Towards More Reliable Estimation of GPS Positioning Accuracy

Transcription

1 Towards More Reliable Estimation of GPS Positioning Accuracy J. Krynski, Y.M. Zanimonskiy Institute of Geodesy and Cartography, 7 Modzelewskiego St., -679 Warsaw, Poland krynski@igik.edu.pl, yzan@poczta.onet.pl; Fax: , Tel.: Abstract Uncertainty of vector components estimation, obtained from processing GPS data using either commercial or scientific software, represents rather an internal consistency than the accuracy of positioning. Short period biases, including non-modelled effects, some with unstable amplitudes, that affect GPS solutions are as systematic-type terms not reflected in uncertainty estimation. Therefore, calculated uncertainty is usually much too optimistic as the accuracy estimate. In the routine GPS data processing one observing session provides a single solution. Data that form such observing session can be represented by a series of overlapped sessions. Comparison of solutions obtained from processing overlapped sessions leads to more reliable accuracy estimation than the one based on uncertainty estimation of individual solutions. The strategy of GPS solutions quality analysis based on the concept of overlapping sessions with optimum length and temporal resolution is presented. The strategy was verified with use of data from the EPN stations processed with both Bernese and Pinnacle software packages. Keywords: Global Positioning System (GPS), Positioning accuracy, Statistical analysis, Spectral analysis 1. Introduction It is well known that the solutions for vector components or coordinates obtained from processing precise GPS data from different observing sessions vary usually much stronger than what their precision estimate indicates. Standard deviations of GPS solutions provided by processing software reflect only the internal consistency of data processed and the internal precision. They do not, however, indicate the actual accuracy of GPS positioning in the real scale (e.g. Gandolfi et al., ). An extensive research is conducted to improve the precision of GPS solutions by using better models for GPS observations, and to improve the precision estimate (e.g. Teunissen, ). Time series of GPS solutions are particularly suitable for such investigations (Krynski and Zanimonskiy, ). The tools of statistical analysis and spectral analysis are useful for separation factors causing variations in GPS solutions and estimation of their magnitudes. Time series of GPS solutions do not exactly represent a random process. Besides a random part that is mainly due to observation noise the variations in GPS solutions also contain components of a chaotic character as well as biases (Krynski et al., a). Model errors, non-modelled effects and varying satellite configuration, including multipath cause systematic variations in GPS solutions. In addition, due to non-linearity of the system, data noise generates biases in computed results. Missed cycles in integer ambiguity resolution process and sudden changes in satellite configuration due to a rise of a satellite or satellite s repair are the main sources of chaotic errors. The choice of the method used to suppress disturbances in time series corresponds to their character, i.e. random or chaotic. Noise, external with respect to measuring system, is filtered out using the smoothing procedure that employs a rectangular window the size of which becomes a parameter to be determined. The optimum size of the window can be estimated at each filtering stage. The analysis of such time series indicates the existence of a number of periodic components and trends not modelled in data processing stage (Bruyninx, 1; Poutanen et al., 1; Krynski et al., a). Major part of the power spectrum of the variations is concentrated in diurnal and longer periods. In particular, periodic variations with dominating 1h and h periods (Krynski et al., ) are distinguished. Few hours long periods in the spectrum are most probably the artefacts (King et al., ) caused by the effects dominated by random noise due to non-linearity of the system (Krynski and Zanimonskiy, ).

