Three-dimensional tomography of ionospheric variability using a dense GPS receiver array

Authors

  • Jeffrey K. Lee,

    1. Department of Electrical and Computer Engineering and the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
    2. Now at Applied Physics Laboratory, Johns Hopkins University, Laurel, Maryland, USA.
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  • Farzad Kamalabadi,

    1. Department of Electrical and Computer Engineering and the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
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  • Jonathan J. Makela

    1. Department of Electrical and Computer Engineering and the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
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Abstract

[1] This paper investigates the localized 3D imaging of equatorial ionospheric depletions and mesoscale traveling ionospheric disturbances (MS-TIDs) with a dense ground-based GPS receiver array. A tomographic forward model is presented using dual-frequency GPS measurements from receivers in the Japanese GPS Earth Observation Network (GEONET). The application of inversion techniques to the discrete observation matrix allows a 3D electron density volume to be reconstructed from the GPS total electron content (TEC) measurements. Because the inverse problem is ill-posed and ill-conditioned, regularization functionals are used to constrain the solution in the presence of noise. To capture the abrupt height and density variations of the structures, a technique using nonconvex penalty constraints is proposed. The reconstructions have 40 to 50-km horizontal and 30-km vertical resolution. Time series of reconstructed images show the evolution and movement of ionospheric structures through the reconstruction region. The results are analyzed and compared with independent ionosonde measurements and vertical TEC (VTEC) maps constructed from the observed GPS data.

1. Introduction

[2] The terrestrial ionosphere exhibits unpredictable short-term effects and localized anomalies in electron density. In the mid- and tropical latitudes, equatorial plasma depletions and traveling ionospheric disturbances (TIDs) have been studied using ground-based instruments such as radars and airglow cameras, space-based instruments, and in situ measurements (e.g., reviews by Makela [2006], Hunsucker [1982], and Hocke and Schlegel [1996]). These structures are of practical interest to the study of space weather since plasma structures and secondary growth of ionospheric irregularities along sharp gradients affect many types of radio wave signals.

[3] Equatorial depletions, or plumes, result from severe plasma instabilities generated at the magnetic equator and are commonly referred to as equatorial spread F (ESF) [Kelley, 1989]. They can grow vertically at the magnetic equator to altitudes over 1500 km at times. The primary instability mechanism driving the ESF is the Rayleigh-Taylor instability (RTI), first proposed in Dungey [1956]. The instability is triggered during post-sunset when the lower F-layer recombines with the neutral atmosphere, causing a very sharp vertical gradient in electron density, and when the eastward electric field generated by coupling processes between the E and F regions at the terminator pushes the F-region plasma vertically upward [Farley et al., 1986]. Density irregularities from the upward growing plume create strong electric fields at multiple scales that map along the magnetic field lines, affecting the local, off-equator ionosphere and extending the perturbation in a wedge-like manner. For large-scale electric fields (>10 km in extent), the magnetic field lines act as near-perfect conductors [Farley, 1959; Saito et al., 1995]. The higher altitude field lines have footprints at increasingly high latitudes, and thus, the effects of the irregularities can be measured off the equator [Makela, 2006]. These depletions can extend poleward 15°–20° or more and are often seen to bifurcate and tilt to the west over time [e.g., Mendillo and Tyler, 1983; Huang and Kelley, 1996; Makela and Kelley, 2003].

[4] TIDs are propagating, quasi-periodic fluctuations of the electron density and layer height. They were first observed by Munro [1950] using a network of ionosondes distributed over southeastern Australia. While large-scale TIDs (LS-TIDs) are known to be generated by high-latitude F-layer storm effects (see the review by Prölss [1995]), the physics behind the seeding and growth rates of mesoscale TIDs (MS-TIDs) are still poorly understood. Kelley and Miller [1997] have suggested that daytime and nighttime MS-TIDs have different triggering mechanisms. Daytime MS-TIDs could be driven by neutral atmospheric gravity waves via the mechanisms described in Miller and Kelley [1997]. Electrodynamic processes, such as electric fields, could have a role in generating nighttime MS-TIDs because at night, the plasma density in the E layer recombines and the E-region conductivity is significantly reduced.

