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SCAR Report 23

Current Results on the Investigation of
GPS Positioning Accuracy and Consistency

J. Krynski, Y.M. Zanimonskiy
Institute of Geodesy and Cartography, Modzelewskiego 27, PL 02-679 Warsaw, Poland;

E-mail: krynski@igik.edu.pl, yzan@poczta.onet.pl;
Fax: +48 22 3291950, Tel.: +48 22 3291904

Abstract

Considerable progress observed in geodynamics research is mainly the result of development of measuring techniques. The qualitative results on crustal movements presented in some publications seem, however, to be at the level of their accuracy determination. A realistic estimation of the potential of the experiment is necessary to avoid false conclusions that may describe the non-existent occurrences (artefacts), especially when experiment is difficult or very expensive. 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. The problem of reliable accuracy estimation of GPS positioning concerns all fields of surveying practice including GPS positioning for geodynamics.

The strategy of GPS solutions quality analysis based on the concept of overlapped sessions with optimum length and temporal resolution is presented. The strategy was verified with use of data from the Antarctic and European permanent GPS stations processed with both Bernese and Pinnacle software packages. Numerical examples are given. Keywords: Global Positioning System (GPS) – Positioning accuracy – Statistical analysisIntroduction
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 their precision estimate indicates. Standard deviations of GPS solutions provided by processing software reflect the internal consistency of data processed and the internal precision. In general they do not, however, indicate the actual accuracy of GPS positioning in the real scale (e.g. Dubbini et al., 2003).

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, 2002). One approach is to study the variations of GPS solutions using data from permanent GPS stations. Time series of GPS solutions are particularly suitable for such investigations (Krynski and Zanimonskiy, 2002). The tools of statistical analysis and spectral analysis are useful to separate factors causing variations in GPS solutions and to estimate their magnitudes.
Time series of GPS solutions do not exactly represent a random process. The variations in GPS solutions have a complex structure. Besides a random part that is mainly due to observation noise they contain also components of a chaotic character as well as biases (Krynski et al., 2002a). 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 and sudden changes in satellite configuration due to a rise of a new 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 simply filtered out using the smoothing procedure that employs a rectangular window. The size of the window becomes a parameter to be determined. According to the classical procedure of time series processing, 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 that are not modelled in data processing stage (Bruyninx, 2001; Poutanen et al., 2001; Krynski et al., 2002). Major part of the power spectrum of the variations is concentrated in diurnal and larger periods. In particular, periodic variations with dominating 12h and 24h periods (Krynski et al., 2000) are distinguished. Few hours’ long periods in the spectrum are most probably the artefacts (King et al., 2002) caused by the effects that are dominated by random noise processes due to non-linearity of the system (Krynski and Zanimonskiy, 2002).

GPS solutions based on processing of as long as 24h sessions, that are common for establishing and maintaining geodetic reference frame and for geodynamic applications, are considered as ones smoothed off for daily and sub-daily periodic biases. The use of shorter observing sessions with preserving high quality of GPS positioning as well as the improvement of real-time GPS positioning performance 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 (jerks) in satellite configuration occur with a period of one sidereal day. They cause specific variations in time series of the components of computed vectors from GPS data with periods significantly smaller than one sidereal day, i.e. even of the order of 1h. Variations in such a time series 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, 2002; Cisak et al., 2002).

In spite of jerks in GPS solutions the continuous change of satellite constellation causes smooth changes of parameters of measuring process, i.e. signal to noise ratio, atmospheric delays, multipath, orbit corrections, etc. High regularity of changes in satellite constellation makes all those variations periodic with a half of sidereal day period. Spectral and correlation analysis of time series of GPS solutions shows the existence of such period (Krynski et al., 2002). Periodic terms with periods of half of sidereal day and one sidereal day occur in informative parameters such as vector components as well as in non-informative parameters, like standard deviations of GPS solutions provided by processing software, cross-correlation coefficients of vector components, number of single measurements taken to the solution, etc.

