Single-frequency users of a satellite-based augmentation system (SBAS) rely on ionospheric models to mitigate the delay due to the ionosphere. The ionosphere is the major source of range and range rate errors for users of the Global Positioning System (GPS) who require high-accuracy positioning. The purpose of the present study is to develop a tomography model to reconstruct the total electron content (TEC) over the low-latitude Indian region which lies in the equatorial ionospheric anomaly belt. In the present study, the TEC data collected from the six TEC collection stations along a longitudinal belt of around 77 degrees are used. The main objective of the study is to find out optimum pixel size which supports a better reconstruction of the electron density and hence the TEC over the low-latitude Indian region. Performance of two reconstruction algorithms Algebraic Reconstruction Technique (ART) and Multiplicative Algebraic Reconstruction Technique (MART) is analyzed for different pixel sizes varying from 1 to 6 degrees in latitude. It is found from the analysis that the optimum pixel size is 5° × 50 km over the Indian region using both ART and MART algorithms.
 Tomographic imaging of the electron density has evolved during the past decade to become an important ionospheric diagnostic. Efforts have been directed by a number of research groups [Raymund, 1995; Raymund et al., 1993, 1990; Austen et al., 1988; Sutton and Na, 1996] to verify and establish the technique as a tool that can be readily used for many purposes like routine monitoring of the ionosphere, correcting the ionospheric delay experienced by the Global Positioning System (GPS) signals, geophysical studies of the plasma etc. It is important to establish an algorithm that will produce reliable reconstruction of the whole range of possible ionospheric density distribution. The ray tomography technique involves use of one-dimensional information to reconstruct a two-dimensional image. The data consists of measurements of the line integral of electron density in the ionosphere for many different paths. The line integral paths can be considered as many unique rays traversing the plane of the reconstruction region. Region of interest can be gridded into small areas or pixels. It is assumed that the electron density will remain constant in each pixel. The line integral of electron density over some path is the total electron content (TEC). Mathematically it can be expressed as below:
where, Ne(l) is the variable electron density along the signal path “l,” and the line integration is along the signal path from the satellite S to the receiver R [Klobuchar, 1996].
 Ray tomography includes two-dimensional as well as three-dimensional tomography. An alternative approach extends the tomographic idea into imaging the entire ionospheric region by considering full three-dimensional voxel based (i.e., volume pixel based) tomography. Meggs et al. [2002, 2004], developed an inversion program called the MultiInstrument Data Analysis System (MIDAS), for determining the electron density distribution and consequently the TEC values in each voxel. Their main conclusion was that on an hour by hour basis, imaging the ionospheric TEC (or determining delay) using a full inversion is more reliable than using a thin shell model. In two-dimensional tomography a chain of stations receive the signals from the visible GPS satellites, and the geometry can be created in such a way that the TEC measurements can be used to reconstruct a vertical slice or vertical cross section of the ionospheric electron density in a vertical plane between the satellite and the ground stations [Mitchell et al., 1997].
 To reconstruct the ionosphere, the TEC can be used with the tomography technique for the two-dimensional imaging of electron density distribution in the ionosphere. The estimated TEC for a chain of receivers are inverted to obtain the electron density distribution as a function of latitude and altitude over a given longitude. Inversion, by nature being an ill-posed mathematical problem which may not have a unique set of solution. As a result, various mathematical algorithms are used for this inversion [Pryse et al., 1998; Raymund, 1995]. When these algorithms are used in ionospheric tomography to reconstruct the electron density profiles, almost all of them suffer from the basic inability to correctly estimate the vertical profiles of these distributions. This problem to a large extent is also due to the geometry of the whole system, as one cannot have the TEC information from large projection angles from a ground based receiver. Also the completeness of the data is limited, as the receivers do not lock onto a satellite until it is at least a few degrees above the horizon.
 The accuracy of the reconstructed image generally depends on many factors like satellite receiver configuration, the raypath modeling, grid intersections and finally the reconstruction algorithm. There is a direct relationship between the information contained in the measured TEC data and the accuracy of the reconstructed image. Thus the proper choice of receiving stations which could optimize the information contend in a given longitudinal plane is one of the key factors for obtaining accurate images from any ionospheric tomography network [Thampi et al., 2004].
 In this paper, a novel approach to the tomography model is adopted which will produce the path length matrix (PLM) required in the reconstruction technique and some iterative approaches such as, Algebraic Reconstruction Technique (ART) and Multiplicative Algebraic Reconstruction Technique (MART), for the reconstruction of electron density and the TEC.
