Ionospheric tomography via iterative cross‐entropy minimization

Kuklinski, Walter S.

doi:10.1029/97rs00108

Cited by 8 publications

(8 citation statements)

References 9 publications

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“…In addition to this, there is another way to incorporate the additional information into the system via a prior model that includes the information about the desired solution [ Shieh et al , 2006]. These are certainly useful [ Kuklinski , 1997; Cornely , 2003] to improve the speed of the iteration for a huge data set, especially to do a 3‐D ionospheric reconstruction. In the present study, the background ionosphere is used as an initial state which takes care of the ill‐posedness of the problem by the algorithm.…”

Section: Tomography Modelmentioning

confidence: 99%

Two‐dimensional ionospheric tomography over the low‐latitude Indian region: An intercomparison of ART and MART algorithms

Das

Shukla

2011

Radio Science

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[1] Single-frequency users of a satellite-based augmentation system (SBAS) rely on ionospheric models to mitigate the delay due to the ionosphere. The ionosphere is the major source of range and range rate errors for users of the Global Positioning System (GPS) who require high-accuracy positioning. The purpose of the present study is to develop a tomography model to reconstruct the total electron content (TEC) over the low-latitude Indian region which lies in the equatorial ionospheric anomaly belt. In the present study, the TEC data collected from the six TEC collection stations along a longitudinal belt of around 77 degrees are used. The main objective of the study is to find out optimum pixel size which supports a better reconstruction of the electron density and hence the TEC over the low-latitude Indian region. Performance of two reconstruction algorithms Algebraic Reconstruction Technique (ART) and Multiplicative Algebraic Reconstruction Technique (MART) is analyzed for different pixel sizes varying from 1 to 6 degrees in latitude. It is found from the analysis that the optimum pixel size is 5°× 50 km over the Indian region using both ART and MART algorithms.Citation: Das, S. K., and A. K. Shukla (2011), Two-dimensional ionospheric tomography over the low-latitude Indian region: An intercomparison of ART and MART algorithms, Radio Sci., 46, RS2005,

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Section: Tomography Modelmentioning

confidence: 99%

Two‐dimensional ionospheric tomography over the low‐latitude Indian region: An intercomparison of ART and MART algorithms

Das

Shukla

2011

Radio Science

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“…The two dimensional problem of reconstructing ionospheric electron density from ground‐based receiver to TEC measurements is challenging because of the limited range of ray path angles associated with this geometry. Assuming absolute TEC data are available [ Austen , 1996], a major issue associated with the development of an ionospheric tomography algorithm is the trade‐off between the manner in which additional information is used to obtain a unique ionospheric electron density reconstruction from limited angle TEC measurements and the computational complexity of the resulting algorithm [ Kuklinski , 1997]. Three‐dimensional ionospheric reconstruction problems differ from two‐dimensional ones because, in addition to providing a unique density solution from additional data, there are usually more voxels thereby leading to larger problems where direct inversion methods, and direct transform methods may not always be practical.…”

Section: Results and Conclusionmentioning

confidence: 99%

“…In addition, the A matrix is quite parse having on the order of 0.01–0.02% nonzero elements. A computationally efficient iterative image reconstruction algorithm, for problems with nonnegative densities and measurement matrices for a two dimensional geometry [ Kuklinski , 1997], was modified to produce a three‐dimensional band‐limited iterative minimum cross entropy reconstruction algorithm. The modified algorithm uses a scaled version of the TEC model in : where y is the scaled TEC data, P the scaled ray path distance matrix and x the scaled electron density.…”

Section: Algorithm Developmentmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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Three‐dimensional ionospheric tomography via band‐limited constrained iterative cross‐entropy minimization

Cornely

Kuklinski

2005

Radio Science

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[1] The problem of reconstructing ionospheric electron density from ground-based receiver to satellite total electron content (TEC) measurements is formulated as an underdetermined discrete linear inverse problem. If receivers and satellite orbit are coplanar, then a single two-dimensional (2-D) imaging plane can be used as a geometrical model. In most cases, receiver locations are determined by convenience and availability of sites, and thus a 3-D imaging volume is required in order to capture the part of the electron density solution that is associated with gaps between receiver stations within a chain. It is well understood that spurious features, or ''wings,'' associated with these gaps can be produced in tomographic image reconstructions where ray path coverage from individual stations is lost or minimal. In this paper, a 3-D reconstruction system that takes advantage of a multiple-chain data acquisition geometry and provides a solution consistent with available TEC data everywhere within a receiver chain is presented. 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 the 3-D imaging volume. With parameterized ionospheric model (PIM)-generated data as a nonnegative prior estimate of the electron density, the reconstruction algorithm uses constraints based on prior knowledge of the 3-D spatial Fourier transform of the prior electron density as a smoothing mechanism in the tomographic reconstruction process. Consequent to the underlined Fourier transform formulation, a unique solution is produced at locations that contributed to the measured TEC data, and a solution is interpolated within the remainder of the imaging volume. The band-limited BISMART algorithm has been evaluated using a multiple-receiver chain TEC data acquisition system, under known ionospheric conditions. The algorithm satisfactorily reconstructs density solutions, consistent with small-scale enhancements, irregularities, and troughs in the auroral ionosphere, from the available TEC data. The quality of the density reconstructions, coupled with the computational efficiency of this algorithm, indicates the potential utility of this technique for real-time three-and four-dimensional ionospheric tomography.Citation: Cornely, P.-R. J., and W. S. Kuklinski (2005), Three-dimensional ionospheric tomography via band-limited constrained iterative cross-entropy minimization, Radio Sci., 40, RS5S90,

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“… Kuklinski [1997] set up a three‐dimensional image grid in longitude, latitude, and altitude providing voxels (a voxel is a 3‐D pixel and can be in any coordinate frame) of size 0.5° × 0.5° × 45 km. TEC measurements, recorded during the Russian‐American tomography experiment (see section 4), were then used to reconstruct a tomographic image in the plane defined by the receiver chain and the satellite orbit.…”

Section: Theory: Two‐dimensional Imagingmentioning

confidence: 99%

History, current state, and future directions of ionospheric imaging

Bust

Mitchell

2008

Reviews of Geophysics

245

210

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With the current data availability from both ground‐ and space‐based sources, the network of ground‐based Global Positioning System (GPS) receivers, GPS occultation receivers, in situ electron density sensors, and dual‐frequency beacon transmitters, the time is right for a comprehensive review of the history, current state, and future directions of ionospheric imaging. A brief introduction and history of ionospheric imaging is presented, beginning with computerized ionospheric tomography. Then, a comprehensive review of the current state of ionospheric imaging is presented. The ability of imaging algorithms to ingest multiple types of data and use advanced inverse techniques borrowed from meteorological data assimilation to produce four‐dimensional images of electron density is discussed. Particular emphasis is given to the mathematical basis for the different methods. The science that ionospheric imaging addresses is discussed, and the scientific contributions that ionospheric imaging has made are described. Finally, future directions for this research area are outlined.

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Ionospheric tomography via iterative cross‐entropy minimization

Cited by 8 publications

References 9 publications

Two‐dimensional ionospheric tomography over the low‐latitude Indian region: An intercomparison of ART and MART algorithms

Two‐dimensional ionospheric tomography over the low‐latitude Indian region: An intercomparison of ART and MART algorithms

Three‐dimensional ionospheric tomography via band‐limited constrained iterative cross‐entropy minimization

History, current state, and future directions of ionospheric imaging

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