[5] While much work has been done in understanding the general characteristics of these structures, imaging the variability would provide further insights on their behavior. Because the ionospheric electron density is an important parameter for monitoring and characterizing the ionosphere, tomographic techniques have been used to produce multi-dimensional electron density profiles to study ionospheric structures. 2D ground-based radio tomography, first proposed in Austen et al. [1988], has been used to image high-latitude troughs [Kersley et al., 1997] and traveling ionospheric disturbances (TIDs) [Pryse et al., 1995; Bernhardt et al., 1998]. In Comberiate et al. [2007], space-based UV tomography is used to image plasma depletions in the low-latitudes; this is a 2D approach that requires the reconstruction area to be invariant over 10° of magnetic latitude, which is unsuitable for imaging MS-TIDs or non-field-aligned structures. A 3D variational data assimilation technique (3DVAR) proposed in Bust et al. [2004] has been used to image and track high-latitude patches and blobs [Bust and Crowley, 2007]. This approach relies primarily on GPS and low Earth-orbit (LEO) radio beacon measurements to capture variations against a physics-based background model.

[6] One problem with current 3D tomographic methods is the difficulty in imaging ionospheric structures such as turbulent flow and plasma instabilities. For instance, physics-based data-assimilative schemes, such as those described in Bust et al. [2004], Hajj et al. [2004], and Schunk et al. [2004], include a physics-based background model or solve for a set of parameterized equations that describe some governing physics regarding the electron density. However, the accuracy of the reconstructions suffers if the physics behind these phenomena is not completely understood. Also, methods that rely on constraints such as global smoothness or adherence to “average” ionospheric profiles may be effective for certain scenarios, but have limited applicability to situations where constraints do not represent the phenomena being studied.

[7] In this paper, we report on a new technique for obtaining 3D electron density images of structured ionosphere. The approach is data-driven in the sense that reconstructed ionospheric variations are not constrained by physical models or empirically derived basis sets. The data sources used in our technique are GPS ground-based absolute slant total electron content (TEC) measurements and ionosonde data. The algorithm we propose uses regularization functionals to stabilize the solution in the presence of noise. The local structure is preserved by using nonconvex, edge-preserving regularization functionals.

[8] The experimental results focus on the local ionosphere over the Japan region. We take advantage of the dense GPS receiver coverage in the area to reliably obtain 40 to 50-km horizontal and 30-km altitude resolution. Images are compared at different times to track the evolution of structures as they pass through the reconstruction region. Because receiver networks in general tend to be sporadically clustered or constrained by physical boundaries, we investigate the choice of grid discretization and placement to fully exploit the available GPS measurement geometry.

2. Ground GPS and Ionosonde Data

[9] The GPS Earth Observation Network (GEONET) [Miyazaki et al., 1997] is a dense collection of over 1000 GPS dual-frequency receivers spread throughout Japan. The GPS data were processed using techniques described in Saito et al. [1998] and Otsuka et al. [2002]. The processing involves calculating the code- and phase-measurement TEC, fitting the code to the phase, correcting for cycle slips, removing the satellite biases using post-processed estimates, and removing the receiver biases. It has been used successfully in the past to study the 2D structure of a variety of ionospheric phenomena [Saito et al., 2001, 2002; Tsugawa et al., 2003].

[10] At any given time, there are usually at least three active ionosondes in the Japanese region available for obtaining vertical electron density profiles. However, they are far apart, so typically only one ionosonde will be near the reconstruction region being considered here. Manually scaled factor values calculated on an hourly basis from ionograms are available from the World Data Center (WDC) for the Ionosphere at the National Institute of Information and Communication Technology (NICT), Tokyo (http://wdc.nict.go.jp/). Figure 1 shows the ionosonde network and the GEONET GPS receivers. The ionosonde stations are Wakkanai (45.39°N 141.7°E), Kokubunji (35.7°N 139.5°E), Yamagawa (31.2°N 130.6°E), and Okinawa (26.3°N 127.8°E) (pre-2001). The Okinawa station was moved northeast by about 50 km in November 2001 to Okinawa/Ogimi (26.7°N 128.2°E) (post-2001).

Figure 1.

Distribution of GEONET GPS receivers (circles) and Japanese ionosondes (squares).

2.1. Noise Model and Plasmaspheric Contribution

[11] It is assumed that the bias for each individual receiver is constant over several days for all GPS measurements. Sardón and Zarraoa [1997] studied the biases and found that the satellite bias errors are small (±0.43 TEC units (TECU)) compared to those of the receiver bias (±1.52 TECU). However, some additional variance of these bias estimates is to be expected due to residual errors in fitting the code and phase measurements to form the TEC estimate. We assume that after post-processing, the bias errors are independent and identically distributed (iid) Gaussian for all observations.

