Attenuation by atmospheric gases

Recommendation ITU-R P.676-0 (09/03) Attenuation by atmospheric gases P Series Radioave propagation

ii Rec. ITU-R P.676-0 Foreord 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 ithout limit of frequency range on the basis of hich 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 of Resolution ITU-R. Forms to be used for the submission of patent statements and licensing declarations by patent holders are available from http://.itu.int/itu-r/go/patents/en here the Guidelines for Implementation of the Common Patent Policy for ITU-T/ITU-R/ISO/IEC and the ITU-R patent information database can also be found. Series of ITU-R Recommendations (Also available online at http://.itu.int/publ/r-rec/en) Series BO BR BS BT F M P RA RS S SA SF SM SNG TF V Title Satellite delivery Recording for production, archival and play-out; film for television Broadcasting service (sound) Broadcasting service (television) Fixed service Mobile, radiodetermination, amateur and related satellite services Radioave propagation Radio astronomy Remote sensing systems Fixed-satellite service Space applications and meteorology Frequency sharing and coordination beteen fixed-satellite and fixed service systems Spectrum management Satellite nes gathering Time signals and frequency standards emissions Vocabulary and related subjects Note: This ITU-R Recommendation as approved in English under the procedure detailed in Resolution ITU-R. Electronic Publication Geneva, 03 ITU 03 All rights reserved. No part of this publication may be reproduced, by any means hatsoever, ithout ritten permission of ITU.

Rec. ITU-R P.676-0 RECOMMENDATION ITU-R P.676-0 Attenuation by atmospheric gases (Question ITU-R 0/3) (990-99-99-997-999-00-00-007-009-0-03) Scope This Recommendation 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 - 000 GHz; and b) a simplified approximate method to estimate gaseous attenuation that is applicable in the frequency range -30 GHz. The ITU Radiocommunication Assembly, considering a) the necessity of estimating the attenuation by atmospheric gases on terrestrial and slant paths, recommends that, for general application, the procedures in Annex be used to calculate gaseous attenuation at frequencies up to 000 GHz; that, for approximate estimates of gaseous attenuation in the frequency range to 30 GHz, the computationally less intensive procedure given in Annex be used. Annex Line-by-line calculation of gaseous attenuation Specific attenuation The specific attenuation at frequencies up to 000 GHz due to dry air and ater 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 ater vapour, together ith small additional factors for the non-resonant Debye spectrum of oxygen belo 0 GHz, pressure-induced nitrogen attenuation above 00 GHz and a et continuum to account for the excess ater vapourabsorption found experimentally. Figure shos the specific attenuation using the model, calculated from 0 to 000 GHz at GHz intervals, for a pressure of 03 hpa, temperature of C for the cases of a ater-vapour density of 7. g/m 3 (Curve A) and a dry atmosphere (Curve B).

Rec. ITU-R P.676-0 Near 60 GHz, many oxygen absorption lines merge together, at sea-level pressures, to form a single, broad absorption band, hich is shon in more detail in Fig.. This figure also shos the oxygen attenuation at higher altitudes, ith the individual lines becoming resolved at loer 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 30 GHz, in cases here high accuracy is not required, simplified algorithms are given in Annex for restricted ranges of meteorological conditions. The specific gaseous attenuation is given by: γ = γ γ 0.80 f N" ( f ) db/km () o = here γ o and γ are the specific attenuations (db/km) due to dry air (oxygen, pressure-induced nitrogen and non-resonant Debye attenuation) and ater vapour, respectively, and here f is the frequency (GHz) and N ( f ) is the imaginary part of the frequency-dependent complex refractivity: N" ( f ) = S F N" ( f ) () i i i D 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 8.70343 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 = ); 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 = = 7 a 0 p b 0 e 3 θ 3. θ exp [ a ( θ) ] exp for oxygen [ b ( θ) ] for ater vapour (3) here: p : dry air pressure (hpa) e : ater vapour partial pressure in hpa (total barometric pressure p tot = p e) θ = 300/T T : temperature (K).

