International Telecommunication Union ITU-R Radiocommunication Sector of ITU Recommendation ITU-R P.676-9 (02/2012) Attenuation by atmospheric gases P Series Radiowave propagation Foreword The role of the Radiocommunication Sector is to ensure the rational, equitable, efficient and economical use of the radio-frequency spectrum by all radiocommunication services, including satellite services, and carry out studies without limit of frequency range on the basis of which Recommendations are adopted. The regulatory and policy functions of the Radiocommunication Sector are performed by World and Regional Radiocommunication Conferences and Radiocommunication Assemblies supported by Study Groups. Policy on Intellectual Property Right (IPR) ITU-R policy on IPR is described in the Common Patent Policy for ITU-T/ITU-R/ISO/IEC referenced in Annex 1 of Resolution ITU-R 1. 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Series of ITU-R Recommendations (Also available online at http://www.itu.int/publ/R-REC/en) BO: Satellite delivery BR: Recording for production, archival and play-out; film for television BS: Broadcasting service (sound) BT: Broadcasting service (television) F: Fixed service M: Mobile, radiodetermination, amateur and related satellite services P: Radiowave propagation RA: Radio astronomy RS: Remote sensing systems S: Fixed-satellite service SA: Space applications and meteorology SF: Frequency sharing and coordination between fixed-satellite and fixed service systems SM: Spectrum management SNG: Satellite news gathering TF: Time signals and frequency standards emissions V: Vocabulary and related subjects Note: This ITU-R Recommendation was approved in English under the procedure detailed in Resolution ITU-R 1. Electronic Publication Geneva, 2012 © ITU 2012 All rights reserved. No part of this publication may be reproduced, by any means whatsoever, without written permission of ITU. RECOMMENDATION ITU-R P.676-9 Attenuation by atmospheric gases (Question ITU-R 201/3) (1990-1992-1995-1997-1999-2001-2005-2007-2009-2012) Scope Recommendation ITU-R P.676 provides methods to estimate the attenuation of atmospheric gases on terrestrial and slant paths using: a) an estimate of gaseous attenuation computed by summation of individual absorption lines that is valid for the frequency range 1-1 000 GHz, and b) a simplified approximate method to estimate gaseous attenuation that is applicable in the frequency range 1-350 GHz. The ITU Radiocommunication Assembly, considering a) the necessity of estimating the attenuation by atmospheric gases on terrestrial and slant paths, recommends 1 that, for general application, the procedures in Annex 1 be used to calculate gaseous attenuation at frequencies up to 1 000 GHz; 2 that, for approximate estimates of gaseous attenuation in the frequency range 1 to 350 GHz, the computationally less intensive procedure given in Annex 2 be used. Annex 1 Line-by-line calculation of gaseous attenuation 1 Specific attenuation The specific attenuation at frequencies up to 1 000 GHz due to dry air and water vapour, can be evaluated most accurately at any value of pressure, temperature and humidity by means of a summation of the individual resonance lines from oxygen and water vapour, together with small additional factors for the non-resonant Debye spectrum of oxygen below 10 GHz, pressure-induced nitrogen attenuation above 100 GHz and a wet continuum to account for the excess water vapour absorption found experimentally. Figure 1 shows the specific attenuation using the model, calculated from 0 to 1 000 GHz at 1 GHz intervals, for a pressure of 1 013 hPa, temperature of 15° C for the cases of a water-vapour density of 7.5 g/m³ (Curve A) and a dry atmosphere (Curve B). Near 60 GHz, many oxygen absorption lines merge together, at sea-level pressures, to form a single, broad absorption band, which is shown in more detail in Fig. 2. This figure also shows the oxygen attenuation at higher altitudes, with the individual lines becoming resolved at lower pressures. Some additional molecular species (e.g. oxygen isotopic