2 Processing of as long as h GPS sessions leads to solutions smoothed off for daily and sub-daily periodic biases. The use of shorter observing sessions with preserving high quality of GPS positioning requires the investigation of periodic biases, their detection, their source specification and an attempt towards their modelling. Due to a large amount of information contained in time series of GPS solutions based on overlapped sessions it becomes possible to apply statistical tests to detect outliers. Numerous sudden changes in satellite configuration occur with a period of one sidereal day. They cause specific variations (jerks) in time series of GPS solutions, with periods significantly smaller than one sidereal day, i.e. even of the order of 1h. Such variations correspond rather to a chaotic process then a random one. Jerks can be suppressed by optimising the length of overlapping sessions and eliminating the disturbing results (Krynski and Zanimonskiy, ). Continuous and regular change of satellite constellation causes smooth variations of parameters of measuring process, i.e. signal to noise ratio, atmospheric delays, multipath, orbit corrections, etc., that are periodic with a half of sidereal day period (Krynski et al., b). Terms with periods of half and one sidereal day occur in informative parameters such as vector components as well as in non-informative parameters, like software-provided standard deviations of GPS solutions, cross-correlation coefficients of vector components, number of measurements accepted for the solution, etc. In time series, including those of GPS solutions, there exists a corresponding to the sampling interval periodicity that should be interpreted as the artefact. The shorter sampling interval the finer is time resolution, the higher is the Nyquist frequency and the less distinguished become spectral leakage effects in time series. A careful estimation of an optimum length of a session used to calculate positions from GPS data is needed due to jerk-type variations in GPS solutions corresponding to a satellite rising or its descending. The optimum length of a session does not necessarily correspond to longer ones. With the increase of the session length an internal accuracy of the output data increases but at the same time the growth of spectral leakage is observed. Thus the extension of a session length used for computing GPS data reduces the estimated uncertainty of the solution but it simultaneously decreases Nyquist frequency. The sum of those two counteracting effects depends also on spectrum of noise and the signal itself. The increase of the Nyquist frequency can be accomplished by using overlapping sessions. Correlation accompanying time series of GPS solutions obtained from overlapping sessions is significantly smaller than the one in the classical random process series. For example, the correlation coefficient in time series based on solutions from the sessions with 7% overlap is at the level of.5 (Krynski et al., b) while such a coefficient for a wideband random process reaches.5 in case of 5% overlap (Harris, 197). Predictability of the reaction of the GPS measuring system (both receiver and processing software) on disturbances and inadequacy of models used could be viewed experimentally by statistical and correlation analysis of time series of GPS solutions.. Numerical experiments The problem of reliable accuracy estimation of GPS positioning can be investigated using time series of GPS solutions obtained from sessions of different lengths for vectors of different length, located in different geographic regions. Practical needs, potentiality of accessible data processing infrastructure, and the experience gained in GPS research contributed towards making the choice of GPS data for processing and determining its strategy for numerical experiments. As an example, GPS data from 1 provided by two EPN Stations BOGO and JOZE was used to generate time series of BOGO-JOZE vector (4 km length) components with the Bernese v.4. and Pinnacle software packages. With the Bernese software the GPS solutions were obtained from processing h, h, 4h, and 6h sessions over 19 days in August, with an overlap (1h shift), h sessions with.5h overlap (m shift), and h sessions with h overlap over 4 months (February-May). GPS solutions were obtained with Pinnacle from processing h, h, 4h, 5h, 6h, h, 9h, 1h, 11h, 1h, 1h, 14h, 17h, 1h, 19h, h, 1h, h, h, 5h, 6h, 7h, and h with 1h time resolution (1h shift), over 15 days in August. Chosen data represent two different periods of seasonal atmospheric dynamics, i.e. winter-spring corresponds to quiet atmosphere while summer to a disturbed one. Time series of GPS solutions generated were then the subjects of statistical analysis. The dispersion of GPS solutions as well as their averaged combinations (mean values m, standard deviations σ z and the estimated standard deviations σ m of the mean) together with their precision estimates (software-provided standard deviations σ w ) was analysed. The conceptual scheme of forming groups of GPS solutions for further statistical analysis is shown in Fig. 1.