[12] The TEC measurements contain both the ionospheric and plasmaspheric contribution. Because the plasmaspheric contribution is always positive, we model it as Gaussian noise with nonzero mean. We use Figures 2 and 3 in Lunt et al. [1999] to estimate the mean plasmaspheric TEC values during the appropriate times in the solar cycle and for the magnetic dip angles. The dip angles at the grid location (generally between 40° and 50°) are taken from the International Geomagnetic Reference Field (IGRF) website (http://modelweb.gsfc.nasa.gov/models/igrf.html). The plasmaspheric contribution for each ray path is calculated as

equation image

where TECps is the plasmaspheric contribution, TECps,90° is the mean vertical plasmaspheric contribution, re is the radius of the Earth, ϕel is the elevation angle, and Hp is the height at the lower edge of the plasmasphere, which is set to the upper edge of the reconstruction grid. This value is then subtracted from the total TEC to obtain the TEC in the ionosphere. The sum of all the errors is assumed to have 50% noise confidence limits of ±3 to 4 TECU.

3. Theory and Methodology

[13] The observed slant TEC along the satellite-to-receiver ray path is given by

equation image

for i = 1 … I, j = 1 … J, where Ne is the electron density, the first term in the right hand side is the slant TEC along the satellite-to-receiver ray path, t is the time, Bi is the receiver bias for the ith receiver, Bj is the satellite bias for the jth satellite, I and J are the number of receivers and satellites, and ri, sj are the ground receiver and satellite positions, respectively.

[14] A parametric representation of (2) using basis functions is written as

equation image

where Ne(sk, t) is the electron density at a sampling point sk, {ak}1K is the set of weights at the sampling points, and nij is the measurement and model error. We use weights corresponding to the ray-path lengths through an elliptical grid where the voxels are divided into longitude, latitude, and altitude, and then locally rotated around a vertical axis to align with the receiver geometry. Voxels are used as the basis set to more naturally represent sharp spatial gradients.

[15] The set of ray-path equations can be written as a discrete algebraic problem, given as

equation image

where y is the TEC data, H is the observation matrix with weights corresponding to the ray-path lengths, x is the electron density in each voxel, and n is the observation noise.

3.1. Tomographic Reconstruction

[16] The goal of tomographic reconstruction is to extract the electron density x from the observation vector y. For limited-angle tomography problems, the instability of the solution is linked to the data being “incomplete”; the TEC measurements are limited to certain directions. For ill-conditioned problems, the noise error can overwhelm the solution, so small errors in the data can lead to large errors in the solution. The only viable methods for solving x are those that regularize the problem in some way.

[17] We consider a formal framework for regularization that incorporates a priori information about the underlying structure of the ionosphere. The general form of the regularization cost equation used in this paper is

equation image

where the first term is the least-squares residual norm and the other terms are the penalty constraints. In the penalty terms, α1 and α2 are the regularization parameters, D1 and D2 are the regularization matrices, ϕ1 and ϕ2 are the regularization functions, x0 is a reference image, and l1, l2 refer to the individual elements of the vector inside the brackets.

[18] The structure of the regularization matrix encapsulates the a priori information. We want to incorporate the fact that the ionosphere is typically smoothly stratified without relying on physics-driven estimates. The choice of a discrete approximation of a derivative for the regularization matrix captures the notion of a roughness penalty (smoothness constraint). The first derivative gradient constraint between adjacent voxels is written as D[x(l)] = x(l) − x(l + 1), where l is the voxel index along a single horizontal or vertical dimension. The second derivative gradient constraint takes the form D[x(l)] = −x(l − 1) + 2x(l) − x(l + 1). Because the ionosphere is characteristically different in the horizontal and vertical directions, D1 is mainly used to preserve horizontal smoothness using first derivative gradients and D2 is mainly used to preserve vertical stratification using second derivative gradients.

[19] For imaging structures, we use nonconvex cost norms for ϕ(·) that increase less rapidly than the quadratic form for sufficiently large arguments, thus penalizing them less in the cost equation. Depletions are characterized by sharp electron density gradients in an otherwise smoothly varying ionosphere. The first penalty term in (5) can be used to produce a smooth background ionosphere with sharp edges forming the boundaries of the depletions. MS-TIDs are typically smooth, wavelike structures with minimal change in electron density, but with appreciable fluctuations in height. The second penalty term constrains the smooth background ionosphere to a reference image while allowing for significant height variation within the structured regions.

[20] In this paper, the nonconvex cost norms are derived from a general class of ϕ functions, proposed in Delaney and Bresler [1998], that is based on a family of ρ functions: ρ = 1/[1 + (t/T2)q] for some positive T and q, where ρ(t) = ϕ′(equation image). For q → 0 or for small values of t/T, ϕ(t) becomes the (scaled) equation image2 norm used in Tikhonov (quadratic) regularization. For q = 0.5 and T = 0.5, ϕ(t) closely resembles the equation image1 cost function used in TV regularization [Vogel and Oman, 1996; Karl, 2000]. For q > 0.5, equation image is nonconvex; as q increases, ϕ becomes flatter, so large gradient values are less penalized.

