Scattering from rain is known to be a source of possible interference between communications systems sharing the same frequency bands. A new model has been developed for estimating the transmission loss due to bistatic rain scatter between an Earth station and a terrestrial station. This model has particular application in the process of coordination of Earth stations with terrestrial stations operating in the same frequency bands, whereby detailed interference studies are carried out only for stations within an area beyond which harmful interference may be considered negligible. The new model includes the Earth station elevation angle as an input parameter, together with the most recent information on rainfall rates and surface water vapor densities included in the recommendations of the International Telecommunications Union (ITU), and is more flexible yet neutral in its overall impact when applied to the process of coordination as defined in Appendix 7 of the ITU Radio Regulations.
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 Efficient utilization of the radio spectrum generally requires a degree of sharing of the available spectrum between services, and it is then necessary to minimize the likelihood of interference. This can be accomplished through two methods: (1) the imposition of limits to the amount of power that can be transmitted and (2) the process of coordination, whereby an area is established around a receiver or transmitter beyond which the likelihood of harmful interference may be considered negligible. Within this area, potential interfering or victim systems can be identified and detailed studies of interference between these can then be carried out, based on a knowledge of the system characteristics and the path between the two systems.
 The International Telecommunications Union (ITU) specifies methods for establishing the coordination area, on the basis of, inter alia, the propagation conditions between two locations, and for the purpose of coordination around an Earth station, interfering with or being interfered by a terrestrial station, two methods are included in Appendix 7 of the ITU Radio Regulations, one for clear air propagation and the other for interference through scattering from rain. These two procedures are derived from Recommendation ITU-R P.620, which provides methods for assessing the worst case levels of interference, as a function of distance, between one station with known characteristics and another station with unspecified and hence generalized characteristics. Given the basic transmission loss between two such systems which is necessary to provide the required degree of protection from harmful interference, the methods yield the distance beyond which this basic transmission loss will be exceeded, and hence the interference be negligible. The method currently specified in Appendix 7 for rain scatter interference—also known as propagation mode (2)—was developed with a number of simplifying assumptions, one of which was that the Earth station antenna has an elevation angle of 20°. With increasing deployments of satellite-based communications systems at low latitudes, this simplification is no longer satisfactory, and the World Radiocommunications Conference in 2000 requested the International Telecommunications Union Radiocommunication Sector (ITU-R), in Resolution 74, to refine the propagation mode (2) method to address the elevation angle dependency.
 This paper describes the development of the new method, which has now been incorporated into Recommendation ITU-R P.620-5 [International Telecommunications Union (ITU), 2002]. In addition to incorporating the appropriate elevation dependency, the model now includes meteorological input from world maps which have also recently been developed by the ITU-R. The model has been evaluated using the ITU Radiocommunications Bureau database of Earth stations, and is shown to be more flexible while being neutral in overall change, compared with the previous method in Recommendation ITU-R P.620-4 [ITU, 1999].
2. Derivation of the Bistatic Radar Equation
 Hydrometeor scatter is usually described using the Bistatic Radar Equation (BRE) [Crane, 1974], the geometry of which is illustrated schematically in Figure 1, for scattering in the vertical plane.
 If the transmitting power from a transmitting station Tx is Pt and the antenna gain is Gt (in linear units), the flux (power per unit area) arriving at a volume element δV, at a distance rt from the transmitter is and the scattered power is given by ηδV, where η is the scattering cross section per unit volume. Assuming isotropic scattering, the flux density at a receiving station Rx, at a distance rr, is ηδV. Now, the power received by an isotropic antenna is s, where s is the flux density and λ is the wavelength. The power δPr received by an antenna with linear gain Gr can then be expressed as ηδVGr. If the attenuation along the path is A, then the total power received at the receiving station Rx is
on transforming to polar coordinates, with the receiving station at the origin and with the angles of the receiving antenna θ, ϕ defined as in Figure 1.
