Localized three-dimensional ionospheric tomography with GPS ground receiver measurements


  • Jeffrey K. Lee,

    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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  • 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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[1] This paper describes a technique for 3-D tomographic imaging of the ionosphere with high spatial resolution (75 to 95 km in latitude/longitude and 30 km in altitude). The total electron content (TEC) values are derived from dual-frequency measurements obtained from GPS satellites by ground-based receivers. When available, ionosonde data are used to construct a priori vertical profiles modeled using Chapman functions. Two regularization algorithms are investigated for tomographic image reconstruction: Tikhonov and Total Variation (TV), corresponding to quadratic and l1 norm minimizations of the penalty constraint, respectively. The TV method is used because it generally preserves discontinuities in the image effectively and is more resistant to noise. By contrast, Tikhonov, or quadratic, regularization tends to oversmooth image structures and discontinuities. However, a closed-form solution of the TV method does not exist and so performance depends heavily on numerical optimization techniques, which are nontrivial to implement because the inverse problem is both ill-posed and ill-conditioned. We also apply regularization parameter-selection methods to demonstrate their applicability in our study. The algorithms are demonstrated using real GPS TEC measurements from an observation geometry centered in southern California. We demonstrate the performance of these techniques under quiet, midlatitude conditions. The resulting reconstructions reasonably determine the shape of the ionospheric profile. Artifacts can potentially appear near voxels with no raypath information, which is directly related to the sparseness and nonuniform distribution of the GPS raypaths. We discuss some methods to constrain the solution to realistic bounds.

1. Introduction

[2] Ionospheric imaging using 2-D computerized tomography was first proposed in Austen et al. [1988]. In ground-based radio tomography, coherent radio beacon transmissions, typically at 150 and 400 MHz, from low Earth orbit (LEO) satellites are tracked by an array of ground receivers. The image reconstruction problem is typically formulated using a series expansion (weighted sum) of basis functions. The inverse problem using the raypath geometry is ill-posed with the limited number of viewing angles of the measurement geometry being the fundamental difficulty in its solution. This geometry is dictated by the number and spacing of the available ground receivers and satellite orbits. Furthermore, the ionospheric layers and other horizontal structures are difficult to reconstruct without horizontal view angles. Some of the theoretical limitations of ionospheric tomography from limited data are described by Yeh and Raymund [1991] and Raymund et al. [1994].

[3] If there are insufficient raypath angles, a unique solution does not exist. Even if the number of equations is larger than the number of unknowns (overdetermined), the matrix still can be ill-conditioned in the sense that small measurement errors can lead to large perturbations in the solution. Since raypaths close to the horizontal direction are not available, some form of a priori information is necessary to improve the resolution of the reconstructed vertical profile.

[4] Early work used iterative techniques commonly encountered in the medical imaging literature [Austen et al., 1988; Kak and Slaney, 1988; Natterer, 1986]. Later studies used linear least squares methods on a stochastic or semistochastic formulation of the inverse problem [Fremouw et al., 1992; Nygrén et al., 1997]. In stochastic inversion, both the observations and unknown electron densities are treated as stochastic variables. This formulation is used for Bayesian approaches such as maximum likelihood (ML) or maximum a posteriori (MAP) estimation [Kamalabadi et al., 1999, 2002]. The semistochastic formulation assumes that unknown electron densities are deterministic, but the observations are stochastic. The covariance matrix of the observation noise can be modeled using both formulations.

[5] For ill-conditioned systems, least squares methods require matrix inversions that are numerically unstable. To avoid numerical instabilities in the presence of noise, regularization using a priori information is needed. In Nygrén et al. [1996, 1997], the regularization functional incorporates variance information using peak value, height, and thickness of the electron density as a function of altitude. In Fehmers et al., [1998], a model-independent approach is taken using constraints based on ionospheric physics. Horizontal smoothness, vertical stratification, and convex constraints are imposed as a reference model; the solution space is also constrained to a convex set.

[6] In addition to transmitting radio beacon signals, LEO satellites have been used to obtain occultation data from GPS satellites. In this case, the LEO satellite takes the place of the ground-based receiver. The GPS radio occultation technique was tested for the first time using GPS/MET, a receiver on a small research satellite [Rocken et al., 1997].

[7] More recently, slant TEC measurements from ground receivers observing multiple GPS satellites have been used for 3-D ionospheric reconstruction. For 3-D ionospheric tomography, the traditional tomographic algorithm proposed by Kaczmarz [Kaczmarz, 1937], which forms the basis of ART used in Austen et al. [1988], will not converge to a viable solution. Because of the ill-posedness of the problem, the only viable methods are those that regularize the problem in some way. A 3-D ionospheric tomography method has been recently described by Mitchell and Spencer [2003] with experimental results presented by Yin et al. [2004]. Their method uses orthogonal basis function sets to limit the solution space, and is based on the 2-D tomographic imaging work by Fremouw et al. [1992]. A spherical harmonic expansion is used to create the horizontal profile, and empirically determined orthogonal functions (EOF) are used to generate the vertical profile. The inversion is accomplished using singular value decomposition (SVD). Using the same tomographic method, IRI model and ionosonde data were used to constrain the results in Cillers et al. [2004]. Their algorithm is sensitive to the choice of basis functions since the EOFs only allow a certain range of possible solutions and will work better on certain images than for others.