 The whole organization of the paper is as follows: In section 2, a short view on ray tomography is given. The ionospheric model, the reconstruction technique with the algorithms of ART and MART and a new method for determining the initial guess are discussed in section 3. In section 4, the validation methodology is presented. Analysis and results using ART and MART have been provided in section 5. Conclusion is provided in section 6.
2. Ionospheric Tomography
 Ionospheric tomography is a technique for remotely sensing ionospheric electron density which is done by passing through finite rays through the space of consideration and observing the spatial distribution of its physical quantity. There are mainly three fundamental problems associated with ionospheric tomography [Thampi et al., 2004]. First one is that the observation may be inconsistent. Receiver difference, timing problems, changes in the ionosphere during a pass, and so long all contribute to inconsistencies in the observations. The second problem is that the observations do not contain enough information to uniquely specify the ionospheric density distribution. Last, only relative TEC can be observed as most of the popular systems only broadcast a dual frequency and so an unknown constant offsets the TEC observed at each station [Idenden et al., 1998; Eftaxiadis et al., 1999]. Some of these problems can be removed partially by correcting the data received from several stations and using some background ionosphere containing the information about the vertical distribution as the initial guess of iterative process like ART and MART.
 One of the main limitations of ionospheric tomography results from the lack of near-horizontal “satellite to ground” raypaths. This geometrical restriction leads to lack of information in the data set about the vertical ionization distribution. To overcome this limitation, one can use the iterative algorithms like ART [Gordon et al., 1970; Pryse et al., 1998; Andersen and Kak, 1984] or MART [Pryse et al., 1998; Raymund, 1995; Raymund et al., 1993, 1990]. In this paper an additional information, inserted in the form of a background ionosphere taken from a standard ionospheric model that initialized the reconstruction, is used.
3. Tomography Model
 The problem of ionospheric TEC estimation and reconstruction over the Indian region is formulated as follows: Considering the vertical plane of around 77 degrees longitude over the Indian region, with six ground stations around that vertical slice as Trivandrum (8.48°N, 76.92°E), Bangalore (12.95°N, 77.68°E), Hyderabad (17.45°N, 78.47°E), Bhopal (23.28°N, 77.34°E), Delhi (28.58°N, 77.21°E) and Shimla (31.08°N, 77.07°E). Locations of these TEC collection stations are given in Figure 1. In that vertical plane, altitude variation is taken from 200 km to 700 km with latitude from 5 degrees to 40 degrees. The plane is divided into the chosen region with small pixels of different sizes say 6° × 50 km, 5° × 50 km, 4° × 50 km, 3° × 50 km, 2° × 50 km and 1° × 50 km where 1° × 50 km sizes describe the pixel having dimension 1 degree in latitude and 50 km in altitude. The purpose of the present study is to develop a model to reconstruct the TEC over the Indian region where ionospheric anomaly is present and also to understand the effect of the pixel dimension in TEC reconstruction.
3.1. Basic Approach to the Problem
 As shown in Figure 2, the whole region of consideration is divided into two-dimensional pixels which are labeled from 1 to N. By assuming that the electron densities are constant throughout in a pixel at a given time, the electron density coefficients are discretized for N number of pixels. Let the electron density coefficients for the pixel i as xi where i varies from 1 to N. For instance, one snapshot of ionospheric tomography system is illustrated in Figure 3, with one ray piercing a 3 × 3 degree grid. Here Xk, (k = 1, 2, … 9) indicates the electron density in the kth pixel and dk, (k = 1, 2, … 9) indicates the length of the ray traversed through the pixel k. The TEC value for the ray can be given as,
In general, the equation can be written as
Assuming that dk = 0, if the ray cannot pierce the pixel k. N is the total number of pixels. For the present illustration N is taken as 9.The entire ray tomography reconstruction algorithm can be divided into two parts say, (1) tomography model and (2) inversion procedure.
3.2. Tomography Model
 In this section, we demonstrate a novel approach to compute the PLM of the rays received by the ground stations. The PLM “A” is of order m × N where m represent the total number of rays received at a given time and N the total number of pixels in the grid. In other words, each row of the matrix corresponds to a ray received at the given time and each column represents a pixel. A common element say Aij of the PLM is the intersection length of the ith ray with jth pixel. A ray entering into the vertical layer with angle θ made with positive side of x axis (along latitude) is shown in Figure 4. The ray has been received by the ground station say G with ΦG (degree) latitude and hG (km) altitude.