 For high-gain antennas, as in the case of an Earth station, it is convenient to use the so-called “narrow beam” (or “pencil beam”) approximation, in which the antenna beam width is much narrower at the common volume than the beam width of a lower-gain antenna that would be deployed at a terrestrial station, and the volume illuminated by the Earth station is correspondingly much smaller than the volume illuminated by the terrestrial station, since this is generally at a greater distance from the beam intersection. In this approximation, the spreading loss from the scattering volume to the high-gain antenna is cancelled by the antenna gain. The dimension of the common volume in the direction of the Earth station antenna along rr is in this case much greater than in the plane perpendicular to rr. Then, to a first approximation, Gt, η, A and rt are all independent of θ and ϕ, and, considering the Earth station (with the high-gain antenna) to be the receiver, the received power simplifies to
 As an approximation, the receiving station antenna gain pattern can be represented by a Gaussian beam shape, that is,
where θ3 dB is the half power (−3 dB) beam width, and since
the received power can thus be written as
 Now, from Figure 1, drr = dψ, where α is the scattering angle, and the received power becomes
 This is the basic form of the bistatic radar equation for scattering between a terrestrial station with a wide-beam (lower gain) antenna and an Earth station with a narrow-beam (higher gain) antenna. Each station can be either transmitting or receiving. It is relevant to note that this method does not apply to the bidirectional case, where both transmitting and receiving stations have high-gain antennas.
 Considering scattering by raindrops, the scattering cross section, η in m2/m3, is described in terms of the bistatic radar reflectivity, Z, as follows:
where m is the complex permittivity of water, and the wavelength λ is in meters. Customarily, the radar reflectivity Z is given in dB relative to 1 mm6/m3. The term M is a polarization decoupling factor, included to allow for the fact that rain scatter is anisotropic and that the polarization of the scattered signal may differ from the polarization of the receiving station antenna.
 The integration in equation (6) is carried out over all angles between the boresight of the terrestrial station antenna and the common volume. It is more convenient, however, to integrate over the height h of the beam intersection within the rain cell, and to consider only scattering in the vertical plane, that is, forward or back scattering only. So, if dl is the incremental length within the common volume, along the boresight of the Earth station antenna, then dl can be expressed as
where ɛ is the elevation angle of the Earth station antenna boresight.
 Assuming that the radar reflectivity term depends on height, that is, Z(h) = ZRF(h), where ZR is the reflectivity due to rain at the ground, the bistatic radar equation can be written as
where C is now the effective scatter transfer function, and includes all dependencies on height:
 The Earth station antenna gain pattern is assumed to be Gaussian, and thus the product Grmaxθ3dB2 is a constant, = 4 ln 2 · ηe, where ηe is the antenna efficiency. Finally, the permittivity term, for water at frequencies near 10 GHz, is ∣∣2 ≈ 0.93. Collecting together all the constants, the bistatic radar equation simplifies to
where f = c/λ is the frequency and c is the speed of light, c = 2.9979 · 108 m/s. Expressing the range rt and the height element dh, in km, the frequency in GHz, the constant K becomes
 Thus, assuming a value of ηe = 0.6 for the antenna efficiency, the bistatic radar equation can be written in terms of the transmission loss in dB as
where r is now the horizontal distance from the terrestrial station to the rain cell, which is approximately equal to the distance along the ray path to the rain cell, for low elevation angles.
3. Rain Scatter Loss Model
 The bistatic radar equation is the basis for the model for rain scatter loss between a terrestrial station and an Earth station, from which the coordination distance may be estimated. Several further approximations and additional terms are included, for completeness, in order fully to take account of all losses along the paths and because of various system aspects which may be of relevance. First, the radar reflectivity at the ground due to rain, ZR, is determined from the rainfall rate, R in mm/h, through the relation
 It is assumed that the gain of the terrestrial antenna is constant throughout the rain cell, and is hence removed from the integral in equation (10). Additional terms are included to account for the fact that, at frequencies above about 10 GHz, Rayleigh scattering no longer applies and the scattering deviates from being isotropic, while the effects of gaseous attenuation are additionally included.