[21] Efficient algorithms designed for minimizing convex cost functions will not necessarily work well for nonconvex functions. To solve (5), Delaney and Bresler [1998] derive a deterministic relaxation technique to minimize a succession of convex minimization problems that will globally converge to a local minimum for ϕ functions meeting certain conditions. The deterministic relaxation technique iteratively minimizes a set of cost equations, written in the general form as

equation image

where J0 is an approximation to the original cost function that is convex in x. Each iteration requires a guess for x, defined as equation image. For each element l in the summation terms,

equation image

Thus, the algorithm consists of minimizing J0(x(b+1); e1(x(b)), e2(x(b))) for b ≥ 0 starting with an initial guess x(0). A popular minimization technique that can be used on (6) is the conjugate gradient method [e.g., Bertsekas, 1999].

[22] Once the algorithm converges to a fixed-point solution, the right-hand side of (6) can be written as

equation image

where Wk is a diagonal matrix with elements ek,l(·). Taking the derivative of (8) and setting it to zero gives the minimum solution, equation image, which satisfies

equation image

[23] In our study, since the general location of a feature of interest can be inferred though the VTEC maps, it makes sense to focus the reconstruction at or near the feature itself. For a given set of grid boundaries, a smaller grid (hereby referred to as the “inner grid”) is created. Depending on the scenario, this inner grid can be placed around a structure or over the densest concentration of receivers. Generally, the inner grid will focus on the 350-km VTEC pierce points, especially if the region of interest is around that altitude in the F-region peak. For simplicity, we keep the altitude voxel discretization constant throughout the entire grid.

[24] If ray paths with inner grid VTEC pierce points pass through the outer grid, the assumption is that those ray paths will generally pass through the upper and lower altitude voxels of the outer grid where the electron densities are lower; the relative contribution of the boundary voxels will be small. Thus, to allow for greater peak height perturbations from the reference image, we strongly enforce the reference constraint along the outer grid voxels and less strongly in the inner grid. Also, to prevent the image from having faulty boundary values in the upper and lower altitude voxels, we use additional smoothing and boundary constraints to enforce continuity with the reference image.

3.2. Estimating the Ionosonde Profile Using Manually Scaled Ionosonde Values

[25] The electron density profile at an ionosonde location can be derived by calculating the peak F2 electron density (NmF2) and height (hmF2) from the ionosonde data and using these values with an estimated scale height in a Chapman profile [Chapman, 1931]. The peak electron density can be calculated directly from the ionosonde measurement by the equation NmF2 = (f0F2/8.980)2 electrons/m3, where f0F2 is the measured critical frequency of the F2 layer.

[26] Following several authors [Maruyama et al., 2004; Xu et al., 2004], the hmF2 value is calculated using empirically derived correction terms. Let M(3000)F2 be defined as the ratio of the maximum usable frequency (MUF) at a distance of 3000 km to the F2 layer critical frequency, f0F2 (i.e., MUF(3000)/f0F2). A relationship between hmF2 and the propagation factor M(3000)F2 was first proposed by Shimazaki [1955] as

equation image

which uses a parabolic distribution of electron density. A correction term for the retardation due to ionization in the lower layers was added in Bradley and Dudeney [1973] as

equation image

where ΔM = (0.18)/((f0F2/f0E) − 1.4).

[27] In Bilitza et al. [1979], based on data from ionosondes and incoherent scatter radars, a different expression for ΔM was proposed as

equation image

where

equation image

In addition to the ratio f0F2/f0E, (12) depends on the solar activity, R, (i.e., R12, which is the 12-month average sunspot number) and the geomagnetic latitude, Φ. We use Bilitza's method for estimating hmF2 in this work. With the Japan ionosonde data, f0E data often are only available during midday, since the scaling of f0E is often affected by sporadic E and interferences. Following Maruyama et al. [2004], we use empirically determined values of f0E [Muggleton, 1973] when the observed values are not reliable or can not be discerned from the ionograms.

4. Regularization Parameter and Resolution Selection

[28] An important task in using our reconstruction method is selecting the optimal values of the regularization parameters. Parameter selection techniques are generally divided into methods that depend on knowledge or estimates of the observation error norm, ||n||, and methods that seek to extract that information. The methods we examined are representative of these classes: Unbiased Predictive Risk Estimator (UPRE) [Mallows, 1973], Generalized Cross Validation (GCV) [Golub et al., 1979], and L-curve [Hansen and O'Leary, 1993]. The latter two methods do not require an estimate of the noise variance. The GCV method aims to find the regularization parameter that can best predict missing data values. If an arbitrary observation, yi, is left out, then the regularized solution should be able to predict this missing observation from the remaining observations in y. The L-curve is a heuristic graphical method that plots the regularization penalty norms verses the residual norm and looks for a characteristic “elbow” in the curve that marks the compromise between the minimization of these two terms.

