C. T Liu1, L. C Tsai2, 1, and W. H. Tsai 1
1 Institute of Space Science, National Central University (NCU), Chung-Li, Taiwan
2 Center for Space and Remote Sensing Research, NCU, Chung-Li, Taiwan
We have implemented the Multiplicative Algebraic Reconstruction Technique (MART) algorithm to the ionospheric electron density inversion from measured total electron contents (TECs) through radio observation of the Global Positioning System (GPS) signals and the Naval Navigation Satellite System (NNSS) transit signals to reconstruct two-dimensional ionospheric structures. We are also going to compare with the tomography results and show good agreement for both of the Global Positioning System / Meteorology (GPS/MET) and the Low-latitude Ionospheric Tomography Network (LITN) programs.
The mathematical justification for tomographic reconstructions have been deeply rooted in the Radon transform [Radon, 1917;Dean, 1983;Kak and Slaney, 1988]. In a tomographic application, projections in as many directions as practicable and feasible are usually measured. The Fourier slice theorem [Bracewell, 1956] tells us how the two dimensional Fourier transform space of the object function is being filled by each such measured projection. Only when the Fourier space is completely filled can one hope to find a unique object function. Combine the theory of image reconstruction and the computerized processing ability, we may get the high-resolution image. Methods of computerized ionospheric tomography (CIT) from satellite radio measurements have been under development in more than ten years. The earlier experiments were conduced by receiving satellite signals from ground-based stations. In June 1994, National Central University also built up the Low-altitude Ionospheric Tomography Network (LITN). Furthermore, a recent mission termed the Global Positioning System/Meteorology (GPS/MET) program used a low Earth orbiting (LEO) satellite (the MicroLab-1) to receive multi-channel GPS carrier phase signals (~1.5GHz and ~1.2GHz) and demonstrate active limb sounding of the Earth’s atmosphere and ionosphere.
In this paper, the LITN and GPS/MET programs will be described and some initial ionospheric tomography result will be presented. In section 2, section 3, the technical details of GPS/MET and LITN will be discussed. In section 4, the multiplicative algebraic reconstruction technique (MART) will be described and some of derived tomography results will be presented. Some future works for imaging the ionosphere will be summarized in section 5.
2. The basic technique of GPS/MET
Since the mid-1960s, the radio occultation technique has been used to study the properties and structure of the atmospheres of Venus, Mars, some other outer planets and many of their moons [Kliore, et. Al, 1965; Lindal, et. Al, 1979, 1981, and 1987]. In 1993 the University Corporation for Atmospheric Research (UCAR) organized a proof-of-concept experiment on a 735-km low Earth orbiting (LEO) satellite (the MicroLab-1 satellite) to receive GPS signals and demonstrate active limb sounding of the Earth’s atmosphere and ionosphere by radio occultation techniques. In the geometrical optics approximation as shown as Figure 1, a ray passing through the ionosphere is refracted according to Snell’s law due to the vertical gradient of electron density and hence the refractive index n. The overall effect of the atmosphere can be characterized by a total bending angle a, an impact parameter a, and a tangent radius rt as defined in Figure 1. During an occultation, the variation of a with the impact parameter a can be given by Snell’s law when local spherical symmetry is assumed and can be expressed by
And then, using the Abel integral transformation under a spherically symmetrical assumption, the corresponding refractivity at a tangent radius rt can be expressed in term of a(a) and the impact parameter a as
where at ( =n(rt )rt) is the impact parameter for the ray whose tangent radius is rt.
Actually, the GPS frequencies bending in the ionosphere is so small. Even during the daytime and near solar maximum, the absolute magnitude of the bending angle does not exceed 0.03° for both of L1 and L2 GPS frequencies [Hajj and Romans, 1998; Schreiner et al., 1999] in the F-region. Applying the Abel transformation, as is similarly done with inversions through bending angles with an assumption of local spherical symmetry, the electron density can then be given by the following integral equation:0
We note that the derived electron density from the Abel integral transform can be used an initial condition for the MART algorithm described in later section.
Figure 1. Illustration of the geometry of the GPS-LEO occultation problem for ionosphere observations, where p1 is an occulting LEO point, p2 is an auxiliary LEO point with the same impact distance of p1, a is the bending angle, rt is the ray’s tangent radius, and a is the impact parameter.