 To identify the pierce points with horizontal layers, we consider the following: Let the pierce point on the ith horizontal layer is Ai(Φi, hi) with the known altitude hi = 200 + 50 · i and the latitude
Here s is the conversion factor from degree to the length. Note that the ray will be able to pierce the selected grid, only if 5 ≤ Φi ≤ 40 for some i.
 To identify the pierce points with vertical layers, we do the following: Let the pierce point on the jth vertical layer are Bj(Φi, hi) with the known latitude Φj = 5 + d · j for d° × 50 km pixel size, and the altitude
Four cases demonstrated in Figure 4 are discussed as below:
 In case 1, the ray pierces the grid at the topmost horizontal layer where hi = 700 km and 5 ≤ Φi ≤ 40, so that it will be piercing all the horizontal layers one after another.
 In case 2, the ray pierces the grid at the leftmost vertical layer where Φi = 5 degree and 200 < hi < 700, so that it misses a few top horizontal layers.
 In case 3, the ray pierces the grid at the rightmost vertical layer where Φi = 40 and 200 < hi < 700, so that it misses again a few top horizontal layers.
 In case 4, the ray cannot pierce the grid at all. It may happen if hi ≤ 200 for Φi = 5 and Φi = 40. Though these rays lie in the considered vertical region, we do not consider them as a candidate as they fail to pierce any of the grids and all the path length matrix elements corresponding to these rays are zero.
 After identifying all the pierce points with both the horizontal and vertical layers, we find out all the pixels in which the ray entered and compute the corresponding intersection length of the ray occupied by that pixel. These intersection length values go to the matrix PLM of path length in its appropriate position. All other entries of that row corresponding to the ray considered are zero. After determining the path-intersection lengths of each ray, the PLM of path lengths was obtained, which is the main outcome of this section.
 To ensure the availability of significant number of rays for the tomographic model, a longitudinal range of 77 ± 5 degrees was considered. Also to avoid the measurement error on the TEC data due to multipath, an elevation cutoff of 20 degree has been considered.
3.3. Reconstruction Procedure
 By using the PLM generated in section 3.2, a system of algebraic equations can be expressed as below:
In a compact form, above equation may be represented as, Ax = b where A is the matrix of path lengths, rows (m) represent the number of rays and columns (N) represent the number of pixels with unknown electron density coefficients, denoted by the column matrix x and b is the column matrix of measured TEC values.
 Let xe be the exact solution of the system and an initial guess say x0 which is an approximate solution to the system. Let E be the error term due to the approximation. Mathematically, it can be expressed as below:
Noting that b, x0 and E are column vectors of TEC, electron density at each pixel and the error vector, respectively. The reconstruction algorithms uses an iterative approach to solving the above system where resulting b0 is calculated from the initial guess and then the difference (in case of ART) or the ratio (in case of MART) between b and b0 is used to modify x0, creating x1. In the whole process, the error term E is expected to be reduced as the cycle is repeated until the desired accuracy is achieved.
 As the iterative process required a suitable initial guess, a new idea for determining the initial guess has been used in the iterative procedures of ART and MART. Also we choose proper convergence criteria to terminate the process of iteration.
3.4. The Reconstruction Algorithms: ART and MART
3.4.1. ART Algorithm
 Step 0: Assume some suitable initial guess x0.
 Step k: Compute the (k + 1)th iteration using the following equation.
where bi represents the TEC measured along the ray i.
 The input term (bi − Aimxmk) in the above expression is the difference between the measured TEC and the corresponding value calculated through the current ionosphere. The current and modified electron densities in pixel j are xjk and xjk+1, respectively. the weighting term and the denominator ensure that the shape of the vertical profile of the background ionosphere was retained throughout the reconstruction. λ is the relaxation parameter which may assume values between 0 and 1. In this paper, λ has been chosen as 0.02 which is the optimum value [Pryse et al., 1998].
3.4.2. MART Algorithm
 Step 0: Assume some suitable initial guess x0.
 Step k: Compute the (k + 1)th iteration using the following equation.
Here bi represents the TEC measured along the ray i. The current and modified electron density in pixel j are xjk and xjk+1, respectively. Amax is maximum path pixel intersection length in the grid. In this case, λ is taken as 0.2 whish is the optimum value in case of MART [Pryse et al., 1998].
