## 1. Introduction

[2] The problem of reconstructing ionospheric electron density, from ground-based receiver to satellite total electron content (TEC) measurements, has been addressed in a number of previous studies. The algorithms used to perform these inversions have included the simultaneous iterative reconstruction technique (SIRT) [*Austen et al.*, 1988], the multiplicative algebraic reconstruction technique (MART) [*Raymund et al.*, 1990], singular value decomposition (SVD) null-space methods [*Raymund et al.*, 1994b], transform methods [*Fremouw et al.*, 1992; *Na and Lee*, 1991], along with maximum entropy techniques [*Fougere*, 1995]. The majority of these reconstruction algorithms have assumed that the ionospheric electron density can be represented as a linear combination of basis functions used to convert the integral relationship between TEC measurements and ionospheric electron density to a set of linear algebraic equations. These algorithms also have various mechanisms that allow the use of prior information to resolve the ambiguity that is associated with the ground-based receiver to satellite geometry [*Na and Lee*, 1990; *Raymund et al.*, 1994a]. Bayesian reconstruction algorithms are also used and they are known 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. Difficulties arise when the spatial coordinate of the prior model is held fixed relative to the spatial coordinate system of the reconstruction [*Hanson and Wecksung*, 1983]. A technique that has been used to represent variations in an image being restored for a general class of distortion has been reported by *Hanson* [1992]. In his work, the prior is given an inherent geometrical flexibility. This flexibility, which is achieved through a warping of the coordinate system of the prior into that of the reconstruction, allows some of the geometrical characteristics, inherent in the reconstructed density solution, to be captured. *Cornely* [2003] used the flexible prior model approach to account for geometrical variations in electron density not captured in state of the art prior models such as the parameterized ionospheric model (PIM) and the International Reference Ionosphere (IRI). This flexible prior model approach has been tested and successfully adapted to tomographic reconstruction with independent verification from European Incoherent Scatter (EISCAT) radar data. A number of three and four dimensional tomographic reconstruction techniques have also been proposed [e.g., *Howe et al.*, 1998; *Hernández-Pajares et al.*, 1998; *Spencer and Mitchell*, 2001]. Howe et al. propose a four-dimensional stochastic model of ionosphere perturbations based on data from GPS and LEO. A Kalman filter is used to assimilate data into a time-dependent model in the form of a first-order Markov process. However, the computational complexity of this method, when the dimension of the state vector is considerably large, is not known. Hernández-Pajares et al. also propose a temporal evolution of a three-dimensional electron density by means of Kalman filtering. The technique uses a combination of real-time and LEO GPS data with independent verification from IRI. As in the previous case, the computational complexity of this technique, when the dimension of the state vector is large, is not known. Mitchell's multi-instrument data analysis system (MIDAS) produces free electron density maps from slant TEC values and from ionosonde in conjunction with LEO-GPS. This method shows significant improvements only when precise vertical ionization profiles on the topside density are needed. For the past three to four years a collaborative effort, to update off-line and real-time models such as IRI, PIM and its real-time cousin, the parameterized Real Time Ionospheric Specification Model (PRISM), has been underway. This effort has produced the Global Assimilation of Ionospheric Measurements (GAIM) [*Schunk et al.*, 2004]. GAIM uses a physics-based ionosphere-plasmasphere wind model and a Kalman filter as a basis for assimilating a diverse set of near-real-time measurements from various sources (LEO, GPS, ionosondes and DMSP satellites). Because of the large dimension of the electron density state, GAIM's implementation of a full Kalman filter is not computationally feasible. Therefore GAIM implements a band-limited Kalman filter in which a full time propagation of the covariance matrix is performed but only a portion of the matrix is retained. There is a trade off between, how sparse or full the covariance matrix is over repeated runs, resulting in a tradeoff between the accuracy of the derived electron density state and the computational complexity of the algorithm. The preceding discussion suggests that the specific technique, used to assimilate data into the reconstruction process, has a significant impact on the computational complexity of the algorithm and the accuracy of the derived solution. A useful class of algorithms for discrete linear inverse problems generally referred to as expectation maximization maximum likelihood (EMML) algorithms [*Byrne*, 1993] allow closed form iterative solutions for problems where each element of the unknown density, measurement matrix, and observational measurements are positive quantities. One of these algorithms, the simultaneous multiplicative algebraic reconstruction technique (SMART), allows the use of prior information, a user-controlled method of regularizing the solution to reduce the effects of noise, and a structure that can be easily implemented in parallel form. In this paper, we propose a three-dimensional modified version of the two-dimensional SMART [*Kuklinski*, 1997]. A BISMART method is implemented where the cross-entropy minimization is applied separately to two subsets of equations. One subset represents the relation between the electron density and the TEC data while the second subset represents the band-limited characteristics of the prior that satisfy the available TEC data. Since prior estimates of ionospheric electron density obtained from PIM data are well modeled as three-dimensional (3-D) band-limited functions, the Fourier transform of the electron density restorations were constrained to be zero outside the corresponding 3-D band-limited region. The constraints are formulated as M additional linear equations, where M is the number of discrete frequencies in Fourier space where the prior had essentially zero energy. This additional information is used to produce a unique solution, at locations that contributed to the measured TEC data, and interpolate a solution within the remainder of the imaging volume. The band-limited BISMART method is used to implement a three-dimensional ionospheric reconstruction system with two key properties. First, the reconstruction system exploits the fact that the structure of the BISMART algorithm, allowing subsets of data corresponding to the available TEC data and the band-limited constraints to be solved separately and efficiently, can be used as a platform for data assimilation from other sources such as GPS, and ionosondes. Second, the reconstruction system exploits the fact that gaps, created by longitudinally unaligned receivers within a chain, can be captured as additional planes of constant longitudes within a three-dimensional imaging volume. The technique can be extended to include gaps between chains of receivers. In the consistent case, BISMART converges to a solution for any configuration of the measured TEC and band-limited data subsets. In the inconsistent case, BISMART produces a limit cycle, from which an approximate solution can be obtained using a “feedback” approach. The algorithm has been evaluated using TEC data from a multiple-chain receiver acquisition system, to produce restorations for a volume of up to 1.31 × 10^{9} voxels under known ionospheric conditions.