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Reconstructing Regional Ionospheric Electron Density: A Combined Spherical Slepian Function and Empirical Orthogonal Function Approach

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Abstract

The computerized ionospheric tomography is a method for imaging the Earth’s ionosphere using a sounding technique and computing the slant total electron content (STEC) values from data of the global positioning system (GPS). The most common approach for ionospheric tomography is the voxel-based model, in which (1) the ionosphere is divided into voxels, (2) the STEC is then measured along (many) satellite signal paths, and finally (3) an inversion procedure is applied to reconstruct the electron density distribution of the ionosphere. In this study, a computationally efficient approach is introduced, which improves the inversion procedure of step 3. Our proposed method combines the empirical orthogonal function and the spherical Slepian base functions to describe the vertical and horizontal distribution of electron density, respectively. Thus, it can be applied on regional and global case studies. Numerical application is demonstrated using the ground-based GPS data over South America. Our results are validated against ionospheric tomography obtained from the constellation observing system for meteorology, ionosphere, and climate (COSMIC) observations and the global ionosphere map estimated by international centers, as well as by comparison with STEC derived from independent GPS stations. Using the proposed approach, we find that while using 30 GPS measurements in South America, one can achieve comparable accuracy with those from COSMIC data within the reported accuracy (1 × 1011 el/cm3) of the product. Comparisons with real observations of two GPS stations indicate an absolute difference is less than 2 TECU (where 1 total electron content unit, TECU, is 1016 electrons/m2).

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References

  • Afraimovich EL, Pirog OM, Terekhov AI (1992) Diagnostics of large-scale structures of the high-latitude ionosphere based on tomographic treatment of navigation-satellite signals and of data from ionospheric stations. J Atmos Terr Phys 54(10):1265–1273

    Article  Google Scholar 

  • Al-Fanek OJS (2013) Ionospheric imaging for Canadian polar regions, Ph.D. dissertation, Department of Geomatics Engineering UCGE reports no. 20383, University of Calgary

  • Alizadeh MM (2013) Multi-dimensional modeling of the ionospheric parameters, using space geodetic techniques. Ph.D. thesis, Vienna University of Technology, Geowissenchaftliche Mitteilungen Heft Nr. 93, ISSN 1811-8380

  • Alizadeh MM, Schuh H, Todorova S, Schmidt M (2011) Global ionosphere maps of VTEC from GNSS, satellite altimetry and formosat-3/COSMIC data. J Geodesy 85(12):975–987

    Article  Google Scholar 

  • Austen JR, Franke SJ, Liu CH, Yeh KC (1986) Application of computerized tomography techniques to ionospheric research. In: Presented at the international beacon satellite symposium on radio beacon contribution to the study of ionization and dynamics of the ionosphere and to corrections to geodesy and technical workshop, pp 25–35

  • Beggan C, Saarimaki J, Whaler K, Simons F (2013) Spectral and spatial decomposition of lithospheric magnetic field models using spherical Slepian base functions. Geophys J Int 193:136–148

    Article  Google Scholar 

  • Bilitza D, McKinnell L, Reinisch B, Fuller-Rowell T (2011) The international reference ionosphere today and in the future. J Geodesy 85:909–920

    Article  Google Scholar 

  • Chambodut A, Panet I, Mandea M, Diament M, Holschneider M, Jamet O (2005) Wavelet frames: an alternative to spherical harmonic representation of potential fields. Geophys J Int 163:875–899

    Article  Google Scholar 

  • Ciraolo L, Azpilicueta F, Brunini C, Meza A, Radicella SM (2007) Calibration errors on experimental slant total electron content (TEC) determined with GPS. J Geodesy 81(2):111–120

    Article  Google Scholar 

  • COSMIC Program Office Website (2013). Retrieved from http://cdaac-www.cosmic.ucar.edu/cdaac/products.html

  • Daniell R, Brown L, Anderson D, Fox M, Doherty P, Decker D, Sojka J, Schunk R (1995) Parameterized ionospheric model: a global ionospheric parameterization based on rst principles models. Radio Sci 30:1499–1510

