Computerized Ionospheric Tomography (CIT)
  • Tomographic imaging of the ionosphere using GPS data.
  • Computerized Ionospheric Tomography (CIT) is a method to reconstruct ionospheric electron density image by computing Total Electron Content (TEC) values from the recorded GPS signals. Due to the multi-scale variability of the ionosphere and inherent biases and errors in the computation of TEC, CIT constitutes an underdetermined ill-posed inverse problem.
  • Estimation of electron density distribution in the ionosphere as a function of space and time is a problem with important application areas. GPS satellites and receivers provide Total Electron Content (TEC) measurements along a network of lines connecting satellites to the receivers. Therefore, a line-projection relates the electron density distribution to the available measurements resulting in a tomographic set up for the estimation problem. However, the classical tomographic reconstruction techniques fail to provide reliable results with the limited number of available line-projections. In addition, the time varying nature of the electron density distribution creates further complications.
  • Ionospheric imaging of electron density distribution has four dimensions in Latitude, Longitude, Height, Time.
  • Computerized Ionospheric Tomography is of utmost interest recent years. Various approaches for the solution of the CIT include serial expansion of electron density into two dimensional basis functions, iterative algebraic reconstruction methods, neural network and statistical analysis methods.
  • For basis function expansion, the following bases can be chosen:
  • Squeezed Legendre Polynomials
  • Truncated Legendre Polynomials
  • Haar Wavelets
  • SVD based basis functions
  • Optimum number of basis functions are determined by computing the error in the reconstruction.
  • Tomographic reconstruction algorithms are
    • Regularized Least Squares (RLS)
    • Needs regularization
    • Requires matrix inversion and matrix multiplication, thus higher computational complexity compared to other techniques
    • Total Least Squares (TLS)
    • No regularization
    • Requires SVD computation
    • Useful for linear system equations, Ax=b, in which A and b include error
    • Truncated SVD (TSVD)
    • Method for regularization of ill-posed linear Least Squares problem
    • Requires SVD computation
    • Algebraic Reconstruction Technique (ART)
    • Iterative algorithm and independent of basis functions
    • Performance is very sensitive to the initial state
    • Hybrid Reconstruction Technique (HART)
    • Initial state is obtained by TLS and iterative ART is used for reconstruction
    • No regularization is needed
    • Lower reconstruction error than TLS alone
    • More efficient in implementation, lower computational complexity
  • IONOLAB Contributions :
    • Tomographic reconstructions of 2-D IRI based surfaces are obtained using Basis Function Expansion (BFE) method using
      • Squeezed Legendre Polynomials
      • Truncated Legendre Polynomials
      • Haar Wavelets
      • SVD based basis functions
      • RLS, TLS, TSVD, ART and HART are implemented and performances are compared over 2-D IRI surfaces.
      • It is observed that SVD bases provides an optimum basis both for latitude and height with minimum number of basis functions to represent the variations.
      • Determination of the initial state for algebraic reconstruction techniques is of extreme importance in the convergence and reduction of reconstruction error.
    • Tomographic Imaging with Random Field Priors
      • CIT is performed by using a Bayesian approach with Gaussian random field priors.
      • The electron density distribution can be estimated by the minimum mean squared error algorithm given the prior distribution of the random field and the measurements
      • The 3-D mean and the covariance of the assumed Gaussian random field priors can either be obtained from ionospheric models such as IRI or they can be estimated by an iterative algorithm from the GPS measurements.
      • Given sparse and non-uniform TEC measurements, the electron field is obtained from mean square estimation where the Gaussian random field structure provides regularization.
      • Geographical and temporal variations of ionosphere can be observed by obtaining tomographic reconstructions of electron density distribution from Earth-based GPS stations for both quiet and disturbed days of ionosphere.
      • 2-D slices that will be obtained from 3-D reconstructions can be compared with the model based reconstructions or with the available Global Ionospheric Maps from IGS centers.
      • The random field theory is applied to the ionospheric electron density reconstruction problem for the first time and the field is estimated using the minimum mean squared error algorithm.