[29] In our experiments, we found that the UPRE method generally outperforms the other regularization selection methods when the noise variance can be reasonably estimated. The UPRE method is based on minimizing the predictive risk, defined as Lequation image||p||2 = equation image||equation imageytrue||2, where ytrue = Hxtrue, equation image = Hequation image for some regularized solution equation image , and m is the dimension of y. Although ytrue cannot be directly computed, it can be estimated based on the properties of the observation noise. If the receiver and satellite biases are assumed to be iid Gaussian noise with zero mean and variance σn2, the UPRE estimator is defined as

equation image

where the expected value of U[L] is equal to the expected value of the predictive risk. The matrix M is sometimes referred to as the influence matrix because Hequation image = My, and so M describes how well Hequation image predicts y. In our algorithm, we use M = H(HTH + α1D1TW1(x)D1 + α2D2TW2(x, x0)D2)−1HT; the derivation can be found in Lee et al. [2007]. The regularized solution will typically change little with small changes in α [Vogel, 2002]. Thus, to reduce the computational load of the parameter search, we can estimate a range of possible values for the regularization parameter through noise statistics or numerical experiments.

[30] The grid resolution is determined using a class of methods expressed as a minimization problem,

equation image

where S is a function, v is the model order, V is the set of candidate resolutions, and equation image(v) is the “optimal” regularization parameter for a given resolution. To solve (14), we first solve for the optimal α using the UPRE method at each resolution. Sharif and Kamalabadi [2005] investigated various model order selection approaches to determine optimal resolutions using the UPRE, GCV, and a Bayesian approach. For instance, with the UPRE method, (13) is used for S in (14). The optimal discretization resolution using the Bayesian framework is written as

equation image

This method differs from the UPRE and GCV methods because it can assign a probability to the resolutions considered. For instance, using a uniform p(v) removes the last term in (15). Other heuristic methods where higher complexities are penalized are discussed in Rissanen [1989, 1996] and MacKay [1992].

[31] In this study, we examined different horizontal resolutions of the inner grid while keeping the vertical resolution fixed. Because the vertical resolution does not change, we can fix the value for α2, which controls the influence of the reference image based on the reliability of the a priori information. As long as the image is generally smooth or piecewise smooth in the horizontal direction, the optimal regularization parameter should not vary too significantly between different resolutions. Using 30-km altitude voxels, we found that the resolution selection methods usually agree to within 10 km of each other in the horizontal direction. In our experiments, the optimal horizontal resolution corresponds to 40 to 50-km length voxels in the inner grid.

5. Numerical Experiments Using Model Data

[32] To examine the effectiveness of our reconstruction method, we designed test structures by modifying simulated electron density volumes taken from the International Reference Ionosphere (IRI) 2001 model [Bilitza, 2001]. These structures are models (i.e., phantoms) used to test the algorithm's ability to capture horizontal and vertical variations. The equatorial-originating depletion is modeled as an elongated region pointing towards the geomagnetic pole with the poleward edge extending to 24.5° geomagnetic north. This region is the effect of the density irregularities from the equatorial depletion creating strong electric fields that map along the magnetic field lines, affecting the local, off-equator ionosphere. The depletion is 75-km wide, occupies the area between 200 to 800-km magnetic altitude, and contains 10% of the background electron density. The depletion is modeled with a backwards C-shaped figure to simulate the tilt that develops as it drifts through the ionosphere. In the northern hemisphere, MS-TIDs have been observed traveling in the southwest direction [e.g., Garcia et al., 2000; Saito et al., 2001] and appear to be conjugate in nature [Otsuka et al., 2004]. The MS-TID is simulated as an elongated structure 125-km wide in the northeast to southwest direction and 1000-km long. The height modulation is modeled using a half-sinusoid vertical shift; the peak height fluctuation is 60 km, which is consistent with the radar observations in Behnke [1979].

[33] The ray-path geometries from the corresponding real scenarios are used to calculate the TEC values. The reference image is constructed by using the density profile from the ionosonde at Kokubunji. We focus on the southernmost tip of Japan to image the depletion since it is rare for these structures to travel farther north than this. However, the MS-TID travels over most of mainland Japan, so we use a different grid that is locally rotated and centered over the densest concentration of receivers. Figure 2 shows the grids used in the reconstruction. The 50% confidence limits of the receiver noise is set to ±1.5 TECU and the receiver-independent noise and residual error is set to ±2 TECU.

