Reciprocity, and the designation of terminals
The terms “transmitting antenna” and “receiving antenna”, or more briefly just “transmitter” and “receiver”, are used to distinguish the two terminals. This is convenient for the purposes of description.
The method, however, is symmetrical. Which terminal is designated the “transmitter” makes no difference to the result. By convention the “transmitter” is at the start of the terrain profile.
Iteration
Some parts of the method require iterative calculations. Explicit iteration procedures are described which have been found to be efficient and stable. However, these are not necessarily optimal. Other iterative methods can be used if they are shown to give very similar results.
Organization of the Recommendation
The inputs, and the symbols used to denote them, are described in § 2.
Preliminary calculations, including obtaining various radio-climatic parameters, are described in § 3. Climatic parameters, and values derived from the inputs, are listed in approximately alphabetical order of their symbols in Table 3.1. Many of these parameters are used in more than one place in the overall method, and all symbols in Table 3.1 are unique within this Recommendation.
Section 4 describes the four main sub-models into which the method is divided. The following subsections describe the calculation of these sub-models, most of which apply to a group of propagation mechanisms. These descriptions refer extensively to appendices which define various blocks of calculation. The sub-models in WRPM are independent of each other and each calculates results over the range 0% to 100%.
Section 5 describes how the final prediction is obtained by combining results from the four main sub‑models. The combination method takes account of the statistical correlation properties between the sub-models. Two alternative methods are given. One is appropriate when a direct calculation of the overall basic transmission loss is required for a given value of time percentage. This method involves an approximate treatment of uncorrelated statistics. The second method is appropriate when the wide-range propagation model (WRPM) is used within a Monte-Carlo simulator. In this case, the uncorrelated statistics can be modelled more accurately by combining the sub-models within the Monte-Carlo method.
Style of description
The method is described in a step-by-step manner, that is, expressions are given in the order in which they should be evaluated. Equations are sometimes followed by a “where”, but this is limited to a few lines. Long lists of “where”s are avoided.
Symbols appearing within the Appendices and which do not appear in Table 3.1 should be considered re-usable. They are defined close to where they are used, or cross-referenced if appropriate.
Logarithms are to the base 10 by default. That is, log(x) = log10(x). Natural logarithms where used are indicated as ln(x) = loge(x).
Inputs
The inputs to the model consist of a terrain profile, described in § 2.1, and other inputs described in § 2.2.
Terrain profile
A terrain profile giving heights above sea level of the Earth’s surface, whether land or water, at points along the great-circle radio path, must be available. Information is also required on the distances over sea or a large body of water, and over low-lying coastal land or areas with many lakes, according to the zones defined in Appendix D, § D.1.
In principle, the terrain profile consists of arrays each having the same number of values, n, as follows:
di: distance from transmitter of i‑th profile point (km) (2.1a)
hi: height of i‑th profile point above sea level (m) (2.1b)
where:
i: 1, 2, 3 ... n = index of the profile point
n: number of profile points.
It is convenient to define an additional array holding zone codes as part of the profile:
zi: zone code at distance di from transmitter (2.1c)
where the z values are codes representing the zones in Table D.1.
The profile points must be spaced at equal intervals of distance. Thus d1 = 0 km, and dn = d km where d is the overall length of the path. Similarly, di = (i − 1) d / (n − 1) km.
It is immaterial whether an array di is populated with distances, or whether di is calculated when needed.
There must be at least one intermediate profile point between the transmitter and the receiver. Thus n must satisfy n ³ 3. Such a small number of points is appropriate only for short paths, less than of the order of 1 km.
Only general guidance can be given as to the appropriate profile spacing. Typical practice is a spacing in the range 50 to 250 m, depending on the source data and the nature of terrain.
However, it is stressed that equally-spaced points should be included for the complete path, even where it passes over water. Expressions in this method assume that this is so. For instance, it is not acceptable to have zero-height points only at the start and end of a section over sea when the length of the section exceeds the point spacing. Horizon points must be located taking Earth curvature into account, and omitting points in such a manner could result in the misinterpretation of a profile.
Other inputs
Table 2.2.1 lists the other inputs which must be provided by the user, in addition to the geographic information, including the terrain profile, described in § 2.1 above. The symbols and units given here apply throughout this Recommendation.