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 even more distinguishably to its descending as well as to ionospheric storms. 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 increase 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 Nyquist frequency can be accomplished by using overlapping sessions. Correlation accompanying time series of solutions obtained from overlapping GPS sessions is significantly smaller than the one in the classical time of a random process. For example, the correlation coefficient in time series based on solutions from the sessions with 87% overlap is at the level of 0.5 (Krynski et al., 2002) while such a coefficient for a wideband random process reaches 0.5 in case of 50% overlap (Harris, 1978).

Predictability of the reaction of the GPS measuring system (both receiver and processing software) on disturbances and inadequacy of models used is difficult due to user’s limited access to the algorithms applied to data processing. That reaction could, however, 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 provided by two EUREF Permanent Network Stations BOGO and JOZE from 2001 was used to generate time series of BOGO-JOZE vector (42 km length) components with the Bernese v.4.2 and Pinnacle software. With the Bernese software the GPS solutions were obtained from processing 1h, 2h, 3h, 4h, and 6h sessions over 19 days in August, with overlap (1h shift), 3h sessions with 2.5h overlap (30m shift), and 24h sessions with 23h overlap over 4 months (February-May). GPS solutions were obtained with Pinnacle from processing 2h - 28h sessions with 1h time resolution (1h shift), over 15 days in August. Chosen data represent two different periods of seasonal atmospheric dynamics in Europe, i.e. winter-spring corresponds to quiet atmosphere while summer to a disturbed one. Data set from the Antarctic stations used covered the period of extremely active ionosphere (October and November 2001) as well as the period of quiet ionosphere in July 2001. Time series of GPS solutions generated were then the subjects of statistical analysis. The dispersion of GPS solutions as well as their averaged combinations together with their precision estimates was analysed. The conceptual scheme of forming groups of GPS solutions for further statistical analysis is shown in Fig. 1.

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 those dispersions for vertical component of the vector calculated with the Bernese software using 3hsessions with 0.5h shift over 3 months are given in Fig. 2a and Fig. 2b, respectively.

In both cases shown in Fig. 2, the change of the variation rate of dispersion is observed around data window of 12h. For longer windows the change of the variation rate of dispersion becomes substantially less significant. The mechanisms that affect GPS solutions obtained from sessions shorter than 12h differ from those observed in the solutions from longer sessions. Solutions from sessions shorter than 12h are mainly affected with noise and periodic biases due to varying GPS satellite constellation. Fig. 2b also shows that vertical component of 42 km vector can be determined from 12h GPS data with accuracy of 6-7 mm at one sigma level.

Fig. 1. The scheme of forming groups of GPS solutions and estimating their statistics

Fig. 2a

Fig. 2b

Fig. 2. The rms of vertical component from single sessions in the group of n sessions (a) and rms of average solutions in the groups of n sessions (b) versus the length of data window. Grey line at (a) corresponds to the rms in 3 months long group of sessions. Dashed line at (b) corresponds to the accuracy (3_ level) of average solution from 3 months data

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 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 by combining and averaging GPS solutions obtained using short data window. The effect of smoothing GPS-derived vertical component of the vector obtained from 3h data window, by applying running average with 12,5h window may be seen by comparison of Fig. 3a and Fig. 3b, respectively.

Fig. 3a

Fig. 3b

Fig. 3. Time series of vertical component of the vector obtained from processing 3h sessions with 30m shift (a), and running average of vertical component of the vector obtained from processing 3h sessions with 30m shift with a window of 12.5h (averaging current groups of 20 solutions from single sessions) (b)

The results shown so far indicate the external accuracy estimate of GPS solutions that is based on analysis of repeatability with use of regular 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 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 obtained using the Bernese and Pinnacle software from sessions of different length with 1h shift, are given in Fig. 4 and Fig. 5, respectively.

Fig. 4. Standard deviations of the GPS-derived vector lengths provided by processing software and estimated by statistical analysis of time series of GPS solutions.