[8] Another 3-D tomographic method using neural networks is described by Ma et al. [2005]. The neural network minimizes a set of squared residual functions estimating the electron density as well as the receiver and satellite biases. This method trains the neural network using observed TEC and density values at particular points on the ionosonde data. The implementation and robustness of the training phase is critical in obtaining suitable weights for the neural network.

[9] In this paper, we limit the TEC data sources to absolute slant TEC measurements at ground-based receivers from GPS satellites. In contrast to the 3-D studies discussed above, we employ regularization methods to stabilize the solution in presence of noise. In addition to commonly encountered Tikhonov (quadratic) regularization, we also examine the method of Total Variation (TV) for reproducing ionospheric profiles for localized tomographic reconstruction. This method is used because it generally preserves discontinuities in the image more effectively and is more resistant to noise. By contrast, Tikhonov or quadratic regularization tends to oversmooth image structures. A closed-form solution of the TV method does not exist, and so performance depends heavily on the algorithmic formulation. We present results from both simulated and real TEC measurements. We use regularization parameter-selection methods and discuss their applicability to this study. Lastly we discuss the implications for future incorporation of additional raypaths and occultation-derived vertical profiles.

2. Forward Model

[10] The general approach of calculating the TEC from dual-frequency information is described by Lanyi and Roth [1988]. With dual-frequency GPS signals, using the carrier phase measurements at the two frequencies currently transmitted by each GPS satellite gives a precise relative TEC. The code, or range measurements, are much noisier, and give a less precise absolute TEC measurement. After correcting for cycle slips and fitting the phase measurements to the code measurement (e.g., using least squares fitting or a Kalman filter), the resulting error is assumed to be on the order of ±0.01 TEC units (TECU). However, there are biases caused by a differential delay between the two GPS frequencies in both the receiver and satellite hardware which must be accounted for to reach this level of accuracy. The integrated ionospheric density in the slant-range direction is given by

equation image

for i = 1…I, j = 1…J, where Ne is the electron density, s is the path along the satellite-to-receiver raypath, t is 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 receivers and satellites positions, respectively.

[11] The satellite biases are assumed to be constant on the order of days and can easily be subtracted out. The receiver biases can be mitigated by using measurements from different receivers, usually taken at night to minimize differences in ionospheric delays. The measured delay is assumed independent of the elevation angle, excluding low angles. In addition, the receiver biases are assumed constant up to several days. Using different time and elevation-angle dependencies between the measurements, the biases can be solved simultaneously and the differences averaged out [Mannucci et al., 1998]. Our data was processed using the MIT Automated Processing of GPS (MAPGPS) software package [Rideout and Coster, 2006]. This software calculates the code- and phase-measurement TECs, fits the code to the phase, removes the satellite biases using post-processed estimates, and removes the receiver biases. It has been used successfully in the past to study the 2-D structure of a variety of ionospheric phenomena [Foster et al., 2002; Coster et al., 2003; Nicolls et al., 2004].

[12] In this work, we are primarily interested in reconstructing the ionosphere from 100 km to 800 km. Note that the line integral in (1) includes the TEC contribution along the entire raypath from satellite to receiver, including the plasmasphere. The plasmaspheric contribution is fairly uniform as a function of time of day, with typical values for the latitude range of interest on the order of 1-3 TECU [Lunt et al., 1999a, 1999b]. For our experiments, we focus on daytime and nighttime conditions near solar minimum and estimate the plasmaspheric contribution as Gaussian noise with nonzero mean. A parametric representation of (1) using basis functions is written as

equation image

where Ne(sk, t) is the electron density at a sampling point k, {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 raypath lengths through an elliptical grid where the voxels are divided in longitude, latitude, and altitude.

[13] The set of raypath equations in (2) can be written as a discrete algebraic problem, given as

equation image

where yequation imageequation imagem×1 is the absolute slant TEC from GPS observations, Hequation imageequation imagem×n is the observation (weighting) matrix corresponding to the discrete grid, xequation imagen×1 is the electron density at each voxel, and n is the noise. After the GPS data are processed, n can be represented as Gaussian noise with covariance matrix Σn.

3. Three-Dimensional Model Formulation

[14] LEO satellites have orbital periods of roughly 90 min, so these satellites can cover the entire angular span of the observation plane in a short period of time. Thus, for the 2-D problem using radio beacons where the reconstruction grid is on the observation plane, the raypath angles and coverage density is primarily determined by the number and spacing of the receivers. However, this setup does not allow for an arbitrary placement of the observation plane; the placement is dependent on the orbital trajectory of the satellite.