    Figure 5.1 IRI Image of Ionospheric Electron Density in Latitude and Height

     

    Figure 5.2 Ionospheric electron density is reconstructed with RLS and SVD basis functions both in height and latitude. Normalized reconstruction error for 3 SVD basis both in latitude and height is epsilon (3,3) = 0.1934

     

      • SVD Based Tomographic imaging
      • In this work, to improve the reliability of the obtained 3-D estimates, we propose an SVD based tomographic reconstruction technique, where the IRI-2001 model is used as an a priori source of information.
      • To improve the performance of the estimation, we form the SVD basis by using IRI-2001 model results for the location and the time of interest. Also, to account for variation as a function solar activity, we consider IRI-2001 model results with similar sun-spot number index.
      • We investigated variations in the obtained SVD basis as a function of spatial and temporal location in detail showing that there is significant change with respect to both variables. Therefore, a reconstruction based on a fixed basis would have limited applicability around the earth.
      • We also investigated reconstruction quality of the proposed technique both on synthetic and real measurements showing that robust estimation of the ionospheric electron density distribution that fits to the observed data as well as the IRI-2001 model is possible.
    • Projects:
      • Scientific and Technical Research Council of Turkey, TUBITAK EEEAG 105E171: Computer Imaging of Electron Distribution Using GPS Measurements, Jun. 2006 - Jun. 2008.
    • Book Chapters:
      • O. Arikan, F. Arikan, C.B.Erol, `3-D Ionospheric Tomography with Random Field Priors,' in Mathematical Methods in Engineering, K. Tas, J.A. Tenreiro Machado and D. Baleanu (Eds), Springer, Netherlands, 335-334, 2007.
    • Journal Publications:
      • O. Arikan, F. Arikan, and C.B. Erol, `Computerized ionospheric tomography with the IRI model', Journal of Advances in Space Research, doi:10.1016/j.asr.2007.02.078
    • Conference Publications:
      • E. Yavuz, F. Arikan,, O. Arikan, ve C.B. Erol, `Model Based Comparison of Basis Functions and Reconstruction Algorithms in Computerized Ionospheric Tomography, (in Turkish) Symposium of Signal Processing and Applications, (13. Sinyal I{\c sleme and Uygulamalari Kurultayi) SIU 2005, Erciyes University, Kayseri, 16-18 May 2005.
        • E. Yavuz, F. Arikan, O. Arikan, `A Hybrid Reconstruction Algorithm for Computerized Ionospheric Tomography, Proceedings of RAST-2005, Recent Advances in Space Research, Harbiye Askeri Muze, Hava Harb Okulu, Istanbul, Turkey, 782-787, 9 - 11 June 2005.
        • E. Yavuz, O. Arikan, F. Arikan, C.B. Erol, `Computerized Ionospheric Tomography with the IRI Model, Abstracts Booklet of IRI-2005, Observatori de l'Ebre, Roquetes, Spain, 41, 27 June - 1 July 2005.
        • E. Yavuz, F. Arikan, O. Arikan and C.B. Erol, `Algorithms and Basis Functions in Tomographic Reconstruction of Ionosphere Electron Density, Proceedings of EUSIPCO'2005, 13th European Signal Processing Conference, Antalya, Turkey, 4-8 September 2005.
        • O. Arikan, F. Arikan and C.B. Erol, `3-D Ionospheric tomography with random field priors, Abstract Booklet of MME'06, Mathematical Methods in Engineering, Cankaya University, Ankara, 27-29 April 2006.
        • O. Erturk, O. Arikan, F. Arikan, `Tomographic Reconstruction of the Ionospheric Electron Density as a function of Space and Time, Abstracts Booklet of IRI/COST 296-2007, Institute of Atmospheric Physics, Prague, Czech Republic, 10 - 14 July 2007.