 This results in the following expression for the loss due to rain scatter between a terrestrial station and an Earth station:
range from the terrestrial station to the scattering volume, associated with the location of the rain cell;
frequency, in GHz;
rainfall rate, in mm/h;
scatter transfer function;
gain of the terrestrial station, in dB (for the purpose of determining the coordination distance, a value of GT = 42 dB is generally assumed);
efficiency of the antenna (typically, a value of ηe = 0.6 is assumed);
additional attenuation along the transmission path due to atmospheric gases, in dB;
factor to account for the deviation from Rayleigh scattering at frequencies above 10 GHz, given by
where αT is the main beam azimuth, measured at the scattering volume (αT = 0° for forward scatter, αT = 180° for back scatter). These terms are now considered in the following sections.
4. Effective Scatter Transfer Function C
4.1. Basic Geometry of Rain Scatter
 The geometry of scattering between a transmitting station with elevation angle ϕ and a receiving station with elevation angle ɛ is illustrated schematically in Figure 2.
 The point of intersection between the two main beams of the transmitting and receiving stations is represented by the point D, and the distance between the transmitting station and the surface beneath this intersection is given by the arc AC = r. The angular separation between the transmitting station and this point C is defined by δ:
where rE is the effective radius of the Earth, rE = 8500 km.
 Then, from the sine rule, the height of the beam intersection, hm, is given by
 The distances from the transmitting station to the point of intersection, AD, and the point of intersection to the receiving station, DB, are determined from the cosine rule:
 Finally, the distance along the surface from the point C to the receiving station is given by
 For the case of a terrestrial station interfering with an Earth station, as required for the rain scatter interference model in Recommendation ITU-R P.620 [ITU, 1999, 2002], the terrestrial station antenna elevation angle is defined as ϕ = 0°, assuming that the terrestrial station in this case is the transmitter. The height of the beam intersection then becomes
while the distance from the terrestrial station to the beam intersection simplifies to
 This defines the basic geometry of the rain scatter process. It is pertinent to note that the same formulation applies when the Earth station is the transmitter, interfering with a terrestrial receiving station.
4.2. Hydrometeor Scatter Geometry Between a Terrestrial Station and an Earth Station
 In order to describe the interference introduced by scattering from rain between the beams of a terrestrial station, with an antenna elevation angle of 0°, and an Earth station with an elevation angle of ɛ, it is convenient to consider the edge of the rain cell aligned with the intersection of the beams. The geometry of this case is shown in Figure 3.
 The rain cell has a diameter dependent on the rainfall rate, given by
 The effective scatter transfer function is given in equation (10) in terms of the gain of the terrestrial antenna as a function of height within the rain cell, together with the attenuation along the paths within the rain cell and the height variation of the scattering cross section. Some simple approximations are made, in order to derive an analytic expression suitable for application in Recommendation P.620 [ITU, 1999, 2002]:
 1. The gain of the terrestrial antenna is assumed to be constant throughout the rain cell, and can thus be removed from the integral.
 2. The scattering cross section, for scattering in rain, is similarly assumed to be constant throughout the rain cell, for heights below the rain height.
 3. Above the rain height, the scattering cross section is assumed to decay by −6.5 dB/km.
 With these approximations, the scatter transfer function reduces to an integral of the attenuation through the rain cell, between the limits defined by the intersection of the Earth station antenna main beam axis with the edges of the rain cell:
 The attenuation along the Earth station main beam axis will depend on whether the scattering element is below or above the rain height, and two regions are defined, for scattering by rain below the rain height and scattering by the melting layer above the rain height:
where Cb,a are the scatter transfer functions for scattering below and above the rain height:
where γR = kRα is the specific attenuation due to rain, in dB/km, and the constant 0.23 is required to change the units of attenuation from dB to nepers. The frequency- and polarization-dependent coefficients k and α are given in Recommendation ITU-R P.838 [ITU, 2003b]. The path length within the rain cell is given by y(h).
 Γb,a represents the attenuations of the incident and scattered signals by rain outside the common volume, for scattering below and above the rain height, respectively.