[15] For 3-D imaging, the current GPS satellite configuration does not allow a similar level of uniform coverage as in the 2-D case. GPS satellites have orbital periods of slightly less than 12 hours and are located in six orbital planes. Depending on the grid resolution, there can be voxels without line integral information inside the grid. The regularization equation relies on neighboring voxels to constrain the solution, so we would want to minimize the number of voxels with little or no raypath information.

[16] We need to choose the grid size and placement for a particular receiver-satellite geometry. Generally, that means using the minimum grid size for a set number of viewable satellites. While a larger grid would have more TEC raypaths and angles due to more viewable satellites and usable receivers, the limited number of viewable satellites means less uniform raypath coverage over the entire grid. With a larger grid, the middle grid voxels will contain more raypath angles, but surrounding edge voxels will have fewer raypath angles. This is problematic when using smaller voxels for high resolution imaging.

[17] It is also preferable to maximize the number of raypaths in areas of high ionospheric variation. For instance, the grid placement can be changed to allow more raypaths to pass through areas with more ionospheric structure. The tradeoff between better local coverage density versus smaller coverage gaps is influenced by how well the models or reference profiles can be used to accurately describe regions with less raypath information.

[18] The ionosphere does not typically change significantly during a 5–10 min time window. Hence, observations from multiple time steps are used simultaneously to increase the number of raypath angles intersecting the grid. However, having too many raypaths intersecting the grid can lead to noise artifacts appearing in the solution because the tomography matrix inversion resembles a least squares solution using a matrix with a large condition number. So even though the GPS receiver data samples are typically available every 30 s, the actual sample rate used in the reconstructions may be reduced to avoid this problem. In addition, because the ground receivers are arbitrarily spaced, some receivers might be located very close to one another. We avoid including too many receivers along a particular line geometry so as not to over-constrain the corresponding voxels in the reconstruction.

4. Regularized Tomographic Reconstruction

[19] Because the inverse problem is ill-conditioned, the validity of least squares solutions quickly degrades as the noise increases. The instability of the solution is linked to the data being “incomplete;” the TEC measurements are limited to certain directions. Our algebraic setup using only ground receivers is severely ill-conditioned; the singular values of the forward matrix decay exponentially. In algebraic reconstruction, there are a variety of methods designed to cope with the ill-conditioning of the forward matrix, including truncating the singular values in the SVD reconstruction [Bertero and Boccacci, 1988]. A general framework for regularization is formulated using a cost function written as

equation image

where W is an appropriate weighting matrix, and Ci and γi are the ith regularization functional and regularization parameter, respectively. The weighted norm is defined as ∥xW2 = xTWx. The first term controls the data fidelity by minimizing the least squares residual. The second term, also called the penalty or constraint term, controls how well the reconstruction matches the a priori knowledge of the solution.

[20] The most common type of regularization has a quadratic penalty function, C(x) = ∥Dx22, where D is an appropriately chosen regularization matrix. The structure of the regularization matrix encapsulates the a priori information. For example, the choice of a discrete approximation of a derivative for a regularization matrix captures the notion of a roughness penalty (smoothness constraint). Kamalabadi et al., [1999] have shown the stochastic interpretation of quadratic regularization by formulating it as a statistical estimation problem. Assuming Gaussian statistics for both unknown image and noise, quadratic regularization can be expressed as the maximum a posteriori (MAP) estimate

equation image

where Σn and Σx are the noise and prior covariances, respectively.

[21] Because the ionosphere is characteristically different in the horizontal and vertical directions, we use two penalty terms to incorporate the a priori information. The regularization cost functional is written as

equation image

where α1 and α2 are the regularization parameters, D1 and D2 are the respective regularization matrices, and x0 is a reference profile or image. We define the lp norm as ∥xp = (∑∣xp)1/p. Solving the cost equation with p = 2 is quadratic regularization. Using p = 1 with regularization matrices that are not identity matrices is known as Total Variation (TV) regularization.

4.1. Tikhonov Regularization

[22] Tikhonov regularization, first introduced in Tikhonov [1963], uses a quadratic penalty functional to constrain the solution. We formulate the cost equation using a vertical reference profile derived from a Chapman profile. It may not be possible to completely omit the constraint using the reference profile (i.e., ∥D2(xx0)∥pp) from the cost equation, depending on the distribution of receivers and viewable satellites required for a given resolution. The Tikhonov cost equation minimizes

equation image

where the terms are defined in (6).