3.4.3. A Novel Approach for the Initial Guess
 The initial guess plays a very important role in proper evaluation of any reconstruction technique. A suitable initial guess reflects the gross estimate of the initial state of the ionosphere at the time of reconstruction. Some previous studies use the initial guess [Censor, 1983; Raymund et al., 1990] which is constant in space and time. In these studies the vertical structure and the daytime variation of the ionosphere was not taken care of. The initial guess should be able to produce high temporal and spatial variability of the ionosphere. This fact is more important for the Indian region which falls in equatorial ionospheric anomaly (EIA) belt [Rama Rao et al., 1997]. In addition to this, there is another way to incorporate the additional information into the system via a prior model that includes the information about the desired solution [Shieh et al., 2006]. These are certainly useful [Kuklinski, 1997; Cornely, 2003] to improve the speed of the iteration for a huge data set, especially to do a 3-D ionospheric reconstruction. In the present study, the background ionosphere is used as an initial state which takes care of the ill-posedness of the problem by the algorithm.
 In this paper, information about the electron density contributions in each pixel carried out by the signals/rays received at that time along with the standard background ionosphere generated by using International Reference Ionosphere (IRI) [Bilitza and Reinisch, 2008] is explored for the initial guess. The information retrieved from the signal/rays is sensitive to the temporal variability of the ionosphere where as the background ionosphere preserves its vertical structure. This way, both the time-space variation as well as vertical structure of the ionosphere is taken into consideration. This approach makes the initial guess more adaptive and robust and as per authors' knowledge is a novel way for selecting the initial guess.
 Mathematically, let the initial guess for the pixel j be given as below
where ω1 + ω2 = 1 and 0 ≤ (ω1, ω2) ≤ 1,
bj is the background standard model ionosphere.
 If aj = 0, we assume ω1 = 0 and ω2 = 1, i.e., if no ray intercepted the grid box, assign the total weight to the background ionosphere. To implement the above approach, some standard ionosphere profile is required, which may be a model ionosphere preserving only the vertical structure of it over any time frame.
4. Validation Methodology
 Cross validation has been done for different pixel sizes, i.e., 6 × 50, 5 × 50, 4 × 50, 3 × 50, 2 × 50 and 1 × 50 (first object is in degrees and the second is in km) by deleting one ray and reconstructing the TEC for that ray with the help of the remaining rays. The procedure has been repeated for each ray and the TEC is reconstructed. After reconstructing the TEC, the root- mean-square error (RMS error) in the TEC for the different pixel sizes at different times of the day is obtained. A discussion on the comparison of RMS error in the TEC is made on the basis of the obtained results. The following formula has been used to calculate the RMS error in the TEC values [Shukla et al., 2010].
 The RMS error in slant TEC (STEC) (RMSTEC) is obtained as below:
Here TECM is the measured TEC, TECr is the reconstructed TEC and m is the total number of raypaths.
 Analysis is done for the geomagnetic quiet days of March 2005. Choice of quiet days is done on the basis of geomagnetic Ap index. In this paper Ap index less than 50 is considered for the quiet days which includes the days up to the minor storms and excludes major and severe storms [Shukla et al., 2010, 2009].
5. Results and Discussion
 Multilayer ray tomography model using reconstruction algorithms ART and MART has been developed. For this, a novel approach for the model development and the initial guess is proposed in this paper. Software in “C language” has been developed. Analysis has been done for different pixel sizes for all the geomagnetic quiet days of March 2005 (i.e., 2, 4, 12, 15, 22, 23, 28, and 29 March), using the TEC data collected from the selected stations as described in section 3. First, the tomography images and their vertical profiles obtained from both ART and MART algorithms have been compared. Afterward, comparison between the reconstructed and the measured TEC for the different pixel sizes for the same time instants has been done. Monthly averages of RMS errors for different pixel sizes are also estimated using both ART and MART algorithms. In the last, convergence of the considered algorithms is discussed.
