade slope: for θ_{w} = 70° (the average value for Sparsholt), the calculated values would be more than 10 times too small.

[65] To express the dependence on wind speed and direction in the comparison of this section, the effective wind speed, derived by trial and error, is defined as

Figure 10 also includes the values of *σ*_{ζr}^{2} calculated with *v*_{w,eff} substituted for *v*_{w}, a constant value θ_{l} + 90° substituted for θ_{w}, and the rest of the parameters the same (dashed lines), as functions of the value of θ_{w} used in equation (28). In Sparsholt, during 70% of all rainy time θ_{w} was between 20° and 100°. Figure 10 shows that in this region, the result using *v*_{w,eff} is very similar to that using the true *v*_{w}; for ∣θ_{w}–θ_{l}∣ > 10° (±180°), the relative error is less than 5%. This means that for the validity check with the rain attenuation measurements in Sparsholt, the dependence of *σ*_{ζr}^{2} on the effective wind speed *v*_{w,eff} can be used to represent the dependence on both *v*_{w} and θ_{w}. Note that equation (28) is valid for the specific parameter values of the Sparsholt link only. Note also that this parameter is only for the sake of comparison in this paper; it is not needed in the application of the model.

[66] Now, first the dependence of *σ*_{ζr}^{2} on *v*_{ver} (an indication of rain type) will be verified with the measurements. In order to check any correlation between this parameter and other meteorological parameters, the average *h*_{r} and the average *v*_{w,eff} as functions of *v*_{ver} were evaluated during the measured attenuation time on the link in Sparsholt and for ∣θ_{w}–θ_{l}∣ > 10° (±180°). The results showed that both are fairly constant over the range of *v*_{ver}, with overall averages of *h*_{r} = 2550 m and *v*_{w,eff} = 13.6 m/s. This first result is a little surprising, considering that convective rainstorms usually are much higher than stratiform storms [*Rogers and Yau*, 1989], so a correlation between *h*_{r} and *v*_{ver} would be expected. Possibly this means that during convective rain, the rain height cannot really be assumed to be 360 m above the 0°C height. However, in the absence of a better rain height model, this result is kept as it is.

[67] The measured long-term *σ*_{ζr}^{2} was calculated concurrent with different values of *v*_{ver} and for ∣θ_{w}–θ_{l}∣ > 10°, using the procedure described in section 2 (with *A* > *A*_{th}). Figure 11 shows the result as a function of *v*_{ver}. Also included is the result of the theoretical expression, with *h*_{r} = 2550 m and *v*_{w,eff} = 13.6 m/s. Figure 11 shows that the model agrees well with the measurements, both on average and in their variation with *v*_{ver}.

[68] Next, the theoretical *σ*_{ζr}^{2} is verified as dependent on *h*_{r}. It was checked that there is not much correlation between this parameter and *v*_{w,eff}: in the most significant region (more or less between 1700 and 3300 m), the average *v*_{w,eff} is fairly constant with respect to *h*_{r}, around the long-term average of 13.6 m/s.

[69] The measured long-term *σ*_{ζr}^{2} was calculated concurrent with different values of *h*_{r} and *v*_{ver} and for ∣θ_{w}–θ_{l}∣ > 10°. Figure 12 shows the result as a function of *h*_{r} for different values of *v*_{ver}, and compared to the results of the theoretical expression, with *v*_{w,eff} = 13.6 m/s. Figure 12 shows that the resulting values of the model are on average close to the measured ones, although the expected decrease with *h*_{r} is slightly recognized in only two of the measured curves.

[70] However, it should be noted that the fact that the theoretical values are close to the values observed is already an indication that the dependence of the theoretical model on rain height is in agreement with the measurements. As was seen in Figure 1 and 2, the variance of relative rain rate slope is much larger than that of relative fade slope. If the theoretical model would be based on attenuation calculated from point rain rate, its result for the relative fade slope would be similar to that of rain rate, and hence much larger than the measured results. This can be seen by applying the model using a zero rain height and therefore zero path length. What happens to the result of this can be seen in Figure 12 by the strongly increasing curves for decreasing rain height. It is the integrating effect along the path length which reduces the variance of relative fade slope to the values observed. The results in this paper therefore show that the incorporation of the rain height in the model is well in agreement with measurements.