[23] We want to incorporate the fact that the ionosphere is typically smoothly stratified without relying on non-data-driven estimates. To do so, we use four different types of regularization matrices. Let r, ϕ, and θ be the radial, latitudinal, and longitudinal directions on an elliptical grid. Let Dr be the second derivative gradient matrix in the radial direction and Dϕ,θ = [DϕTDθT]T be the first derivative gradient matrix in the latitude/longitude directions. Furthermore, let Db be a boundary constraint matrix and Dw be a weighing matrix for voxels with ionosonde or other data matching the reference profile. These matrices can be stacked to form D1 and D2. For example, the cost equation in (7) could be written as

equation image

where ηb and ηw are additional optional parameters. In general, there can be multiple boundary and weighting matrices corresponding to different sections of the grid.

[24] It is generally not a good idea to include the first derivative matrix in D2 because using the reference image with this regularization matrix is too restrictive as it forces the entire reconstruction to adhere to the parameters of the reference profile. The reference constraint with the first derivative matrix is particularly sensitive to the peak height and scale height of the reference image. However, the reference constraint with the second derivative matrix is less restrictive than with the first derivative matrix. The second derivative matrix tries to maintain the general shape of the vertical profile, but does not enforce continuity in the horizontal direction. It does not severely penalize reference profiles that underestimate the peak electron density, but follows the general shape of the true vertical profile.

[25] To prevent the image from having faulty boundary conditions in the upper and lower rows of voxels, we enforce additional smoothing constraints to account for the small voxel values and to ensure that the electron density at these layers is generally uniform. A weighting matrix multiplied by a small constant is used to constrain these voxels to a value that is close to, but not necessarily exactly the values from the reference profile. We use this approach instead of forcing the boundary voxels to a set value because the a priori density is generally interpolated or unknown.

[26] Within regions where the electron density is small, the reconstructed electron density can sometimes be negative. We constrain the electron density values to be within realistic bounds by setting the negative values to zero after the regularization algorithm is completed.

[27] The resulting expression that minimizes (7) is

equation image

where equation image is the regularized solution.

4.2. Total Variation Regularization

[29] One of the drawbacks of Tikhonov regularization is that by suppressing the effects of high-frequency noise, it also reduces the high-frequency energy in the image and hence blurs edges or steep gradients. This is problematic for localized tomographic imaging with image structures, high gradient areas in the vertical and horizontal profiles, and areas where the gradient of the reference image does not match that of the true image.

[30] TV regularization is a nonlinear technique that tries to preserve edges or discontinuities in the image [Vogel and Oman, 1996; Karl, 2000]. TV regularization minimizes the cost equation

equation image

where the terms are defined in (6). The l1 norm used in the TV technique does not penalize high frequency energy terms in the image as severely as the quadratic norm used in the Tikhonov method; it is better at preserving jump discontinuities. In addition, the l1 norm is more resistant to noise and incorrect peak or scale height in the image.

[31] Because the cost function is not differentiable at the origin, we use the following smooth approximation to the l1 norm,

equation image

where [Dx]i denotes the i-th element of the vector Dx and β ≥ 0 is a small constant. The gradient operation for the approximation is

equation image

where W(x) is a diagonal matrix with i-th element ([Dx]2i + β)−1/2 and di is the i-th row of D.

[32] Thus, the minimum solution to (10) must satisfy

equation image

where W1(equation image) and W2(equation image, x0) are the weighting matrices in the horizontal and vertical direction, respectively.

[33] The TV algorithm uses a fixed point iteration to solve for x. Using a constant W(xcurrent), an iterative algorithm such as conjugate gradient minimizes the cost equation. A new W(xnext) is calculated and the process is repeated until a suitable convergence condition is satisfied. Some papers include a positivity constraint, or other convex projection inside the fixed point iteration that tries to constrain the image within a valid solution set [e.g., Stark, 1987; Kamalabadi and Sharif, 2005]. This step relies on an a priori convex solution set and does not necessarily lead to a good solution. However, depending on the problem setup, there might be artifacts which are difficult to control using standard regularization techniques. Enforcing positivity inside the fixed point algorithm in this case is a desirable approach.

5. Creating a Reference Image From Ionosonde Data

[34] A commonly used reference profile is the 1-D Chapman profile [Chapman, 1931], expressed as

equation image

where N(h) is the electron density as a function of altitude h, N0 is the maximum electron density (NmF2), h0 is the height of maximum electron density above the Earth's surface (hmF2), and Hs is the effective scale height. The peak electron density, N0, can be estimated 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.

[35] While incoherent scatter radar (ISR) can give both topside and bottomside ionosphere profiles, the number of ISR locations and the frequency of observations are limited. Ionosondes are more numerous and can operate year-round, but only give enough information to reconstruct the bottomside profile of the F2 layer. Standard “true height” inversion programs, such as POLAN and ARTIST, are available to calculate the bottomside profile [Titheridge, 1985, Reinisch and Huang, 1983].