Rec. ITU-R P.676-0 3 FIGURE Specific attenuation due to atmospheric gases, calculated at GHz intervals, including line centres 0 0 4 0 3 0 0 0 0 0 3 Standard Dry 0 00 00 300 400 00 600 700 800 900 000 Frequency (GHz) P.067 6-0 Specific attenuation (db/km)

4 Rec. ITU-R P.676-0 FIGURE Specific attenuation in the range 0-70 GHz at the altitudes indicated 0 0 0 0 0 0 0 0 3 0 4 6 8 60 6 64 66 68 70 Frequency (GHz) P.0676-0 Specific attenuation (db/km) Local values of p, e and T measured profiles (e.g. using radiosondes) should be used; hoever, in the absence of local information, the reference standard atmospheres described in Recommendation ITU-R P.83 should be used. (Note that here total atmospheric attenuation is being calculated, the same-ater vapour partial pressure is used for both dry-air and ater-vapour attenuations.)

Rec. ITU-R P.676-0 The ater-vapour partial pressure, e, may be obtained from the ater-vapour density ρ using the expression: ρt e = (4) 6.7 The coefficients a, a are given in Table for oxygen, those for ater vapour, b and b, are given in Table. The line-shape factor is given by: Δf δ ( fi f ) ( f f ) Δf Δf δ ( f ) ( ) i f fi f Δf f Fi = () fi i here f i is the line frequency and Δf is the idth of the line: Δf = = a3 b3 4 0 ( p 4 0 ( p (0.8 θ b θ 4 a4 ) b b θ 6 e ). e θ) for oxygen for ater vapour (6a) The line idth Δf is modified to account for Doppler broadening: Δf = Δf = 0.3Δf. 0 6 0.7Δf.36 0 θ f i for oxygen for ater vapour (6b) δ is a correction factor hich arises due to interference effects in oxygen lines: δ = ( a θ) 0 ( p e) θ = 0 The spectroscopic coefficients are given in Tables and. a 6 4 0.8 for oxygen for ater vapour (7)

6 Rec. ITU-R P.676-0 TABLE Spectroscopic data for oxygen attenuation f 0 a a a 3 a 4 a a 6 0.4744 0.97 9.6 6.690 0.0.66 6.80 0.98774.9 8.63 7.70 0.0.46 6.800.03360 6.93 7.709 7.640 0.0.947 6.79.049 4.30 6.89 8.0 0.0.667 6.640.448 3.40.983 8.80 0.0.388 6.6 3.066934 64.90.0 9.060 0.0.349 6.06 3.977 4.600 4.474 9.0 0.0.7.08 4.300 7.300 3.800 9.960 0.0 3.70 3.70 4.6780 389.700 3.8 0.370 0.0 3.8.64.384 67.00.68 0.890 0.0.60.9.7838 94.300.09.340 0.0.7 6.3 6.64774 43.400 0.04 7.030 0.0 3. 0.978 6.363399 33.800.64.890 0.0.378 6.47 6.968 746.600..30 0.0 3.4 6.4 7.6486 0.00 0.90.60 0.0.46 6.06 8.33877 363.700 0.6.90 0.0.93 0.436 8.44688 44.00 0.083 4.90 0.0 6.768.73 9.6404 379.900 0.387 3.30 0.0 6.6.309 9.90983 090.700 0.07 4.080 0.0 6.97 0.776 60.30606 03.400 0.07 4.0 0.0 6.39 0.699 60.434778 438.000 0.386 3.390 0.0 6.34.8 6.06 479.00 0.6.90 0.0.04 0.84 6.8008 7.900 0.90.630 0.0.04 6.69 6.40 9.400..70 0.0 3.09 6.79 6.4863 03.000 0.083.30 0.0 4.499 0.844 6.997984 490.00.64.740 0.0.86 6.67 63.686 078.000.08.340 0.0 0.68 6.39 64.777 78.700.67 0.880 0.0 3.036.89 64.67890 46.300 3.8 0.380 0.0 3.968.90 6.4078 74.000 3.800 9.960 0.0 3.8 3.680 6.764779 3.000 4.473 9.0 0.0.48.00 66.30096 80.400.00 9.060 0.0.660 6.09 66.836834 39.800.98 8.80 0.0.680 6.393 67.36960 8.60 6.88 8.0 0.0.96 6.47 67.900868 8.7 7.708 7.640 0.0.6 6.4 68.43006 3.397 8.6 7.70 0.0.49 6.600 68.9603.334 9.60 6.690 0.0.773 6.60 8.70334 940.300 0.00 6.640 0.0 0.439 0.079