 The limit at which the integration changes between scattering from rain below the rain height and scattering from the melting layer above the rain height is determined by a parameter hc, which depends on the position of the beam axis intersections with the edges of the rain cell relative to the rain height hR, and is given by
 The path length within the rain cell, y(h), includes two paths, shown in Figure 3: from the edge of the rain cell at point A to the integration element dx at point B, and from point B along a horizontal path to point C for the case of forward scatter or to point D for the case of back scatter. The distance AB is given by , while the distance BD is , and the distance BC is (d − BD). It can be shown that the backscatter geometry exhibits the lower transmission loss, and hence represents the worst case scenario. In this case, the path length through the rain cell, for any arbitrary point dx, at a height h above ground, is given by
The scatter transfer function can then be readily integrated symbolically:
 In the previous model for rain scatter in Recommendation P.620-4 [ITU, 1999], a constant elevation angle of 20° was assumed for the Earth station, and the integration carried out over the entire Earth station main beam path within the rain cell, that is, from hm to hm + dc tan 20°:
where ds = ≈ 3.5R−0.08. This is essentially the expression which was used in Recommendation P.620-4 [ITU, 1999] (and is still employed in both Recommendation SM.1448 [ITU, 2000] and Appendix 7 of the Radio Regulations), where the additional multiplicative term ds is incorporated into the 10logZ term in equation (12).
 The scatter transfer function for scattering above the rain height, equation (27), can similarly be solved analytically, yielding the following expression:
 This term was originally neglected, as a conservative assumption, in the development of the rain scatter model for Recommendation P.620 [see ITU, 1986]. However, an additional term E was later included to account empirically for scattering above the rain height, on the basis of the decrease in attenuation with height above the melting layer of −6.5 dB/km (note that this term of −6.5 appears in equation (32) multiplied by 0.23, to convert dB to nepers). The resultant correction for coupling loss due to scattering from the melting layer was given, in dB, by
 It is instructive to examine the interrelationship between the two components in the scatter transfer function, Cb and Ca. These are illustrated in Figures 4a and 4b as a function of range, equivalent to the height within the rain cell, for two frequencies and two values of the rain height. The contribution from scattering from rain within the rain cell remains constant as the height of the scattering volume increases within the rain cell, until the scattering volume moves through the rain height, and decreases to zero at heights above hR.
 While the scattering volume is fully contained within the rain cell, the contribution from scattering above the rain height (from the “melting” layer), Ca, is zero, but this increases to a maximum when the scattering volume is centered at the rain height, decreasing above this at a rate of −6.5 dB/km. The range at which this occurs depends, of course, on the value of hR. In the examples shown in Figure 4, the range at which the height on the scattering volume, hm, becomes equal to the rain height, hR, is about 185 km for hR = 2 km and about 290 km for hR = 5 km. For distances greater than these ranges, the scatter transfer function for scattering below the rain height, Cb, becomes zero. It is also clear that scattering from the melting layer is independent of frequency, whereas the coupling loss, in dB, due to scattering from rain increases as a function of frequency.
 In the model described above, the rain is assumed to be confined to a cell with diameter dc, dependent on the rain rate, and the common volume between the two antennas is similarly assumed to be contained within this cell. However, in practice, rain will extend well beyond the common volume, introducing additional signal attenuation, and the terms Γb and Γa provide an estimate of this additional attenuation for scattering below and above the rain height, respectively. These terms each comprise two components—for the horizontal component of the rain attenuation outside the common volume toward the Earth station, Γ1, and for the (near-horizontal path) rain attenuation outside the common volume toward the terrestrial station, Γ2. A simple model is assumed in which the rain intensity outside the common volume decreases exponentially with distance, with a scaling distance rm defined by
 This scaling distance is that distance which, when multiplied by the specific rain attenuation within the cell, γR = kRα, is assumed to account for all the rain attenuation outside the rain cell.