Figure 2.

Grid geometries used in reconstructions of (a) depletion, and (b) MS-TID.

[34] Figures 3 and 4show the longitudinal cross sections of the simulated images and reconstructions. For each reconstruction scenario, 30 different noise realizations are used to generate the mean reconstructed image. The black voxels are areas without ray-path information. The outer boundary grid voxels at the upper and lower latitudes are not expected to match the simulated results well. However, the reconstructions in the inner grid are generally consistent with the simulated images. One of the disadvantages of the GEONET is that the receiver network is physically surrounded by water, which means that away from the grid center, the ray-path angle density suffers. With the depletion results, the lower latitudes are essentially off-network, so the edges will start to smear and become more ambiguous. But since the structure is largely depleted in the vertical direction and has sharp horizontal gradients, it is possible to achieve a satisfactory level of latitude/longitude location accuracy in the reconstruction even when there are only a few ray-path angles passing through various sections of the depletion. The MS-TID reconstruction is more sensitive to grid placement because the ray-path angle density plays an important role in resolving plasma height variations, although using a rotated rectangular grid helps in resolving the height variations in all the inner grid longitudinal cross sections. We define the normalized error as equation image. In the inner grid, the normalized errors for the depletion and MS-TID reconstructions are 20% and 10%, respectively, which is an improvement over the normalized errors of 30% and 18% with the reference images constructed from the IRI profile at the ionosonde location.

Figure 3.

Depletion: (a) simulation, and (b) reconstruction. The longitudinal cross sections are in the inner grid.

Figure 4.

MS-TID: (a) simulation, and (b) reconstruction. The top and bottom longitudinal cross sections are of the outer boundary grid. The jagged black areas are voxels without ray-path information.

6. Reconstructions Using Actual Observation Data

[35] In this section, we discuss the application of our algorithm to actual GPS observations. We examine three different times corresponding to different ionospheric conditions: quiet conditions on March 15, 2000 (Day 075), a plasma depletion on April 7, 2002 (Day 097), and an MS-TID on July 27, 2004 (Day 209). The GPS observation data for each reconstruction are taken from only one time instance, which allows us to capture snapshots of moving structures. The ionosonde values are available hourly. Although there are four ionosonde stations in Japan, we use profiles at Kokubunji for the reference image.

6.1. Baseline Case

[36] As a baseline case, we examine a quiet time (Kp = 0) ionosphere on March 15, 2000 at 0400 UT. Figure 5 shows the grid setup over the VTEC map and the longitudinal cross sections of the reconstruction. The optimal resolution when imaging an ionospheric region without significant horizontal gradients is generally coarser than when imaging ionospheric structures. The reference image is derived from ionosonde data from Kokubunji, which is located inside the inner grid boundaries, but the reconstructed bottomside profile at the ionosonde location is not forced to match the reference profile. The peak electron density error of the reconstruction at Kokubunji is within 12% of the measured ionosonde data. Figure 6 compares the ionosonde-derived Chapman profiles with the closest reconstructed profiles. It is important to note that the ionosonde stations at Wakkanai and Okinawa are outside the reconstruction region, and the good match seen in Figure 6 is a result of the lack of spatial gradients in the quiet time ionosphere over Japan. The VTEC maps derived from the reconstructions (not shown) are also consistent with those derived from the data; the average difference between the two maps in the inner and entire grid is 2.2% and 4.0%, respectively.

Figure 5.

Quiet time: (a) nonuniform grid discretization overlayed over of VTEC map, and (b) reconstruction at 0400 UT. The jagged black areas are voxels without ray-path information.

Figure 6.

Comparison of closest reconstructed profiles with ionosonde-derived Chapman profiles during quiet time ionosphere.

6.2. Structure Imaging

[37] Figure 7 shows the grid setup and reconstructions during the depletion event on April 7, 2002 at 1325 UT. The grid setup is shown overlayed over the observation VTEC map. The reference image was derived from the Kokubunji ionosonde at 1300 UT. Figure 8 shows the time progression of the depletion from 1300 UT to 1335 UT in the bottom center longitudinal slice in Figure 2a. The results show the depletion moving in the eastward direction; the average drift velocity is about 75 to 100 m/s. The depletion edges are tilted to the west with increasing altitude above the F-region peak. Since depletion boundaries are field-aligned, the westward tilt with altitude is consistent with the VTEC maps which show a westward horizontal tilt with increasing poleward latitude.