Rec. ITU-R P.676-0 7 TABLE (end) f 0 a a a 3 a 4 a a 6 368.49846 67.400 0.048 6.400 0.0 0.000 0.000 44.76300 637.700 0.044 6.400 0.0 0.000 0.000 487.4973 37.400 0.049 6.000 0.0 0.000 0.000 7.3990 98.00 0.4 6.000 0.0 0.000 0.000 773.839490 7.300 0.4 6.00 0.0 0.000 0.000 834.446 83.00 0.4 4.700 0.0 0.000 0.000 TABLE Spectroscopic data for ater-vapour attenuation f 0 b b b 3 b 4 b b 6.3080 0.30.43 8..69 4.800.00 67.803960 0.00 8.73 8.8.69 4.930.8 9.99940 0.0008 8.36 9.48.70 4.780.79 83.3009.400.668 30.0.64.300.8 3.644 0.0483 6.8 3.03.67 4.690.4 3.99.4990.40 7.83.68 4.80.74 336.60 0.00 9.89 6.93.69 4.740.6 380.9737.00.048 8.73.4.380.89 390.3408 0.0046 7.30..63 4.80. 437.346667 0.060.00 8.4.60 4.30.48 439.08 0.98 3.96.00.63 4.90. 443.089 0.976.00 8.60.60 4.30.0 448.0007 0.300.40 6.3.66 4.840.67 470.888947 0.397 3.99..66 4.70.6 474.6897.60.38 3..6 4.60.64 488.4933 0.0.83 6.0.69.040.7 03.683 0.0390 6.733 6..6 3.980.43 04.4869 0.030 6.733 6..6 4.00.4 47.676440 9.700.4 6.00.70 4.00.00.00960 4.7700.4 6.00.70 4.00.00 6.93600 487.4000.9 3.0.69 4.0.00 60.700807.00.00 4.38.7 4.680.68 64.866 0.073 8.80 8.00.60 4.000.0 68.0080 0.30 7.80 3.0.69 4.40.00 7.0337 39.6000.396 30.60.68 4.090.84 84.03973 0.040 8.80.90.33.760.4 89.9633 0.47 7.989 30.60.68 4.090.84

8 Rec. ITU-R P.676-0 TABLE (end) f 0 b b b 3 b 4 b b 6 899.30667 0.060 7.97 9.8.68 4.30.90 90.6673 0.046 8.43 8.6.70.00.9 906.073 0.876. 4.08.70 4.700.3 96.78 8.3400.44 6.70.70 4.780.78 93.847 0.0869 0.0 9.00.70.000.80 970.30 8.970.90.0.64 4.940.67 987.96764 3.000.8 9.8.68 4.0.90 780.000000 300.0000.9 76.0.0 30.00.00 The dry air continuum arises from the non-resonant Debye spectrum of oxygen belo 0 GHz and a pressure-induced nitrogen attenuation above 00 GHz.. 6.4 0.4 0 p θ N" D ( f ) = f p θ (8). f.9 0 f d d here d is the idth parameter for the Debye spectrum: 0. 8 ( p e) 4 =.6 0 θ d (9) Path attenuation. Terrestrial paths For a terrestrial path, or for slightly inclined paths close to the ground, the path attenuation, A, may be ritten as: here r 0 is path length (km).. Slant paths ( γ γ ) db A = γ r 0 = o r 0 (0) 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 ith any geometrical configuration ithin 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.83 may be used, either