species, oxygen vibrationally excited species, ozone, ozone isotopic species, and ozone vibrationally excited species, and other minor species) are not included in the line-by-line prediction method. These additional lines are insignificant for typical atmospheres, but may be important for a dry atmosphere. For quick and approximate estimates of specific attenuation at frequencies up to 350 GHz, in cases where high accuracy is not required, simplified algorithms are given in Annex 2 for restricted ranges of meteorological conditions. The specific gaseous attenuation is given by: γ = γo + γw = 0.1820 f N"(f) dB/km (1) where γo and γw are the specific attenuations (dB/km) due to dry air (oxygen, pressure-induced nitrogen and non-resonant Debye attenuation) and water vapour, respectively, and where f is the frequency (GHz) and N"(f) is the imaginary part of the frequency-dependent complex refractivity: N"(f) = ∑_i S_i F_i + N"_D(f) (2) S_i is the strength of the i-th line, F_i is the line shape factor and the sum extends over all the lines (for frequencies, f, above 118.750343 GHz oxygen line, only the oxygen lines above 60 GHz complex should be included in the summation; the summation should begin at i = 38 rather than at i = 1); N"_D(f) is the dry continuum due to pressure-induced nitrogen absorption and the Debye spectrum. The line strength is given by: S_i = a_1 × 10^-7 p θ^3 exp[ a_2 (1 - θ) ] for oxygen = b_1 × 10^-1 e θ^3.5 exp[ b_2 (1 - θ) ] for water vapour (3) where: p: dry air pressure (hPa) e: water vapour partial pressure in hPa (total barometric pressure p_tot = p + e) θ = 300/T T: temperature (K). FIGURE 1 Specific attenuation due to atmospheric gases, calculated at 1 GHz intervals, including line centres (Standard: 7.5 g/m³; Dry: 0 g/m³) [Graph showing specific attenuation in dB/km from 10^-3 to 10^5 versus frequency from 0 to 1 000 GHz for Standard atmosphere and Dry atmosphere]. FIGURE 2 Specific attenuation in the range 50-70 GHz at the altitudes indicated (0 km, 5 km, 10 km, 15 km and 20 km) [Graph showing fine structure and resonant peaks of oxygen specific attenuation in dB/km from 10^-3 to 10^2 across frequencies 50 to 70 GHz at various altitudes]. Local values of p, e and T measured profiles (e.g. using radiosondes) should be used; however, in the absence of local information, the reference standard atmospheres described in Recommendation ITU-R P.835 should be used. (Note that where total atmospheric attenuation is being calculated, the same-water vapour partial pressure is used for both dry-air and water-vapour attenuations.) The water-vapour partial pressure, e, may be obtained from the water-vapour density ρ using the expression: e = (ρ T) / 216.7 (4) The coefficients a_1, a_2 are given in Table 1 for oxygen, those for water vapour, b_1 and b_2, are given in Table 2. The line-shape factor is given by: F_i = (f / f_i) [ (Δf - δ(f_i - f)) / ((f_i - f)^2 + Δf^2) + (Δf - δ(f_i + f)) / ((f_i + f)^2 + Δf^2) ] (5) where f_i is the line frequency and Δf is the width of the line: Δf = a_3 × 10^-4 ( p θ^(0.8 - a_4) + 1.1 e θ ) for oxygen = b_3 × 10^-4 ( p θ^b_4 + b_5 e θ^b_6 ) for water vapour (6a) The line width Δf is modified to account for Doppler broadening: Δf = sqrt( Δf^2 + 2.25 × 10^-6 ) for oxygen = 0.535 Δf + sqrt( 0.217 Δf^2 + (2.1316 × 10^-12 f_i^2) / θ ) for water vapour (6b) δ is a correction factor which arises due to interference effects in oxygen lines: δ = (a_5 + a_6 θ) × 10^-4 (p + e) θ^0.8 for oxygen = 0 for water vapour (7) The spectroscopic coefficients are given in Tables 1 and 2. TABLE 1 Spectroscopic data for oxygen attenuation f0 (GHz) | a1 | a2 | a3 | a4 | a5 | a6 50.474238 | 0.94 | 9.694 | 8.90 | 0.0 | 2.400 | 7.900 50.987749 | 2.46 | 8.694 | 9.10 | 0.0 | 2.200 | 7.800 51.503350 | 6.08 | 7.744 | 9.40 | 0.0 | 1.970 | 7.740 52.021410 | 14.14 | 6.844 | 9.70 | 0.0 | 1.660 | 7.640 52.542394 | 31.02 | 6.004 | 9.90 | 0.0 | 1.360 | 7.510 53.066907 | 64.10 | 5.224 | 10.20 | 0.0 | 1.310 | 7.140 53.595749 | 124.70 | 4.484 | 10.50 | 0.0 | 2.300 | 5.840 54.130000 | 228.00 | 3.814 | 10.70 | 0.0 | 3.350 | 4.310 54.671159 | 391.80 | 3.194 | 11.00 | 0.0 | 3.740 | 3.050 55.221367 | 631.60 | 2.624 | 11.30 | 0.0 | 2.580 | 3.390 55.783802 | 953.50 | 2.119 | 11.70 | 0.0 | -1.660 | 7.050 56.264775 | 548.90 | 0.015 | 17.30 | 0.0 | 3.900 | -1.130 56.363389 | 1344.00 | 1.660 | 12.00 | 0.0 | -2.970 | 7.530 56.968206 | 1763.00 | 1.260 | 12.40 | 0.0 | -4.160 | 7.420 57.612484 | 2141.00 | 0.915 | 12.80 | 0.0 | -6.130 | 6.970 58.323877 | 2386.00 | 0.626 | 13.30 | 0.0 | -2.050 | 0.510 58.446590 | 1457.00 | 0.084 | 15.20 | 0.0 | 7.480 | -1.460 59.164207 | 2404.00 | 0.391 | 13.90 | 0.0 | -7.220 | 2.660 59.590983 | 2112.00 | 0.212 | 14.30 | 0.0 | 7.650 | -0.900 60.306061 | 2124.00 | 0.212 | 14.50 | 0.0 | -7.050 | 0.810 60.434776 | 2461.00 | 0.391 | 13.60 | 0.0 | 6.970 | -3.240 61.150560 | 2504.00 | 0.626 | 13.10 | 0.0 | 1.040 | -0.670 61.800154 | 2298.00 | 0.915 | 12.70 | 0.0 | 5.700 | -7.610 62.411215 | 1933.00 | 1.260 | 12.30 | 0.0 | 3.600 | -7.770 62.486260 | 1517.00 | 0.083 | 15.40 | 0.0 | -4.980 | 0.970 62.997977 | 1503.00 | 1.665 | 12.00 | 0.0 | 2.390 | -7.680 63.568518 | 1087.00 | 2.115 | 11.70 | 0.0 | 1.080 | -7.060 64.127767 | 733.50 | 2.620 | 11.30 | 0.0 | -3.110 | -3.320 64.678903 | 463.50 | 3.195 | 11.00 | 0.0 | -4.210 | -2.980 65.224071 | 274.80 | 3.815 | 10.70 | 0.0 | -3.750 | -4.230 65.764772 | 153.00 | 4.485 | 10.50 | 0.0 | -2.670 | -5.750 66.302091 | 80.09 | 5.225 | 10.20 | 0.0 | -1.680 | -7.000 66.836830 | 39.46 | 6.005 | 9.90 | 0.0 | -1.690 | -7.350 67.369598 | 18.32 | 6.845 | 9.70 | 0.0 | -2.000 | -7.440 67.900867 | 8.01 | 7.745 | 9.40 | 0.0 | -2.280 | -7.530 68.431005 | 3.30 | 8.695 | 9.20 | 0.0 | -2.400 | -7.600 68.960311 | 1.28 | 9.695 | 9.00 | 0.0 | -2.500 | -7.650 118.750343 | 945.00 | 0.009 | 16.30 | 0.0 | -0.360 | 0.090 368.498350 | 67.90 | 0.049 | 19.20 | 0.6 | 0.000 | 0.000 424.763124 | 638.00 | 0.044 | 19.30 | 0.6 | 0.000 | 0.000 487.249370 | 235.00 | 0.049 | 19.20 | 0.6 | 0.000 | 0.000 715.393150 | 99.60 | 0.145 | 18.10 | 0.6 | 0.000 | 0.000 773.839675 | 671.00 | 0.130 | 18.20 | 0.6 | 0.000 | 0.000 834.145330 | 180.00 | 0.147 | 18.10 | 0.6 | 0.000 | 0.000 TABLE 2 Spectroscopic data for water-vapour attenuation f0 (GHz) | b1 | b2 | b3 | b4 | b5 | b6 22.235080 | 0.1130 | 2.143 | 28.11 | 0.69 | 4.800 | 1.00 67.803960 | 0.0012 | 8.735 | 28.58 | 0.69 | 4.930 | 0.82 119.995940 | 0.0008 | 8.356 | 29.48 | 0.70 | 4.780 | 0.79 183.310091 | 2.4200 | 0.668 | 30.50 | 0.64 | 5.300 | 0.85 321.225644 | 0.0483 | 6.181 | 23.03 | 0.67 | 4.690 | 0.54 325.152919 | 1.4990 | 1.540 | 27.83 | 0.68 | 4.850 | 0.74 336.222601 | 0.0011 | 9.829 | 26.93 | 0.69 | 4.740 | 0.61 380.197372 | 11.5200 | 1.048 | 28.73 | 0.54 | 5.380 | 0.89 390.134508 | 0.0046 | 7.350 | 21.52 | 0.63 | 4.810 | 0.55 437.346667 | 0.0650 | 5.050 | 18.45 | 0.60 | 4.230 | 0.48 439.150812 | 0.9218 | 3.596 | 21.00 | 0.63 | 4.290 | 0.52 443.018295 | 0.1976 | 5.050 | 18.60 | 0.60 | 4.230 | 0.50 448.001075 | 10.3200 | 1.405 | 26.32 | 0.66 | 4.840 | 0.67 470.888947 | 0.3297 | 3.599 | 21.52 | 0.66 | 4.570 | 0.65 474.689127 | 1.2620 | 2.381 | 23.55 | 0.65 | 4.650 | 0.64 488.491133 | 0.2520 | 2.853 | 26.02 | 0.69 | 5.040 | 0.72 503.568532 | 0.0390 | 6.733 | 16.12 | 0.61 | 3.980 | 0.43 504.482692 | 0.0130 | 6.733 | 16.12 | 0.61 | 4.010 | 0.45 547.676440 | 9.7010 | 0.114 | 26.00 | 0.70 | 4.500 | 1.00 552.020960 | 14.7700 | 0.114 | 26.00 | 0.70 | 4.500 | 1.00 556.936002 | 487.4000 | 0.159 | 32.10 | 0.69 | 4.110 | 1.00 620.700807 | 5.0120 | 2.200 | 24.38 | 0.71 | 4.680 | 0.68 645.866155 | 0.0713 | 8.580 | 18.00 | 0.60 | 4.000 | 0.50 658.005280 | 0.3022 | 7.820 | 32.10 | 0.69 | 4.140 | 1.00 752.033227 | 239.6000 | 0.396 | 30.60 | 0.68 | 4.090 | 0.84 841.053973 | 0.0140 | 8.180 | 15.90 | 0.33 | 5.760 | 0.45 859.962313 | 0.1472 | 7.989 | 30.60 | 0.68 | 4.090 | 0.84 899.306675 | 0.0605 | 7.917 | 29.85 | 0.68 | 4.530 | 0.90 902.616173 | 0.0426 | 8.432 | 28.65 | 0.70 | 5.100 | 0.95 906.207325 | 0.1876 | 5.111 | 24.08 | 0.70 | 4.700 | 0.53 916.171582 | 8.3400 | 1.442 | 26.70 | 0.70 | 4.780 | 0.78 923.118427 | 0.0869 | 10.220 | 29.00 | 0.70 | 5.000 | 0.80 970.315022 | 8.9720 | 1.920 | 25.50 | 0.64 | 4.940 | 0.67 987.926764 | 132.1000 | 0.258 | 29.85 | 0.68 | 4.550 | 0.90 1 780.000000 | 22 300.0000 | 0.952 | 176.20 | 0.50 | 30.500 | 5.00 The dry air continuum arises from the non-resonant Debye spectrum of oxygen below 10 GHz and a pressure-induced nitrogen attenuation above 100 GHz. N"_D(f) = f p θ^2 [ (6.14 × 10^-5) / (d [1 + (f/d)^2]) + (1.4 × 10^-12 p θ^1.5) / (1 + 1.9 × 10^-5 f^1.5) ] (8) where d is the width parameter for the Debye spectrum: d = 5.6 × 10^-4 (p + e) θ^0.8 (9) 2 Path attenuation 2.1 Terrestrial paths For a terrestrial path, or for slightly inclined paths close to the ground, the path attenuation, A, may be written as: A = γ r_0 = (γo + γw) r_0 dB (10) where r_0 is path length (km). 2.2 Slant paths This section gives a method to integrate the specific attenuation calculated using the line-by-line model given above, at different pressures, temperatures and humidities through the atmosphere. By this means, the path attenuation for communications systems with any geometrical configuration within and external to the Earth's atmosphere may be accurately determined simply by dividing the atmosphere into horizontal layers, specifying the profile of the meteorological parameters pressure, temperature and humidity along the path. In the absence of local profiles, from radiosonde data, for example, the reference standard atmospheres in Recommendation ITU-R P.835 may be used, either for global application or for low (annual), mid (summer and winter) and high latitude (summer and winter) sites. Figure 3 shows the zenith attenuation calculated at 1 GHz intervals with this model for the global reference standard atmosphere in Recommendation ITU-R P.835, with horizontal layers 1 km thick and summing the attenuations for each layer, for the cases of a moist atmosphere (Curve A) and a dry atmosphere (Curve B). The total slant path attenuation, A(h, ϕ), from a station with altitude, h, and elevation angle, ϕ, can be calculated as follows when ϕ ≥ 0: A(h, ϕ) = ∫_h^∞ (γ(H) / sin Φ) dH (11) where the value of Φ can be determined as follows based on Snell's law in polar coordinates: Φ = arccos( c / [ (r + H) × n(H) ] ) (12) where: c = (r + h) × n(h) × cos ϕ (13) where n(h) is the atmospheric radio refractive index, calculated from pressure, temperature and water-vapour pressure along the path (see Recommendation ITU-R P.835) using Recommendation ITU-R P.453. On the other hand, when ϕ < 0, there is a minimum height, h_min, at which the radio beam becomes parallel with the Earth's surface. The value of h_min can be determined by solving the following transcendental equation: (r + h_min) × n(h_min) = c (14) This can be easily solved by repeating the following calculation, using h_min = h as an initial value: h'_min = c / n(h_min) - r (15) Therefore, A(h, ϕ) can be calculated as follows: A(h, ϕ) = ∫_h_min^∞ (γ(H) / sin Φ) dH + ∫_h_min^h (γ(H) / sin Φ) dH (16) In carrying out the integration of equations (11) and (16), care should be exercised in that the integrand becomes infinite at Φ = 0. However, this singularity can be eliminated by an appropriate variable conversion, for example, by using u^4 = H - h in equation (11) and u^4 = H - h_min in equation (16). A numerical solution for the attenuation due to atmospheric gases can be implemented with the following algorithm. To calculate the total attenuation for a satellite link, it is necessary to know not only the specific attenuation at each point of the link but also the length of path that has that specific attenuation. To derive the path length it is also necessary to consider the ray bending that occurs in a spherical Earth. Using Fig. 4 as a reference, a_n is the path length through layer n with thickness δ_n that has refractive index n_n. α_n and β_n are the entry and exiting incidence angles. r_n are the radii from the centre of the Earth to the beginning of layer n. a_n can then be expressed as: a_n = -r_n cos β_n + (1/2) sqrt( 4 r_n^2 cos^2 β_n + 8 r_n δ_n + 4 δ_n^2 ) (17) The angle α_n can be calculated from: α_n = π - arccos[ (-a_n^2 - 2 r_n δ_n - δ_n^2) / (2 