Three-dimensional ionospheric tomography via band-limited constrained iterative cross-entropy minimization

Authors


Abstract

[1] The problem of reconstructing ionospheric electron density from ground-based receiver to satellite total electron content (TEC) measurements is formulated as an underdetermined discrete linear inverse problem. If receivers and satellite orbit are coplanar, then a single two-dimensional (2-D) imaging plane can be used as a geometrical model. In most cases, receiver locations are determined by convenience and availability of sites, and thus a 3-D imaging volume is required in order to capture the part of the electron density solution that is associated with gaps between receiver stations within a chain. It is well understood that spurious features, or “wings,” associated with these gaps can be produced in tomographic image reconstructions where ray path coverage from individual stations is lost or minimal. In this paper, a 3-D reconstruction system that takes advantage of a multiple-chain data acquisition geometry and provides a solution consistent with available TEC data everywhere within a receiver chain is presented. The reconstruction system exploits the fact that gaps, created by longitudinally unaligned receivers within a chain, can be captured as additional planes of constant longitudes within the 3-D imaging volume. With parameterized ionospheric model (PIM)–generated data as a nonnegative prior estimate of the electron density, the reconstruction algorithm uses constraints based on prior knowledge of the 3-D spatial Fourier transform of the prior electron density as a smoothing mechanism in the tomographic reconstruction process. Consequent to the underlined Fourier transform formulation, a unique solution is produced at locations that contributed to the measured TEC data, and a solution is interpolated within the remainder of the imaging volume. The band-limited BISMART algorithm has been evaluated using a multiple-receiver chain TEC data acquisition system, under known ionospheric conditions. The algorithm satisfactorily reconstructs density solutions, consistent with small-scale enhancements, irregularities, and troughs in the auroral ionosphere, from the available TEC data. The quality of the density reconstructions, coupled with the computational efficiency of this algorithm, indicates the potential utility of this technique for real-time three- and four-dimensional ionospheric tomography.