Rec. ITU-R P.676-0 9 for global application or for lo (annual), mid (summer and inter) and high latitude (summer and inter) sites. Figure 3 shos the zenith attenuation calculated at GHz intervals ith this model for the global reference standard atmosphere in Recommendation ITU-R P.83, ith horizontal layers 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 ith altitude, h, and elevation angle, ϕ, can be calculated as follos hen ϕ 0: ( ) ( ) γ H A h, ϕ = dh () h sin Φ here the value of Φ can be determined as follos based on Snell s la in polar coordinates: here: c Φ = arccos ( r H ) n( H ) () c = ( r h) n( h) cos ϕ (3) here n(h) is the atmospheric radio refractive index, calculated from pressure, temperature and ater-vapour pressure along the path (see Recommendation ITU-R P.83) using Recommendation ITU-R P.43. On the other hand, hen ϕ < 0, there is a minimum height, h min, at hich the radio beam becomes parallel ith the Earth s surface. The value of h min can be determined by solving the folloing transcendental equation: ( r h ) n( h ) c (4) min min = This can be easily solved by repeating the folloing calculation, using h min = h as an initial value: c h' min = r () n Therefore, A(h, ϕ) can be calculated as follos: (, ϕ) A h = ( h ) min γ ( H ) h γ ( H ) dh dh (6) h sin Φ h sin Φ min In carrying out the integration of equations () and (6), care should be exercised in that the integrand becomes infinite at Φ = 0. Hoever, this singularity can be eliminated by an appropriate variable conversion, for example, by using u 4 = H h in equation () and u 4 = H h min in equation (6). A numerical solution for the attenuation due to atmospheric gases can be implemented ith the folloing algorithm. To calculate the total attenuation for a satellite link, it is necessary to kno 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. min

0 Rec. ITU-R P.676-0 Using Fig. 4 as a reference, a n is the path length through layer n ith 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: The angle α n can be calculated from: a n = rn cos βn 4 rn cos βn 8 rn δn 4 δn (7) a δ δ α = π n rn n n n arccos (8) an rn an δn β is the incidence angle at the ground station (the complement of the elevation angle φ). β n can be calculated from α n using Snell s la that in this case becomes: here n n and n n are the refractive indexes of layers n and n. n β = n n arcsin sin αn (9) nn Equation (9) may become invalid at very lo elevation angles (φ < ) hen radiosonde data from certain regions of the orld susceptible to ducting conditions are used as input. In such cases, air layers ith radio refractivity gradients smaller in magnitude than 7 N/km are present and the ray-tracing algorithm (equations (7) to (9)), based on geometrical optics, is no longer applicable. The arcsine function in equation (9) becomes complex under these anomalous conditions since its argument is then slightly larger than. It should be noted that equation (9) is valid for all elevation angles hen the reference standard atmospheres described in Recommendation ITU-R P.83 are used as input, since these idealized atmospheres clearly ithout strong negative refractivity gradients do not support such anomalous propagation conditions.

Rec. ITU-R P.676-0 FIGURE 3 Zenith attenuation due to atmospheric gases, calculated at GHz intervals, including line centres 0 0 4 0 3 0 0 0 0 0 3 Standard Dry 0 00 00 300 400 00 600 700 800 900 000 Frequency (GHz) P.0676-03 Zenith attenuation (db) The remaining frequency dependent (dispersive) term has a marginal influence on the result (around %) but can be calculated from the method shon in the ITU-R Handbook on Radiometeorology.

Rec. ITU-R P.676-0 The total attenuation can be derived using: k A = γ db (0) gas a n n = here γ n is the specific attenuation derived from equation (). To ensure an accurate estimate of the path attenuation, the thickness of the layers should increase exponentially, from 0 cm at the loest layer (ground level) to km at an altitude of 00 km, according to the folloing equation: n i δi = 0.000 exp km () 00 9 i = i from i = to 9, noting that δ 9.0 km and δ 00 km. For Earth-to-space applications, the integration should be performed at least up to 30 km, and up to 00 km at the oxygen line-centre frequencies. FIGURE 4 Layer Layer β β a β a 3 a3 α α n r r n r 3 3 Dispersive effects The effects of dispersion are discussed in the ITU-R Handbook on Radiometeorology, hich 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