TABLE 2.2.1
Other inputs
| Symbol | Description |
| f (GHz) | Frequency |
| Tpol | Code indicating either horizontal or vertical linear polarization |
| fre, rn (degrees) | Longitude, latitude, of receiver |
| fte, tn (degrees) | Longitude, latitude, of transmitter |
| htg, rg (m) | Height of electrical centre of transmitting, receiving antenna above ground. |
| Tpc (%) | Percentage of average year for which the predicted basic transmission loss is not exceeded |
| Gt, Gr (dBi) | Gain of transmitting, receiving, antenna in the azimuthal direction of the path towards the other antenna, and at the elevation angle above the local horizontal of the other antenna in the case of a line-of-sight (LoS) path, otherwise of the antenna’s radio horizon, for median effective Earth radius. |
Longitudes and latitudes in this method are positive east and north.
Constants
Table 2.3.1 gives values of constants used in the method.
TABLE 2.3.1
Constants
| Symbol | Value | Description |
| c (m/s) | 2.998 ´ 108 | Speed of propagation |
| Re (km) | 6 371 | Average Earth radius |
| erland | 22.0 | Relative permittivity for land |
| ersea | 80.0 | Relative permittivity for sea |
| sland (S/m) | 0.003 | Conductivity for land |
| ssea (S/m) | 5.0 | Conductivity for sea |
Integral digital products
Only the file versions provided with this Recommendation should be used. They are an integral part of the Recommendation. Table 2.4.1 gives details of the digital products used in the method.
TABLE 2.4.1
Digital products
| Filename | Ref. | Origin | Latitude (rows) | Longitude (columns | ||||
| First row (ºN) | Spacing (degrees) | Number of rows | First col (ºE) | Spacing (degrees) | Number of cols | |||
| DN_Median.txt | § 3.4.1 | P.2001 | 1.5 | 1.5 | ||||
| DN_SupSlope.txt | § 3.4.1 | P.2001 | 1.5 | 1.5 | ||||
| DN_SubSlope.txt | § 3.4.1 | P.2001 | 1.5 | 1.5 | ||||
| dndz_01.txt | § 3.4.2 | P.453-10 | 1.5 | 1.5 | ||||
| Esarain_Pr6_v5.txt | § C.2 | P.837-5 | 1.125 | 1.125 | ||||
| Esarain_Mt_v5.txt | § C.2 | P.837-5 | 1.125 | 1.125 | ||||
| Esarain_Beta_v5.txt | § C.2 | P.837-5 | 1.125 | 1.125 | ||||
| h0.txt | § C.2 | P.839-4 | 1.5 | 1.5 | ||||
| Surfwv_50_fixed.txt(1) | Appx F | P.836-4 (corrected) | 1.5 | 1.5 | ||||
| FoEs50.txt | Appx G | P.2001 | 1.5 | 1.5 | ||||
| FoEs10.txt | Appx G | P.2001 | 1.5 | 1.5 | ||||
| FoEs01.txt | Appx G | P.2001 | 1.5 | 1.5 | ||||
| FoEs0.1.txt | Appx G | P.2001 | 1.5 | 1.5 | ||||
| TropoClim.txt | § E.2 | P.2001 | 89.75 | 0.5 | –179.75 | 0.5 | ||
| (1) The file “surfwv_50_fixed.txt” is a corrected version of the file “surfwv_50.txt” associated with Recommendation ITU-R P.836-4. “surfwv_50.txt” has one column less than expected according to the “surfwv_lat.txt” and “surfwv_lon.txt” files provided with the data. It has been assumed that the column corresponding to a longitude of 360° was omitted from the file, and this has been corrected in “surfwv_50_fixed.txt. |
The “First row” value is the latitude of the first row.
The “First col” value is the longitude of the first column. The last column is the same as the first column (360° = 0°) and is provided to simplify interpolation.
“Spacing” gives the latitude/longitude increment between rows/columns.
Except for file “TropoClim.txt”, a parameter value at a particular latitude/longitude should be obtained by bilinear interpolation using the four nearest grid points, as described in Rec. ITU‑R P.1144.
TropoClim.txt contains integer zone identifiers rather than continuous meteorological variables. Consequently, the values should not be interpolated to obtain a value at a particular latitude/longitude. Instead the value at the closest grid point should be taken. For this file, note that a) the grid is offset by half a pixel compared to the other files, b) the values in the last column are not a duplicate of the first column. Consequently, the latitudes of the rows range from 89.75°N to 89.75°S, and the longitudes of the columns range from 179.75°W to 179.75°E.
The files are contained in the zip file R-REC-P.2001-1-201309-I!!ZIP-E.
Preliminary calculations
The following subsections describe the calculation of important parameters derived from the inputs. These parameters are listed in Table 3.1.