1. Introduction

[2] The problem of reconstructing ionospheric electron density, from ground-based receiver to satellite total electron content (TEC) measurements, has been addressed in a number of previous studies. The algorithms used to perform these inversions have included the simultaneous iterative reconstruction technique (SIRT) [Austen et al., 1988], the multiplicative algebraic reconstruction technique (MART) [Raymund et al., 1990], singular value decomposition (SVD) null-space methods [Raymund et al., 1994b], transform methods [Fremouw et al., 1992; Na and Lee, 1991], along with maximum entropy techniques [Fougere, 1995]. The majority of these reconstruction algorithms have assumed that the ionospheric electron density can be represented as a linear combination of basis functions used to convert the integral relationship between TEC measurements and ionospheric electron density to a set of linear algebraic equations. These algorithms also have various mechanisms that allow the use of prior information to resolve the ambiguity that is associated with the ground-based receiver to satellite geometry [Na and Lee, 1990; Raymund et al., 1994a]. Bayesian reconstruction algorithms are also used and they are known to substantially improve the accuracy of the reconstruction obtained from limited data, if the object under study does not differ very much in size, shape or position from the assumed model. Difficulties arise when the spatial coordinate of the prior model is held fixed relative to the spatial coordinate system of the reconstruction [Hanson and Wecksung, 1983]. A technique that has been used to represent variations in an image being restored for a general class of distortion has been reported by Hanson [1992]. In his work, the prior is given an inherent geometrical flexibility. This flexibility, which is achieved through a warping of the coordinate system of the prior into that of the reconstruction, allows some of the geometrical characteristics, inherent in the reconstructed density solution, to be captured. Cornely [2003] used the flexible prior model approach to account for geometrical variations in electron density not captured in state of the art prior models such as the parameterized ionospheric model (PIM) and the International Reference Ionosphere (IRI). This flexible prior model approach has been tested and successfully adapted to tomographic reconstruction with independent verification from European Incoherent Scatter (EISCAT) radar data. A number of three and four dimensional tomographic reconstruction techniques have also been proposed [e.g., Howe et al., 1998; Hernández-Pajares et al., 1998; Spencer and Mitchell, 2001]. Howe et al. propose a four-dimensional stochastic model of ionosphere perturbations based on data from GPS and LEO. A Kalman filter is used to assimilate data into a time-dependent model in the form of a first-order Markov process. However, the computational complexity of this method, when the dimension of the state vector is considerably large, is not known. Hernández-Pajares et al. also propose a temporal evolution of a three-dimensional electron density by means of Kalman filtering. The technique uses a combination of real-time and LEO GPS data with independent verification from IRI. As in the previous case, the computational complexity of this technique, when the dimension of the state vector is large, is not known. Mitchell's multi-instrument data analysis system (MIDAS) produces free electron density maps from slant TEC values and from ionosonde in conjunction with LEO-GPS. This method shows significant improvements only when precise vertical ionization profiles on the topside density are needed. For the past three to four years a collaborative effort, to update off-line and real-time models such as IRI, PIM and its real-time cousin, the parameterized Real Time Ionospheric Specification Model (PRISM), has been underway. This effort has produced the Global Assimilation of Ionospheric Measurements (GAIM) [Schunk et al., 2004]. GAIM uses a physics-based ionosphere-plasmasphere wind model and a Kalman filter as a basis for assimilating a diverse set of near-real-time measurements from various sources (LEO, GPS, ionosondes and DMSP satellites). Because of the large dimension of the electron density state, GAIM's implementation of a full Kalman filter is not computationally feasible. Therefore GAIM implements a band-limited Kalman filter in which a full time propagation of the covariance matrix is performed but only a portion of the matrix is retained. There is a trade off between, how sparse or full the covariance matrix is over repeated runs, resulting in a tradeoff between the accuracy of the derived electron density state and the computational complexity of the algorithm. The preceding discussion suggests that the specific technique, used to assimilate data into the reconstruction process, has a significant impact on the computational complexity of the algorithm and the accuracy of the derived solution. A useful class of algorithms for discrete linear inverse problems generally referred to as expectation maximization maximum likelihood (EMML) algorithms [Byrne, 1993] allow closed form iterative solutions for problems where each element of the unknown density, measurement matrix, and observational measurements are positive quantities. One of these algorithms, the simultaneous multiplicative algebraic reconstruction technique (SMART), allows the use of prior information, a user-controlled method of regularizing the solution to reduce the effects of noise, and a structure that can be easily implemented in parallel form. In this paper, we propose a three-dimensional modified version of the two-dimensional SMART [Kuklinski, 1997]. A BISMART method is implemented where the cross-entropy minimization is applied separately to two subsets of equations. One subset represents the relation between the electron density and the TEC data while the second subset represents the band-limited characteristics of the prior that satisfy the available TEC data. Since prior estimates of ionospheric electron density obtained from PIM data are well modeled as three-dimensional (3-D) band-limited functions, the Fourier transform of the electron density restorations were constrained to be zero outside the corresponding 3-D band-limited region. The constraints are formulated as M additional linear equations, where M is the number of discrete frequencies in Fourier space where the prior had essentially zero energy. This additional information is used to produce a unique solution, at locations that contributed to the measured TEC data, and interpolate a solution within the remainder of the imaging volume. The band-limited BISMART method is used to implement a three-dimensional ionospheric reconstruction system with two key properties. First, the reconstruction system exploits the fact that the structure of the BISMART algorithm, allowing subsets of data corresponding to the available TEC data and the band-limited constraints to be solved separately and efficiently, can be used as a platform for data assimilation from other sources such as GPS, and ionosondes. Second, the reconstruction system exploits the fact that gaps, created by longitudinally unaligned receivers within a chain, can be captured as additional planes of constant longitudes within a three-dimensional imaging volume. The technique can be extended to include gaps between chains of receivers. In the consistent case, BISMART converges to a solution for any configuration of the measured TEC and band-limited data subsets. In the inconsistent case, BISMART produces a limit cycle, from which an approximate solution can be obtained using a “feedback” approach. The algorithm has been evaluated using TEC data from a multiple-chain receiver acquisition system, to produce restorations for a volume of up to 1.31 × 109 voxels under known ionospheric conditions.