Rec. ITU-R P.676-0 3 communication systems operating ith bandidths of up to a fe hundred MHz over short ranges (for example, less than about 0 km), especially in the indo 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 ill constrain operating frequencies further to the indo regions, here both atmospheric attenuation and the corresponding dispersion are lo. Annex Approximate estimation of gaseous attenuation in the frequency range -30 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. Specific attenuation The specific attenuation due to dry air and ater vapour, from sea level to an altitude of 0 km, can be estimated using the folloing simplified algorithms, hich are based on curve-fitting to the lineby-line calculation, and agree ith the more accurate calculations to ithin an average of about ±0% at frequencies removed from the centres of major absorption lines. The absolute difference beteen the results from these algorithms and the line-by-line calculation is generally less than 0. db/km and reaches a maximum of 0.7 db/km near 60 GHz. For altitudes higher than 0 km, and in cases here higher accuracy is required, the line-by-line calculation should be used. For dry air, the attenuation γ o (db/km) is given by the folloing equations: For f 4 GHz: γ o = f For 4 GHz < f 60 GHz: 7. r 0.34.8 t rp.6 t r (4 f ) 0.6ξ.6ξ 3 0.83 ξ f rp 0 3 (a) ln γ ln γ8 ln γ60 γ = ( 4)( 60) ( 4)( 8) o exp ( 8)( 60) 4 f f f f f 4 8 f (b) For 60 GHz < f 6 GHz: 60 ( 60 ) f γo = γ60 γ6 γ (c) For 6 GHz < f 66 GHz: ln γ ln γ64 ln γ66 γ = ( 6)( 66) ( 6)( 64) o exp ( 64)( 66) 6 f f f f f 8 4 8 f (d)

4 Rec. ITU-R P.676-0 For 66 GHz < f 0 GHz: 3.8 4 3. 0.83rt 0.0ξ6[ 0.063ξ7 ( f 66)] 3 γo = 3.0 0 t f r 0.6.4346ξ p ( f 8.7).9r ( 66) 4 p rt f.ξ For 0 GHz < f 30 GHz: r (e) ith: γ o 3.0 0 =.9 0 γ γ ξ ξ ξ ξ 4 f. ( f 0.83r 8.7) 0.3 t.9r.6 p rt f 3. rp rt 0 3 δ (f) = ϕ( r,,0.077,.83,0.06,.6) (g) p rt = ϕ( r,,0.46, 4.6368, 0.9,.746) (h) p rt ξ = ϕ( r,,0.344, 6.8,0.30, 8.84) (i) 3 p rt = ϕ( r,, 0.0,0.009, 0.033, 0.0009) (j) 4 p r t = ϕ( r,,0.70,.79, 0.306, 4.033) (k) p r t ξ = ϕ( r,,0.44,.99,0.04, 8.079) (l) 6 p r t ξ = ϕ( r,, 0.833,6.89, 0.40,6.3) (m) 7 p r t =.9ϕ( r,,.886,.9487,0.40,.809) (n) 4 p rt γ =.9 ϕ ( r p, r,.004,3.60,0.88,.834) (o) 8 t γ =.0ϕ( r p, r,0.9003,4.33,0.047,.6088) (p) 60 t γ = 4.8 ϕ ( r p, r,0.9886,3.476,0.87,.349) (q) γ 6 t = 6.89ϕ( r,,.430,0.68,0.377, 0.94) (r) 64 p r t =.908ϕ( r,,.077, 4.404,0.490, 4.878) (s) 66 p rt δ = 0.00306ϕ( r,,3., 4.94,.83, 6.37) (t) p r t here: a b ( p t p t p t ϕ r, r, a, b, c, d) = r r exp[ c( r ) d( r )] (u) f : frequency (GHz) r p = p tot /03, here p tot represents total air pressure r t = 88/(73 t) p : pressure (hpa)

Rec. ITU-R P.676-0 t : temperature ( C), here mean temperature values can be obtained from maps given in Recommendation ITU-R P.0, hen no adequate temperature data are available. For ater vapour, the attenuation γ (db/km) is given by: γ 3.98η exp[.3( rt )].96η exp[0.7( rt )] = g( f,) ( f.3) 9.4η ( f 83.3).4η 0.08η exp[6.44( r )] 3.66η exp[.6( rt )] ( f 3.6) 6.9 ( f 3.3) 9.η t η.37η exp[.09( rt )] 7.4η exp[.46( rt )] ( f 380) ( f 448) 844.6η exp[0.7( rt )] 90η exp[0.4( rt )] g( f,7) g( f,7) ( f 7) ( f 7) 4 8.338 0 η exp[0.99( r )] t g( f, 780) f ( f 780). rt ρ 0 4 (3a) ith: 0.68 η = 0.9r prt 0. 006ρ (3b) 0. 4 0 p t t η =.73 r r 0. 033 r ρ (3c) f f (, ) i g f fi = f fi (3d) here ρ is the ater-vapour density (g/m 3 ). Figure shos the specific attenuation from to 30 GHz at sea-level for dry air and ater vapour ith a density of 7. g/m 3. Path attenuation. Terrestrial paths For a horizontal path, or for slightly inclined paths close to the ground, the path attenuation, A, may be ritten as: here r 0 is the path length (km). A= γ r 0 = γ γ ) r db (4) ( o 0