a_n r_n + 2 a_n δ_n) ] (18) β_1 is the incidence angle at the ground station (the complement of the elevation angle φ). β_{n+1} can be calculated from α_n using Snell's law that in this case becomes: β_{n+1} = arcsin( (n_n / n_{n+1}) sin α_n ) (19) where n_n and n_{n+1} are the refractive indexes of layers n and n + 1. Equation (19) may become invalid at very low elevation angles (φ < 1°) when radiosonde data from certain regions of the world susceptible to ducting conditions are used as input. In such cases, air layers with radio refractivity gradients smaller in magnitude than -157 N/km are present and the ray-tracing algorithm (equations (17) to (19)), based on geometrical optics, is no longer applicable. The arcsine function in equation (19) becomes complex under these anomalous conditions since its argument is then slightly larger than 1. It should be noted that equation (19) is valid for all elevation angles when the reference standard atmospheres described in Recommendation ITU-R P.835 are used as input, since these idealized atmospheres – clearly without strong negative refractivity gradients – do not support such anomalous propagation conditions. FIGURE 3 Zenith attenuation due to atmospheric gases, calculated at 1 GHz intervals, including line centres (Standard: 7.5 g/m³ at sea level; Dry: 0 g/m³) [Graph of zenith attenuation in dB from 10^-3 to 10^5 across 0 to 1 000 GHz for Standard and Dry atmospheres]. The remaining frequency dependent (dispersive) term has a marginal influence on the result (around 1%) but can be calculated from the method shown in the ITU-R Handbook on Radiometeorology. The total attenuation can be derived using: A_gas = ∑_{n=1}^k a_n γ_n dB (20) where γ_n is the specific attenuation derived from equation (1). To ensure an accurate estimate of the path attenuation, the thickness of the layers should increase exponentially, from 10 cm at the lowest layer (ground level) to 1 km at an altitude of 100 km, according to the following equation: δ_i = 0.0001 exp[ (i - 1) / 100 ] km (21) from i = 1 to 922, noting that δ_922 ≅ 1.0 km and ∑_{i=1}^922 δ_i ≅ 100 km. For Earth-to-space applications, the integration should be performed at least up to 30 km, and up to 100 km at the oxygen line-centre frequencies. FIGURE 4 [Diagram showing ray path geometry through atmospheric layers 1 and 2, depicting earth radii r1, r2, r3, layer thicknesses δ1, δ2, path segment lengths a1, a2, a3, entry angles α1, α2, and exit angles β1, β2, β3]. 3 Dispersive effects The effects of dispersion are discussed in the ITU-R Handbook on Radiometeorology, which contains a model for calculating dispersion based on the line-by-line calculation. For practical purposes, dispersive effects should not impose serious limitations on millimetric terrestrial communication systems operating with bandwidths of up to a few hundred MHz over short ranges (for example, less than about 20 km), especially in the window regions of the spectrum, at frequencies removed from the centres of major absorption lines. For satellite communication systems, the longer path lengths through the atmosphere will constrain operating frequencies further to the window regions, where both atmospheric attenuation and the corresponding dispersion are low. Annex 2 Approximate estimation of gaseous attenuation in the frequency range 1-350 GHz This Annex contains simplified algorithms for quick, approximate estimation of gaseous attenuation for a limited range of meteorological conditions and a limited variety of geometrical configurations. 