Flexible Prior Models: Three-dimensional ionospheric tomography

Authors


Abstract

[1] A three-dimensional ionospheric reconstruction system is presented that models ionospheric dynamics and accounts for well-known limitations in the available data by using geometrically transformed prior models based on available models, such as the Parameterized Ionospheric Model (PIM) or the International Reference Ionosphere (IRI), to produce density solutions consistent with available Total Electron Content (TEC) measurement data. It is known that current state of the art prior density models, such as PIM or IRI, can only simulate a statistical mean ionosphere. As a result, sudden variations in ionospheric electron density will not be represented in these models. This paper focuses on a three-dimensional ionospheric density reconstruction system that uses a set of geometrical transformations to produce Flexible Prior Models (FPM). The Flexible Prior Model process allows means to include flexibility in a prior model in the form of geometrical variations that are not well represented in current state-of-the-art ionospheric prior density models. The updated priors are used as initial solution models for a set of (A, B) Multiplicative Algebraic Reconstruction Technique (ABMART) algorithms, that can easily incorporate additional information and provide a spatially constrained density solution, consistent with the available data and our current knowledge of ionospheric dynamics. The ultimate goal is to create a three-dimensional adaptive ionospheric electron density reconstruction system that would make it possible to generate real-time ionospheric maps supported by available TEC data. Such maps would be of significant utility in predicting and correcting the impact of electron density gradients and irregularities on radio waves.

1. Flexible Prior Models

1.1. Introduction

[2] Many methods of incorporating variations in ionospheric electron density have been attempted over the years. These methods have ranged from real time modeling [Anderson et al., 1989; Daniell et al., 1995] to stochastic representations of a nonstationary ionosphere [Fremouw et al., 1992]. In this section, a new procedure for incorporating variations in ionospheric electron density will be presented. This procedure has the following characteristics:

[3] 1. It will include in the reconstruction electron density perturbations that are not well represented by state-of-the-art prior models. Special attention is paid to geometrical transformation models consistent with the available data, additional prior information, and known features of ionospheric electron density.

[4] 2. It will produce a flexible prior background density to be subsequently utilized as a starting point for a specific class of iterative algorithms. These iterative algorithms allow one to use the starting point as part of incorporating additional information to the problem formulation.

[5] Bayesian reconstruction methods have been shown to substantially improve the accuracy of the reconstruction obtained from limited data if the object under study does not differ very much in size, shape or position from the assumed model. However, difficulties arise when the prior model is held fixed relative to the spatial coordinate system of the reconstruction [Hanson and Wecksung, 1983; Hanson, 1992]. A deterministic parameter modeling approach is proposed here in which, the prior model for the object under study is allowed to both alter its characteristics, and include additional information in a manner that can still accommodate the available data. The available data and the updated prior can then be used in conjunction with carefully selected iterative Bayesian algorithms to produce updated density solutions. Combettes [1993] suggests a set-theoretic framework where reliable and optimal reconstruction systems can be built by exploiting known information through successive projections onto convex sets, rather than imposing a subjective notion of optimality. Hanson [1992] suggests that the prior can acquire geometrical flexibility based on the available data. This flexibility is achieved by allowing a warping of the coordinate system of the prior into that of the reconstruction, which makes it possible for some of the geometrical characteristic features, inherent in the density solution, to be included. Geometrical transformations when judiciously applied to a prior model can accommodate differences in size, position, shape, and orientation of the model. The nature of the resulting solutions is fundamentally different, in the sense that these solutions will be supplied with a mechanism, which accommodates variations in the geometrical differences in the solution, known to exist, but not at all inherent to state-of-the-art prior models such as PIM or IRI. Combettes and Trussel [1990] and Hanson [1992] together provide a framework within which the parameters needed to specify a desired transformation, and the use of additional information, can be factored as part of an overall reconstruction system.