    Article  Google Scholar 

  • Davies K (1990) Ionospheric radio (No. 31). IET

  • Erdogan E, Schmidt M, Seitz F, Durmaz M (2017) Ann Geophys 35 263–277. https://doi.org/10.5194/angeo-35-263-2017, https://www.ann-geophys.net/35/263/2017/angeo-35-263-2017.pdf

  • Erturk O, Arikan O, Arikan F (2009) Tomographic reconstruction of the ionospheric electron density as a function of space and time. Adv Space Res 43(11):1702–1710

    Article  Google Scholar 

  • Etemadfard H, Hossainali MM (2015) Application of Slepian theory for improving the accuracy of SH-based global ionosphere models in the Arctic region. J Geophys Res Space Phys. https://doi.org/10.1002/2015JA021811

    Google Scholar 

  • Etemadfard H, Hossainali MM (2016) Spherical Slepian as a new method for ionospheric modeling in arctic region. J Atmos Solar Terr Phys 140:10–15. https://doi.org/10.1016/j.jastp.2016.01.003

    Article  Google Scholar 

  • Feng M (2010) Detection of high-latitude ionospheric irregularities from GPS radio occultation. M.Sc. diss., University of Calgary

  • Forootan E (2014) Statistical signal decomposition techniques for analyzing time-variable satellite gravimetry data. Ph.D. thesis, University of Bonn. http://hss.ulb.uni-bonn.de/2014/3766/3766.pdf

  • Fremouw EJ, Secan JA, Howe BM (1992) Application of stochastic inverse theory to ionospheric tomography. Radio Sci 27(5):721–732

    Article  Google Scholar 

  • Garcia-Fernandez M, Hernandez-Pajares M, Juan J, Sanz J (2005) Performance of the improved abel transform to estimate electron density pro_les from GPS occultation data. GPS Solut 9:105–110

    Article  Google Scholar 

  • Hofmann-Wellenhof B, Lichtenegger H, Wasle E (2008) GNSS—global navigation satellite systems—GPS, GLONASS, Galileo & more. Springer, Wien

    Google Scholar 

  • Jakowski N, Hoque M, Mayer C (2011) A new global TEC model for estimating transionospheric radio wave propagation errors. J. Geodesy 85:965–974

    Article  Google Scholar 

  • JPL (2011) JPL-NASA, GAIM introduction. http://iono.jpl.nasa.gov/gaim/intro.html

  • Klobuchar J (1986). Design and characteristics of the GPS ionospheric time-delay algorithm for single-frequency users. In PLANS’86—position location and navigation symposium, Las Vegas, Nevada, pp 280–286

  • Lei J, Syndergaard S, Burns AG, Solomon SC, Wang W, Zeng Z et al (2007) Comparison of COSMIC ionospheric measurements with ground-based observations and model predictions: preliminary results. J Geophys Res 112:A07308

    Google Scholar 

  • Liou YA, Pavelyev AG, Liu SF, Pavelyev AA, Yen N, Huang CY et al (2007) FORMOSAT-3/COSMIC GPS radio occultation mission: preliminary results. IEEE Trans Geosci Remote Sens 45(11):3813–3826

    Article  Google Scholar 

  • Liu Z (2004) Ionospheric tomography modeling and application using global positioning system (GPS) measurements. Ph.D. dissertation, Department of Geomatics Engineering, University of Calgary

  • Liu Z, Gao Y (2003) Ionospheric TEC predictions over a local area GPS reference network. GPS Solut 8(1):23–29

    Article  Google Scholar 

  • Liu Z, Gao Y (2004) Development and evaluation of a new 3-D ionospheric modeling method. Navigation 51(4):311–329

    Article  Google Scholar 

  • Liu Z, Skone S, Gao Y (2006a) Assessment of ionosphere tomographic modeling performance using GPS data during the October 2003 geomagnetic storm event. Radio Sci. https://doi.org/10.1029/2004RS003236