[36] A reference image can be estimated from the ionosonde data even though the scale height is not directly given by the ionosonde data. Another method to introduce a scale height for the topside profile is to use an α-Chapman equation with variable scale height Hs(h) [Rishbeth and Garriott, 1969]:

equation image

where NB(h) is the bottomside profile, Hm is the scale height at the F2 peak, and Hs(h) is the variable scale height. Then, the topside density profile can be obtained by using the shape of the profile at the F2 peak to derive an estimate of the topside scale height, HT. Huang and Reinisch [2001] show how to obtain Hs(h) and argue that HT = Hm is a good estimate of the topside scale height.

6. Noise Model

[37] 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 TECU) compared to those of the receiver bias (±1.52 TECU). The total bias error is then expected to be approximately ±2 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. In our technique, the sum of the bias errors is assumed to have 50% noise confidence limits of ±3 to 4 TECU. The confidence limits specify the endpoints of a confidence interval where the probability that a given parameter lies within that range.

[38] Because the plasmaspheric contribution is always positive, we model it as Gaussian noise with nonzero mean. This should be a valid assumption over the time- and spatial-scales of this experiment. We use Figures 2 and 3 in Lunt et al. [1999a] to estimate the TEC values during solar minimum at our location. The dip angles for our grid location (between 57° and 63°) are taken from the IGRF website (http://modelweb.gsfc.nasa.gov/models/igrf.html). The plasmaspheric TEC along the elevations of interest is between 1 to 3.3 TECU, assuming no geomagnetic storm activity. In our experiment, we estimate the plasmaspheric contribution as Gaussian noise with mean 2 TECU and standard deviation of 1 TECU. A more detailed noise model would correlate the mean depending on satellite look angle, but given the small amount of the plasmaspheric contribution, this level of detail should not be consequential for this study. In summary, after taking account of all the bias errors, residual errors, and plasmaspheric contribution, the TEC noise, n, is estimated as Gaussian with mean 2 TECU and a 50% noise confidence interval between ±3.5 to 4.5 TECU.

7. Regularization Parameter Selection

[39] An important task in using our reconstruction model is selecting the optimal value of the regularization parameter, especially for operational use. In selecting the regularization parameter in (6), we investigated three commonly used methods: Unbiased Predictive Risk Estimator (UPRE) [Mallows, 1973], L-curve [Hansen and O'Leary 1993, and Generalized Cross Validation (GCV) [Golub et al., 1979]. The parameter search is complicated because there are two regularization parameters (α1, α2), but the selection methods are designed for finding one variable. The ratio between the parameters are not image invariant, nor is there an optimal value as a function of the two variables. Either the ratio or one of the parameters has to be set beforehand.

[40] For the parameter selection methods, we need to express the regularized solution, equation image, in terms of the data y through a linear operator K. Also, let M = HK. This matrix is sometimes referred to as the influence matrix because Hequation image = My, and so M describes how well Hequation image predicts y.

[41] The regularization cost equation in (6) can be written in terms of a single penalty function with simple substitution. Under the MAP framework in (5), the cost equation can be expressed as

equation image

where xN(0, Σx1)N(x0, Σx2) under Gaussian assumptions. The statistics for x can be rewritten as [N(x0, Σx1 + Σx2)]x=0N(xb, Σx), where the first term is evaluated at x = 0, xb = (Σx1−1 + Σx2−1)−1Σx2−1x0, and Σx = (Σx1−1 + Σx2−1)−1. Let nequation image(0, σn2I), where σn2 is the variance of the observation noise, and Σxi = equation image(DiTDi)−1 for i = 1, 2. Minimizing (16) is equivalent to minimizing (6) for p = 2 if we define αi = (σnγi)2. The quadratic approximation of the TV regularization method uses equation image.

[42] The cost equation in (16), which is equivalent to

equation image

can be transformed to standard form (i.e., (5)) by substituting equation image = x - xb and equation image = yHxb. Thus, the influence matrix for equation image can be written as M = H(HTΣn−1H + Σx−1)−1HT.

7.1. Selection Methods

[43] The three regularization parameter methods investigated here are representative of two classes with different assumption of the error norm, ∥n∥, where n is the noise term in (3). These methods either rely on a good estimate of ∥n∥, or seek to extract this information if the noise statistics are not known. Also, two of the three methods are analytical while the other is a graphical approach. We describe each method below and comment on their applicability to this problem.

[44] The UPRE finds the regularization parameter that minimizes the predictive risk, defined as

equation image

where ytrue = Hxtrue. 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 becomes

equation image

where equation image = Hequation image and m is the length of y. The UPRE method returns a weighted sum of the residual error and the trace of the influence matrix. In our experiments, we found that the UPRE method gives a reasonable estimate of the regularization parameter when the 50% noise confidence interval is set to the noise variance; the confidence interval is estimated to be between ±3.5 to 4.5 TECU. This method does seem to slightly underestimate the TV regularization parameter for reasonable model orders, but the method is generally robust and consistent.

