The mathematical problem of how to reconstruct a function from its projections was originally solved by Radon [1917], but the first practical application was not published until 1956, when the tomographic method was applied to radio astronomy [Bracewell (1956)]. Recent interest in tomographic imaging began with the invention of the X-ray computerized tomography scanner by Hounsfield in 1972. This original medical application, the CAT (computer aided tomography) scanner, took measurements of the attenuation of X-rays passing through a human body from many different angles. By converting these measurements directly into digital impulses and feeding them into a computer, a two-dimensional, cross-sectional image of the body was obtained. More recent developments in the medical field have seen the technique applied to nuclear medicine, magnetic resonance imaging, ultrasound and microwave imaging.

Put into simple terms, tomography is a technique for finding unknown numbers inside a grid. Imagine that you have a square grid divided into four sections each containing an unknown number (Fig. 2) and you are only allowed to know the sums of the numbers along certain paths. So take, for example, the case where the sums are those four shown by the thick grey lines in the figure. Now suppose that the sums of numbers along each of the four paths labelled a to c is equal to ten and d is equal to 5. Then you can easily set up four simultaneous equations and find a solution where each of the original numbers was equal to five. Then to check it — five plus five along each direction is equal to ten. This is the simplest case for tomography — the measurements (sums) have no error and you have all the measurements you need to find the solution.

In reality the problem is not quite so simple, although the underlying principles remain the same. The first difference is that you need to account for the length of each path through each section (pixel). In the case of Fig. 2, if each pixel has sides of unit length then the lengths of each path a, b and c through each pixel would be one, but ray'd' which would have length root 2. So in reality the measurement'd' would be root 2 times 5. The measurement 'a' would be two times one times five. The second difference is that sometimes the measurements cannot be taken from all angles. Returning to Fig. 2, if you only have measurements a and b, then you cannot discover unique values for all of the four unknown numbers. To overcome this problem you can try to obtain other information.

Fig. 2 Diagram showing a simple tomographic system with four unknown numbers (*) to be found from four measurements a, b, c and d.

For example, if you know that all of the numbers are equal and that the sum along path 'a' is ten, then it is clear that all of the numbers must be equal to 5. In the case of ionospheric imaging this is a big problem; how do you compensate for the limited measurements? The satellite-to-ground geometry omits rays passing horizontally through the ionosphere. Fortunately the missing information that is needed can be partly compensated for by bringing in some realistic assumptions about the distribution of electron density. Another problem in tomography is coping with systematic errors and noise on the measurements. In the ionospheric imaging case signals are often temporarily lost and regained causing discrete jumps in the measurement record. It is important to distinguish these jumps from real changes in the ionosphere so that actual ionospheric features can be imaged and not artificial features caused by these jumps in the phase records. A record of the changes in total electron content (TEC) in the ionosphere between a low-Earth-orbit satellite and ground-based receiver (the line integral of the electron density that we need to image) is shown in Fig. 3. Luckily this particular record had no phase jumps in it.

Geographic latitude of satellite (°N)

Fig. 3 Total electron content (X 1016m~2) record in January 1998 from a single receiver close to Rome in Italy. The undulations in the record are characteristic of ionospheric waves called travelling ionospheric disturbances.

Geographic latitude of satellite (°N)

Fig. 3 Total electron content (X 1016m~2) record in January 1998 from a single receiver close to Rome in Italy. The undulations in the record are characteristic of ionospheric waves called travelling ionospheric disturbances.

For ionospheric imaging, dual-frequency radio signals can be recorded by ground-based receivers to obtain relative phase shift and delay. The dispersive nature of the ionospheric component allows the ionospheric delay to be determined separately from effects caused by propagation through the non-ionised part of the atmosphere. This provides information that can be related directly to TEC (Fig. 3). The first experimental result showing a tomographic image of a slice of the ionosphere was published by Andreeva et al. [1990]. These authors, from the Moscow State University, used TEC data collected at three receivers located at Murmansk, Kem and Moscow. They made use of radio transmissions from Russian navigation satellites. For such preliminary tomographic results no other local measurements of the ionospheric electron density were available for comparison with the reconstruction and it was not until 1992 (Pryse and Kersley) that a tomographic image with independent verification was published. These images used data from the US Navy Navigation Satellite System (NNSS), the predecessor to the GPS (Global Positioning System). The verification for these early images over Scandinavia was provided by a scanning experiment of the European Incoherent Scatter (EISCAT) radar (Fig. 1). Subsequent co-ordinated studies between tomographic imaging and the EISCAT radar have contributed hugely to the general acceptance of the tomographic technique.

Many ionospheric features have been imaged using tomography. Beautiful images show snapshots of waves called travelling ionospheric disturbances (TIDs) [Pryse et al. (1995); Cook and Close (1995)]. These close relations of ocean waves are the manifestation in the ionosphere of internal atmospheric gravity waves. An example of an image showing a TID on Boxing Day of 1992 is shown in Fig. 4. Mitchell et al. [1995] have presented tomographic images of magnetic-field-aligned irregularities and E-region enhancements in the auroral ionosphere above northern Scandinavia. This region is particularly interesting to physicists because it is where particles from the Sun, having travelled though space in the solar wind, are able to enter the Earth's upper atmosphere. These high-speed particles whiz down the Earth's magnetic field lines like corkscrewing bullets, eventually colliding with atoms causing impact ionization and give up their energy in exchange for fantastic displays of the northern lights or aurora borealis.

Kersley et al. [1997] demonstrated that tomography could be used to make images of large-scale ionisation depletions known as troughs, generally found on the night-side auroral mid-latitude boundary. Results from the polar cap have revealed ionospheric signatures of processes occurring

40 45 SO SS 60 65

Fig. 4 Tomographic image of TIDs at 14:54 UT on 26 December, 1992. Contours show the electron density in units of X 1011m—3.

40 45 SO SS 60 65

LATITUDE (degrees)

Fig. 4 Tomographic image of TIDs at 14:54 UT on 26 December, 1992. Contours show the electron density in units of X 1011m—3.

further out in space, such as magnetic reconnection events [Walker et al. (1998)]. A novel idea by Bernhardt et al. [1996] proposed the inclusion of measurements taken from natural extreme ultraviolet emissions in the ionosphere into tomographic inversions. These satellite-based observations can provide vertical profiles of ionized oxygen, which are essentially the same as the electron-density profiles at F-layer heights. More recently, Materassi et al. [2001] have applied tomographic techniques to measurements recorded from southern Italy and the Mediterranean to image and study the large-scale enhancement known as the equatorial anomaly. These great lumps of ionisation are produced by the 'fountain effect' where the plasma rises like a fountain over the geomagnetic equator and falls along the magnetic field lines forming distinct peaks on either side. An image of the northern peak of this peculiar structure is shown in Fig. 5.

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