3 z DOY 1 1 m=.6 mm σz =17 mm σ m =. mm σ w = mm m =.4 mm σ z = 15 mm σ m =.5 mm σ w = mm m = 1.4 mm σ z = mm σ m =.5 mm σ w = mm m = -.9 mm σ z = 17 mm σ m =.57 mm σ w = mm m = -1.7 mm σ z = 1 mm σ m =.4 mm σ w = mm m =.55 mm σ z =. mm σ m = 1. mm σ w = mm Fig. 1. The scheme of forming groups of GPS solutions and example of their estimated statistics Variations of GPS solutions in the investigated time series are typical for the solutions based on data from numerous EPN stations. Standard deviations of those solutions that are the measure of internal accuracy are all at the same level. Thus mean standard deviation σ w in the group of solutions in Fig. 1 and their combinations are all equal to mm. Estimators of m, σ z and σ m that are used to estimate external accuracy of GPS solutions take different values in different groups. Averaging solutions that represent the groups (statistics in lowest rectangle in Fig. 1) provides similar result to the one based on all data included in the groups considered (statistics in upper rectangle in Fig. 1) but its accuracy estimate becomes more realistic (estimate of internal accuracy σ w will correspond to the estimate of external accuracy σ z ). Dispersion of GPS solutions grows with the size of sample, i.e. with a number of solutions that form the group investigated; that also corresponds to the length of data window used. On the other hand the dispersion of the average solution from the group of sessions decreases with growing number of sessions in the group investigated. The plots illustrating σ z and σ m dispersions for vertical component of the vector calculated with the Bernese software using h sessions with.5h shift over months are given in Fig. a and Fig. b, respectively. σ length of data window in the group [h] (a) number ( n ) of sessions in the group σ m length of data window in the group [h] (b) number ( n) of sessions in the group Fig.. Mean st. dev. σ z in the group of n consecutive solutions for vertical component of the vector (a) and mean st. dev. of the mean of n consecutive solutions in the group (b). Grey line in (a) corresponds to the st. dev. in months long group of sessions. Dashed line in (b) corresponds to the accuracy of average solution from months data The change of the variation rate of dispersion (Fig. a and b) is observed around data window of 1h. For longer windows the variation rate of dispersion becomes substantially less significant. The mechanisms that affect GPS solutions obtained from sessions shorter than 1h differ from those observed in the solutions from longer sessions. Solutions from sessions shorter than 1h are mainly affected with noise and periodic biases due to GPS satellite constellation. Fig. b also shows that vertical component of 4 km vector can be determined from 1h GPS data with accuracy of 6-7 mm. The increase of the length of session (data window) used to generate GPS solutions, results in reduction of dispersion of those solutions, mainly due to averaging both noise and periodic biases. Time series of GPS solutions based on longer sessions is much smoother as compared with the one derived from short sessions. The effect of such a smoothing procedure could be simulated with combining and averaging GPS solutions obtained using short data window. The effect of smoothing GPS-derived vertical component of the vector obtained from h sessions, by applying running average with 1.5h window may be seen when comparing Fig. a with Fig. b. The results shown so far indicate the external accuracy estimate of GPS solutions that is based on analysis of repeatability with use of time series of high temporal resolution. Such an accuracy estimate does not coincide with standard deviations provided by GPS processing software that reflect an internal

4 accuracy of the system. Short period biases, including non-modelled effects, some with unstable amplitudes, that affect GPS solutions are as systematic-type terms not reflected in uncertainty estimation. Therefore, calculated uncertainty is usually much too optimistic as the accuracy estimate. Comparison of external with internal accuracy of determination of the length and vertical component of the vector investigated using the Bernese and Pinnacle software from sessions of different length with 1h shift, is given in Fig. 4 and Fig. 5, respectively σ z = 15. mm DOY 1 (a) σ z = 5.7 mm (b) DOY 1 Fig.. Time series of vertical component of the vector obtained from processing h sessions with m shift (a), and running average with a window of 1.5h (averaging the current groups of solutions from single sessions) (b) σ dd [mm] number (n) of sessions in the group Bernese σ dd [mm] σ z (from single session) σ m (from the group of n sessions) σ w (from single session) number (n) of sessions in the group Pinnacle Fig. 4. St. dev. σ of the GPS-derived vector lengths provided by processing softwares, and estimated by statistical analysis of time series of GPS solutions σ number (n) of sessions in the group Bernese σ σ z (from single session) σ m (from the group of n sessions) σ w (from single session) number(n) of sessions in the group Pinnacle Fig. 5. St. dev. σ of the GPS-derived vertical component provided by processing