## 1. Introduction

[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].

[4] To calculate the rain fade slope from measured time series, the rapid component due to tropospheric scintillation is filtered out; this component is unpredictable and not useful in the design of FMTs [*Poiares Baptista and Davies*, 1994]. Commonly used filter bandwidths are in the order of 0.01–0.02 Hz [*Poiares Baptista and Davies*, 1994]. The resulting fade slope of the filtered signal is dependent on the filter bandwidth used [*van de Kamp and Clérivet*, 2004; *ITU*, 2005].

[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.