3 Diffraction over a spherical Earth The additional transmission loss due to diffraction over a spherical Earth can be computed by the classical residue series formula. A computer program GRWAVE, available from the ITU, provides the complete method. A subset of the outputs from this program (for antennas close to the ground and at lower frequencies) is presented in Recommendation ITU-R P.368. The following subsections describe numerical and nomogram methods which may be used for frequencies 10 MHz and above. For frequencies below 10 MHz, GRWAVE should always be used. Section 3.1 gives methods for over-the-horizon paths. Section 3.1.1 is a numerical method. Section 3.1.2 is a nomogram method. Section 3.2 is a method applicable for the smooth earth case for any distance and for frequencies 10 MHz and above. This utilizes the numerical method in § 3.1.1. 3.1 Diffraction loss for over-the-horizon paths At long distances over the horizon, only the term of the residue series is important. Even near or at the horizon this approximation can be used with a maximum error around 2 dB in most cases. This first term can be written as the product of a distance term, F, and two height gain terms, GT and GR. Sections 3.1.1 and § 3.1.2 describe how these terms can be obtained from simple formula or from nomograms. 3.1.1 Numerical calculation 3.1.1.1 Influence of the electrical characteristics of the surface of the Earth The extent to which the electrical characteristics of the surface of the Earth influence the diffraction loss can be determined by calculating a normalized factor for surface admittance, K, given by the formulae: in self-consistent units: for horizontal polarization (11) and for vertical polarization (12) or, in practical units: (11a) (12a) where: ae : effective radius of the Earth (km); ε : effective relative permittivity;  : effective conductivity (S/m); f : frequency (MHz). Typical values of K are shown in Fig 2.
(28) (29) where: h : height of the top of the obstacle above the straight line joining the two ends of the path. If the height is below this line, h is negative; d1 and d2 : distances of the two ends of the path from the top of the obstacle; d : length of the path; : angle of diffraction (rad); its sign is the same as that of h. The angle is assumed to be less than about 0.2 rad, or roughly 12º; 1 and 2 : angles in radians between the top of the obstacle and one end as seen from the other end. 1 and 2 are of the sign of h in the above equations. NOTE 1 – In equations (26) to (29) inclusive h, d, d1, d2 and should be in self-consistent units. FIGURE 8 Geometrical elements Figure 9 gives, as a function of , the loss J() (dB). J() is by: (30) where C() and S() are the real and imaginary parts respectively of the complex Fresnel integral F() defined in § 2.7. For greater than –0.78 an approximate value can be obtained from the expression: (31) FIGURE 9 Knife-edge diffraction loss 4.2 Single rounded obstacle The geometry of a rounded obstacle of radius R is illustrated in Fig 8c). Note that the distances d1 and d2, and the height h the baseline, are all measured to the vertex where the projected rays intersect above the obstacle. The diffraction loss for this geometry may be calculated as: (32)
– Set p = u + 1. – If both p<v and hp>hp1, then increase p by 1 and repeat. – Set q = v – 1. – If both q>u and hq>hq–1, then decrease q by 1 and repeat. If p = q then the sub-path obstruction loss is set to 0. Otherwise the calculation proceeds as follows. It is now necessary to find the minimum value of normalized clearance, CF, given by hz / F1, where in self-consistent units: hz: height of ray above profile point; F1: radius of Fresnel zone. The minimum normalized clearance may be written: (107) where: (hz)i = (hr)i – (ht)i (108) (109) (hr)i, the height of the ray above a straight line joining sea level at u and v at the i-th profile point is given by: (hr)i = (hu · div+hv · dui) / duv (110) (ht)i, the height of the terrain above a straight line joining sea level at u and v at the i-th profile point is given by: (ht)i =hi+dui · div / 2ae (111) The minimum value of normalized clearance is used to compute the knife-edge diffraction geometrical parameter for the most significant sub-path obstruction: (112) The sub-path diffraction loss L" is now obtained from equation (31) or Fig 9. For some applications it may be undesirable to include sub-path diffraction enhancements. In this case a value of L" should be set to zero when it would otherwise be negative. ________________