Attenuation by atmospheric gases which is entirely caused by absorption depends mainly on frequency, elevation angle, altitude above sea level and water vapour density (absolute humidity). At frequencies below 10 GHz, it may normally be neglected. Its importance increases with frequency above 10 GHz, especially for low elevation angles. Annex 1 of Recommendation ITU R P.676 gives a complete method for calculating gaseous attenuation, while Annex 2 of the same Recommendation gives an approximate method for frequencies up to 350 GHz.
At a given frequency the oxygen contribution to atmospheric absorption is relatively constant. However, both water vapour density and its vertical profile are quite variable. Typically, the maximum gaseous attenuation occurs during the season of maximum rainfall (see Recommendation ITU R P.836).
## 2.2 Attenuation by precipitation and clouds ### 2.2.1 Prediction of attenuation statistics for an average year
The general method to predict attenuation due to precipitation and clouds along a slant propagation path is presented in § 2.2.1.1.
If reliable long-term statistical attenuation data are available that were measured at an elevation angle and a frequency (or frequencies) different from those for which a prediction is needed, it is often preferable to scale these data to the elevation angle and frequency in question rather than using the general method. The recommended frequency-scaling method is found in § 2.2.1.2.
Site diversity effects may be estimated with the method of § 2.2.4.
#### 2.2.1.1 Calculation of long-term rain attenuation statistics from point rainfall rate
The following procedure provides estimates of the long-term statistics of the slant-path rain attenuation at a given location for frequencies up to 55 GHz. The following parameters are required:
* **R*_{0.01}_{ }: point rainfall rate for the location for 0.01% of an average year (mm/h)
* h*_{s}_{ }: height above mean sea level of the earth station (km)
_{ }: elevation angle (degrees)
_{ }: latitude of the earth station (degrees)
*f*_{ }: frequency (GHz)
* R*_{e}_{ }: effective radius of the Earth (8 500 km).
If local data for the earth station height above mean sea level is not available, an estimate can be obtained from the maps of topographic altitude given in Recommendation ITU-R P.1511.
The geometry is illustrated in Fig. 1.
*Step 1:* Determine the rain height, *h*_{R}, as given in Recommendation ITU-R P.839.
*Step 2:* For 5 compute the slant path length, *L*_{s}, below the rain height from:
(1)
For 5, the following formula is used:
(2)
If *h*_{R}* – h*_{s} is less than or equal to zero, the predicted rain attenuation for any time percentage is zero and the following steps are not required.
*Step 3:* Calculate the horizontal projection, *L*_{G}, of the slant path length from:
*L*_{G} *L*_{s} cos km (3)
*Step 4:* Obtain the rainfall rate, *R*_{0.01}, exceeded for 0.01% of an average year (with an integration time of 1 min). If this long-term statistic cannot be obtained from local data sources, an estimate can be obtained from the maps of rainfall rate given in Recommendation ITU R P.837. If *R*_{0.01} is equal to zero, the predicted rain attenuation is zero for any time percentage and the following steps are not required.
*Step 5:* Obtain the specific attenuation, _{R}, using the frequency-dependent coefficients given in Recommendation ITU R P.838 and the rainfall rate, *R*_{0.01}, determined from Step 4, by using:
_{R} *k* (*R*_{0.01})^{} dB/km (4)
*Step 6:* Calculate the horizontal reduction factor, *r*_{0.01}, for 0.01% of the time:
(5)
*Step 7:* Calculate the vertical adjustment factor, *v*_{0.01}, for 0.01% of the time:
For ,
Else,
If | | 36, 36 – | | degrees
Else, 0 degrees
*Step 8:* The effective path length is:
*L*_{E} *L*_{R} _{0.01} km (6)
*Step 9:* The predicted attenuation exceeded for 0.01% of an average year is obtained from:
*A*_{0.01} _{R} *L*_{E} dB (7)
*Step 10:* The estimated attenuation to be exceeded for other percentages of an average year, in the range 0.001% to 5%, is determined from the attenuation to be exceeded for 0.01% for an average year:
If *p* 1% or | | 36: 0
If *p* < 1% and | | < 36 and 25: –0.005(| | – 36)
Otherwise: –0.005(| | – 36) + 1.8 – 4.25 sin
(8)
This method provides an estimate of the long term statistics of attenuation due to rain. When comparing measured statistics with the prediction, allowance should be given for the rather large year-to-year variability in rainfall rate statistics (see Recommendation ITU R P.678).
#### 2.2.1.2 Long-term frequency and polarization scaling of rain attenuation statistics
The method of § 2.2.1.1 may be used to investigate the dependence of attenuation statistics on elevation angle, polarization and frequency, and is therefore a useful general tool for scaling of attenuation according to these parameters.
If reliable attenuation data measured at one frequency are available, the following empirical formula giving an attenuation ratio directly as a function of frequency and attenuation may be applied for frequency scaling on the same path in the frequency range 7 to 55 GHz:
(9)
where:
(10a)
(10b)
*A*_{1} and *A*_{2} are the equiprobable values of the excess rain attenuation at frequencies *f*_{1} and *f*_{2} (GHz), respectively.
Frequency scaling from reliable attenuation data is preferred, when applicable, rather than the prediction methods starting from rain data.
When polarization scaling is required, it is more appropriate to use directly the parameters *k* and as given in Recommendation ITU R P.838. These parameters also provide a radiometeorological basis for frequency scaling.
System planning often requires the attenuation value exceeded for a time percentage, *p*_{w}, of the worst month. The following procedure is used to estimate the attenuation exceeded for a specified percentage of the worst month.
*Step 1:* Obtain the annual time percentage, *p*, corresponding to the desired worst-month time percentage, *p*_{w}, by using the equation specified in Recommendation ITU R P.841 and by applying any adjustments to *p* as prescribed therein.