[45] Hansen and O'Leary [1993] describe a graphical method for choosing the regularization parameter, called the L-curve method, which consists of plotting the regularization penalty norms and the residual norm, r = ∥Hequation imagey2, for different realizations of λ. The plot typically has a characteristic L-shaped appearance. The point of maximum curvature on the L-curve, or “corner,” provides the optimal regularization parameter. The L-curve method does not require knowledge of the noise statistics. In our experiments, we found that an L-shaped curve does appear, but the corner is not very sharp. The reconstruction changes significantly in the range of values on the L-curve. While this method is useful in constraining the range of possible parameter values, it is not precise enough to pick an optimal value.

[46] The generalized cross-validation (GCV) method looks 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 regularized parameter should also be independent of an orthogonal transformation on y. The GCV method minimizes the expression

equation image

This method also does not require an estimate of σ2n. Also, unlike ordinary cross validation, it is invariant to linear transformations and is less computationally intensive [Golub and Loan, 1996]. In our experiments, the GCV method gives parameter values that are generally lower than the UPRE method. If the problem is too ill-conditioned, the GCV is known to give very low values. We prefer the UPRE method for its robustness and flexibility in incorporating our noise variance estimate.

7.2. Resolution Selection

[47] The grid resolution can be determined using a class of methods expressed as a joint minimization problem,

equation image

where S is a function, v is the model order, and Vm is the set of candidate resolutions. The UPRE and GCV methods can be adapted to search through both regularization and resolution parameter spaces [Sharif and Kamalabadi, 2005]. These methods, which calculate the predictive risk, can be used to compare different grid geometries and resolutions. The downside is the computational difficulty. The function S may possess many local minima, complicating the search for a global minimum. However, the regularized solution will typically changes little with small changes in α [Vogel, 2002]. Thus, for a given resolution, we can estimate a range of possible values for the regularization parameter through noise statistic or numerical experiments. In our experiments, we looked at different horizontal resolutions while keeping the vertical resolution fixed. Because the vertical resolution does not change, we can fix the value for α2. As long as the image is generally smooth or piece-wise smooth in the horizontal direction, the optimal regularization parameter should not vary too significantly between different resolutions.

[48] Using fixed grid boundaries and real data, we found the optimal horizontal resolution to be between 8 × 8 to 12 × 12. In our reconstructions, the reconstruction scheme is fairly robust to different resolutions (to within 2–3 horizontal voxel dimensions). The selection curves, generated by solving (21) over different resolutions, generally become flat at higher resolutions due to increasing boundary constraints. As the resolution increases, more coverage gaps appear, so the number of boundary voxels constrained by the reference image increases. If the model order is too large, the regularization parameter selection methods give oversmoothed solutions; the artifacting is so prevalent that the only stable solution is an oversmoothed reconstruction.

8. Reconstructions Using GPS TEC Measurements

[49] This section presents the reconstruction results using post-processed GPS TEC measurements collected on April 2, 2005 (Day 92) during two 5-min spans. The first (2235 to 2240 UT) corresponds to local daytime conditions, while the second (0845 to 0850 UT) corresponds to nighttime. Ionospheric conditions were quiet at these times; the Kp index was around 1. The GPS data are obtained from a worldwide network of receivers available on the web. Data from many U.S. based receivers are available on the Continuously Operating Reference Stations (CORS) website (http://www.ngs.noaa.gov/CORS/).

[50] The line integrals of electron density (TEC) values are calculated for GPS receivers centered around southern California. This location was picked because of the high concentration of receivers in the area, the majority monitoring the San Andreas fault line. The TEC information is taken from at least the four highest elevation GPS satellites at each receiver. When selecting the location of the grid boundaries, all available receivers in the region were considered. The grid is defined in the WGS-84 coordinate frame. The altitude range of interest is 100 km to 800 km. Valid raypaths have to intersect the grid boundaries below 100 km and above 750 km. This eliminates raypaths with significant contribution of the ionosphere outside the reconstruction volume. In addition, satellites lower than 30° in elevation are ignored because of potential multipath effects in the calculated TEC data. In Figure 1, we show the distribution of available receivers and the grid boundaries used for the reconstruction; the number of receivers with valid line-of-sight information is smaller.

Figure 1.

Receiver network and grid boundaries, southern California region.

[51] Unfortunately, different satellite elevation angles and the nonuniformity of the receiver positions makes it nontrivial to calculate the optimal grid layout. An exhaustive search computing the condition number over all grid sizes and model orders is computationally prohibitive. Instead, we use a heuristic method to find the grid size. The starting point is to calculate the minimum latitude/longitude range for a grid that contains raypaths from the q highest satellites to a selected receiver point. It is preferable to place this point around the largest concentration of receivers; the grid size can then be adjusted because the raypath information is redundant around a large concentration of receivers. In our experiments, we use q = 4 satellites.