 The horizontal path attenuation outside the common volume toward the Earth station is then scaled by the horizontal distance between the rain cell and the Earth station, taking account of an appropriate reduction in attenuation when the beam intersection is above the rain height, that is, when part of the path is above the rain and hence not attenuated:
while the attenuation outside the common volume toward the terrestrial station is similarly scaled by the distance from the rain cell to that station:
 The overall path attenuation for scattering below the rain height is then given by
while the overall path attenuation for scattering above the rain height is determined by
 For comparison, these attenuation factors were much simplified for the previous rain scatter model in Recommendation P.620-4 [ITU, 1999], with the attenuation for scattering above the rain height being neglected and the remaining attenuation reducing to a term dependent only on rain rate and frequency:
5. Gaseous Attenuation
 In equation (12), a term is included to account for the attenuation due to atmospheric gases along the paths from the terrestrial station to the beam intersection point and from the beam intersection point to the Earth station. Simple expressions are included in Recommendation P.620 for the specific attenuation, in dB/km, due to dry air and to water vapor:
with the water vapor attenuation being evaluated, for all locations, at a water vapor density of ρ = 7.5 g/m3.
 These specific attenuations are multiplied by distances which are intended to reflect the fact that, as the altitude increases, and hence the molecular density and atmospheric pressure decrease, the specific attenuation decreases. The effective path length by which the specific attenuation is multiplied will therefore be shorter than the actual path length. Recommendation P.620-4 [ITU, 1999] included the following simple expressions for the effective path length for dry air and water vapor attenuation, respectively:
 These effective path lengths have been re-examined in the new model presented here, using calculations of gaseous attenuation based on the full line-by-line procedure in Recommendation ITU-R P.676-5 [ITU, 2001a], and using the standard reference atmosphere from Recommendation ITU-R P.835-3 [ITU, 2001b]. The calculations, taking account of refractive ray bending through the atmosphere, were carried out using an effective Earth radius of 8500 km, with the attenuation being determined for each layer of the atmosphere, starting with the thinnest, at ground level, of thickness 10 cm, and increasing exponentially up to an altitude of 20 km. Summing the contributions for each layer yields the total path attenuation as a function of path length, and dividing this attenuation by the specific attenuation at ground level gives a value for the effective path length, by which the specific attenuation must be multiplied to obtain the path attenuation. Using this procedure, a re-evaluation of the algorithms for the effective path lengths has been carried out, resulting in the following expressions for dry air and water vapor attenuation:
 These new algorithms are shown in Figure 5 to yield a closer approximation to the effective path lengths calculated at 20 GHz, and demonstrate an improvement over the previous model in Recommendation P.620-4 [ITU, 1999].
 In the previous model, gaseous attenuation was included only for the path from the terrestrial station to the rain cell, and no account was taken of the gaseous attenuation along the path from the rain cell to the Earth station. Although this is generally likely only to be of any significance at the higher frequencies, where gaseous attenuation becomes greater, this is included in the model described here, for completeness. Following the same procedure as above, the path attenuation has been calculated using Annex 1 of Recommendation P.676-5 [ITU, 2001a], for elevation angles from 10° to 90°, and the resultant effective path lengths are shown in Figures 6a and 6b for dry air and water vapor attenuation, respectively.
 The rapid decrease in water vapor attenuation along the path, due to the decrease in water vapor density, is clearly apparent. For simplicity, however, a simple linear model is derived, with the following form for the effective path lengths for gaseous attenuation along the path to the Earth station:
 The gaseous attenuation is then given by
6. Deviation From Rayleigh Scattering
 Rayleigh scattering is isotropic at low frequencies for scattering from raindrops and assumed to be isotropic for scattering from the melting layer, above the rain height, at all frequencies. At frequencies above about 10 GHz, however, scattering from raindrops starts to deviate from Rayleigh scattering. The deviation depends, inter alia, on the scattering angle, on the polarizations of incident and scattered waves and on the shape of the scattering particles, and is generally not simple to calculate. As an approximation, however, the deviation can be taken into account to first order by the correction factor S, defined in equation (15). As in the previous model in Recommendation P.620-4 [ITU, 1999], this includes the simplification that only the scattering angle in the horizontal plane is considered, and not the angle between the two antenna beams. For the purpose of coordination, it is appropriate to consider worst case conditions, and hence the approximation is also made to consider only the case of backscattering in the development of the scatter transfer function. Similarly, only the backscatter component of the deviation from Rayleigh scattering is included.
 Since this deviation is applicable only to scattering from rain, that is, below the rain height, it is not appropriate to include the correction when the common volume between the two antenna beams is above the rain height. This condition is found when the scatter transfer function for scattering below the rain height has a value of zero.