Figure 7.

Depletion: (a) nonuniform grid discretization overlayed over VTEC map, and (b) reconstruction at 1325 UT. The longitudinal cross sections are in the inner grid.

Figure 8.

Time progression of depletion from 1300–1335 UT in bottom center longitudinal slice in Figure 2a.

[38] Figure 9 shows the grid setup and reconstructions during the MS-TID event on July 27, 2004 at 1325 UT. Ionosonde data at Kokubunji were not available so the reference image parameters were estimated as being between the values from the Yamagawa and Wakkanai stations. The results show an elongated, SW-facing region with peak height fluctuations of 30 to 60 km. Correlation studies between 630.0-nm airglow emission observations and GPS TEC measurements have shown that the electron density at the MS-TID height uplifts is lower than the directly adjacent regions [Ogawa et al., 2002]. The airglow intensities are strongly dependent on F-region densities in part because the 630.0-nm emissions are due to the dissociative recombination of molecular oxygen ions O2+; these ions exponentially decrease with altitude. Ogawa et al. [2002] showed a clear spatial coincidence between the bright (dark) airglow regions and positive (negative) TEC fluctuation regions during an MS-TID event over Japan, suggesting that upward MS-TID fluctuations have lower electron densities. Figure 10 shows the time progression of the MS-TID in the bottom center longitudinal cross section in Figure 2b. The MS-TID was determined to be moving in the southwestward direction with a horizontal phase speed of 100 to 150 m/s and has a wavelength of ∼200 km, consistent with airglow observations of similar events [Garcia et al., 2000; Shiokawa et al., 2003].

Figure 9.

MS-TID: (a) nonuniform grid discretization overlayed over VTEC map, and (b) reconstruction at 1325 UT. The jagged black areas, mostly in the upper altitudes, are voxels without ray-path information.

Figure 10.

Time progression of MS-TID from 1300–1335 UT in the bottom center longitudinal slice in Figure 2b.

[39] Figure 11 shows a reconstructed cross section that is parallel to the MS-TID propagation direction. The image is interpolated using Newton's interpolation formula for vector arguments in 3D. If a gravity wave was propagating along the direction of the MS-TID, then mapping the gravity wave shear to the magnetic field lines would result in a MS-TID sinusoid that was tilted, or sheared. Unfortunately, the original reconstruction in Figure 11 below 600 km is too coarse to show that shearing effect. Above 600 km, the GPS ray path geometry does not have enough ray paths angles and the electron density is too small for our algorithm to reliably reconstruct ionospheric structures. The same issues occurred with another reconstruction (not shown) using a grid locally rotated along the MS-TID propagation direction. While this is a drawback to our approach, understanding the resolution limitation is important in being able to definitively answer questions related to ionospheric behavior.

Figure 11.

Reconstruction of MS-TID using interpolated cross section aligned along MS-TID propagation path.

[40] Figure 12 compares the corresponding ionosonde and reconstructed profiles during the structure events. Only 3 of the 4 ionosondes in the network had data during each event. The reference images were derived from ionosonde data at Kokubunji; the other ionosondes are independent measurements. Because the outer grid voxels are not necessarily good indicators of the reconstruction quality, we try to compare the ionosonde data with the closest reconstructed profiles in the inner grid. We make one exception by using the outer grid SW corner profile in the MS-TID comparisons. This is because the VTEC values near the Yamagawa site are very different from the nearest inner grid values; the corner profile seems more indicative of the ionosonde profile. We exclude comparisons with Wakkanai during the structure scenarios because of the large distance from the station to the receivers used in the reconstructions.

Figure 12.

Comparison of ionosonde-derived Chapman profiles with relevant grid profiles during (a) depletion, and (b) MS-TID, both at 1300 UT.

6.3. Validation

[41] To confirm the reliability of our method, we compare the reconstructions with ionosonde data and VTEC maps constructed from the observation data. Table 1 shows the average percentage differences between the observed and reconstruction-derived VTEC values. The values in the inner grid are within 6%. Most of the errors in the entire grid comparisons can be attributed to the oversmoothing of significant density gradients in the large corner boundary voxels during the structure events.

Table 1. Average Percentage Differences Between Observed and Reconstructed VTEC Values
VTEC DifferenceQuiet (3/15/00) 0400 UTDepletion (4/7/02) 1325 UTMS-TID (7/27/04) 1325 UT
inner grid2.22.95.4
entire grid4.05.617.2

[42] Table 2 shows the percentage differences between the NmF2 and hmF2 values from the reconstructed profiles and ionosonde locations, calculated as equation image · 100 for some parameter β (either NmF2 or hmF2). Despite the spatial differences, the comparisons show that the hmF2 and NmF2 results are bounded by ionosonde data that are north and south of the reconstruction region. Also, these values are consistent with ionosonde data at similar latitudes. Lastly, Table 3 shows the percentage differences between the parameters from the reconstructed profiles and reference profile.