[52] In addition, to avoid overweighing the voxels along a particular line geometry, we allow no more than 1 receiver within a d-km radius. A heuristic algorithm is used: for each receiver, we find all the receivers that are within a d-km radius. Afterwards, we remove the elements of the largest group except one. This is repeated until each remaining receiver is more than d-km away from every other receiver. In our experiments, we use d = 10 km.

[53] The grid is divided into 25 × 10 × 10 voxels (radial/latitude/longitude). The grid altitudes are 80 to 830 km; the top and bottom layers are used as boundary constraints. The voxel sizes are 30 km in altitude and between 75 to 95 km in horizontal length. The grid is located at 32° to 40° N and 115° to 124° W. In choosing the grid placement, we looked to minimize the average squared estimation error, given by E[eTe] divided by the number of voxels, where e = (xequation image) for some regularized solution equation image. This approach assumes statistics for the unknown image and noise, and only pertains to voxels with raypath information. We also wanted to maintain a fairly uniform coverage of raypaths and raypath angles throughout the grid. Figures 2 and 3 show the raypath geometry corresponding to the grid placement in Figure 1.

Figure 2.

Grid setup showing reduced resolution and number of raypaths, daytime (22:35 UT).

Figure 3.

Grid setup showing reduced resolution and number of raypaths, nighttime (8:45 UT).

[54] In our data experiments, Point Arguello, CA is used as the ionosonde site; the site is located inside the grid boundaries at 35.6° N, 120.6° W. Reference images are constructed using a Chapman profile derived from the ionosonde data at 8:45:5 UT and 22:35:5 UT.

8.1. Boundary Constraints and Nested Gridding

[55] A major obstacle in our reconstructions is the nonuniform coverage of the GPS raypaths. We would like to solve only for the voxels with raypath information to avoid using model-based estimates of the electron density inside the coverage gaps. For horizontal and vertical rows with large coverage holes, the smoothness constraints are not as effective near these gaps, which can lead to artifacts in the image. This is especially true near the reconstruction perimeter, though at higher grid resolutions, these coverage gaps become more pronounced and can occur throughout the reconstruction. The TV algorithm is more adversely affected by these coverage gaps than the Tikhonov method.

[56] To avoid this problem, we include the boundary voxels in the coverage gaps in our reconstruction. These coverage gap voxels have no raypath information, so it is necessary to assign values to these voxels; the simplest option is to constrain these voxels with the corresponding voxels from the reference image. For a quiet nighttime ionosphere with little change in NmF2 or hmF2 over the region of interest, it may be sufficient to rely on a horizontally invariant reference image, such as a Chapman profile derived from ionosonde data. For ionospheres with horizontal density gradients, we use a nested-gridding technique. This technique iteratively increases the reconstruction resolution by using lower resolution images to generate higher resolution reconstructions. After each iteration, the voxels at the ionosonde site are adjusted to match the known bottomside profile measured by the ionosonde. Voxels without any raypath information are assigned the closest latitude voxel value. This image is then upsampled and used as the reference image for the next, higher resolution reconstruction.

[57] To incorporate the boundary constraints in the cost equation framework, we solve (6) for the coverage gap and neighboring boundary voxels. The coverage gap voxels are constrained to the reference profile values using a weighing matrix multiplied by a small constant, similar to the approach described in section 4.1. Smoothness constraints are used between the coverage gap and adjacent boundary voxels, also multiplied by a small constant to mitigate the effects of incorrect reference values in the coverage gaps.

[58] We explored whether it was possible to reconstruct the image without boundary constraints at the coverage gaps. The Tikhonov method can give reasonable images, but the TV method often completely fails because of artifacts. The accuracy of the parameter choice methods was influenced by wayward artifacts and the GCV method often completely failed due to the severe ill-conditioning of the inverse problem. Thus, we concluded that having boundary constraints at the coverage gaps is necessary for this problem.

8.2. Simulated Results

[59] We first test the algorithm using line-of-sight TEC measurements generated using simulated high-resolution images from the PIM model [Daniell et al., 1995]. Using the noise model developed in section 6, we add zero-mean Gaussian noise to the simulated TEC values; the 50% noise confidence limits are set to ±3.5 TECU. The grid and raypath geometry is the same as the experimental setup. The reference image uses a Chapman profile fitted to the bottomside profile at the ionosonde location; the topside profile is modeled using a slightly higher scale height. Boundary constraints are placed on the very high and low altitudes where the electron density is small and fairly uniform horizontally, and along the coverage gaps.

[60] The latitudinal cross-sections of the PIM simulated images are shown in Figure 4. Figure 5 shows the reference images used in this experiment. For both reconstructions, a 5 × 5 (latitude/longitude) resolution image is generated with the Tikhonov algorithm. Figures 6 and 7show the Tikhonov and TV reconstructions based on the experimental geometry with the UPRE method selecting the parameters. The black voxels are voxels without raypath information. The reconstructions are a compromise between the reference image and the true image. The extent that the reconstruction matches the true image depends on the geometry of the raypaths and the ill-posedness of the inverse problem. For the simulated images, we define the normalized error as equation image. The normalized error for the daytime and nighttime reconstructions were 14% and 8%, respectively. The normalized error for the Chapman reference images were 23% and 14%, respectively.