1 Specific attenuation The specific attenuation due to dry air and water vapour, from sea level to an altitude of 10 km, can be estimated using the following simplified algorithms, which are based on curve-fitting to the line-by-line calculation, and agree with the more accurate calculations to within an average of about ±10% at frequencies removed from the centres of major absorption lines. The absolute difference between the results from these algorithms and the line-by-line calculation is generally less than 0.1 dB/km and reaches a maximum of 0.7 dB/km near 60 GHz. For altitudes higher than 10 km, and in cases where higher accuracy is required, the line-by-line calculation should be used. For dry air, the attenuation γo (dB/km) is given by the following equations: For f ≤ 54 GHz: γo = [ (7.2 r_t^2.8) / (f^2 + 0.34 r_p^2 r_t^1.6) + (0.62 ξ_3) / ((54 - f)^1.16ξ_1 + 0.83 ξ_2) ] f^2 r_p^2 × 10^-3 (22a) For 54 GHz < f ≤ 60 GHz: γo = exp[ ((ln γ_54) / 24)(f - 58)(f - 60) - ((ln γ_58) / 8)(f - 54)(f - 60) + ((ln γ_60) / 12)(f - 54)(f - 58) ] (22b) For 60 GHz < f ≤ 62 GHz: γo = γ_60 + (γ_62 - γ_60) * ((f - 60) / 2) (22c) For 62 GHz < f ≤ 66 GHz: γo = exp[ ((ln γ_62) / 8)(f - 64)(f - 66) - ((ln γ_64) / 4)(f - 62)(f - 66) + ((ln γ_66) / 8)(f - 62)(f - 64) ] (22d) For 66 GHz < f ≤ 120 GHz: γo = { 3.02 × 10^-4 r_t^3.5 + (0.283 r_t^3.8) / ((f - 118.75)^2 + 2.91 r_p^2 r_t^1.6) + (0.502 ξ_6 [1 - 0.0163 ξ_7 (f - 66)]) / ((f - 66)^1.4346ξ_4 + 1.15 ξ_5) } f^2 r_p^2 × 10^-3 (22e) For 120 GHz < f ≤ 350 GHz: γo = [ (3.02 × 10^-4) / (1 + 1.9 × 10^-5 f^1.5) + (0.283 r_t^0.3) / ((f - 118.75)^2 + 2.91 r_p^2 r_t^1.6) ] f^2 r_p^2 r_t^3.5 × 10^-3 + δ (22f) with: ξ_1 = ϕ(r_p, r_t, 0.0717, -1.8132, 0.0156, -1.6515) (22g) ξ_2 = ϕ(r_p, r_t, 0.5146, -4.6368, -0.1921, -5.7416) (22h) ξ_3 = ϕ(r_p, r_t, 0.3414, -6.5851, 0.2130, -8.5854) (22i) ξ_4 = ϕ(r_p, r_t, -0.0112, 0.0092, -0.1033, -0.0009) (22j) ξ_5 = ϕ(r_p, r_t, 0.2705, -2.7192, -0.3016, -4.1033) (22k) ξ_6 = ϕ(r_p, r_t, 0.2445, -5.9191, 0.0422, -8.0719) (22l) ξ_7 = ϕ(r_p, r_t, -0.1833, 6.5589, -0.2402, 6.131) (22m) γ_54 = 2.192 ϕ(r_p, r_t, 1.8286, -1.9487, 0.4051, -2.8509) (22n) γ_58 = 12.59 ϕ(r_p, r_t, 1.0045, 3.5610, 0.1588, 1.2834) (22o) γ_60 = 15.0 ϕ(r_p, r_t, 0.9003, 4.1335, 0.0427, 1.6088) (22p) γ_62 = 14.28 ϕ(r_p, r_t, 0.9886, 3.4176, 0.1827, 1.3429) (22q) γ_64 = 6.819 ϕ(r_p, r_t, 1.4320, 0.6258, 0.3177, -0.5914) (22r) γ_66 = 1.908 ϕ(r_p, r_t, 2.0717, -4.1404, 0.4910, -4.8718) (22s) δ = -0.00306 ϕ(r_p, r_t, 3.211, -14.94, 1.583, -16.37) (22t) ϕ(r_p, r_t, a, b, c, d) = r_p^a r_t^b exp[ c(1 - r_p) + d(1 - r_t) ] (22u) where: f: frequency (GHz) r_p = p_tot / 1013, where p_tot represents total air pressure r_t = 288 / (273 + t) p: pressure (hPa) t: temperature (°C), where mean temperature values can be obtained from maps given in Recommendation ITU-R P.1510, when no adequate temperature data are available. For water vapour, the attenuation γw (dB/km) is given by: γw = { (3.98 η_1 exp[2.23(1 - r_t)]) / ((f - 22.235)^2 + 9.42 η_1^2) g(f, 22) + (11.96 η_1 exp[0.7(1 - r_t)]) / ((f - 183.31)^2 + 11.14 η_1^2) + (0.081 η_1 exp[6.44(1 - r_t)]) / ((f - 321.226)^2 + 6.29 η_1^2) + (3.66 η_1 exp[1.6(1 - r_t)]) / ((f - 325.153)^2 + 9.22 η_1^2) + (25.37 η_1 exp[1.09(1 - r_t)]) / ((f - 380)^2) + (17.4 η_1 exp[1.46(1 - r_t)]) / ((f - 448)^2) + (844.6 η_1 exp[0.17(1 - r_t)]) / ((f - 557)^2) g(f, 557) + (290 η_1 exp[0.41(1 - r_t)]) / ((f - 752)^2) g(f, 752) + (8.3328 × 10^4 η_2 exp[0.99(1 - r_t)]) / ((f - 1780)^2) g(f, 1780) } f^2 r_t^2.5 ρ × 10^-4 (23a) with: η_1 = 0.955 r_p r_t^0.68 + 0.006 ρ (23b) η_2 = 0.735 r_p r_t^0.5 + 0.0353 r_t^4 ρ (23c) g(f, f_i) = 1 + ((f - f_i) / (f + f_i))^2 (23d) where ρ is the water-vapour density (g/m³). Figure 5 shows the specific attenuation from 1 to 350 GHz at sea-level for dry air and water vapour with a density of 7.5 g/m³. 2 Path attenuation 2.1 Terrestrial paths For a horizontal path, or for slightly inclined paths close to the ground, the path attenuation, A, may be written as: A = γ r_0 = (γo + γw) r_0 dB (24) where r_0 is the path length (km). FIGURE 5 Specific attenuation due to atmospheric gases [Graph of specific attenuation in dB/km from 10^-3 to 10^2 versus frequency from 1 to 350 GHz at sea-level for Total, Dry air, and Water vapour; pressure: 1 013 hPa, temperature: 15° C, water vapour density: 7.5 g/m³]. 