Table 2. Percentage Differences Between Closest Inner Grid Reconstructions and Ionosonde Profiles
SiteQuiet 0400 UTDepletion 1300 UTMS-TID 1300 UT
NmF2hmF2NmF2hmF2NmF2hmF2
  • a

    Comparisons with Wakkanai relevant for quiet time only.

  • b

    Ionosonde data not available.

  • c

    Estimated ionosonde data.

  • d

    Comparison with outer grid corner reconstruction.

Wakkanai8.406.62N/AaN/AaN/AaN/Aa
Kokubunji11.91.5023.7−2.028.10b,c4.12bc
YamagawaN/AbN/AbN/AbN/Ab−0.75d−6.46d
Okinawa−2.24−2.23−32.36.37−27.6d−8.4d
Table 3. Percentage Differences Between Closest Inner Grid Reconstructions and Reference Profile
SiteQuiet 0400 UTDepletion 1300 UTMS-TID 1300 UT
NmF2hmF2NmF2hmF2NmF2hmF2
  • a

    Comparisons with Wakkanai relevant for quiet time only.

  • b

    Ionosonde data not available.

  • c

    Comparison with outer grid corner reconstruction.

Wakkanaia8.401.50N/AaN/AaN/AaN/Aa
YamagawaN/AbN/AbN/AbN/Ab24.0c−3.78c
Okinawa22.21.5010.66.7524.0c−3.78c

7. Conclusion and Future Work

[43] We have presented a tomographic technique for imaging ionospheric structures using GPS and ionosonde data. This work generalizes and extends the methodology in Lee et al. [2007], which focused on quiescent ionospheric conditions. We have developed a robust form of a cost equation that is suitable for imaging both quiescent and structured ionospheres, and does not require modification for different scenarios. For future applications, depending on the a priori information, the cost equation could be modified for different structures if identification is not a priority. Another addition to the method of Lee et al. [2007] is the use of an inner grid to take advantage of the unique physical layout of the GEONET receivers in order to avoid artifacts due to the sparseness, nonuniform distribution, and limited-angle nature of the GPS ray paths.

[44] We examined two classes of ionospheric structures: equatorial-originating depletions and MS-TIDs. The depletion reconstructions were very promising. The westward tilt in altitude and latitude were consistent with previous radar studies. More impressively, the algorithm could effectively locate and reconstruct the sharp edges of the depletion even in areas with few GPS ray-path angles. We also demonstrated that the algorithm could capture appreciable height fluctuations of MS-TIDs. The location and scale sizes of these upward lifts were consistent with radar and airglow studies. In validating the results, the figures and tables showed that the reconstructions were reasonably bounded by the independent ionosonde measurements and observed VTEC values.

[45] There are a number of ways this algorithm can be enhanced. Additional data sources such as GPS occultation measurements (e.g., from the COSMIC constellation) and satellite EUV/FUV limb scans could be included. More sophisticated ionosonde data processing, for instance through true height inversion programs, can give better estimates of the hmF2 and bottomside scale height. Also, other background models showing horizontal variation could be considered for the initial background field. In areas of dense ionosonde coverage, multiple ionosonde profiles could be incorporated in the reference image. A general source of error is that the plasmaspheric model uses extrapolated values. Since the plasmaspheric contribution is diurnally invariant, the relative error to the GPS data is more severe at night. A more sophisticated model is desired for evaluating the plasmaspheric effect, especially in the lower latitudes.

[46] The algorithm described in this paper is different from previous 3D methods in that significant small-scale ionospheric gradients can be recovered from the TEC data. However, in some regions, the reconstruction cannot be considered entirely reliable due to the GPS ray path geometry, for instance, such as in high altitudes or other areas with limited number of ray-path angles. The next step would be to consider how to combine our methodology in this work with model-based or physics-driven reconstruction algorithms. A combined approach could provide a more reliable overall solution based on a priori information, but also a solution that does not wash away small-scale details. This paper described a procedure to help achieve the latter, and could be used as an intermediate step in developing such a hybrid-type approach.

Acknowledgments

[47] This work was supported in part by the National Science Foundation under grant ATM 01-35073 and the Navel Research Laboratory under grant N00173-05-G904 to the University of Illinois, and supported in part by the NASA Jet Propulsion Laboratory. The GPS data and processing were provided by Akinori Saito.

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