softwares, and estimated by statistical analysis of time series of GPS solutions The dispersion of GPS solutions obtained using the Bernese software is larger than that from the Pinnacle software, in particular when short sessions were processed (Fig. 4 and Fig. 5). That phenomenon is more distinct for vector length than for the vertical component. In spite of large dispersion of GPS solutions that might be explained by the use of QIF strategy for processing data for 4 km vector with the Bernese software, the softwareprovided standard deviations of the solutions obtained are at the very low level. Both vertical component and the length of a vector examined were determined with the accuracy of about 5 cm from h sessions while precision of the solution provided by the Bernese software was at the level of single millimetres. In case of solutions based on h sessions the discrepancy between the external and internal estimate of accuracy drops down to the level of a few millimetres holding their ratio at the same level. The effect of noise and periodic biases on GPS solutions can also be reduced by averaging the solutions over the groups of sessions, e.g. the mean of n h sessions, that form the group. Such simple

5 averaging does not remove all effects that are eliminated when processing with the Bernese software one session of length corresponding to the length of the respective group of short sessions. The external accuracy based on analysis of groups of sessions is thus overestimated although its trend remains similar to the one related to single sessions. For GPS solutions obtained using the Pinnacle software being less sophisticated in terms of GPS observations modelling than the Bernese one, the main trends for the external accuracy getting improved with growing session length are preserved. Different image has, however, the mutual relationship of the external and internal accuracy. The internal accuracy estimate given by Pinnacle is more realistic than in case of the Bernese software. Vertical component of the vector examined was determined from h sessions with the Pinnacle software with the accuracy of about.5 cm while software-provided precision of the solution was about cm. For the length of the vector investigated the external and internal accuracy was cm and 1.5 cm, respectively. Moreover, for a certain length of session, internal accuracy coincides with the external one. With further growing session length the internal accuracy of the solutions becomes overestimated. Such a singularity corresponds to 1h and 4h sessions in case of vertical component and vector length determination, respectively. GPS solutions from the Pinnacle software, averaged over the groups of sessions, practically coincide with those corresponding to the respective single sessions. It particularly concerns vertical component. σz [mm] dd: σ z = 9.5 * σ w -.4 mm dh: σ z = 6.5 * σ w -. mm σ w [mm] Fig. 6. Correlation between the external accuracy σ z and the internal accuracy σ w of GPS solution provided by the Bernese software The discrepancy between the external accuracy σ z and internal consistency σ w of GPS solutions obtained using the Bernese software was investigated for numerous vectors of length from a few tens to a few thousand kilometres. h and h GPS sessions covering 4 months of 1 were processed. The correlation between the external and the internal accuracy estimates is given in Fig. 6. Internal accuracy differs from the external one by a scale factor of about 7 and 1 for a vertical component and vector length, respectively.. Strategy of GPS solution quality analysis The strategy developed for detecting and modelling biases in time series of GPS solutions is given in Fig. 7. Input data in consecutive blocks GPS processing software Series of GPS output solutions (biased, chaotic and noisy) Spectral analysis PSD at Nyquist freq. significant Yes No Optimum time series with more clearly reflected biases, chaos and noise LONG SHORT Filtering & modeling in time domain Correlation & modeling External data, i.e. TROPO & IONO Need of re-processing with overlapped sessions Filtering & modeling in stacked time domain Correlation & modeling No Shorter sessions acceptable Need of re-processing with shorter sessions Fig. 7. Flowchart of the strategy of GPS solutions quality analysis Temporal resolution of a series of GPS solutions is determined by a sampling rate that corresponds to the length of session when data is processed in consecutive blocks. The longer the processed GPS sessions the smoother become solutions and consequently time series obtained. Smoothing obviously reduces random effects but also some periodic biases. Solutions based on shorter sessions are thus affected by larger biases. To study biases in GPS solutions the examination of time series with sufficient temporal resolution is required. Thus time series of GPS solutions obtained from short sessions is preferable despite of increased noise level. Shortening the session is, however, limited by the length of the vector determined. To increase temporal resolution of time series the overlapped sessions need to be processed. Overlapping causes an increase of correlation between consecutive GPS solutions that has to be carefully considered when estimating statistical parameters of such time series. It allows, however, for efficient detection and separation of chaotic effects from biases. Power spectrum density at the Nyquist frequency can be used as an indicator of need of a Yes

