## 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 (F_{10.7}) are well known. Variations in SSN create shifts of peak electron density with latitude. Variations in Kp and F_{10.7} 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 *p*_{i} 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 *N*_{e}(*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 *p*_{i} 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 *p*_{i}, can be written as

where, *c*_{i} is the TEC along path *p*_{i}. The actual density *N*_{d}(*r*, θ, ϕ) is approximated by using *K* basis functions from the set {*a*_{j}(*r*, θ, ϕ)} as follows:

where, *d*_{j} is the weighting coefficient for {*a*_{j}(*r*, θ, ϕ)}. Substituting (2) into (1) yields:

where the *A*_{ij}'*s* are the elements of a matrix that depends only on the choice of paths and basis functions, i.e.,

Let *n*_{i} be an error term that is due to the approximate nature of the series expansion and inaccuracy in the measurements. Then,

or, in a matrix notation,

where *c*^{a} is the absolute TEC data. In (6), *c*^{a} is a (*N* × 1) vector of TEC data, *d* is a (*K* × 1) vector of electron density, *A* a (*N* × *K*) matrix with elements *A*_{ij}. The elements *A*_{ij} 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

Once the basis functions are chosen and the ray paths are known, (1.3.4) can be used to obtain *A*_{ij}. 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 *d*^{o}(*r*, θ, ϕ) and consider a density function *d*^{m}(*f*_{1}(*r*, τ_{r}), *f*_{2}(θ, τ_{θ}), *f*_{3}(ϕ, τ_{ϕ})) to vary as a function of several functions *f*_{1}(*r*, τ_{r}), *f*_{2}(θ, τ_{θ}) and *f*_{3}(ϕ, τ_{ϕ}) of independent variables (*r*, τ_{r}), (θ, τ_{θ}), (ϕ, τ_{ϕ}) respectively. We further assume that the variations of interest are accurately modeled by only using the function (*f*_{1}(*r*, τ_{r}), and define it to be (*f*_{1}(*r*, τ_{r}) = *r* − *m*(), a function of a vector model altitude shift function *m*() such that

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

and where, *m*() is a family of polynomials of continuous vector variable and *U* is the number of terms in the expansion of (10). In (9), for *U* = 0, *m*() = τ_{0}θ^{0} = τ_{0} (an altitude shift, with = τ_{0}); for *U* = 1, *m*() = τ_{0}θ^{0} + τ_{1}θ^{1} (a linear gradient with = [τ_{0}, τ_{1}]). A discrete representation of (8) can be written as

The shift modeled by *m*(), ( the discrete τ_{r} counterpart) in (10) is captured using a linear convolution by taking the three dimensional Discrete Fourier Transform of both sides of (10) as

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 d^{m}(*n*_{r}, *n*_{θ}, *n*_{ϕ}) can also be rewritten by taking the Discrete Inverse Fourier Transform of (11)

Equation (12) can be viewed as a filtering operation. Using *F* and *F*_{i} as the 3-D Fourier Transform matrices representative of the Discrete Fourier Transform operations, *S*, as the system matrix and *d*^{0} and *d*^{m} as the original and modeled density vectors, the expression in (12) can be rewritten in its matrix formulation as

where

*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*() = τ_{0}θ^{0} = τ_{0}, and *S* can be written aswhere the *B*_{τ0}'s are block diagonal matrices analogous to the one on the upper left hand corner of the matrix, *S*, defined in (16).

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

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

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

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

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 *F*_{i}*S**F*. 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 *d*_{h}^{0}, and its shifted counterpart by *d*_{h}^{m}, a linearly shifted profile *d*_{h}^{m} can be written in terms of the original profile *d*_{h}^{0} by using (12) as

Then, at any point *t*, including noninteger ones, we can estimate *d*^{m}_{h}(*t*) as

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.

### 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

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:

with the assumption that *Aa*_{n} < *c*_{n} < *Ab*_{n} for all *n*, *k* and that *a*_{k} < d_{k}^{0} < *b*_{k} for all *k*. Then, for each *k* = 0, 1, ..., *q* = 0, 1, ..., and *n* = *m*(mod *N*) + 1 we have

with

and

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

where Π^{n} denotes the product over those indices, *n*, that are in the block *B*_{n}, and *Ad*_{n} = *A*_{nk}*d*_{k}. All terms in (23) to (28) are positive. Each term of the iterative sequence {*d*_{k}^{q}} is a convex combination of *a*_{k} and *b*_{k}, 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

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*.