6 Rec. ITU-R P.676-0 0 FIGURE Specific attenuation due to atmospheric gases 0 Total Specific attenuation (db/km) 0 Total Dry air 0 Water vapour 0 3 0 0 3. Frequency (GHz) Pressure: 03 hpa Temperature: C Water vapour density: 7. g/m 3

Rec. ITU-R P.676-0 7. 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 hich the specific attenuation calculated in 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 0 km. The resulting zenith attenuations are accurate to ithin ±0% for dry air and ±% for ater vapour from sea level up to altitudes of about 0 km, using the pressure, temperature and ater-vapour density appropriate to the altitude of interest. For altitudes higher than 0 km, and particularly for frequencies ithin 0. GHz of the centres of resonance lines at any altitude, the procedure in Annex should be used. Note that the Gaussian function in equation (b) describing the oxygen equivalent height in the 60 GHz band can yield errors higher than 0% at certain frequencies, since this procedure cannot reproduce the structure shon in Fig. 7. The expressions belo ere derived from zenith attenuations calculated ith the procedure in Annex, integrating the attenuations numerically over a bandidth of 00 MHz; the resultant attenuations hence effectively represent approximate minimum values in the 0-70 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: here: ith the constraint that: t 3 6. ho = ( t. t t 0.7 r p 3 ) (a) 4.64 f 9.7 t = exp (b).3 0.066 r p.87.4 exp ( 7.9 r p ) t 0.4 exp (. rp) = (c) ( f 8.7) 0.03 exp(. r ) 0.04 0.047 0.000f.6 0 f = f (d) 7 3 0.4 r 0.069 f 4. 0 f 3. 0 f.6 p and for ater vapour, the equivalent height is: 0.3 ho 0.7 rp hen f < 70 GHz (e) p 6 h =.66.39σ ( f.3).6σ 3.37σ ( f 83.3) 4.69σ.8σ ( f 3.).89σ (6a) for f 30 GHz σ.03 = exp[ 8.6 ( r p 0.7)] (6b) The zenith attenuation beteen 0 to 70 GHz is a complicated function of frequency, as shon 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 should be used.

8 Rec. ITU-R P.676-0 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 ith altitude. Note that scale heights for both dry air and ater vapour may vary ith latitude, season and/or climate, and that ater vapour distributions in the real atmosphere may deviate considerably from the exponential, ith corresponding changes in equivalent heights. The values given above are applicable up to altitudes of about 0 km. The total zenith attenuation is then: A = γ h γ h db (7) o o Figure 6 shos the total zenith attenuation at sea level, as ell as the attenuation due to dry air and ater vapour, using the mean annual global reference atmosphere given in Recommendation ITU-R P.83. Beteen 0 and 70 GHz greater accuracy can be obtained from the 0 km curve in Fig. 7 hich as derived using the line-by-line calculation as described in Annex... Elevation angles beteen and 90... Earth-space paths For an elevation angle, ϕ, beteen and 90, the path attenuation is obtained using the cosecant la, as follos: For path attenuation based on surface meteorological data: A = o A A db (8) sin ϕ here Ao = ho γo and A = h γ and for path attenuation based on integrated ater vapour content: Ao A ( P) A( P) = db (9) sin ϕ here A (P) is given in.3.... Inclined paths To determine the attenuation values on an inclined path beteen a station situated at altitude h and another at a higher altitude h, here both altitudes are less than 0 km above mean sea level, the values h o and h in equation (8) must be replaced by the folloing h o ' and h ' values: h o = h o h [ e / ho h / e ho ] km h [ e / h h / e h ] km ' (30) h ' = h (3) it being understood that the value ρ of the ater-vapour density used in equation (3) is the hypothetical value at sea level calculated as follos: ρ = ρ exp / (3) h ( ) here ρ is the value corresponding to altitude h of the station in question, and the equivalent height of ater vapour density is assumed as km (see Recommendation ITU-R P.83). Equations (30), (3) and (3) use different normalizations for the dry air and ater-vapour equivalent heights. While the mean air pressure referred to sea level can be considered constant around the orld (equal to 03 hpa), the ater-vapour density not only has a ide range of