1.2. Geometrical Variations With PIM Input Parameters

[6] In this section we review the type of geometrical variations that can be derived from variations in PIM input parameters. The nature of the variations in electron density taking place as we vary the following parameters: The sunspot number (SSN), the magnetic index (Kp) and the instantaneous solar flux (F10.7) are well known. Variations in SSN create shifts of peak electron density with latitude. Variations in Kp and F10.7 create density bending and stretching effects at midlatitudes. These variations represent behaviors not well represented in PIM or IRI. The Flexible Prior Model and the ABMART iterative processes, to be discussed, are utilized as a means of incorporating geometrical variations in models such as PIM and IRI, and adding additional information to the overall reconstruction process. The Flexible Prior Model is therefore used as a means of adding flexibility to the prior, while the ABMART iterative reconstruction process is used as a regularization procedure; a smoothing mechanism and a means of adding additional information to the problem formulation.

1.3. Flexible Prior Models Theory

[7] An easily measured parameter from the ionosphere is the total electron content (TEC), which is the number of free electrons in a column of a unit cross sectional volume. In order to measure the TEC, it is assumed that the necessary corrections for the receiver biases have been taken into account so that absolute TEC measurements are available. In this paper, the TEC are not corrected for the angle between the path and zenith but are the “slant TEC” values. The techniques for measuring the TEC, such as Farraday rotation and differential Doppler, are well known; for the following discussion, it is assumed that an appropriate technique has been used to measure the TEC. We assume that the carrier frequencies used to measure the TEC are high enough so that the path pi is a straight line joining the source to the receiver. Computerized Tomography (CT) is the process of reconstructing an image from a series of its projections. Bates et al. [1983] and Scudder [1978] review the development of CT, its principles, current techniques, and applications. The algorithms used for CT can be divided into two classes: transform methods and finite series expansion methods. These methods are reviewed by Lewitt [1983] and Censor [1983], respectively. Since the ionospheric imaging problem does not involve a geometry that lends itself to a direct application of the Transform Methods, a finite-series expansion technique, independent of the geometry of the problem, is used here. Let Ne(r, θ, ϕ), where r, θ and ϕ (Height, Longitude, Latitude) are coordinates in a three-dimensional coordinate system with the origin at the center of Earth, represent the electron density in a volume. In addition, let pi be the path that ray i takes through the ionosphere and assume that all the paths go through the volume. The TEC measurements, defined as the line integral through the ionosphere along paths pi, can be written as

equation image

where, ci is the TEC along path pi. The actual density Nd(r, θ, ϕ) is approximated by using K basis functions from the set {aj(r, θ, ϕ)} as follows:

equation image

where, dj is the weighting coefficient for {aj(r, θ, ϕ)}. Substituting (2) into (1) yields:

equation image

where the Aij's are the elements of a matrix that depends only on the choice of paths and basis functions, i.e.,

equation image

Let ni be an error term that is due to the approximate nature of the series expansion and inaccuracy in the measurements. Then,

equation image

or, in a matrix notation,

equation image

where ca is the absolute TEC data. In (6), ca is a (N × 1) vector of TEC data, d is a (K × 1) vector of electron density, A a (N × K) matrix with elements Aij. The elements Aij are the lengths of the ray path for the nth TEC measurement through the kth voxel of a three-dimensional density, or the kth cell of a two-dimensional density, and n is a random noise vector. The representation is valid for both two and three-dimensional structures, allowing two-dimensional plane or three-dimensional volume electron density reconstructions. The region being imaged is divided into unit volume like regions in a spherical coordinate system called voxels, and the number of electrons in each voxel is the density of the object in that region. To represent the voxels, a set of basis functions can be defined such that

equation image

Once the basis functions are chosen and the ray paths are known, (1.3.4) can be used to obtain Aij. The problem is then reduced to solving (6) where d is the unknown and c is the known data, subject to some constraint on d and the error term n. There are many different techniques for solving (6) [Raymund et al., 1994]. In using these techniques, if the approximate shape of an object to be reconstructed is known beforehand, Bayesian methods of reconstruction can incorporate the known structural characteristics and produce substantially improved reconstructions from limited data. However, if the assumptions about the structural characteristics of the object differ only slightly, the prior that results from such assumptions can lead to very poor reconstructions [Hanson and Wecksung, 1983]. In addition, if the mean of the desired reconstruction varies from that of the prior, the fixed mean of the prior will prevent these structural characteristics to be reflected in the reconstruction. This situation is remedied by allowing the prior model for the object being reconstructed to alter its characteristics to accommodate the data. The structural characteristics of the prior can be altered through geometrical transformations, but these transformations need to be consistent with ionospheric physics. We can use linear transformations to accommodate structure changes in size, position and orientation of the prior. Structure changes in shape are allowed with nonlinear transformations.