    Google Scholar 

  • Liu L, Wan W, Ning B, Pirog OM, Kurkin VI (2006b) Solar activity variations of the ionospheric peak electron density. J Geophys Res 111:A08304. https://doi.org/10.1029/2006JA011598

    Article  Google Scholar 

  • Mautz R, Ping J, Heki K, Schaffrin B, Shum CK, Potts L (2005) Efficient spatial and temporal representations of global ionosphere maps over Japan using B-spline wavelets. J Geodesy 78:660–667

    Article  Google Scholar 

  • Mitchell C, Cannon P (2002) Multi-instrumental data analysis system (MIDAS) imaging of the ionosphere. Technical report, University of Bath, United States Air Force European Office of Aerospace Research and Development

  • Mitchell CN, Spencer PSJ (2003) A three-dimensional time-dependent algorithm for ionospheric imaging using GPS. Ann Geophys 46(4):687–696

    Google Scholar 

  • Nava B, Radicella SM, Leitinger R, Coïsson P (2006) A near real-time model-assisted ionosphere electron density retrieval method. Radio Sci 41(6)

  • Nohutcu M, Karslioglu MO, Schmidt MB (2010) B-spline modeling of VTEC over Turkey using GPS observations. J Atmos Solar Terr Phys 72:617–624

    Article  Google Scholar 

  • Percival DB, Walden AT (1993) Spectral analysis for physical applications, multitaper and conventional univariate techniques. Cambridge Univ. Press, New York

    Book  Google Scholar 

  • Pryse SE, Kersley L (1992) A preliminary experimental test of ionospheric tomography. J Atmos Terr Phys 54(7–8):1007–1012

    Article  Google Scholar 

  • Radicella S (2009) The NeQuick model genesis, uses and evolution. Ann Geophys 52:417–422

    Google Scholar 

  • Raymund TD, Pryse SE, Kersley L, JaT Heaton (1993) Tomographic reconstruction of ionospheric electron density with European incoherent scatter radar verification. Radio Sci 28(5):811–817

    Article  Google Scholar 

  • Schaer S (1999) Mapping and predicting the Earth’s ionosphere using the global positioning system. Ph.D. thesis, Bern University, Switzerland

  • Schaer S, Gunter W, Feltens J (1998) Ionex: the ionosphere map exchange format version 1. In: Dow JM, Kouba J, Springer T (eds) 233. 247, Proceeding of the IGS AC Workshop, Darmstadt, Germany

  • Sharifi MA, Farzaneh S (2017) The ionosphere electron density spatio-temporal modeling based on the Slepian basis functions. Acta Geod et Geophys 52(1):5–18

    Article  Google Scholar 

  • Schmidt M (2007) Wavelet modelling in support of IRI. Adv Space Res 39(5):932–940

    Article  Google Scholar 

  • Schmidt M, Bilitza D, Shum CK, Zeilhofer C (2007a) Regional 4-D modeling of the ionospheric electron content. Adv Space Res 42:782–790

    Article  Google Scholar 

  • Schmidt M, Fengler M, Mayer-Gurr T, Eicker A, Kusche J, Sanchez L, Han SC (2007b) Regional gravity modeling in terms of spherical base functions. J Geodesy 81:17–38

    Article  Google Scholar 

  • Schmidt M, Karslioglu M, Zeilhofer C (eds) (2008) Regional multidimensional modeling of the ionosphere from satellite data. TUJK Annual Scienti_c Meeting, Ankara

  • Schmidt M, Dettmering D, Mößmer M, Wang Y, Zhang J (2011a) Comparison of spherical harmonic and B spline models for the vertical total electron content. Radio Sci. https://doi.org/10.1029/2010RS00460

    Google Scholar 

  • Schmidt M, Dettmering D, Mossmer M, Wang Y, Zhang J (2011b) Comparison of spherical harmonics and B-spline models for the vertical total electron content. Radio Sci 46:RS0D11

