## 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.