[8] To allow structures changes and build flexibility into the prior, consider the original density model of (2), and denote it by do(r, θ, ϕ) and consider a density function dm(f1(r, τr), f2(θ, τθ), f3(ϕ, τϕ)) to vary as a function of several functions f1(r, τr), f2(θ, τθ) and f3(ϕ, τϕ) of independent variables (r, τr), (θ, τθ), (ϕ, τϕ) respectively. We further assume that the variations of interest are accurately modeled by only using the function (f1(r, τr), and define it to be (f1(r, τr) = rm(equation image), a function of a vector model altitude shift function m(equation image) such that

equation image

where we also assume the model function m(equation image) to be of the general form

equation image

and where, m(equation image) is a family of polynomials of continuous vector variable equation image and U is the number of terms in the expansion of (10). In (9), for U = 0, m(equation image) = τ0θ0 = τ0 (an altitude shift, with equation image = τ0); for U = 1, m(equation image) = τ0θ0 + τ1θ1 (a linear gradient with equation image = [τ0, τ1]). A discrete representation of (8) can be written as

equation image

The shift modeled by m(equation image), (equation image the discrete τr counterpart) in (10) is captured using a linear convolution by taking the three dimensional Discrete Fourier Transform of both sides of (10) as

equation image

Equation (11) assumes that the sampling issues are addressed; that is, the density profiles are modeled as bandlimited functions (functions limited in extent in the frequency domain). This assumption is not satisfied by the available density profiles and consequently the density profiles are padded appropriately, to satisfy both the bandlimited requirements and as required by the ionospheric physics boundary constraints. In cases where the density profiles display a sudden drop at their tails, Gibbs effects will be reflected in their Fourier transforms and the padding is used; a two points moving average is employed to add a few more values at each end of the profile. The basic idea is that by averaging values locally, only lower frequency variations are preserved, corresponding to a smoothing of the tails. In addition, this operation is consistent with the physical behavior of the Chapman Layer profiles at the altitudes of interest (150–600 Km) and as such, it will not change the nature of the electron density solutions obtained. The parameterized model density dm(nr, nθ, nϕ) can also be rewritten by taking the Discrete Inverse Fourier Transform of (11)

equation image

Equation (12) can be viewed as a filtering operation. Using F and Fi as the 3-D Fourier Transform matrices representative of the Discrete Fourier Transform operations, S, as the system matrix and d0 and dm as the original and modeled density vectors, the expression in (12) can be rewritten in its matrix formulation as

equation image

where

equation image

S is a block diagonal system matrix representative of the linear shift, gradient or polynomial gradient operations, and the subscripts L, l and h in (14) represent the number of longitudes, latitudes and altitudes respectively. In (13), for U = 0, m(equation image) = τ0θ0 = τ0, and S can be written aswhere the Bτ0's are block diagonal matrices analogous to the one on the upper left hand corner of the matrix, S, defined in (16).

Figure 1.

(continued)

[9] For L planes of constant longitude as shown in Figure 1, the vector density will be made of L density blocks, DL and the problem becomes one of finding the parameter model vector m(equation image) that will satisfy the available measurement data, i.e., find m(equation image) such that

equation image

A least squares formulation to the parameter estimation problem can be written as

equation image

The formulation in (17) reduces to optimizing the objective function f[m(equation image)] as

equation image

The objective function f[m(equation image)] can be rewritten to incorporate the model function m(equation image) in the form of the system matrix S that depends on m(equation image) by using (14)

equation image

It is important to note at this point that for the specific family of model shift functions used here, the shift function mapping formulation of (13) and (17) was possible because it uses a system matrix, S, that diagonalizes FiSF. The general geometrical transformation problem may therefore require a system matrix that diagonalizes the particular mapping of interest. There may be other classes of geometrical transformations possessing the diagonalization property, which may also have many other interesting and useful additional properties that could add to the flexibility of the FPM process. The FPM process also has the ability to implement a noninteger shift. In the implementation of the noninteger shift, since the density profiles are modeled as bandlimited functions, in any imaging plane of constant longitude we can describe the variations in electron density with altitude for any specific latitude within the imaging plane. If we denote a given density profile by dh0, and its shifted counterpart by dhm, a linearly shifted profile dhm can be written in terms of the original profile dh0 by using (12) as

