Effiziente Umsetzung der Integration der Elektronendichte innerhalb der Ionosphäre entlang des Signalweges

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1 Effiziente Umsetzung der Integration der Elektronendichte innerhalb der Ionosphäre entlang des Signalweges (DFG-Projekt MuSIK) Marco Limberger 1, Urs Hugentober 1, Michael Schmidt 2, Denise Dettmering 2, Wenjing Liang 2 1 Technische Universität München (TUM) 2 Deutsches Geodätisches Forschungsinstitut (DGFI), München, Germany marco.limberger@bv.tum.de Geodetic Week, Nürnberg

2 Outline Modeling Integration Analysis Conclusion Geodetic Week, Nürnberg

3 The geometry free linear combination from double frequency carrier phase GNSS observations can be used to eliminate geometry, i.e. satellite orbits, station positions, clock errors and tropospheric delays with : Slant Total Electron Content : Inter-frequency differential delays : Carrier phase bias : Noise Geodetic Week, Nürnberg

4 STEC is defined as the integral of the space- and time-dependent electron density along the ray path between satellite and receiver wherein, : Geocentric position vector : Latitude, longitude : Time : Geocentric distance The height dependency of the electron density can be modelled by the physics-motivated Chapman function for the F2-layer in combination with a plasmasphere profile Geodetic Week, Nürnberg

5 with : F2 peak electron density : Plasmasphere basis density : Peak height : Scale height : Height Geodetic Week, Nürnberg

6 The linearization of the derived Chapman/plasmasphere-function leads to the final observation equation for STEC which considers a background model and correction terms with respect to five unknown target parameters. Background model Correction part Geodetic Week, Nürnberg

7 stepwidth c stepwidth b stepwidth a Geodetic Week, Nürnberg

8 Certain number of integration Pre-definitions Layer boundaries steps due to all 3 layers Quadrature order Integration stepsize Numerical Integration with N-point Gauss-Legendre Quadrature Geodetic Week, Nürnberg

9 Objective: Approximation of an integral as a sum of given sample values at nodal points with positive-valued weights Nodal points are derived from zeros of Legendre polynomials and are required to calculate the corresponding weights Gauss-Legendre quadrature formulas with polynomials of degree exactly nodal points can integrate Geodetic Week, Nürnberg

10 Hardware Information Ubuntu Natty Narwhal 2048 MB Ram CPU: Intel Core 2.26 GHz Software-Development Programming language: C++ Math Libraries: Armadillo with LAPACK (open source), Standard Cmath Geodetic Week, Nürnberg

11 Reference solution 5 stations of Sistema de Referencia Geocéntrico para Las Américas (SIRGAS) Integration stepwidth 1/1/1 km Quadrature order N=9 (9 weighted nodal points per integration step) Database: UT, 2002/07/01, 30s sampling rate, GPS only Geodetic Week, Nürnberg

12 Data processing Constant layer boundaries at 80/200/1200/2000 km height Pre-defined vertical Integration stepwidth transformation to slant direction by wherein : slant stepwidth : vertical stepwidth : elevation angle [deg] Varying quadrature order N (2,3,,9) Focus on differences to calculated reference solution in comparison to the processing duration Geodetic Week, Nürnberg

13 Processing Duration [s] RMS of absolute Differences [TECU] Modeling Integration Analysis Conclusion RMS of reference data : TECU Duration 20/20/20 Duration 40/40/40 Duration 60/60/60 Duration 80/80/ RMS 20/20/20 RMS 40/40/40 RMS 60/60/60 RMS 80/80/80 0,2 4 min 26 s 0, , min 35 s 0,16 0,14 2 min 43 s 0, ,1 1 min 52 s min 12 s 59 s 0, s 0, , , , , , Quadrature Order N 31 s 0,08 0,06 0,04 0,02 0 Geodetic Week, Nürnberg

14 An efficient processing strategy is required in order to deal with the complex formulas behind this ionospheric modeling approach especially when different observation techniques such as GNSS, altimetry and LEO GPS will be considered in future parallel computing! The Gauss-Legendre quadrature method has been succesfully applied Further tests with real observation data will show how the setting parameters (e.g. layer height, integration stepwidth and quadrature level) have to be chosen in order to yield a compromise between acceptable processing time and desired accuracy A 24 h GNSS data set (2002/07/01) w.r.t. to the 5 SIRGAS stations already covers 138,240 observations. Introducing vsw = 80/60/80 km and N = 5, the process duration yields 36 min. Geodetic Week, Nürnberg

15 Thank you for your attention Acknowledgement: This work is funded by the Deutsche Forschungsgemeinschaft (DFG), Bonn Geodetic Week, Nürnberg