Rec. ITU-R P.676-0 9 climatic variability but is measured at the surface (i.e. at the height of the ground station). For values of surface ater-vapour density, see Recommendation ITU-R P.836... Elevation angles beteen 0º and º... Earth-space paths In this case, Annex of this Recommendation should be used. The same Annex should also be used for elevations less than zero.... Inclined paths The attenuation on an inclined path beteen a station situated at altitude h and a higher altitude h (here both altitudes are less than 0 km above mean sea level), can be determined from the folloing: A = γ o h o R e h F( x ) e cos ϕ h / ho R e h F( x cos ϕ ) e h / ho here: γ R e : h R e h F( x' ) e cos ϕ h / h R e h F( x' ) e cos ϕ h / h db (33) effective Earth radius including refraction, given in Recommendation ITU-R P.834, expressed in km (a value of 8 00 km is generally acceptable for the immediate vicinity of the Earth's surface) ϕ : elevation angle at altitude h F : function defined by: F( x ) = (34) 0.66 x 0.339 x. R ϕ = arccos e h cos ϕ (3a) Re h R h x = tan ϕ e i i i for i =, (3b) ho R h x = tan ϕ e i i ' i for i =, (3c) h it being understood that the value ρ of the ater vapour density used in equation (3) is the hypothetical value at sea level calculated as follos: ( ) ρ = ρ exp h / (36) here ρ is the value corresponding to altitude h of the station in question, and the equivalent height of ater vapour density is assumed as km (see Recommendation ITU-R P.83). Values for ρ at the surface can be found in Recommendation ITU-R P.836. The different formulation for dry air and ater vapour is explained at the end of..

0 Rec. ITU-R P.676-0 0 3 FIGURE 6 Total, dry air and ater-vapour zenith attenuation from sea level 0 0 Zenith attenuation (db) 0 Total Dry air 0 Water vapour 0 3 0 0 30 Frequency (GHz) Surface pressure: 03 hpa Surface temperature: C Surface ater-vapour density: 7. g/m 3

Rec. ITU-R P.676-0 FIGURE 7 Zenith oxygen attenuation from the altitudes indicated, calculated at intervals of 0 MHz, including line centres (0 km, km, 0 km, km and 0 km) 0 3 0 0 0 0 0 km 0 0 0 4 6 8 60 6 64 66 68 70 Frequency (GHz) Zenith attenuation (db)

Rec. ITU-R P.676-0.3 Zenith path ater-vapour attenuation The above method for calculating slant path attenuation by ater vapour relies on the knoledge of the profile of ater-vapour pressure (or density) along the path. In cases here the integrated ater vapour content along the path, Vt, is knon, an alternative method may be used. The total ater-vapour attenuation can be estimated as: here: f : A ( P) 0.073Vt ( P) γ ( f, pref, ρv, ref, tref ) = db (37) γ ( f, p, ρ, t ) frequency (GHz) ref ref v, ref ref f ref : 0.6 (GHz) p ref = 780 (hpa) ρ v,ref = V t (P) (g/m 3 ) 4 0. V ( ) t ref = 4 ln t P 3 ( C) 4 V t (P): integrated ater vapour content at the required percentage of time (kg/m or mm), hich can be obtained either from radiosonde profiles, radiometric measurements, or Recommendation ITU-R P.836 (kg/m or mm) γ W (f, p, ρ, t): specific attenuation as a function of frequency, pressure, ater-vapour density, and temperature calculated from equation (3a) (db/km).