[71] Figure 12 also shows that the fact that during convective rain the calculated rain height might be too small (see above), may have some effect on the result. The bottom two measured curves are on average slightly to the left of the theoretical ones, so the rain height assigned to the measured results could be slightly too small.

[72] Finally, the model is verified as dependent on effective wind speed *v*_{w,eff}. The measured long-term *σ*_{ζr}^{2} was calculated concurrent with different values of *v*_{w,eff} and *v*_{ver} and for ∣θ_{w}–θ_{l}∣ > 10°. Figure 13 shows the result as a function of *v*_{w,eff} for different values of *v*_{ver}, and compared to the results of the theoretical expression, with *h*_{r} = 2550 m. Figure 13 shows again that the average values of the theoretical model are in agreement with the measured values, for the different values of *v*_{ver}, and also, the theoretical values increase with effective wind speed, as do the measured values. However, the theoretical increase with *v*_{w,eff} is stronger than measured.

[73] Especially for small wind speeds, it appears that the measured *σ*_{ζr}^{2} is larger than predicted by the model. This is possibly due to the fact that the frozen storm hypothesis is not entirely correct: for small wind speeds, the variation of rain attenuation cannot entirely be ascribed to the wind moving a stable rainstorm across the propagation path.

[74] Furthermore, it was found that the use of a long-term average *v*_{w,eff} of 13.6 m/s (as in Figures 11 and 12) gave a good agreement with measurements. This suggests that the cause of the discrepancy is that the instantaneous wind speed as obtained by interpolation from ECMWF data, with a 2.5° spatial resolution and a 6-hour time resolution, is not a good representative of the true wind speed on the propagation path during the measured rain event. Given the generally strong variability of wind, it seems likely that meteorological data with a higher resolution would improve the instantaneous correlation between the variance of relative fade slope and wind speed.

### 5. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Measured Attenuation Dynamics and Rain Rate Dynamics
- 3. Theoretical Expression for the Variance of Relative Fade Slope
- 4. Comparison With Measurements
- 5. Conclusions
- Appendix A:: Variance of Relative Fade Slope Related to Rain Rate
- Appendix B:: Autocorrelation of Attenuation
- Acknowledgments
- References
- Supporting Information

[75] A theoretical model has been developed to predict the variance of relative fade slope from rain rate measurements. This model uses the property that the attenuation is a result of the integrated effect of rain along the propagation path. This property strongly reduces the variance of relative fade slope, and is therefore important to be taken into account. The model assumes the frozen storm hypothesis, supposing that all measured dynamic behavior of rain rate is caused by the wind, moving a stable rainstorm across the measurement point. The model uses as inputs two parameters of the autocorrelation of rain rate and several meteorological parameters only.

[76] The theoretical model shows a good agreement with measurements in a comparison as a function of the vertical velocity and of the rain height. As a function of wind speed, the model predicts an increase of the variance of relative fade slope which is stronger than the one observed. This may be partly due to the frozen storm hypothesis, and partly due to the spatial and temporal resolution of the meteorological data. Using a long-term average wind speed, the model is well in agreement with the measurements.

[77] The performance of the model may be improved using meteorological input data with a higher resolution, or a different estimate of the rain height of convective rain. Possibly, the model itself can still be improved by including an offset in the relation between *σ*_{R}_{2} and *m*_{R}_{2} (Figure 5), or in that between *B* and *v*_{w} (Figure 6). However, these alterations are not straightforward, because they imply modifications in two basic assumptions of the model: the Maseng-Bakken model setup and the frozen storm hypothesis, respectively.

[78] The model has been verified with rain attenuation data above a threshold *A*_{th}, to reduce the effect of scintillation and noise. However, it can also be applied to predict attenuations below *A*_{th}, assuming the fade slope is proportional to attenuation similarly as it is for larger attenuation values.

[79] This model allows to predict fade slope statistics from rain gauge data, without the use of rain attenuation data. This can prove very useful for obtaining necessary information for the design of FMT systems at sites where satellite measurements have not been performed, but rain rate measurements, which are much cheaper and easier to do, have.