Figure 4.

PIM generated ionosphere profiles, (a) daytime, and (b) nighttime.

Figure 5.

Simulated reference images (upsampled 5 × 5 (latitude/longitude) resolution): (a) daytime, and (b) nighttime.

Figure 6.

Tikhonov reconstruction using simulated data, (a) daytime, and (b) nighttime.

Figure 7.

TV reconstruction using simulated data, (a) daytime, and (b) nighttime.

8.3. Experimental Results

[61] In this section, we use observed TEC data from approximately 250 GPS receivers in southern California with a vertical profile derived from the ionosonde location. The ionosonde data are taken from the Space Physics Interactive Data Resource (SPIDR) website (http://spidr.ngdc.noaa.gov/spidr/) to create the initial reference image. This is a data-driven reference image, not a model-generated image (e.g., PIM). As part of the nested gridding procedure, we generate a 5 × 5 resolution image to use for the 10 × 10 reconstruction. Figure 8 shows the reduced resolution images used in the reconstructions. Figures 9 and 10 show the Tikhonov and TV reconstructions for daytime and nighttime with parameters selected by the UPRE method. The general trends in these reconstructions correspond to the derived slant VTEC maps and the Global Ionosphere Maps produced by the Center for Orbit Determination in Europe (CODE). For a quiet ionosphere, the Tihkonov and TV methods give similar results. The TV method does contain slightly larger jump discontinuities between neighboring voxels, as expected. This attribute will be useful in the future when imaging ionospheric structures, instead of just a quiet ionosphere.

Figure 8.

Real reference images (upsampled 5 × 5 (latitude/longitude) resolution): (a) daytime, and (b) nighttime.

Figure 9.

Tikhonov reconstructions using real data, (a) daytime, and (b) nighttime.

Figure 10.

TV reconstructions using real data, (a) daytime, and (b) nighttime.

[62] The reconstructions would benefit from additional raypaths, such as from ground receiver to LEO data or GPS-LEO occultation data. At this time, without additional truth information, it is difficult to assess which reconstruction is more accurate. However, as a self-consistency check, we next look at the reconstruction profile at the ionosonde location.

8.4. Removal of Bottomside Vertical Profile at Ionosonde Location

[63] In our experimental reconstructions, a weighting matrix forces the reconstructed bottomside profile at the ionosonde location to match the ionosonde reference profile. As a self-consistency check, if we remove this weighting matrix, the profile at the ionosonde location should be close to the truth data. Figure 11 shows the daytime reconstructed profile using the same TV parameters without the vertical constraint at the ionosonde location. Even without the weighting matrix, the reconstructed profile matches fairly well with the ionosonde profile. Note that the ionosonde-derived Chapman profile is only an approximation with an estimated topside scale height.

Figure 11.

TV reconstruction profile at ionosonde site; voxels not explicitly forced to reference profile values.

9. Conclusion

[64] We have described a technique to reconstruct a localized 3-D electron density volume of the ionosphere using GPS TEC data. The primary data sources are ground-based receivers tracking GPS satellites over southern California. The regularization algorithms, Tikhonov and Total Variation, have the same residual norm but use different penalty norms to regularize the solution. The regularization methods take advantage of the horizontal smoothness and vertical stratification of the ionosphere to constrain the solution. We successfully combined different regularization matrices and ionosonde data to obtain realistic profiles in the presence of instrument biases, observation noise, and model discretization errors. For quiet, midlatitude conditions, both regularization methods behaved similarly. The TV method will have greater applicability in capturing edges when imaging ionospheric structures, an issue we intend to pursue in a follow-up paper.

[65] The initial reconstruction and parameter selection results are very promising. Even with nonuniform raypath coverage, severe ill-posedness of the inverse problem, and limited a priori information, a reasonable reconstruction of the ionosphere can be achieved. We have presented a flexible cost equation framework to account for the amount or types of reliable a priori information that is available. For instance, the ionospheric profile can be constrained at the ionosonde site, but can also be used as a looser constraint for the other vertical profiles. It is very easy to add additional TEC and density observations in this framework. In the future, we also plan to investigate the performance of a nonuniform grid.

[66] We showed how the UPRE method with an accurate noise model can be used to find the regularization parameter. An immediate future application would be to incorporate vertical profiles calculated from LEO occultation geometry. Validation is an important step, and we plan to apply this regularization framework to other platforms using in situ and ISR measurements for independent validation beyond the self-consistency checks used here.


[67] This work was supported in part by the National Science Foundation under grant ATM 01-35073 to the University of Illinois and supported in part by the NASA Jet Propulsion Laboratory. The GPS data were processed using the MAPGPS software package created by W. Rideout and A. J. Coster.