2.2 Slant paths This section contains simple algorithms for estimating the gaseous attenuation along slant paths through the Earth’s atmosphere, by defining an equivalent height by which the specific attenuation calculated in § 1 may be multiplied to obtain the zenith attenuation. The equivalent heights are dependent on pressure, and can hence be employed for determining the zenith attenuation from sea level up to an altitude of about 10 km. The resulting zenith attenuations are accurate to within ±10% for dry air and ±5% for water vapour from sea level up to altitudes of about 10 km, using the pressure, temperature and water-vapour density appropriate to the altitude of interest. For altitudes higher than 10 km, and particularly for frequencies within 0.5 GHz of the centres of resonance lines at any altitude, the procedure in Annex 1 should be used. Note that the Gaussian function in equation (25b) describing the oxygen equivalent height in the 60 GHz band can yield errors higher than 10% at certain frequencies, since this procedure cannot reproduce the structure shown in Fig. 7. The expressions below were derived from zenith attenuations calculated with the procedure in Annex 1, integrating the attenuations numerically over a bandwidth of 500 MHz; the resultant attenuations hence effectively represent approximate minimum values in the 50-70 GHz band. The path attenuation at elevation angles other than the zenith may then be determined using the procedures described later in this section. For dry air, the equivalent height is given by: h_o = (6.1 / (1 + 0.17 r_p^-1.1)) (1 + t_1 + t_2 + t_3) (25a) where: t_1 = (4.64 / (1 + 0.066 r_p^-2.3)) exp[ -((f - 59.7) / (2.87 + 12.4 exp(-7.9 r_p)))^2 ] (25b) t_2 = (0.14 exp(2.12 r_p)) / ((f - 118.75)^2 + 0.031 exp(2.2 r_p)) (25c) t_3 = (0.0114 / (1 + 0.14 r_p^-2.6)) f * (-0.0247 + 0.0001 f + 1.61 × 10^-6 f^2) / (1 - 0.0169 f + 4.1 × 10^-5 f^2 + 3.2 × 10^-7 f^3) (25d) with the constraint that: h_o ≤ 10.7 r_p^0.3 when f < 70 GHz (25e) and for water vapour, the equivalent height is: h_w = 1.66 [ 1 + (1.39 σ_w) / ((f - 22.235)^2 + 2.56 σ_w) + (3.37 σ_w) / ((f - 183.31)^2 + 4.69 σ_w) + (1.58 σ_w) / ((f - 325.1)^2 + 2.89 σ_w) ] (26a) for f ≤ 350 GHz σ_w = 1.013 / (1 + exp[-8.6(r_p - 0.57)]) (26b) The zenith attenuation between 50 to 70 GHz is a complicated function of frequency, as shown in Fig. 7, and the above algorithms for equivalent height can provide only an approximate estimate, in general, of the minimum levels of attenuation likely to be encountered in this frequency range. For greater accuracy, the procedure in Annex 1 should be used. The concept of equivalent height is based on the assumption of an exponential atmosphere specified by a scale height to describe the decay in density with altitude. Note that scale heights for both dry air and water vapour may vary with latitude, season and/or climate, and that water vapour distributions in the real atmosphere may deviate considerably from the exponential, with corresponding changes in equivalent heights. The values given above are applicable up to altitudes of about 10 km. The total zenith attenuation is then: A = γo h_o + γw h_w dB (27) Figure 6 shows the total zenith attenuation at sea level, as well as the attenuation due to dry air and water vapour, using the mean annual global reference atmosphere given in Recommendation ITU-R P.835. Between 50 and 70 GHz greater accuracy can be obtained from the 0 km curve in Fig. 7 which was derived using the line-by-line calculation as described in Annex 1. 2.2.1 Elevation angles between 5° and 90° 2.2.1.1 Earth-space paths For an elevation angle, ϕ, between 5° and 90°, the path attenuation is obtained using the cosecant law, as follows: For path attenuation based on surface meteorological data: A = (A_o + A_w) / sin ϕ dB (28) where A_o = h_o γo and A_w = h_w γw and for path attenuation based on integrated water vapour content: A(P) = (A_o + A_w(P)) / sin ϕ dB (29) where A_w(P) is given in § 2.3. 