equation image

Then, at any point t, including noninteger ones, we can estimate dmh(t) as

equation image

It is also important to note that (9) allows linear, linear gradient and polynomial shifts to be readily implemented. The ability to build flexibility into a prior model reduces the number of eventual solutions that can be derived from basic Bayesian techniques such as: MAP (Maximum A Posteriori), ML (Maximum Likelihood), and EMML (Expected Maximization Maximum Likelihood), by providing mechanisms for changing the mean of the prior before using these techniques. Although changing the mean of the prior model inherently changes the nature of the solutions that are derived, the FPM model discussed above does not provide a mechanism for scaling the density solution obtained. This is of concern because enhancement, depletion and magnetic storm phenomena, are well known to considerably change the value of the density by up to: 2-3 orders of magnitude. Instead of burdening the FPM process with the addition of an extra parameter, a suitable Bayesian iterative method has been used that allows the solutions to include scaling features in electron density solution when they exist. The ABMART (A, B, Multiplicative Algebraic Reconstruction Technique) iterative process [Byrne, 1998] includes upper and lower bounds for the solution.

Figure 1.

Simulated three-dimensional imaging volume. Satellite sampling points Si and chains of receivers Ri.

1.4. A, B Multiplicative Algebraic Reconstruction Technique (ABMART)

[10] The cross-entropy Kullback-Liebler (KL) distance between two nonnegative vectors a and b is

equation image

Several known algorithms for reconstructing tomographic images lead to solutions that minimize certain combinations of the KL distance between convex sets including: EMML and MAP with Gamma distributed priors as well as MART [Byrne, 1993]. This class of algorithm has been offered as a way to increase the smoothness of the reconstructions through the inclusion of a prior into the reconstruction process [Kuklinski et al., 1997]. The ABMART algorithm belongs to that class and it begins in general with an underdetermined set of equations:

equation image

with the assumption that Aan < cn < Abn for all n, k and that ak < dk0 < bk for all k. Then, for each k = 0, 1, ..., q = 0, 1, ..., and n = m(mod N) + 1 we have

equation image

with

equation image
equation image

and

equation image

A block iterative ABMART algorithm is obtained by partitioning the set {n = 1, ..N} into disjoint subsets Bn, {n = 1, …,N} for N > 1 and using the following equation instead of (25),

equation image

where Πn denotes the product over those indices, n, that are in the block Bn, and Adn = equation imageAnkdk. All terms in (23) to (28) are positive. Each term of the iterative sequence {dkq} is a convex combination of ak and bk, as seen from (24), and the iteration proceeds until convergence to a convex combination, which satisfies (24) is found, if such solution exists. If there is no solution of (24), then for the simultaneous case in which N = 1, the algorithm will converge to an appropriate solution satisfying the constraints, and the limit is the unique vector satisfying a < d < b, for which the objective function

equation image

is minimized. The solution, d, obtained is the minimum cross entropy class and is a compromise solution that satisfies a < d < b, the prior, and the available data c.