 With these simplifications, equation (15) reduces to the following expression for the deviation from Rayleigh scattering for frequencies above 10 GHz:
Note that, in the previous model in Recommendation P.620-4 [ITU, 1999], the condition when deviation from Rayleigh scattering is not to be included is determined from the value of the correction term E for scattering by the melting layer, that is, that 10logS (denoted as 10logAb in Recommendation P.620-4 [ITU, 1999]) is zero when E ≠ 0. As shown in section 4.2, the contribution from scattering by the melting layer extends over a range of heights around the rain height, and it is considered more appropriate to base the test on the absence of any contribution from rain scattering, that is, when Cb = 0.
7. Establishment of Coordination Distance and Coordination Area
 The coordination distance is defined as the distance from the Earth station at which the transmission loss, calculated for a given percentage of time, is equal to the minimum permissible basic transmission loss, which depends on system performance and availability requirements. The process is an iterative technique, in which the distance from a hypothetical terrestrial station to the rain cell is decreased, usually in steps of 1 km, starting at the maximum coordination distance (section 9 gives an explanation of why the iteration is in this direction), until the transmission loss is equal to or just less than the minimum permissible basic transmission loss. This distance, known as the calculation distance, is then given by the distance at the step immediately preceding this one.
 The calculation distance thus defines a circle around the location of the rain cell, beyond which there is a negligible likelihood of harmful interference. The location of the center of this circle is then at the distance de, given by equation (20), from the Earth station location along the azimuthal direction of the Earth station antenna, and the coordination distance and coordination area are then fully defined.
8. Meteorological Input Parameters
 Recommendation P.620-4 [ITU, 1999] retained the classification of rainfall climates into rain zones, concatenating the 15 original zones in Recommendation P.837-1 [ITU, 1995] into a reduced set of five rain climates, with algorithms to provide estimates of rainfall rates as a function of the percentage of time exceeded, that is, that percentage of time for which the coordination distance is required. These rain zones have been superseded in Recommendation P.837-3 [ITU, 2001d] by digital world maps of three parameters from which the cumulative distributions of rainfall rates may be constructed, for any latitude and longitude. These maps have a resolution of 1.5° in latitude and longitude, and the rainfall rate at any latitude and longitude may be obtained using a bilinear interpolation from the four closest grid points surrounding the desired location.
 The new model is based on rainfall rates derived from the digital world maps for the required percentage of time and for the latitude and longitude of the Earth station. Note that in the model, the rain cell is actually located at some distance along the main beam axis of the Earth station antenna, at the intersection with the horizon ray from a distant terrestrial station. It is appropriate to consider whether the rainfall rate may differ significantly between the location of the Earth station and the possible location of a rain cell, especially for very low elevation systems, where the rain cell may be located at some distance from the Earth station. The rainfall maps in Recommendation P.837-4 [ITU, 2003a] have been examined in terms of the rain rate gradient, by deriving the percentage change in rain rate over a 1° north-south interval, which corresponds to a great circle distance of ∼111 km. The most significant gradients occur at latitudes below about ±30°, where very low elevation angles are not likely to be utilized, and the displacement of the rain cell from the station will generally be only a few km. The use of rainfall rates at the Earth station location will then be both realistic and justifiable.
 Above 50°N in the northern hemisphere, the highest gradient occurs at 69°N, 33°E, with a change in rainfall rate from 14.5 mm/h at 69°N to 45.9 mm/h at 68°N. For a station operating at a frequency of 3.775 GHz and an elevation angle of 0.5°, with a required transmission loss of 158.9 dB, the coordination distance at 69°N is 266 km, while at 68°N the coordination distance becomes 269 km. If the elevation angle is reduced further to 0.1°, the coordination distances are reduced by 1 km at each location. For comparison, the coordination distance resulting from Recommendation P.620-4 ITU  would be 293 km. Thus use of the rainfall rate at the Earth station location is similarly realistic and justified. In the southern hemisphere, the change in rainfall rate over a 1° change in latitude can approach 100% at latitudes higher than −60°. However, the only landmass at such latitudes is Antarctica, where the need for coordination may not arise.