2.2.1.2 Inclined paths To determine the attenuation values on an inclined path between a station situated at altitude h_1 and another at a higher altitude h_2, where both altitudes are less than 10 km above mean sea level, the values h_o and h_w in equation (28) must be replaced by the following h'_o and h'_w values: h'_o = h_o [ exp(-h_1 / h_o) - exp(-h_2 / h_o) ] km (30) h'_w = h_w [ exp(-h_1 / h_w) - exp(-h_2 / h_w) ] km (31) it being understood that the value ρ of the water-vapour density used in equation (23) is the hypothetical value at sea level calculated as follows: ρ = ρ_1 × exp(h_1 / 2) (32) where ρ_1 is the value corresponding to altitude h_1 of the station in question, and the equivalent height of water vapour density is assumed as 2 km (see Recommendation ITU-R P.835). Equations (30), (31) and (32) use different normalizations for the dry air and water-vapour equivalent heights. While the mean air pressure referred to sea level can be considered constant around the world (equal to 1013 hPa), the water-vapour density not only has a wide range of climatic variability but is measured at the surface (i.e. at the height of the ground station). For values of surface water-vapour density, see Recommendation ITU-R P.836. 2.2.2 Elevation angles between 0° and 5° 2.2.2.1 Earth-space paths In this case, Annex 1 of this Recommendation should be used. The same Annex should also be used for elevations less than zero. 2.2.2.2 Inclined paths The attenuation on an inclined path between a station situated at altitude h_1 and a higher altitude h_2 (where both altitudes are less than 10 km above mean sea level), can be determined from the following: A = γo sqrt(h_o) [ (sqrt(R_e + h_1) F(x_1) exp(-h_1 / h_o)) / cos φ_1 - (sqrt(R_e + h_2) F(x_2) exp(-h_2 / h_o)) / cos φ_2 ] + γw sqrt(h_w) [ (sqrt(R_e + h_1) F(x'_1) exp(-h_1 / h_w)) / cos φ_1 - (sqrt(R_e + h_2) F(x'_2) exp(-h_2 / h_w)) / cos φ_2 ] dB (33) where: R_e: effective Earth radius including refraction, given in Recommendation ITU-R P.834, expressed in km (a value of 8 500 km is generally acceptable for the immediate vicinity of the Earth's surface) φ_1: elevation angle at altitude h_1 F: function defined by: F(x) = 1 / ( 0.661 x + 0.339 sqrt(x^2 + 5.51) ) (34) φ_2 = arccos[ ((R_e + h_1) / (R_e + h_2)) cos φ_1 ] (35a) x_i = tan φ_i sqrt((R_e + h_i) / h_o) for i = 1, 2 (35b) x'_i = tan φ_i sqrt((R_e + h_i) / h_w) for i = 1, 2 (35c) it being understood that the value ρ of the water vapour density used in equation (23) is the hypothetical value at sea level calculated as follows: ρ = ρ_1 ⋅ exp(h_1 / 2) (36) where ρ_1 is the value corresponding to altitude h_1 of the station in question, and the equivalent height of water vapour density is assumed as 2 km (see Recommendation ITU-R P.835). Values for ρ_1 at the surface can be found in Recommendation ITU-R P.836. The different formulation for dry air and water vapour is explained at the end of § 2.2. FIGURE 6 Total, dry air and water-vapour zenith attenuation from sea level [Graph of zenith attenuation in dB from 10^-3 to 10^3 versus frequency from 1 to 350 GHz; surface pressure: 1 013 hPa, surface temperature: 15° C, surface water-vapour density: 7.5 g/m³]. FIGURE 7 Zenith oxygen attenuation from the altitudes indicated, calculated at intervals of 50 MHz, including line centres (0 km, 5 km, 10 km, 15 km and 20 km) [Detailed graph of zenith oxygen attenuation in dB from 10^-2 to 10^3 over the frequency range 50 to 70 GHz at 0, 5, 10, 15, and 20 km altitudes]. 2.3 Zenith path water-vapour attenuation The above method for calculating slant path attenuation by water vapour relies on the knowledge of the profile of water-vapour pressure (or density) along the path. In cases where the integrated water vapour content along the path, V_t, is known, an alternative method may be used. The total water-vapour attenuation can be estimated as: A_w(P) = (0.0173 V_t(P) γw(f, p_ref, ρ_v,ref, t_ref)) / (γw(f_ref, p_ref, ρ_v,ref, t_ref)) dB (37) where: f: frequency (GHz) f_ref: 20.6 (GHz) p_ref = 780 (hPa) ρ_v,ref = V_t(P) / 4 (g/m³) t_ref = 14 ln( (0.22 V_t(P)) / 4 ) + 3 (°C) V_t(P): integrated water vapour content at the required percentage of time (kg/m² or mm), which can be obtained either from radiosonde profiles, radiometric measurements, or Recommendation ITU-R P.836 (kg/m² or mm) γw(f, p, ρ, t): specific attenuation as a function of frequency, pressure, water-vapour density, and temperature calculated from equation (23a) (dB/km).