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## Ray paths

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Suppose that the DOAs of several rays from the target transmitter have been determined. Suppose further that the state of the ionosphere in the general area between the transmitter and the DF is known in some detail. Then it is possible in principle to reverse the ray directions at the point of reception and re-trace the ray paths back to the transmitter, using a suitable computer program to find the paths in the ionosphere. Ideally, all ray paths would ultimately be found to intersect at one point on the earth's surface, possibly after multiple hops. Not only would the DOAs at the DF site be correctly explained; the relative time delays between different paths would also be correctly predicted. That is to say, the versions of a given signal element received over various paths should trace back to a common time-origin as well as a common space-origin. In practice, of course, exact intersections in time and space are not obtained because of measurement errors and of incomplete or imperfect knowledge about the state of the ionosphere. However, the objective of ray re-tracing procedures is to make the best possible use of the available DOAs and ionospheric data. Note that an estimate of the position of the transmitter is now obtained from a single DF site; in effect, a range is determined from elevation angles and ionospheric data, while a direction is determined from the measured azimuth angle or angles. The aim in this chapter is to summarise those parts of ray-tracing theory that are relevant to DF. For further details, textbooks such as those by Budden (1985) and Kelso (1964) should be consulted.

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