2. Algorithm Development

[3] The algorithm presented here uses a set of basis functions that are constant within each voxel of a fixed three-dimensional spherical grid structure and zero elsewhere. These basis functions are used to formulate the ionospheric tomography problem such that the linear relationship between a (L × 1) vector of TEC measurements c and a (J × 1) vector of electron density d can be represented as

equation image

where A is a (L × J) matrix of ray path distances. In a typical case, J > L; that is, there are more electron density values than TEC measurements, yielding an underdetermined system of equations. The measurement geometry typically limits the number of independent measurements so that even if more measurements were available, the rank of A would be less than J, precluding a unique solution without the use of additional information or constraints. There are both spatial and temporal sources of error that may cause these equations to be inconsistent as well. The actual electron density may not be constant within a voxel, and hence two rays passing through the same voxel may produce TEC data that are not consistent with a uniform electron density within that voxel. Since the individual receiver to satellite TEC data are not obtained at the same time, temporal changes in electron density within a given voxel during the data collection period may also contribute to TEC measurement error. The sources of error in the density model of (1), are represented by a (J × 1) noise vector n. The ionospheric tomography problem does have certain characteristics that sets it apart from a general underdetermined linear inverse problem and which can be exploited to produce a computationally efficient inversion algorithm. The ionospheric electron density, grid voxel ray path distances, and resulting TEC measurements are all nonnegative quantities. In addition, the A matrix is quite parse having on the order of 0.01–0.02% nonzero elements. A computationally efficient iterative image reconstruction algorithm, for problems with nonnegative densities and measurement matrices for a two dimensional geometry [Kuklinski, 1997], was modified to produce a three-dimensional band-limited iterative minimum cross entropy reconstruction algorithm. The modified algorithm uses a scaled version of the TEC model in (1):

equation image

where y is the scaled TEC data, P the scaled ray path distance matrix and x the scaled electron density. The algorithm uses the Kullback-Liebler (KL) distance between any two nonnegative vectors a and b which is defined as

equation image

to generate a functional Gs(x)

equation image

Gs(x) is minimized using an iterative alternating projection technique. This technique requires that x and y as defined in (2) be nonnegative vectors. The measurement matrix P must contain only nonnegative elements and have columns sums less than or equal to unity. The constant r(0 < r < 1) controls the tradeoff between the degree to which the solution x satisfies the observed measurements y, quantified by the cross entropy term KL(Px, y) and the degree to which the cross entropy regularization term KL(x, pcm) forces the solution x to be near to the prior estimate pcm. Both of the cross entropy terms use known vectors, y and pcm, respectively, as a and b in the representation KL(a, b). Furthermore, the A matrix associated with the ionospheric tomography problem needs to be scaled to satisfy this column sum convergence requirement. This scaling can be accomplished using a factored representation of A = PDcm.Dcm where Dcm is a diagonal matrix with elements equal to the maximum column sum of A. Dcm can be represented as a coordinate transformation from a native domain, determined by the specific grid structure and satellite-receiver geometry, to a solution domain where the convergence requirements of the iterative projection solution technique are satisfied. Using this factored representation, TEC data are related to electron density as

equation image

Defining x = Dcmd, y = c, and P = ADcm−1, the ionospheric electron density can be obtained by determining the solution domain vector x that minimizes Gs(x) as n approaches infinity of the following iteration:

equation image

where the dots represent vector element-by-element multiplication, the slash means vector element-by-element division, and the subscript T denotes matrix transpose. The indicated log and exponentiations are vector element-by-element operations as well. The superscript n over the sum in (6) represents the number of subblocks chosen in the block iterative formulation, to be discussed later. Also seen in (6), the term r is a regularization constant that allows controlling the degree to which the solution obtained will be dominated by the data rKL(Px, y) or to a penalty function that minimizes the effects of measurement noise (1 − r)KL(x, pcm) on the reconstruction process. In the 3-D reconstruction algorithm presented in this paper, a constrained BISMART technique is used that allows the subsets of equations, satisfying the available TEC data and the band-limited constraint data, to be solved separately. The set of equations in (2) is partitioned into three disjoint subsets and the iterative algorithm processes these subsets, using the iterative formula in (6). The first subset produces the density solution that satisfies the available TEC data:

equation image

Regions of the electron density “seen” by the measurement is considered in (7) while subsets (8) and (9) respectively satisfy the real and complex portions of the Fourier transform of the electron density:

equation image
equation image

The end result is an augmented set of equations consisting of the original set of equations derived from the available data (7) and the additional set of equations derived from the band-limited constraints (8)–(9), where Pblr and Pbli are the matrix representations of the real and complex portions of the 3-D DFT operations. (8)–(9) result in the iterative formulae

equation image
equation image

In (10)–(11), xblr and xbli respectively represent the real and complex parts of the band-limited component of the density solution. The addition of the DC term, in (10)–(11), is necessary to satisfy the positivity requirement in Pblr and Pbli. In addition, computational efficiency is preserved by implementing the matrix multiplication operations Pxblrn, and Pxblin with DFT operations. The algorithm is also designed such that (7), (10) and (11) are balanced and the fraction of the solution, due to the TEC data and the band-limited constraint, is controlled in a flexible manner. The final solution produced by the three-dimensional band-limited block iterative ionospheric tomography reconstruction algorithm is such that, it satisfies the TEC data where available and the Fourier transform of the prior electron density within the remainder of the imaging volume. When there is a nonnegative solution of (2), the BISMART converges to the unique solution x closest to the starting point x0, in the sense that KL(x, x0) is minimized. This happens regardless of the configuration of subsets (7), (10), and (11). If no nonnegative solution of (2) exists, then the BISMART converges to a limit cycle of I distinct vectors, rather than a single vector. How distinct the limit cycle vectors are, depends on the extent to which the equations in (2) are (nonnegatively) inconsistent. In this case we can employ a “feedback” approach, using the vectors of the limit cycle to construct a “pseudodata” vector [Byrne et al., 1997]. A block diagram implementation of BLSMART is shown in Figure 1.

Figure 1.

Block diagram representation and implementation of the band-limited constrained block iterative simultaneous multiplicative algebraic reconstructed technique (BISMART).

3. Results and Conclusions

3.1. Results

[4] An experimental campaign involving both satellite observations for ionospheric tomography and the European Incoherent Scatter (EISCAT) radar was held in May 1995 at the Physics Department of the University of Wales. Radio transmissions from the Navy Navigational Satellite System (NNSS) were monitored by two receiver chains at five sites: Tromso (69.9°N, 19.2°E), Kiruna (67.9°N, 20.4°E), Lycksele (64.6°N, 18.8°E), Karasjok (69.5°N, 25.5°E) and Sodankyla (67.4°N, 26.6°E).