2. Results and Conclusions

2.1. Results

[11] An experimental campaign involving both satellite observations for ionospheric tomography and the European Incoherent Scatter (EISCAT) radar was held in May 1995 at the Physics Department of the University of Wales. Radio transmissions from the Navy Navigational Satellite System (NNSS), were monitored by two receiver chains at five sites: Tromso (69.9N, 19.2E), Kiruna (67.9N, 20.4E), Lycksele (64.6, 18.8), Karasjok (69.5N, 25.5E) and Sodankyla (67.4N, 26.6E) as shown in Figure 1. The differential carrier phase of the received signals was observed, and the absolute total electron content (TEC) values were subsequently obtained using a multistation least squares fit of the equivalent vertical TEC. The EISCAT radar was operated on May 26, 1995 20:55 to 21:14 UT. Details of the experimental methods have been given by Kersley et al. [1993], and the data collection process is described by Mitchell et al. [1997]. The reconstruction process used for the tomographic imaging begins with the creation of the background ionosphere using PIM, corresponding to May 26, 1995, at 20:55 UT, with SSN = 100 and Kp = 3. The FPM process is then used to adjust the peak of the background density and the resulting FPM density serves as a starting density solution to ABMART. The ABMART algorithm is used to reconstruct the distribution of the electron density in an ionospheric region defined by the following boundaries: 100 to 800 km in altitude, 1 to 27 degrees in longitude, 46 to 89 degrees in latitude. As a result of this geometry, the voxel dimension is 7-km altitude, 2.95-km longitude, and 6.02-km in latitude. A satellite crossed 50° N at a longitude of approximately 19° E on May 26, 1995, and was monitored by all five receivers, as shown in Figure 2. The ionospheric volume used for the Wales experiment is made of: 50 altitude planes, 50 latitude planes and 64 longitude planes. One of the goals in the Wales experiment was to study the electron density structuring in the auroral ionosphere. An inspection of Figure 3 reveals two features in the EISCAT measurement data at ≈ (280 km, 63.7° N) and (250 km, 67.5° N). In this example, if only the PIM density is available as a prior density background, the ABMART iterative reconstruction alone is not sufficient to recover the detailed features. To illustrate this point, we use the original three-dimensional PIM background density with ABMART and retrieve from the volume density solution, imaging plane 45, which corresponds to the EISCAT imaging plane. The solution, provided by ABMART using the background PIM density, shows the presence of two structures at about (64.5°, 225 km) and (69°, 175 km), as shown in Figure 4a. However, the altitude and latitude positions, and the magnitude of the structures, are incorrect. If we now take the three dimensional background PIM densities and use the FPM process before using ABMART, the position of the peak electron density is adjusted. The FPM density adjusts the peak of the PIM background using a linear shift (LSFPM) in a manner consistent with the available TEC data. This adjustment makes it possible to create a background density that is no longer related to the monthly mean, but to a mean density that is consistent with the available information for May 26, 1995 at 20:55-21:14 UT.

Figure 2.

Receiver-satellite geometry (top view, see Figure 1 for a 3-D view). Locations of the receiving stations and the satellite trajectory (courtesy of the Physics Department, University of Wales, Aberystwyth).

Figure 3.

EISCAT measurement data at 19.2E (see Figure 2, EISCAT cut) on May 26, 1995, at 21:09 UT (courtesy of the Physics Department, University of Wales, Aberystwyth).

Figure 4.

ABMART density solutions. (a) ABMART solution using the PIM density dAB(p) as an initial solution. (b) EISCAT imaging plane.

[12] The adjustment made in the prior background by FPM allows the position of the peak in the FPM-PIM electron density to be correctly identified, and thus is the ultimate density solution provided by the ionospheric reconstruction system. This feature, provided by the FPM process, is fundamental because it provides a flexible way for incorporating additional information into the problem formulation not available in the original PIM background. If the FPM PIM density is now used as a starting point to ABMART, the resulting solution will reflect the features acquired from the FPM process, as shown in Figure 5a. The FPM-ABMART density solution correctly identifies the position of the peak electron density solution in altitude as expected. However, the latitude position of the density structures in the solution provided is not completely resolved, as expected. This deficiency in the Flexible Prior Model process is hinting that the models developed in this paper may yet lack geometrical flexibility due to the formulation used in their implementation. The Discrete Fourier Transform formulation used, as discussed earlier, although computationally efficient, allows only a limited range of geometrical transformations to be placed on available background densities. Figure 6 shows the TEC curves resulting from FPM-ABMART and EISCAT. The FPM-ABMART TEC results (TEC(dAB(dm)) closely match the available EISCAT TEC data, confirming our assumption that the solution provided by the entire ionospheric reconstruction system is consistent with the available data. Finally, the FPM-ABMART density solution is compared with the available EISCAT density in the mean square sense. The root mean square errors between the density solutions with and without the FPM process are computed. Two error images are produced from these computations. The resulting error images, as shown in Figures 7a and 7b, are quite different. The error image resulting from the ABMART solution, without the a priori FPM process, has a very poor representation of the density structures present in the EISCAT measurements, as shown in Figure 7a. This is primarily because the solution provided contains no information about the position of peak electron density. The solution provided by FPM-ABMART clearly gives a better representation of the density structures seen in the EISCAT measurements, as shown in Figure 7b. These results are significant because the background PIM used was obtained by averaging a collection of PIM densities for May 26, 1995. Therefore, even in cases where a fairly good description of the background density is provided using currently available state-of-the-art prior ionospheric models, the ability to predict the position of the peak electron density solution without using the FPM process is at best average, providing no useful information about the position of peak electron density. On the other hand, the error surface resulting from the FPM-ABMART solution is nearly flat around the density structure regions as, shown in Figure 7b, highlighting that most of the characteristic features contained within the EISCAT density solution are accurately resolved.

