[16] An internationally recognized model for the average annual fade distribution experienced by terrestrial line-of-sight links is provided by *ITU* [2001b]. Separate models exist for raining and nonraining (clear air) conditions, each providing attenuation distributions, *A*_{R}(*p*) and *A*_{CA}(*p*), as a function of exceedance probability *p* = *P*/100, where *P* is the percentage of an average year. The clear-air model also describes enhancements due to multipath and ducting. *Gibbins and Walden* [2003] have proposed a variation to the rain fade component that yields an improved fit to the ITU-T database of link measurements, particularly at frequencies above 40 GHz. *ITU* [2001b] uses a very simple approximation to combine these distributions to yield the total attenuation exceeded in an average year with probability *p*, *A*_{CAR}(*p*):

The rigorously correct mathematical procedure for combining the effects of different attenuation mechanisms uses the inverse of these functions, *F*_{R}(*A*) and *F*_{CA}(*A*); the probability that a given attenuation is exceeded because of rain or clear-air mechanisms, respectively; and the associated probability density functions, *f*_{R}(*A*) and *f*_{CA}(*A*). The probability that a given attenuation is exceeded in an average year is

The second term in the integral is the Bayesian conditional probability that the clear-air attenuation is greater than *A-a* given that the attenuation due to rain is equal to *a*. It is often assumed that clear-air attenuation mechanisms, including ducting and multipath, do not occur at the same time as rain. In this case the integral (12), reduces to

where *P*_{R=0} is the probability of rain and _{CA} and _{R} are the exceedance distributions normalized over periods of no rain and rain, respectively. The expression used by *ITU* [2001b], (11), only approximates the inverse of this CDF when either clear air or rain fade dominates the attenuation, i.e., at very high or very low time percentages.

[17] Attenuation due to variation in atmospheric absorption cannot be assumed to be mutually exclusive with other fading mechanisms. The cool, humid conditions in which atmospheric attenuation is maximized could be correlated with the onset of frontal rain. Until more information is available, we will assume that attenuation due to atmospheric absorption is independent of the other fade mechanisms. The assumption of independence allows the simplification of (12) as a conditional probability is no longer needed. The probability of attenuation *A* being exceeded because of a combination of atmospheric absorption and other mechanisms can then be written

where the function *f*_{AA}(*A* − *a*) = exp (−()^{2}) is obtained by manipulation of (4). Figures 6a and 6b illustrate the average annual fade exceedance predicted by the Gibbins-Walden model for the 54.5- and 56.5-GHz, 5-km links operated at Sparsholt, United Kingdom. These are compared with the prediction including the effects of variation of atmospheric absorption using (14). This increases the predicted fade at 5% of time by approximately 1 dB at 54.5 GHz and 2.5 dB at 56.5 GHz. At time percentages below 1% the change is negligible because of the low probability of extreme atmospheric absorption occurring simultaneously with deep fade because of other mechanisms. The assumption of independence strongly affects the shape of this part of the curve. Also plotted in Figure 6 are the fade exceedance distributions for the two links, measured over the 2-year period 1 October 2002 to 1 October 2004. At time percentages below 0.05% the limited measurement range of the receiver affects the measured incidence of attenuations. This leads to the plateau present in the measured exceedances around 35 dB. In both cases the measured data are a better fit to the predicted curve including variation of atmospheric absorption, particularly in the range 50% to 1% of time.