[5] The EISCAT radar was operated on 26 May 1995, 2055 to 2114 UT. The differential carrier phase of the received signals were observed, and the absolute TEC values were subsequently obtained using a multistation least squares fit of the TEC. Details of the experimental methods have been given by Kersley et al. [1993], and the data collection process is described by Mitchell et al. [1997]. The utility of the BISMART algorithm for, multiple-chain data acquisition, ionospheric tomography was evaluated using the TEC data from the experiment of 26 May 1995, with independent verification from the European Incoherent Scatter (EISCAT) radar data. The reconstruction process used for the tomographic imaging begins with the creation of the background ionosphere using PIM, corresponding to 26 May 1995 at 2055 UT, with SSN = 100 and Kp = 3. A satellite crossed 50°N at a longitude of approximately 19°E on 26 May 1995 and was monitored by all five receivers, as shown in Figure 2. Figure 2 also shows the three-dimensional volume that satisfies the data acquisition geometry. A top view, with a better depiction of the satellite-receiver acquisition geometry, is shown in Figure 3. The BISMART algorithm is used to reconstruct the distribution of the electron density in an ionospheric region defined by the following boundaries: 100 to 800 km in altitude, 1° to 27° in longitude, 46° to 89° in latitude. The three-dimensional volume was partitioned into 16 altitude, 64 longitude and 128 latitude planes for a total of 131,072 voxels. Because of the geometry of the data acquisition system as shown in Figure 3, longitude planes 44 to 50 (17.98°E to 20.41°E) cover regions within the imaging volume corresponding to the first chain, and longitude planes 37 to 43 (14.72°E to 17.57°E) cover the imaging volume corresponding to the second chain. The experimental campaign, involving both satellite observations for ionospheric tomography and independent verification from EISCAT allows correlating, the EISCAT data within the imaging plane at 19.2°E with a corresponding plane within the three-dimensional volume (see Figures 2 and 3). Figure 4 shows the EISCAT imaging plane data at 19.2°E which reveals two enhancement features at ≈(300 km, 63.75°N), ≈(264.25 km, 68.80°N) and a through at ≈(271.35 km, 66.25°N). In this example, if we use the planes within the imaging volume where there are some TEC data, the results should be comparable to the EISCAT imaging plane.

Figure 2.

Three-dimensional data acquisition and reconstruction geometry. The three-dimensional volume is constructed with 16 altitudes from 100 to 800 km, 64 longitudes from 1°E to 27°E, and 128 latitudes from 46°N to 89°N. The three-dimensional volume is made of 131,072 voxels. The satellite path shown is from the Navy Navigation Satellite System (NNSS). These satellite sampling points (Si) are in near-polar orbits at an altitude of about 1100 km, transmitting phase coherent signals on 150 and 400 MHz. The received signals from receivers (Ri) are used to estimate ionospheric TEC.

Figure 3.

Two-dimensional data acquisition and reconstruction geometry. The two-dimensional top view of the three-dimensional geometry with longitude and latitude spans from 1°E to 27°E and from 40°N to 89°N. The EISCAT data are at 19.2°E. Two chains of receivers are shown from 17°E to 21°E and from 25°E to 27°E. A planar view of the satellite pass, where the satellite crosses 50°N at 2055 UT, is also shown (courtesy of the Physics Department, University of Wales, Aberystwyth, UK).

Figure 4.

European Incoherent Scatter (EISCAT) radar-imaging plane at 19.2°E. The imaging plane was of constant longitude where the EISCAT radar data was measured. The EISCAT radar data are measured, by SP-UK-TOMO between on 26 May 1995 2109 UT, on a very limited grid of 8° by 400 km corresponding to regions of auroral ionospheric variations. The three main features of the EISCAT data are shown at ≈(300 km, 63.75°N), ≈(264.25 km, 68.80°N) and at ≈(271.35 km, 66.25°N) (courtesy of the Physics Department, University of Wales, Aberystwyth, UK).

[6] To illustrate this point, we use the original three-dimensional PIM background density as a starting point to BISMART and retrieve from the volume density solution, seven imaging planes of constant longitudes. The seven imaging planes are selected such that planes 44 to 50 (17.98°E to 20.41°E) correspond to the region of the three-dimensional volume where there are some TEC data and planes 37 to 43 (14.72°E to 15.57°E) match the areas where there are virtually no TEC data. As we enter the region within the imaging volume where there are some TEC data, the solution provided by BISMART is comparable to the EISCAT data as shown in Figure 5. However, the altitudes, and latitudes of the structures, depicted in all of these longitude planes, are incorrect. In Figure 5, the peak of the electron density is off by about 30 to 40 km as compared to the EISCAT electron density but depicts all of the prominent features of the EISCAT data throughout all of the interpolated planes. The BISMART solution does not provide the correct position of the peak electron density solution as expected and discussed by Raymund et al. [1993] and Kersley et al. [1993]. However, the technique provides density solutions in very good agreement with those seen by the EISCAT radar data not only at 19.2°E but also in all of the reconstructed planes. The differences in the enhancement (≈34 km and ≈31 km) and through (≈21.37 km) heights are attributed to the PIM prior model used for the creation of the background ionosphere in the reconstruction process, as discussed by Heaton et al. [1995] and Raymund et al. [1993], and comprehensive solutions to this problem are provided by Heaton et al. [1995], Walker et al. [1996], and Cornely [2003].