    Google Scholar 

  • Schmidt M, Dettmering D, Seitz F (2015) Using B-spline expansions for ionosphere modeling. In: Freeden W, Nashed MZ, Sonar T (eds) Handbook of geomathematics. Springer, Berlin, pp 939–983. https://doi.org/10.1007/978-3-642-54551-1_80

    Chapter  Google Scholar 

  • Schunk R (1988) A mathematical model of the middle and high latitude ionosphere. Pure Appl Geophys 128:255

    Article  Google Scholar 

  • Sharifi MA, Farzaneh S (2014) The spatio-spectral localization approach to modeling VTEC over the western part of the USA using GPS observations. Adv Space Res 54(6):908–916

    Article  Google Scholar 

  • Shen Y, Xu P, Li B (2012) Bias-corrected regularized solution to inverse ill-posed models. J Geodesy 86(8):597–608

    Article  Google Scholar 

  • Simons FJ (2010) Slepian base functions and their use in signal estimation and spectral analysis. In: Freeden W, Nashed MZ, Sonar T (eds) Handbook of geomathematics. Springer, Berlin, pp 891–923

    Chapter  Google Scholar 

  • Simons FJ, Dahlen FA, Wieczorek MA (2006) Spatiospectral concentration on a sphere. Soc Ind Appl Math Rev 48:504–536

    Google Scholar 

  • Slepian D (1983) Some comments on Fourier-analysis, uncertainty and modeling. SIAM Rev 25(3):379–393

    Article  Google Scholar 

  • Tikhonov AN (1963) Solution for incorrectly formulated problems and the regularization method. Soviet Math Dokl 4:1035–1038

    Google Scholar 

  • Wang L (2012) Coseismic deformation detection and quantification for great earthquakes using spaceborne gravimetry. The Ohio State University, Ohio

    Google Scholar 

  • Wieczorek MA, Simons FJ (2005) Localized spectral analysis on the sphere. Geophys J Int 162(3):655–675

    Article  Google Scholar 

  • Yang KF, Chu YH, Su CL, Ko HT, Wang CK (2009) An examination of FORMOSAT-3/COSMIC ionospheric electron density profile: data quality criteria and comparisons with the IRI model. Terr Atmos Ocean Sci 20:193–206

    Article  Google Scholar 

  • Yang F, Kusche J, Forootan E, Rietbroek R (2017) Passive-ocean radial basis function approach to improve temporal gravity recovery from GRACE observations. J Geophys Res Solid Earth 122:6875–6892. https://doi.org/10.1002/2016JB013633

    Article  Google Scholar 

  • Zeilhofer C (2008) Multi-dimensional B-spline modeling of spatio-temporal ionospheric signals. Deutsche Geodätische Kommission bei der Bayerischen Akademie der Wissenschaften

  • Zolesi B, Cander LR (2014) Ionospheric prediction and forecasting. Springer, Heidelberg

    Book  Google Scholar 

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Acknowledgements

The authors are grateful to Professor M. Rycroft and two reviewers for their comments, which helped us to improve this manuscript. We would like to acknowledge (1) the NASA’s Archive of Space Geodesy Data (CDDIS, http://cddis.gsfc.nasa.gov) and Instituto Brasileiro de Geografia e Estatística (IBGE, http://www.ibge.gov.br) for the RINEX data, and (2) the NPSO (Taiwan’s National Space Organization) and UCAR (University Center for Atmospheric Research) for access to the COSMIC RO data (http://cdaac-www.cosmic.ucar.edu/cdaac/products.html).

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  1. School of Surveying and Geospatial Engineering, College of Engineering, University of Tehran, Tehran, 113654563, Iran

    Saeed Farzaneh

  2. School of Earth and Ocean Sciences, Cardiff University, Cardiff, CF10 3AT, UK

    Ehsan Forootan

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  1. Saeed Farzaneh
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Correspondence to Saeed Farzaneh.

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Farzaneh, S., Forootan, E. Reconstructing Regional Ionospheric Electron Density: A Combined Spherical Slepian Function and Empirical Orthogonal Function Approach. Surv Geophys 39, 289–309 (2018). https://doi.org/10.1007/s10712-017-9446-y

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