 Recommendation P.620-4 [ITU, 1999] employed a simple model for the rain height, hR, based on latitude, derived from Recommendation P.839-1 [ITU, 1997]. This model has now been replaced, in Recommendation P.839-3 [ITU, 2001e], with digital world maps of the 0°C isotherm, to which is added a height of 0.36 km, to derive estimates of the rain height at any latitude and longitude. The maps have the same resolution of 1.5° in latitude and longitude as those for rainfall rate. The new rain scatter model described here is based on deriving the rain height from these digital maps.
 The remaining meteorological parameter in the previous rain scatter model in Recommendation P.620-4 [ITU, 1999] was the surface water vapor density, which is used in estimating the gaseous attenuation along the transmission path. In the previous model, a fixed value of 7.5 g/m3 was assumed for all regions. WRC 2000 noted in Resolution 74 that the ITU-R is addressing this issue, and new data on the distribution of surface water vapor densities are now available in Recommendation P.836-3 [ITU, 2001c], in the form of world maps, with the same resolution as those maps already noted for rainfall rate and rain height, of the annual values of surface water vapor densities exceeded for various percentages of time, from 1% to 50%. As an average value to be used in estimating the gaseous attenuation along the transmission path, the new model here uses the annual median surface water vapor density, that is, that derived from the maps for 50% exceedence.
9. Comparison of Transmission Loss Predictions
 The principal difference between the new model in Recommendation P.620-5 [ITU, 2002] and the previous model in P.620-4 [ITU, 1999] is in the inclusion of the Earth station elevation angle as an input parameter, and Figures 7a and 7b illustrate some examples of this, for two different ranges, 100 km and 300 km, representing cases where scattering is from (1) below the rain height and (2) above the rain height. Two different behavioral patterns are discernible. For scattering below the rain height, the transmission loss decreases, that is, the coupling increases, as the Earth station elevation angle increases. The reason for this is that, at increasing elevation angles, more of the Earth station beam is enclosed within the rain cell, resulting in increased coupling. Above the rain height, however, the coupling decreases at a rate of −6.5 dB/km, resulting in an increase in transmission loss with increasing elevation angle.
Figures 8a and 8b show the transmission losses at three frequencies for ranges of 100 km and 300 km, respectively, as a function of rainfall rate, for a rain height of hR = 3 km. At 100 km, where the scattering is entirely within the rain cell, there is very little difference between the new model in P.620-5 [ITU, 2002] and the previous model in P.620-4 [ITU, 1999], except perhaps at higher frequencies. Here, the more accurate expressions in the new model for rain losses outside the rain cell, and for gaseous attenuation, are probably having an effect. At 300 km range, where scattering is above the rain height, there are significant differences between the two models, with the new model indicating much higher losses at the lower rainfall rates. The reason for this may lie in the fact that, in the previous model in P.620-4 [ITU, 1999], the rain scatter term is used irrespective of whether scattering is below or above the rain height, and a correction term is added—the term E—to account empirically for the melting-layer coupling. In the new model, the rain scatter term only contributes when the scattering volume is below the rain height.
 The dependence of transmission loss on rainfall rate at high elevation angles is further illustrated in Figures 9a and 9b, for an elevation angle of 70°. Similar behavior at long range is apparent to that observed for scattering above the rain height at low elevation angles, whereas at the shorter range and higher elevation angle, the transmission loss becomes less dependent on the rainfall rate, due perhaps to the compensating effects of longer path lengths through the rain cell and higher rain attenuations.
 The variation of transmission loss with frequency is shown in Figures 10a and 10b for two elevation angles, 20° and 70°, respectively, and for three different ranges from 100 km to 300 km. Again, the new model generally indicates higher transmission losses than Recommendation P.620-4 [ITU, 1999], in accordance with the overall conclusions from the analyses above.
 Finally, the variation of predicted transmission loss with range is illustrated in Figures 11a–11d, for four frequencies.