Figure 5.

Three-dimensional BISMART solutions where there are some TEC data (1e11*e/m3). The imaging planes IP44–IP50, 17.98°E to 20.41°E, are shown from top to bottom and left to right. The imaging plane at 19.2°E is the EISCAT data in Figure 4, shown here for convenience. In all of the imaging planes, the electron density is on the order of 1.6e11 (e/m3), corresponding to the higher contours. The differences in the enhancement (≈34 and ≈31 km) and through (≈21.37 km) heights are attributed to the PIM prior model used for the creation of the background ionosphere.

[7] The BISMART extrapolation therefore allows a flexible means for incorporating additional information, not contained in the available TEC data, into the reconstruction process. If we use the planes within the imaging volume where there is virtually no data, the resulting solution will reflect that fact, as shown in Figure 6. The BISMART density solution contains certain dominant features similar in geometrical characteristics with the ones in the EISCAT data. However, the electron density within the planes where there is no data is significantly different from the EISCAT data solution as would be expected.

Figure 6.

Three-dimensional BISMART solutions where there are no TEC data (1e11*e/m3). The imaging planes IP37–IP43, 14.72°E to 17.57°E, are shown from top to bottom and left to right. The imaging plane at 19.2°E is the EISCAT data in Figure 4, shown here for convenience. In all of the imaging planes, the peak electron density is on the order of 1.6e11 (e/m3), corresponding to the higher contours of electron density.

[8] The electron density in the imaging planes where there are virtually no TEC data, while consistent with a representative ionosphere predicted by a state-of-the-art model such as PIM or IRI, is deficient in sufficient information about the spatial distribution of the electron density. This deficiency in the BISMART process is hinting at the fact that the usefulness of the band-limited reconstruction process developed in this paper is highly dependent on the geometry of the data acquisition system being used. The remedy to this situation, in this specific case, is to add a few more chains of receivers to the data acquisition system between 21.00°E to 25°E, so as to augment the spatial distribution of the TEC data acquisition system. The augmentation in spatial distribution of the collected TEC data will not solve the error in the position of the peak electron density. In this case, we can adjust the peak and the magnitude of the density solution as discussed by Heaton et al. [1995], Walker et al. [1996], and Cornely [2003]. Figure 7 shows the TEC solution curves resulting from BISMART, EISCAT and PIM. The BISMART TEC closely matches the EISCAT TEC data, confirming our assumption that the solution provided by the entire ionospheric reconstruction system is consistent with the available data.

Figure 7.

Total electron content (TEC) solutions. The TEC data from the two chains are shown. The first chain is made of Tromso, Kiruna, and Lycksele (T, KI, L) and the second chain of Karasjok and Sodankyla (KA, S). Each receiver within a chain samples 400 TEC data points, resulting in a total of 2000 data points from the five receivers. The TEC data from EISCAT (EIS), background PIM, and the band-limited BISMART (BL) are shown.

[9] A fit between the TEC data obtained from the band-limited BISMART solution and the TEC data measured by the NNSS system corresponding to the EISCAT data is obtained. The BISMART and EISCAT TEC data are compared versus the PIM TEC data, in the root-mean-square (RMS) sense. Histograms of the RMS errors are shown in Figure 8. The RMS error histograms in Figure 8 depict higher counts at lower RMS error levels for the EISCAT TEC data as compared to the band-limited BISMART TEC data. In particular, the root-mean-square error levels above 1e13 (e/m2) between the EISCAT versus PIM TEC (top) and EISCAT versus BISMART TEC (bottom) are noticeably and considerably different.