Figure 5.

FPM-ABMART density solutions. (a) Density solution with the FPM-PIM, dAB(dm). (b) EISCAT imaging plane.

Figure 6.

TEC solutions from PIM, ABMART, FPM-ABMART from all five receivers Ri (see Figure 2 for receiver locations).

Figure 7.

Root Mean Square Error (RMSE) surfaces. (a) RMSE surface from the ABMART solution without FPM. (b) RMSE surface from the ABMART solution with FPM.

2.2. Conclusions

[13] The utility of the FPM-ABMART reconstruction system was investigated. The primary utility of the FPM models developed in this paper is to make it possible to incorporate additional information about features in the peak electron density solution not available in the prior background density. In addition, the density solution provided will be a statistical mean ionosphere that is consistent with the available data rather than the monthly mean. This feature of the FPM process is significant because it makes the concept of real-time adaptive ionospheric tomography more feasible, accounting for sudden variations in geometrical differences in the resulting density solutions not included in state-of-the-art prior models, such as PIM or IRI. The nature of the minimum cross-entropy solution provided by ABMART, allows the density solutions to acquire correlated features contained in the additional information, while maintaining the average characteristics in the prior background density. The ABMART process removes artifacts from the solutions when they exist by providing a constrained smoothing mechanism, which in addition easily incorporates additional information in the problem formulation, with very little cost in additional complexity. However, the solution will always be close to the prior background in the Kullback-Liebler sense, which limits the extent of the smoothing but guarantees the choice of a solution within a desired set. In addition, the system matrix, which results from the Discrete Fourier Transform formulation is a diagonal matrix and geometrical features, within a class of transformations with system matrices of this type may well provide additional flexibility to the FPM process. The ionospheric response to geomagnetic storms is dependent on latitude and their effects on the ionosphere vary from the equator to the pole. Global Positioning Satellites anywhere on Earth are affected by changes in the ionosphere along the path to the satellite during magnetic storms [Klobuchar, 1995]. Also, auroral irregularities in electron density, such as those displayed in the Wales Experiment, can produce scintillation in varying amounts depending on latitude. The equatorial region is a known site of the greatest ionospheric structure variations [Pryse et al., 1997], even under quiet magnetic conditions, as was the case in the Wales experiment. Unpredictable episodes of density enhancements and depletions in the upper ionosphere can cause radio wave amplitude and phase fluctuations. As a result, radio waves (2-100 MHz) will be reflected or refracted at all latitudes. The ability to create an ionospheric reconstruction system that correctly models the suddenly changing nature of ionospheric irregularities is an important aspect of the ionospheric reconstruction problem discussed in this paper. The FPM-ABMART process clearly illustrates improved solutions to imaging suddenly varying ionospheric structures. The process also provides a framework for the implementation of real-time adaptive ionospheric reconstruction systems.

Acknowledgments

[14] The author wishes to thank Dr. Walter S. Kuklinski from The University of Massachusetts at Lowell, currently at The Mitre Corporation, Dr. Charles L. Byrne from the Department of Mathematics at the University of Massachusetts at Lowell, and Dr. Gary Sales from the Center of Atmospheric Research at the University of Massachusetts at Lowell for their support and useful insights. The author also wishes to thank Dr. Robert E. Daniell from Computational Physics Incorporated, Boston, for his useful remarks and insightful suggestions. The author also wishes to thank Dr. L. Kersley and Dr. Cathryn Mitchell from the Department of Physics, The University of Wales, Aberystwyth, for their help with understanding the EISCAT data set that was used in this paper. Final thanks go to the EISCAT Scientific Association, an International Association supported by Finland (SA), France (CNRS), the Federal Republic of Germany (MPG), Japan (NIPR), Norway (NFR), Sweden (NFR), and the United Kingdom (PPARC).

Ancillary