6 further increase of temporal resolution of a series by processing either shortened sessions or more overlapped sessions. Time series, optimal with respect to the structure of investigated biases, consist of GPS solutions obtained from optimum session length and optimum sampling time. Such a series reflects clearly periodic biases as well as chaotic terms. It is therefore suitable to detect and estimate periodic biases. Similar strategy with 4h window stepped by minutes and followed by a running average procedure was used for investigation of the vertical shift caused by Earth tides (Neumeyer et al., ). Long time series can directly be processed for filtering noise and modelling periodic biases. In case of short time series stacked solutions need to be calculated over e.g. one-day period and then filtered and processed for modelling biases. Finally the models could be correlated with the external data, e.g. troposphere or ionosphere parameters to separate biases and find their sources. Conclusions A dispersion of GPS-derived vector components considered as an external accuracy estimate does not coincide with processing software-provided estimated accuracy that is the internal accuracy estimate. The discrepancy between externally and internally estimated accuracy was investigated using statistical analysis of time series of vector components obtained with the Bernese and Pinnacle software. Internal accuracy provided by the Bernese software differs from the external one, in cases investigated, by a scale factor of about 7 and 1 for a vertical component and vector length, respectively. Internal accuracy estimation provided by the Pinnacle software can be considered as the acceptable rough estimate of accuracy. Accuracy of GPS solutions based on data from permanent stations or from long-term geodynamics campaigns can efficiently be estimated by investigating time series of overlapping solutions using the tools of statistical analysis. The described strategy of quality analysis of GPS solutions besides filtering allows for estimation of biases and chaotic effects. That procedure is not suitable for estimation of accuracy of GPS solutions obtained from single, short time occupation of sites. The majority of noise can be filtered using simplified statistical analysis of overlapped solutions based on sub-intervals of the observed session. Biases and jerks, however, can only be roughly estimated externally using the results of extended statistical analyses. GPS positioning based on shorter than 1h single observing session could not provide the solution for vector components with accuracy below 1 cm, even for short vectors. Acknowledgements The research was partially financed by the Polish State Committee for Scientific Research (grant PBZ-KBN-1/T1/). The authors express special gratitude to Mrs. M. Mank and Mr. L. Zak from the Institute of Geodesy and Cartography, Warsaw, as well as to Dr. P. Wielgosz from the University of Warmia and Mazury, Olsztyn, for processing GPS data. References Bruyninx C., (1): Overview of the EUREF Permanent Network and the Network Coordination Activities, EUREF Publ., Eds. J. Torres, H. Hornik, Bayer. Akad. der Wies., München, Germany, 1, No 9, pp. -. Gandolfi S., Gusella L., Perfetti N., Dubbini M., (): Accuracy and precision vs software and different conditions using Italian GPS fiducial network (IGFN) data, Reports on Geodesy, WUT, Warsaw, No (65), pp Harris F.J., (197): On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform, Proc. IEEE, Vol. 66, January 197, pp King M., Coleman R., Nguyen L.N., (): Spurious Periodic Horizontal Signals in Sub-Daily GPS Position Estimates, J. of Geod., Vol. 77, Nr 1-, May, pp Krynski J., Zanimonskiy Y.M., Wielgosz P., (): Short Time Series Analysis of Precise GPS Positioning, Presented at the XXV Gen. Assembly of EGS, Nice, France, 5-9 April. Krynski J., Cisak J., Zanimonskiy Y.M., Wielgosz P., (a): Variations of GPS Solutions for Positions of Permanent Stations Reality or Artefact, EUREF Publ. No 1, Mitteilungen des Bundesamtes für Kart. und Geodäsie, Frankfurt am Main, Band 9, pp. -5. Krynski J., Zanimonskiy Y.M., Wielgosz P., (b): Modelling biases in GPS Positioning Based on Short Observing Sessions, Presented at the XXVII Gen. Assembly of EGS, Nice, France, 1-6 April. Krynski J., Zanimonskiy Y.M., (): Quality Structure of Time Series of GPS Solutions, Proc. of the NKG Gen. Assembly, 1-5 Sept., Espoo, Finland, pp Neumeyer J., Barthelmes F., Combrink L., Dierks O., Fourie P., (): Analysis of Results from the SG Registration with the Dual Sphere Supercon-ducting Gravimeter at SAGOS (South Africa), BIM 15, 15 July, Paris, pp. 7-. Poutanen M., Koivula H., Ollikainen M., (1): On the Periodicity of GPS Time Series, Proc. of IAG 1 Sc. Assembly, -7 Sept. 1, Budapest, Hungary. Teunissen P.J.G., (): The Parameter Distributions of the Integer GPS Model, J. of Geod., Vol. 76, Nr 1, January, pp