Figure 8.

RMS histograms of band-limited BISMART and PIM versus EISCAT. The band-limited BISMART TEC data are compared with the EISCAT and PIM TEC data in the root-mean-square sense. (top) Root-mean-square histogram for EISCAT versus PIM. (bottom) Root-mean-square histogram for EISCAT versus the band-limited BISMART.

[10] Finally, the band-limited BISMART density solution is compared with the EISCAT density in the root-mean-square sense and the resulting error image is shown in Figure 9. The error image is deficient at the higher altitudes. This is primarily because the solution provided contains insufficient information about the position of the peak electron density as discussed by Raymund et al. [1993] and Kersley et al. [1993]. In addition, it is well known that EISCAT radars must have a calibration method in order to correct for F region peak height (up to 30 km depending on the rage gate used), and electron density (around 10–20%). Very careful analysis of these methods suggests that they are not more accurate than 10–20%. Many modes of EISCAT radars, at F region heights have their range gates set with a resolution accuracy of only 30–40 km. In addition, close examination of other inversion techniques for resolving the peak height of the electron density [e.g., Raymund et al., 1990, 1993; Pryse, 1997] all have accuracies within 30–40 km. Even in the case of occultation techniques as given by Hernández-Pajares et al [1998], usually considered to offer very precise peak density height information, the proposed solution is accurate to only 20–40 km. Figure 5 shows the peak height of the BISMART electron density solution to be about 30 km off from the EISCAT peak density at 300 km, which is within 10% of the EISCAT peak density and well within the error bar of the EISCAT radars. Figure 9 shows the BISMART electron density solution to be within 20 percent of the EISCAT density everywhere, which is within the error bar of the EISCAT radars as well. Therefore the accuracy of the proposed BISMART density solution is comparable with the state of the art.

Figure 9.

Root-mean-square surface between band-limited BISMART and EISCAT. The root-mean-square error surface between the band-limited BISMART density solution and the EISCAT data at 19.2°E is shown. The error surface shows that the error is minimal from 250 to 275 km. The error increases from 275 to 400 km and is maximal between 350 and 400 km, as expected.