*Step 2:* For the path in question obtain the attenuation, *A* (dB), exceeded for the resulting annual time percentage, *p*, from the method of § 2.2.1.1, or from measured or frequency-scaled attenuation statistics. This value of *A* is the estimated attenuation for *p*_{w} per cent of the worst month.
Curves giving the variation of worst-month values from their mean are provided in Recommendation ITU R P.678.
### 2.2.3 Variability in space and time of statistics
Precipitation attenuation distributions measured on the same path at the same frequency and polarization may show marked year-to-year variations. In the range 0.001% to 0.1% of the year, the attenuation values at a fixed probability level are observed to vary by more than 20% r.m.s. When the models for attenuation prediction or scaling in § 2.2.1 are used to scale observations at a location to estimate for another path at the same location, the variations increase to more than 25% r.m.s.
### 2.2.4 Site diversity
Intense rain cells that cause large attenuation values on an Earth-space link often have horizontal dimensions of no more than a few kilometres. Diversity systems able to re-route traffic to alternate earth stations, or with access to a satellite with extra on-board resources available for temporary allocation, can improve the system reliability considerably.
Two concepts exist for characterizing diversity performance: the diversity improvement factor is defined as the ratio of the single-site time percentage and the diversity time percentage, at the same attenuation level. Diversity gain is the difference (dB) between the single-site and diversity attenuation values for the same time percentage. Both parameters are important, depending on the system design approach, and prediction procedures for both are given below.
The procedures have been tested at frequencies between 10 and 30 GHz, which is the recommended frequency range of applicability. The diversity prediction procedures are only recommended for time percentages less than 0.1%. At time percentages above 0.1%, the rainfall rate is generally small and the corresponding site diversity improvement is not significant.
#### 2.2.4.1 Diversity improvement factor
The diversity improvement factor, *I*, is given by:
(11)
where *p*_{1}_{ }and *p*_{2} are the respective single-site and diversity time percentages, and is a parameter depending on link characteristics. The approximation on the right-hand side of equation (11) is acceptable since ^{2} is generally small.
From a large number of measurements carried out in the 10-20 GHz band, and mainly between 11 GHz and 13.6 GHz, it has been found that the value of ^{2} depends basically on the distance, *d*, between the stations, and only slightly on the angle of elevation and the frequency. It is found that ^{2} can be expressed by the following empirical relationship:
(12)
Figure 2 shows *p*_{2} versus *p*_{1} on the basis of equations (11) and (12).
The diversity gain, *G* (dB), between pairs of sites is calculated with the empirical expression given below. Parameters required for the calculation of diversity gain are:
* d* : separation (km) between the two sites
* A* : path rain attenuation (dB) for a single site
*f* : frequency (GHz)
: path elevation angle (degrees)
: angle (degrees) made by the azimuth of the propagation path with respect to the baseline between sites, chosen such that 90.
*Step 1: *Calculate the gain contributed by the spatial separation from:
(13)
where:
* a* 0.78 *A* – 1.94 (1 – e^{–0.11 }^{A})
* b* 0.59 (1 – e^{–0.1 }^{A})
*Step 2: *Calculate the frequency-dependent gain from:
*G*_{f} e^{–0.025 }^{f} (14)
*Step 3: *Calculate the gain term dependent on elevation angle from:
*G*_{} 1 0.006 (15)
*Step 4: *Calculate the baseline-dependent term from the expression:
*G*_{} 1 0.002 (16)
*Step 5: *Compute the net diversity gain as the product:
*G* *G*_{d} · *G*_{f} · *G*_{} · *G*_{} dB (17)
When the above method was tested against the Radiocommunication Study Group 3 site diversity data bank, the arithmetic mean and standard deviation were found to be 0.14 dB and 0.96 dB, respectively, with an r.m.s. error of 0.97 dB.
### 2.2.5 Characteristics of precipitation events
The durations of rain fades that exceed a specified attenuation level are approximately log-normally distributed. Median durations are of the order of several minutes. No significant dependence of these distributions on fade depth is evident in most measurements for fades of less than 20 dB, implying that the larger total time percentage of fades observed at lower fade levels or at higher frequencies is composed of a larger number of individual fades having more or less the same distribution of durations. Significant departures from log-normal seem to occur for fade durations of less than about half a minute. Fade durations at a specified fade level tend to increase with decreasing elevation angle.
For the planning of integrated services digital network (ISDN) connections via satellite, data are needed on the contribution of attenuation events shorter than 10 s to the total fading time. This information is especially relevant for the attenuation level corresponding to the outage threshold, where events longer than 10 s contribute to system unavailable time, while shorter events affect system performance during available time (see Recommendation ITU-R S.579). Existing data
indicate that in the majority of cases, the exceedance time during available time is 2% to 10% of the net exceedance time. However, at low elevation angles where the short period signal fluctuations due to tropospheric scintillation become statistically significant, there are some cases for which the exceedance time during available time is far larger than in the case at higher elevation Earth-space paths.
#### 2.2.5.2 Rates of change of attenuation (fading rate)
There is broad agreement that the distributions of positive and negative fade rates are log-normally distributed and very similar to each other. The dependence of fade rate on fade depth has not been established.
#### 2.2.5.3 Correlation of instantaneous values of attenuation at different frequencies
Data on the instantaneous ratio of rain attenuation values at different frequencies are of interest for a variety of adaptive fade techniques. The frequency-scaling ratio has been found to be log-normally distributed, and is influenced by rain type and rain temperature. Data reveal that the short-term variations in the attenuation ratio can be significant, and are expected to increase with decreasing path elevation angle.
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