3. The Low-Latitude Ionospheric Tomography Network (LITN)
The LITN consists of a chain of six stations. Each station receives and records signals transmitted by the Naval Navigation Satellite System (NNSS). The receiver measures the Doppler shifts of the 400 and 150 MHz signals from NNSS due to the ionosphere, from which total electron content (TEC) can be deduced. As presented in Figure 2, it shows the geographic location of the six stations. The six receiving stations are Manila (121°E, 14.6°N), Baguio (121°E, 16.4°N), Kaohsiung (121°E, 22.5°N), Chungli (121°E, 25°N), Wenzhou (121°E, 28.0°N), and Shanghai (121°E, 31°N). The chain spans a range of 16.4°in latitude within 1° of 121°E longitude or a distance of more than 1800 km along the surface of the Earth. Geomagnetically, the visible region extends from 25° in the north and to just south of the magnetic equator in the south, with the northern equatorial anomaly region completely nested inside.
Referring to Figure 2, for any given path p at any station, the measured phase difference ? between the signals at the two frequencies is related to the slant TEC Cs for that path by (Leitinger et al., 1975)
where Ne is the electron density, F0 the unknown initial phase for a given receiver,and D is a proportional constant. Only when F0 is found, can one obtain the absolute TEC from the measured data Y. To determine F0 [Leitinger et al. 1975] proposed a two-station procedure.
Figure 2. A vertical cross section depicting scanning of radio rays as one Naval Navigation Satellite System (NNSS) satellite passes overhead of six receiving stations on the ground.
4. The MART algorithm for CIT reconstruction and their results
With the initial constants determined, the absolute slant TEC can be obtained from the data. We start by considering the slant TEC along any path p between a transmitter and a receiver. For tomographic applications, the TEC along some path p is approximated by a finite sum of segment of the integral
The TEC along some path p is approximated. This is carried out by dividing the two dimensional ionosphere into a set of n pixels and denoting the electron density in the j-th pixel by xj. Then for the i-th path, equation (1) can be approximated by
or in matrix notation:
where A is a matrix whose element denote the length of the path-pixel intersections for each path. Note that C and X are column vectors for absolute slant TEC and electron density. A is an m n matrix where m is the number of TEC values from all paths at all receiving sites. The elements in A depend on the geometry of the paths and can be computed once the experimental configuration is fixed. The task of CIT is to invert the equation (2) to obtain the electron density vector X.
Raymund(1994) reviewed the CIT reconstruction algorithms proposed by various investigators. One of the most commonly used algorithms is Algebraic Reconstruction Technique, or ART, having an additive correction and first introduced in CIT by Austen et al. (1988). This is an iterative procedure for solving the equation (2). A modified version of ART is the so-called multiplicative ART (MART) algorithm in which the correction in each iteration is obtained by making a multiplicative modification to X as
shown as the following equation (Raymund et al., 1990).
where lk are relaxation parameters such that 0 < lk < 1. In this paper, the MART algorithm has been applied to the data (absolute slant TECs) to reconstruct the ionosphere as the examples of Figures 3 and 4.
Figure 3. The electron density contour reconstructed by LITN on Aug. 28, 1995.
Figure 4. Ionospheric tomography by GPS/MET from 05:00 to 06:39 UT on the 10th of Feb., 1997. We note that the period of GPS satellite orbiting the Earth is about 100 minutes. Furthermore, the left part of this figure (-90°~90° at the x-axis) is dayside tomography; and the right part (90°~-90° along) is night side tomography.
5. Conclusion and future work
We note that, using the GPS/MET to reconstruct the ionosphere, it is hard to observe the Es layer. One of the probable reason is the LEO’s GPS signal observation using a low sampling rate at 0.1 Hz. Furthermore, the initial guess in MART algorithms is very important. The initial value in LITN is used to adopting the prediction of IRI-90 model. However, the GPS/MET occultation observation gives us another way to retrieve the initial electron densities from the Abel integral transform. We hope to get more accurate reconstruction of the ionosphere by the assistant of GPS/MET.
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