3.2. Conclusions

[11] The two dimensional problem of reconstructing ionospheric electron density from ground-based receiver to TEC measurements is challenging because of the limited range of ray path angles associated with this geometry. Assuming absolute TEC data are available [Austen, 1996], a major issue associated with the development of an ionospheric tomography algorithm is the trade-off between the manner in which additional information is used to obtain a unique ionospheric electron density reconstruction from limited angle TEC measurements and the computational complexity of the resulting algorithm [Kuklinski, 1997]. Three-dimensional ionospheric reconstruction problems differ from two-dimensional ones because, in addition to providing a unique density solution from additional data, there are usually more voxels thereby leading to larger problems where direct inversion methods, and direct transform methods may not always be practical. However, three-dimensional ionospheric reconstruction systems have significant potential when compared to their two-dimensional counterparts. They can make full use of multiple-chain data acquisition geometry without being affected by receivers within individual chains removed from their ideal locations. In fact three dimensional reconstruction systems can benefit from such data acquisition geometry. The BISMART three-dimensional ionospheric reconstruction system presented in this paper extrapolates a density solution, within a chain of receivers, without additional cost in computational complexity. The algorithm provides an exact density solution, in the imaging planes where there are some TEC data, while a solution is extrapolated in the remaining planes of the imaging volume where there are virtually no TEC data. In the three dimensional algorithm discussed in this paper, as much as seven planes of constant longitudes (instead of one as per usual in the two dimensional case) were defined within the first receiver chain, all of which except for the position of the peak electron density, provide a very good match to the EISCAT data. Differences in electron density peak height in the BISMART solution are attributed to the PIM prior model used in the creation of the background ionosphere, as discussed by Heaton et al. [1995] and Raymund et al. [1993] and methods for capturing the geometrical features inherent to the density solution have been proposed by Walker et al. [1996] and Cornely [2003]. The peak height of the BISMART electron density solution, proposed in this paper, is about 30 km off from the EISCAT peak density at 300 km, which is within 10% off the EISCAT peak density and well within the error bar of the EISCAT radars. The BISMART electron density solution is within 20 percent of the EISCAT density everywhere, which is within the error bar of the EISCAT radars. Therefore the overall accuracy of the proposed BISMART density solution is comparable with the state of the art. When we move away from where there are some TEC data to where there are virtually no TEC data the provided solution, within these areas of the imaging volume, will eventually converge to the band-limited PIM as expected. The remedy to this situation, in this specific case, is to add a few more chains of receivers to the data acquisition system between 21.00° to 25°E, so as to augment the spatial distribution of the TEC data acquisition system. The reconstruction system exploits the fact that the structure of the BISMART algorithm, allowing subsets of data corresponding to the available TEC data and the band-limited constraints to be solved separately and efficiently, can be used as a platform for data assimilation from other sources such as GPS, and ionosondes. The BISMART algorithm is also designed such as to be computationally efficient at solving the large set of equations that typically results from ionospheric reconstruction problems formulated with a three-dimensional geometry. The algorithm provides very good solutions within three to five iterations in the noise-free case for every starting point and for every choice of data subsets. In the noisy case, the algorithm exhibits cyclic convergence rather than converging to a single vector [Byrne, 1996]. There is no proof in the literature for cyclic convergence of any block iterative methods except for the ones like ART that use the Euclidean distance. Nevertheless, Byrne [1997] has shown that the BISMART algorithm produces high-quality images fast, compared to SMART, and can be stopped after only a few iterations with good results, as is the case in this paper. The absence of convergence proofs for the BISMART, in this case, is therefore not considered an issue. The BISMART three-dimensional reconstruction system can easily be extended to four dimensions. The four dimensional system will provide a solution in near real time, that is approximately every 10 s, to the ionospheric reconstruction problem. Ionospheric response to geomagnetic storms is dependent on latitude and their effects on the ionosphere vary from the equator to the pole. Global Positioning Satellites anywhere on Earth are affected by changes in the ionosphere along the path to the satellite during magnetic storms [Klobuchar, 1991]. In addition, auroral electron density irregularities, such as those displayed in the Wales Experiment can produce scintillation in varying amounts depending on latitude. The equatorial region is a known site of the greatest ionospheric variations [Pryse et al., 1997], even under quiet magnetic conditions, as was the case in this paper. The band-limited BISMART technique, presented in this paper, has been successful in imaging the electron density in the auroral zone. The technique provides density solutions in reasonable agreement with the density solutions provided by the EISCAT radar and results from other authors who used the same EISCAT data set [Walker et al., 1996; Pryse et al., 1997]. Thus the features provided by the BISMART technique can be of significant utility in modeling the effects of traveling ionospheric disturbances and irregularities, in the auroral and equatorial regions, on radio waves.

Acknowledgments

[12] The authors wish to thank Charles L. Byrne from the Department of Mathematics at the University of Massachusetts at Lowell for his useful insights and some spirited discussion about the block iterative techniques and their application to emission tomography. The authors also wish to thank L. Kersley and Cathryn Mitchell from the Department of Physics, University of Wales, Aberystwyth, for their help with understanding the EISCAT data set that was used in this paper. The author also wishes to thank the associate editor, Steven Cummer, for his extremely insightful and useful comments on the accuracy of the solution proposed in this paper. His comments have greatly contributed to strengthening the merits of the results presented in this paper as compared with the state of the art. Final thanks go to the EISCAT scientific association, an international association supported by Finland (SA), France (CNRS), Germany (MPG), Japan (NIPR), Norway (NFR), Sweden (NFR), and the United Kingdom (PPARC).

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