Figure 5 shows the vertical profiles of electron density with respect to altitude for different latitudes at 7.5, 15 and 25 degrees, respectively. Electron density profiles are derived using both ART and MART algorithms. It can be seen from Figure 5 that the reconstruction of the profiles using MART (Figures 5d–5f) are more smooth and closer to the actual profiles. It can further be observed from Figure 5 that maximum electron density remains around 400 km altitude which is the representative altitude of maximum electron density over the Indian region. Figure 6 illustrates the reconstructed tomograms generated by the ART and MART algorithms. It exhibits the EIA structure over the study region with enhanced electron density around 23°N latitude which is the crest location of EIA over INDIA. The EIA structures observed in reconstructed images are in accordance with the observed features of EIA over Indian region with maximum electron density around 350–450 km altitude. Figure 7 shows comparison of the reconstructed and the measured TEC (in TEC units) with the ART algorithm for pixel sizes of 1, 3 and 5 degrees for a particular time. The comparison has been done to illustrate the closeness of reconstructed TEC with the measured TEC. From Figure 7, it can be observed that the reconstructed TEC with pixel size 5 degree is closest to the measured TEC. Further it can be observed that the closeness of reconstructed TEC with measured TEC deteriorates as the pixel size is decreased. Similar observations are seen from Figure 8 using the MART algorithm.
Figure 9 shows the comparison of half-hourly RMS error time averages over the quiet days of March in slant TEC (in TEC units) with different pixel sizes varying from 1 to 6 degree in the steps of 1 degree using ART. It has been observed from Figure 9 that as the pixel size is increased, the RMS error is decreased. From the four plots of Figure 9, it can be clearly observed that the performance of the 5 degree pixel size better than 1, 2, 3 and 4 degrees, whereas with the increased pixel size of 6 degree the RMS error further increases. From Figure 9 it can be further observed that in general, RMS error is least for the pixel size of 5 degree over all other pixel sizes considered. It has also been observed that further increase in pixel size does not improve the results. The reason behind getting 5 degree pixel as optimum size is due to the fact that it creates the balance between the number of empty pixels and discretization error for this pixel size. Analysis has been shown up to the pixel size of 6 degree. Almost similar results have been obtained using MART as shown in Figure 10.
 From Figures 9 and 10, it has also been observed that the maximum error in STEC occurs between 0900 and 1300 h UT that is around 1430–1830 h local time. This is due to the fact that around this time ionospheric anomaly is more prominent over the Indian region. Improvement in the reconstruction in the TEC is also maximum over this period of time. From the whole analysis it has been found that the pixel size of 5 degree is the best performing pixel size using both ART and MART algorithms. In Figure 11, a comparison of half-hourly RMS errors for best performing pixel size that is pixel size of 5 degree using ART and MART is shown separately. The time averages over the quiet days of March in STEC has been considered. From Figure 11, it has been observed that, the RMS error using MART is less in comparison to ART and the maximum RMS error with the ART algorithm is around 20 TEC units whereas with the MART algorithm this error is around 14 TEC units. Also there are no sharp peaks in RMS errors with MART. It can further be observed from Figure 11 that using MART algorithm, the RMS error is 4–5 TECU less for peak ionospheric activity period (i.e., 0800–1500 h UT).
 Since both the techniques ART and MART are iterative in nature, therefore the convergence of the both the techniques has also been discussed. The convergence behavior of both the reconstruction techniques has been analyzed for several arbitrarily selected data sets with different initial guesses. MART converges more rapidly than the ART though the convergence pattern may vary case by case. Results of the analysis are shown in Figure 12. In Figure 12, the horizontal axis shows the number of iterations and the vertical axis shows the difference between the electron density coefficients of current and last estimations. Overall convergence, as may be seen from Figure 12, seems to be satisfactory for both ART and MART. From the overall analysis, it may be concluded that, although both reconstruction techniques are performing well, but the performance of the MART is better than the ART over Indian region for the considered data set.
 Reconstruction of the electron density profiles and the TEC using tomography is done for the Indian region. Multilayer tomography model has been developed and inversion is done using ART and MART algorithms. Analysis is done for different pixel sizes for the geomagnetic quiet days of March 2005 using the TEC data collected by six TEC collection stations over Indian region. A novel approach for model design and selection of initial guess is adopted. Also capabilities of both ART and MART are demonstrated in terms of electron density profile and TEC reconstruction over the Indian region. From the analysis, it may be concluded that, the performance of both ART and MART is fairly well but MART algorithm performs slightly better than ART for the considered data set. It has also been observed that pixel size of 5 degree is the best performing size giving least RMS error in the STEC. It may also be concluded from the convergence analysis that MART achieves a more rapid convergence than ART.
 The authors would like to thank the Director of the Space Applications Centre (ISRO), the Deputy Director of the Remote Sensing Applications Area, and the Group Director of the Meteorology Oceanography Group for encouragement. The valuable and perceptive suggestions of V. K. Garg and all other team members of IITP-17 which helped in improving the quality of the paper are also gratefully acknowledged.