C.2 Preliminary calculations

The preliminary calculations require the following inputs:

− The longitude and latitude for obtaining rain climatic parameters are denoted here as f _n and f _e.

− The heights of the ends of the path for a precipitation calculation are denoted here as h_rainlo and h_rainhi, masl.

− The length of the path for rain calculations, d_rain, km.

The values to use for these five input parameters are given where this section is invoked §§ 4.1 and 4.3.

Obtain P_r ₆, M_T and b for f _n and f _e from the data files, “Esarain_Pr6_v5.txt”, “Esarain_Mt_v5.txt” and “Esarain_Beta_v5.txt”, respectively.

Obtain the mean zero-degree isotherm height rain height h ₀ in km above sea level for f _n and f _e from the data file “h0.txt”.

Calculate mean rain height, h_R, in m above sea level:

masl (C.2.1)

The variation of rain height throughout an average year is taken into account by the discrete probability distribution in 100 m bins given in Table C.2.1.

Calculate the highest rain height given by:

(C.2.2)

where the constant 2400 is the height difference corresponding to the highest bin of the rain-height distribution in Table C.2.1, that is, for n = 49.

The path must be classified as being either subject to rain or to be a “non-rain” path. This classification is used in § C.3.

If either P_r ₆ = 0 or h_rainlo ³ h_Rtop the path is classified as “no-rain”. In this case set F_wvr = 0, Q _{0 ra} = 0 and omit the remaining calculations in this subsection. The meaning of these terms is given below in equations (C.2.4) and (C.2.12).

TABLE C.2.1

Probability distribution of rain height

Index n	Rel. height H metres	Probability Π	Index n	Rel. height H metres	Probability Π
	−2400	0.000555			0.049589
	−2300	0.000802			0.048439
	−2200	0.001139			0.046583
	−2100	0.001594			0.044104
	−2000	0.002196			0.041110
	−1900	0.002978			0.037724
	−1800	0.003976			0.034081
	−1700	0.005227			0.030312
	−1600	0.006764			0.026542
	−1500	0.008617			0.022881
	−1400	0.010808			0.019419
	−1300	0.013346			0.016225
	−1200	0.016225			0.013346
	−1100	0.019419			0.010808
	−1000	0.022881			0.008617
	−900	0.026542			0.006764
	−800	0.030312			0.005227
	−700	0.034081			0.003976
	−600	0.037724			0.002978
	−500	0.041110			0.002196
	−400	0.044104			0.001594
	−300	0.046583			0.001139
	−200	0.048439			0.000802
	−100	0.049589			0.000555
		0.049978

Otherwise the path is classified as “rain”, and the preliminary calculations continue as follows.

Calculate two intermediate parameters given by:

(C.2.3a)

(C.2.3b)

Calculate the percentage of an average year in which rain occurs:

(C.2.4)

Calculate the following three parameters defining the cumulative distribution of rain rate.

(C.2.5a)

(C.2.5b)

(C.2.5c)

Calculate the percentage time approximating to the transition between the straight and curved sections of the rain-rate cumulative distribution when plotted on a logarithmic scale of percentage time:

(C.2.6)

Use the method given in Recommendation ITU-R P.838 to calculate the rain regression coefficients k and a for the frequency, polarization and path inclination. The calculation in Recommendation ITU‑R P.838 requires the following values:

f: Frequency in GHz, which has the same symbol in Recommendation ITU‑R P.838.

Polarization tilt angle, which in Recommendation ITU‑R P.838 has the symbol t, given by:

t = 0 degrees for horizontal linear polarization;

t = 90 degrees for vertical linear polarization.

Path inclination angle, which in Recommendation ITU‑R P.838 has the symbol q, is given by:

radians (C.2.7)

In Recommendation ITU-R P.838, trigonometric functions of t and q are required, and thus the units of these angles must agree with the trigonometric implementation in use. The sign of q in Recommendation ITU‑R P.838 is immaterial, and thus it is safe to derive its value from e _p, noting that this is in milliradians.

Note that the method of Recommendation ITU‑R P.838 is only valid for frequencies of 1 GHz and above. If the frequency is below 1 GHz, the regression coefficients k _{1 GHz} and a_{1 GHz} should be calculated for a frequency of 1 GHz and the values of k and a obtained as:

(C.2.8a)

(C.2.8b)

Limit the path length for precipitation calculations according to:

(C.2.9a)

(C.2.9b)

Calculate modified regression coefficients given by:

(C.2.10a)

(C.2.10b)

The effect of anomalous attenuation in the melting layer on precipitation fading is assessed by considering each 100 m interval of the distribution in Table C.2.1 in turn. During this process two arrays will be assigned:

G_m: attenuation multiplier;

P_m: probability of a particular case.

When these two arrays have been assigned they will both contain the same number, M, of values. M depends on the geometry of the path relative to the melting layer and has a maximum value of 49. The melting layer is modelled by an attenuation multiplier, G, defined by equation (C.4.1). To evaluate the effect of path inclination the melting layer is divided into 12 intervals each 100 m in vertical extent, and a path-averaged multiplier, G, is calculated using the method given in § C.5.

Arrays G_m and P_m are evaluated as follows.

Initialize all P_m to zero.

Initialize G ₁ = 1. This is not normally necessary, but is advisable to guard against a possible situation where the path is classified as “rain”, but in the following loop b) is executed for every value of n.

Initialize an index m to the first members of arrays G and P: m = 1.

For each line of Table C.2.1, for n from 1 to 49, do the following:

a) Calculate rain height given by:

h_T = h_R + H_n masl (C.2.11)

where H_n is the corresponding relative height entry in Table C.2.1.

b) If h_rainlo ³ h_T, repeat from a) for the next value of n.

Otherwise continue from c).

c) If h_rainhi > h_T − 1200 do the following:

i) use the method in § C.5 to set G_m to the path-averaged multiplier for this path geometry relative to the melting layer;

ii) set P_m = Π _n from Table C.2.1;

iii) if n < 49 add 1 to array index m;

iv) repeat from a) for the next value of n.

Otherwise continue from d).

d) Accumulate Π _n from Table C.2.1 into P_m, set G_m = 1, and repeat from a) for the next value of n.

At the end of the above process, set the number of values in arrays G_m and P_m according to:

M = m (C.2.12)

Calculate a factor used to estimate the effect of additional water vapour under rainy conditions given by:

(C.2.13)

where:

(C.2.13a)

The values calculated using this § C.2 for a given path or path segment are those to be used in § C.3 for the corresponding iterative procedure. This applies to the classification “rain” or “non-rain”, and in the “rain” case the parameters a, b, c, d_r, Q _{0 ra}, k_mod, a _mod, the arrays G_m and P_m, and the number of elements in G and P given by M.