TABLE 3.1
Principal parameters
| Symbol | Ref. | Description |
| ae (km) | § 3.5 | Median effective Earth radius |
| AgsurAwrsur,wsur (dB/km) | § 3.10 | Gaseous attenuation, and the water vapour attenuations with and without rain, for a surface path |
| ap (km) | § 3.5 | Effective Earth radius exceeded p% time, limited not to become infinite |
| cp (km–1) | § 3.5 | Effective Earth curvature. Usually positive, but for small p may be zero or negative |
| d (km) | § 3.2 | Path length |
| dlt,lr (km) | § 3.7 | Terminal to horizon distances. For LoS paths set to distances to point with largest knife-edge loss |
| dtcv,rcv (km) | § 3.9 | Terminal to troposcatter common volume distances |
| hcv (masl)(1) | § 3.9 | Height of troposcatter common volume |
| hhi, lo (masl)(1) | § 3.3 | Higher, lower, antenna height |
| hm (m) | § 3.8 | Path roughness parameter |
| hte, re (m) | § 3.8 | Effective transmitter, receiver, height above smooth surface |
| htep, rep (m) | § 3.8 | Effective transmitter, receiver, height above smooth surface fitted to the profile |
| hts, rs (masl)(1) | § 3.3 | Transmitter, receiver, height above mean sea level |
| ilt, lr | § 3.7 | Profile indices of transmitter, receiver, horizons |
| Lbfs (dB) | § 3.11 | Free-space basic transmission loss for the path length and frequency |
| Lbm1 (dB) | § 4.1 | Basic transmission loss associated with sub-model 1, diffraction, clear-air and precipitation fading |
| Lbm2 (dB) | § 4.2 | Basic transmission loss associated with sub-model 2, anomalous propagation |
| Lbm3 (dB) | § 4.3 | Basic transmission loss associated with sub-model 3, troposcatter propagation and precipitation fading |
| Lbm4 (dB) | § 4.4 | Basic transmission loss associated with sub-model 4, sporadic-E propagation |
| Ld (dB) | § 4.1 | Diffraction loss not exceeded p% time |
| Nd1km50 (N-units) | § 3.4.1 | Median value of average refractivity gradient in the lowest 1 km of the atmosphere. Numerically equal to DN as defined in ITU‑R P.452 but with opposite sign |
| Nd1kmp (N-units) | § 3.4.1 | Average refractivity gradient in the lowest 1 km of the atmosphere exceeded for p% of an average year. Normally negative but can be zero or positive |
| Nd65m1 (N-units) | § 3.4.2 | Refractivity gradient in the lowest 65 m of the atmosphere exceeded for 1% of an average year |
TABLE 3.1 (end)
| Symbol | Ref. | Description |
| p (%) | § 3.1 | Percentage of average year for which predicted basic transmission loss is not exceeded, limited to range 0.00001% ≤ p ≤ 99.99999% |
| q (%) | § 3.1 | Percentage of average year for which predicted basic transmission loss is exceeded, given by 100 − p |
| ep (mrad) | § 3.3 | Positive value of path inclination |
| l (m) | § 3.6 | Wavelength |
| fcve, cvn (degrees) | § 3.9 | Troposcatter common volume longitude, latitude |
| ftcve, tcvn (degrees) | § 3.9 | Longitude, latitude, of mid-point of path segment from transmitter to the troposcatter common volume |
| frcve, rcvn (degrees) | § 3.9 | Longitude, latitude, of mid-point of path segment from receiver to the troposcatter common volume |
| fme, mn (degrees) | § 3.2 | Path mid-point longitude, latitude |
| qe (rad) | § 3.5 | Angle subtended by d km at centre of spherical Earth |
| qt, r (mrad) | § 3.7 | Horizon elevation angles relative to the local horizontal as viewed from transmitter and receiver |
| qtpos, rpos (mrad) | § 3.7 | Horizon elevation angles relative to the local horizontal limited to be positive (not less than zero) |
| go (dB/km) | § 3.10 | Sea-level specific attenuation due to oxygen |
| ω | § 3.2 | Fraction of the path over sea |
| (1) masl: metres above sea level. |
Limited percentage times
The percentage of an average year for which the predicted loss is not exceed, Tpc in Table 2.2.1, is allowed to vary from 0% to 100%. The percentage times used for calculation are limited such that they remain in the range 0.00001% to 99.99999%.
Percentage time basic transmission loss not exceeded:
% (3.1.1)
Percentage time basic transmission loss is exceeded:
% (3.1.2)