Influences of different factors on temporal correlations of GNSS observations

Transcription

1 2 T : oscillation period (unknown) C : scaling factor (unknown) ND : determined empirically a priori C,T: least-squares regression on sample ACF ND : first zero point of sample ACF h= ACF h π h ND ND T h : lag (epoch distance) h 2 C exp cos for h ND = C for h = 0 ACF E.g. analytical autocorrelation function (ACF) (Howind 2005) Modelling approach using residual time series Signal processing techniques Site-specific influences (e.g. multipath effects) Physical processes in the atmosphere Impacting factors GNSS observations are temporally correlated! Introduction KIT The Cooperation of Forschungszentrum Karlsruhe GmbH and Universität Karlsruhe (TH) Lag ACF X. Luo, M. Mayer, B. Heck Geodetic Institute Influences of different factors on temporal correlations of GNSS observations Geodetic Week 2009 September 22-24, Karlsruhe S4: Applied Geodesy and GNSS

2 Sample ACF Definition: x, K, x n are observations of a time series { X t }. The sample autocovariance function is γˆ( h ) : = n n h t = ( x The sample autocorrelation function is n t + h x )( x t x ) x t ˆ( γ h ) ˆ ρ ( h ) =, n < h < n. ˆ(0) γ n < h < n, where x = n t =. { X t } (weakly) stationary ˆ ρ ( h ) representative for the dependence in the data Distribution approximation of ˆ ρ ( h ) For large n, the ˆ ρ ( h ) of an iid* sequence with finite variance are approx. iid with distribution ˆ( ρ h ) ~ N (0, ). n * iid: independent and identically distributed 3 Case study: data base Multipath (MP): strong Tab. : GPS processing strategies and data base MP: weak TAAF HEDA Observations Processing interval Obs. weighting Tropospheric model (time span: 5 min) GPS -Hz double differences (phase) DOY2007: 6-8, UT: 5-8 h SNR*-based: w=f(snr) (cut-off: 3 ) Niell (dry) (a priori model) Niell (wet) (mapping function) Ambiguity resolution SIGMA strategy (L5, L3) Antenna correction Individual absolute calibration RATA Residual data base (: 285) Studentised double difference residuals (SDDR) in sidereal time Identical length=3600 epochs ( h) Atmos. conditions Meteorological surface data (DWD**) * SNR: Signal-to-Noise Ratio, ** DWD: Deutscher Wetterdienst ( SIBI Tab. 2: Considered factors impacting temporal correlations Factor Comparison between Multipath HEDA (strong) TAAF (weak) 48/5 Baseline length SIBI (42.5 km) RATA(203.7 km) 48/33 Rel. humidity wet days (83 %) dry days (46 %) 3/50 Fig. : SAPOS network in the area of the state of Baden-Württemberg (Southwest Germany) Wind force windy (4-5 m/s) calm (-2 m/s) 26/38 4

3 5 6 Modelling residual trends (a): Conventional approach (b): Improved approach with additional correlation analysis Fig. 2: Comparison of different approaches for modelling residual trends (conventional vs. improved) Modelling residual trends SDDR: SIBI097(all SDDR) Trend-eliminated SDDR: SIBI097(selected SDDR) Sample ACF Fig. 3: Selected representative example of comparison of trend modelling (conventional vs. improved)

4 7 8 Distribution of ND Fig. 4: Empirical distribution of ND (ND : first zero-point of sample ACF) Tab. 3: Statistical characteristics and hypothesis tests for normal distribution of ND Statistical characteristics Test for normality p-value ** 95% quantile Chi-square Jarque-Bera 0.46 *STD: standard deviation, **: interquartile range Multipath effects (MP) outlier HEDA (MP: strong) TAAF (MP: weak) Fig. 5: Influences of multipath effects on correlation length represented by ND Baseline HEDA Tab. 4: Influences of multipath effects on statistical characteristics of ND MP strong Length [km] Statistical characteristics of ND ** TAAF weak

5 9 0 Baseline length RATA (203.7 km) SIBI (42.5 km) Fig. 6: Influences of baseline length on correlation length represented by ND Baseline RATA Tab. 5: Influences of baseline length on statistical characteristics of ND MP weak Length [km] Statistical characteristics of ND ** SIBI weak Relative humidity (rh) Wet days 66, 73, 76 Dry days 67, 68, 70 Fig. 7: Influences of atmospheric relative humidity on correlation length represented by ND Days Wet Tab. 6: Influences of atmospheric relative humidity on statistical characteristics of ND relative humidity 83% 3 88 Statistical characteristics of ND ** Dry 46%

6 2 Wind force Windy days 72, 73, 77 Calm days 65, 68, 69 Fig. 8: Influences of wind force on correlation length represented by ND Days Windy Tab. 7: Influences of wind force on statistical characteristics of ND wind velocity 4-5 m/s Statistical characteristics of ND ** Calm -2 m/s Conclusions and outlook Temporal correlation analysis Residual-based, using sample ACF Assumption of stationary time series Improved approach for modelling residual trends Correlation length Represented by the first zero-point of sample ACF Asymptotic normal distribution (hypothesis tests) correlation length within this study: approx. 00 s Decreasing with stronger multipath effects and wind force Insignificantly influenced by baseline length and relative humidity Further research work Validating the statements related to atmospheric conditions Analysing the influences of satellite geometry on temporal correlations

7 Questions & comments Thank you very much for your attention! The project Improving the stochastic model of GPS observations by modelling physical correlations (HE433/6-/2) is supported by the Deutsche Forschungsgemeinschaft (DFG). GPS system modernisations + advanced mathematical modelling = accurate and reliable positioning results 3 Geodetic Institute Geodetic Universität Institute: X. Karlsruhe Luo, M. Mayer, (TH) B. Englerstraße Heck luo@gik.uni-karlsruhe.de 7, 763 Karlsruhe, Germany Tel.: +49 (0)72 Influences , of different Fax: +49 factors (0)72 on temporal , correlations home page: of GNSS observations