This chapter discusses ionospheric modes. Both single-hop and multiple-hop echoes from the ionosphere are frequently seen in records made with vertical-incidence sounders (ionosondes) radiating short pulses. It is convenient to regard the region of the ionosphere in which reflection occurs as a well-marked layer; that is to say, the D-, E-, and F-regions of ionospheric physics tend to be visualised as discrete layers in ray-tracing applications. The surface of the earth and the reflecting layer act very much like a pair of parallel mirrors in optics, and the separate returns correspond to multiple images of the transmitter.

In this chapter we shall consider the 'bearing' defined by the normal to a curve of constant phase (CP) at a particular point in a two-ray wave-field. It must be emphasised that 'bearing' is thus being given a very special meaning; it is a geometrical property of the wave-field. In effect, we consider the bearing indicated by a phase-sensitive DF as its aperture shrinks to zero. Because this limiting case is an abstraction, we can continue to postpone any consideration of the behaviour of particular types of instrument; we are still dealing with the inherent properties of the field. Nevertheless, a number of general results can be derived that provide a useful insight into more practical situations.

Although this book is not primarily concerned with DF hardware, it is convenient at this point to summarise the physical sources of measurement errors arising from instrumental and site effects. Re-radiating objects in the vicinity of the receiving site are of particular interest here, because the rays from such objects add to the complexity of the wave-field to be resolved. The sources of error in DF can be broadly divided into four categories: (a) Instrumental errors, which can cause incorrect measurements in even the simplest situation, namely a single plane wavefront arriving at the DF array, (b) Site errors, i.e. distortions and deviations of the arriving wave-fronts caused by features in the neighbourhood of the receiving site such as re-radiating conductors, ground contours and gradients of ground conductivity, (c) Wave-interference errors, when the arrival of two or more rays produces swings in the indicated bearing. (d) Propagation errors, i.e. changes in ray direction produced by ionospheric tilts or magneto-ionic effects.

We now come to the question of how much can be deduced about the structure of a wave-field from an analysis of wave interference effects. The observations might consist of the fluctuations of the indicated bearing of a WADF as a function of time (this Chapter) or a sequence of phase difference measurements on various pairs of elements in an interferometer (Chapter 9). Not surprisingly, the amount of information that can be derived with an orthodox DF, wbich provides only one indicated bearing at any one moment, proves to be very limited. In the numerical simulations for this type of problem, therefore, attention will be confined to two-ray wave fields and the basic question is whether the ray azimuths can be found. In order to distinguish between the zero-aperture bearing defined in Chapter 4 and the indicated bearing from a DF of finite aperture, we shall refer to the former as the tangent bearing; it is the direction normal to the tangent to a curve of constant phase at a specified point in the horizontal plane and is a property of the wave-field. A phase-measuring DF of finite aperture, centred on the same point, would produce an approximation to the same bearing but would average in some way (to be investigated below) over a finite segment of the curve. Thus, the 'segment bearing' for a finite aperture approaches the tangent bearing as the aperture approaches zero.

Signal processing in DF has, until recently, been almost entirely of an analogue nature. The spinning goniometer usually used in a Wullenweber system is an example: the inputs to the goniometer are the signals from a selected arc of elements and the output is fed through a single-channel receiver which determines the narrow frequency band to be examined. The development of multichannel HF receivers in recent years has opened up the possibility of using instead some form of digital processing; the amplitudes and phases of the element signals at a selected frequency can be measured, recorded digitally and processed in a computer program. The experimental and mathematical techniques are referred to collectively as wavefront analysis (WFA). The aim is to resolve the wave-field into its component rays and to print out the ray parameters. Direct analytic or iterative attacks on the equations relating the ray parameters to the measured signals become very unattractive when more than a small number of rays are present. However, a general method of solution, applicable to almost any form of antenna array, can be used with fading signals (Gething, 1971). In this chapter some simple examples of the technique will be given for idealised problems involving neither noise nor measurement errors, in order to explain the principles involved.

The word superresolution is now widely used in place of the term wavefront analysis and the phrase 'the resolution of multicomponent wave fields'. It is used to describe techniques of signal processing that can be applied in many different fields such as radar, sonar, radio astronomy, seismology and spectral analysis, as well as radio direction finding. Some hundreds of papers on superresolution algorithms designed to determine the angular spectrum of the arriving rays have appeared in the literature over the last ten years, many based on concepts described in earlier chapters. In this chapter we describe the main categories of algorithms and their performance against simulated problems. Further developments and refinements of the processing techniques are discussed in Chapter 13. The aim in both chapters is to describe and interpret the important results rather than to reproduce mathematical detail from the research papers.

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.

Azimuth measurements taken on transmitters of known position, i.e. check targets, can be compared with the 'ideal' answer, namely the true g.c. bearing, so that accuracy statistics can be compiled for various purposes described in Section 16.1.2 below. The same procedure cannot be used for measurements of elevation angles, where the 'true' answer is generally unknown. We shall therefore confine our attention in this chapter to azimuth measurements, which will now be referred to for brevity as bearings. It should be noted that bearing errors can, and do, arise from propagation effects as well as from instrumental effects, and that we are not directly concerned here with the physical causes of instrumental errors, which were discussed in Chapter 6.

This appendix gives the parameters used in pattern calculations for circular wide aperture direction finders (WADF). It list these under the headings of: Array dimensions and Goniometer, and lists some simplifying assumptions.

This appendix gives the parameters used in pattern calculations for vertical array of loops.

The covariance matrix of sensor outputs are discussed. Matrix notation, matrix pencil, projection, data model, noise model, asymptotic form, and Toeplitz matrix are also discussed.

This appendix discusses the equations of path integrals for parabolic layers.

We suppose that a Best Point Estimate (BPE) has been calculated from n bearing lines by means of the standard Stansfield theory. The position of the BPE relative to an origin at the (unknown) true position of the transmitter is (X, Y), where X = -(G/A), Y = -(F/B).