 The increase in coupling which occurs when the scattering volume coincides with the rain height becomes increasingly apparent as the frequency increases. Although these calculations were all carried out for a rain height of hR = 3 km, it is also clear that the simple empirical algorithm for scattering from the melting layer included in P.620-4 (the parameter E) [ITU, 1999], does not come into effect at the correct range, when compared with the new model described here, in which the increase in coupling derives from an analytical solution to the integration of path lengths throughout the common volume.
 For many systems, the required transmission loss for a given system performance and availability criterion, will be approximately in the range of ∼ 160–170 dB. From the results presented in Figure 11, it is clear that the dominant effect which will control the coordination distance will be scattering from above the rain height, rather than scattering from below. It is therefore important that this scattering be described as carefully as possible, and not by an empirically derived correction term.
 Because of the increase in coupling in the vicinity of the rain height, the transmission loss does not increase monotonically with distance, especially at the higher frequencies. It therefore becomes necessary to evaluate the coordination distance by iterating down from the maximum coordination distance, until the distance is reached where the predicted transmission loss is just less than the required transmission loss. The coordination distance is then given by the iteration step immediately before this one.
10. Comparison of Coordination Distances
 Extensive comparisons have been carried out by the ITU Radiocommunications Bureau on the coordination distances which result from the new model, when used in Recommendation ITU-R SM.1448 [ITU, 2000], on which Appendix 7 of the Radio Regulations is based. The results were determined using the Bureau's reference Earth stations, with coordination distances calculated at 72 azimuths around 1079 transmitting Earth stations and 591 receiving Earth stations using the method in Recommendation ITU-R SM.1448 [ITU, 2000], replacing the propagation model in the R1448 software by the new model presented here. All other elements of the software such as system parameters and the correction (mitigation) factors were left unchanged. Most of the Earth stations were at latitudes between 30°N and 55°N, and at elevation angles between 20° and 45°.
Figure 12 compares the coordination distances determined using the new model with those from the current model in SM.1448 [ITU, 2000]. The mean value of the 78,984 coordination distances calculated for the transmitting stations was 113 km with both SM.1448 [ITU, 2000] and P.620-5 [ITU, 2002], while for the 42,552 coordination distances for the receiving stations the mean values were 323 km with SM.1448 and 325 km with P.620-5.
 The changes in coordination distances resulting from application of the new model have been further analyzed in terms of latitude, frequency and elevation angle. Figure 13 shows the changes resulting from the new model as a function of latitude. In general, at the lower latitudes there are more stations which result in smaller coordination distances, while at higher latitudes, there are significant numbers of stations which will have larger coordination distances.
 The changes in coordination distances with frequency are shown in Figure 14, suggesting that there is little dependence on frequency, while Figure 15 shows the changes in coordination distance with the elevation angle of the Earth station. Here, there is clearly some evidence that elevation angle plays a role in determining coordination distance. The current model in SM.1448 [ITU, 2000] is based on an Earth station elevation angle of 20°. At higher elevations, the new model shows a significant reduction in coordination distance, while at elevation angles lower than 20°, generally longer distances result.
 This accords with the overall changes with latitude shown in Figure 13, since the higher elevation angles will generally occur at lower latitudes, and indicates that the new method is more appropriate for determining coordination distances for Earth stations located at the lower latitudes.
 A new model for rain scatter interference has been developed, and is now included in Recommendation ITU-R P.620-5, for the determination of coordination distances around an Earth station, in which the elevation angle of the Earth station is taken into account. The new method also includes the updated world maps of rainfall rates contained in Recommendation P.837-4 [ITU, 2003a] and of surface water vapor density in Recommendation P.836-3 [ITU, 2001c]. The model is more realistic than the previous version in describing the rain scatter process, offering a greater degree of flexibility in application and is found to be essentially neutral in its effect on coordination distance.
 The author is extremely grateful to Peter Lundborg of the ITU Radiocommunications Bureau for making available the results of extensive testing of the new method for mode (2) coordination using the ITU-R database of Earth stations and for his helpful suggestions in both its application to the coordination process and in the preparation of this manuscript. The author is also pleased to acknowledge the support for this work by the Radiocommunications Agency, UK, and the helpful comments of the reviewers.