27.9 is a diagram- their inherent complexity. Such programmes run on desktop PC matical representation of the surface of the earth from which the computers (Hitney, 1990) although many research establishments optimum take off angle of radiation from the antenna can be derived have larger and more sophisticated programmers (Dick, 1990). if the distance and refracting layer height are known. A lower take Figure 27.11 shows a typical printout of a link from London to off angle is required for longer distances and for distance of greater than 2000km, two or more hops will normally be required. In practice, a number of different propagation modes may be established on multi-hop links, which makes field strength predic- tions for the receive site difficult and uncertain. Three possible modes are shown in Figure 27.10. A first approximation can be F2 LAYER DOUBLE HOP F2 SINGLE HOP F2 E LAYER DOUBLE HOP E TRANSMITTER RECEIVER Time (hours) Figure 27.10 Possible multi-hop sky wave paths Figure 27.11 Frequencies for an h.f. link from London to Lisbon obtained by calculating the free space loss using Equation 27.7, taking into account increased distance due to ionospheric refraction and applying this to the field strength at the transmitter site. Addi- Lisbon, a great circle distance of 1656 kilometres and a single hop tional losses of up to 20dB will result from ionospheric and ground F2 distance of 1757 kilometres. The programme shows the maxi- reflections, but these are very approximate values. mum usable frequency (MUF) and the lowest usable frequency Actual sky wave refraction losses can be calculated (Braun, (LUF), and at times when the LUF exceeds the MUF, h.f. skywave 1986), as can received field strength (Damboldt, 1975). Computer communication is not possible. The MUF is determined by the propagation prediction programmes are now generally employed degree of layer ionisation whilst the LUF is generally determined for analysing h.f. paths and predicting received signal strength by the multi-hop path attenuation and 