Fig. 5. Standard deviations of the GPS-derived vertical component provided by processing software and estimated by statistical analysis of time series of GPS solution

The dispersion of GPS solutions obtained using the Bernese software is larger than that from the Pinnacle software, when short sessions were processed. That phenomenon is more distinct in case of vector length than of 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 42 km vector with the Bernese software, the software-provided standard deviations of the solutions obtained are at the very low level. That discrepancy is particularly distinguished for solutions based on short sessions. Both height difference and the length of a vector examined was determined with the accuracy of about 5 cm (one sigma level) from 2h sessions while precision of the solution provided by the Bernese software was at the level of single millimetres. The discrepancy between the external and internal estimate of accuracy of GPS solutions decreases with increase of session length while their ratio remains the same. In case of solutions based on 24h sessions it drops down to the level of a few millimetres although their internal accuracy estimated remain a few times better than the external accuracy.

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, e.g. 2h sessions, that form the group. Such simple 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 that is less sophisticated in terms of GPS observations modelling then 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 then in case of the Bernese software. A vertical component of a vector examined was determined with the accuracy of about 3.5 cm (one sigma level) from 2h sessions while precision of the solution provided by the Pinnacle software was about 2 cm. For the vector length the external and internal accuracy was 2 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 12h and 4h sessions in case of vertical component and vector length determination, respectively.

GPS solutions obtained using the Pinnacle software, averaged over the groups of sessions, practically coincide with the ones corresponding to respective single sessions. It particularly concerns vertical component.
The discrepancy between the external accuracy and internal consistency of GPS solutions obtained using the Bernese software was investigated for numerous vectors of European Permanent Network of length from a few tens to a few hundred kilometres and for some vectors of length up to a few thousand kilometres in Antarctic (Cisak et al., 2003a, 2003b). 3h and 24h GPS sessions covering 4 months of 2001 were processed. The correlation between two accuracy estimates, the external and the internal one is given in Fig. 7. Internal accuracy differs from the external one by a scale factor of about 7 and 10 for a vertical component and vector length, respectively.

Fig. 6. Correlation between the external accuracy and the internal accuracy of GPS solution provided by the Bernese software

The observed simple functional relationship between the external and internal estimate of accuracy was fulfilled for the vectors of different length (from a few tens to a few thousand of kilometres) from different geographical regions (mid- and high latitudes).

Fig. 7. Deviations from average values of vectors components vs. internal accuracy of GPS solution

Analysis in detail of the relations of the external and internal estimate of accuracy of GPS solutions is not satisfactorily effective due to the poor discretisation of the software-provided standard deviation that are usually expressed by numbers consisting of one or two digits. Large random errors of GPS solutions present in cases of short observation sessions processed, long vectors calculated, poor satellites configuration visible or ionospheric storms occurred, make the discretisation effect negligible.

The OHIG-MCM4 vector as a long one (3.9 thousand kilometres) and as located at high latitudes (Antarctic) is a good example of coincidence of a number of sources of random errors. Similar effects are observed in the solutions for BOGO-JOZE vector (42 km) based on 3 h sessions. Deviations from the mean for vectors components versus internal accuracy of GPS solutions provided by the Bernese software are shown in Fig. 7.

The larger software-provided standard deviation of GPS solutions the larger is the dispersion of the vector components estimated. GPS solutions for a long vector length may be separated onto two subsets (grey and black marks in Fig. 7). One of them evidently contains a bias. It is possible to separate subsets heuristically by means of analysis of the results in the stacked time domain. Dispersion of vertical component and vector length as well as software-provided standard deviation of the estimated length of the vector versus time of day corresponding to the beginning of 24h session are shown in Fig. 8.

Fig. 8. Dispersion of vertical component and vector length as well as software-provided standard deviation of the estimated length of the vector versus time of day corresponding to the beginning of 24h session (OHIG-MCM4 vector)

The results in Fig. 8 show the non-uniformly weighted data (e.g. due to the choice of reference satellite) in diurnal sessions. A third part of GPS solutions obtained for OHIG-MCM4 vector components (black marks in Fig. 7 and Fig. 8) are non-acceptable due to biases in vector length estimated. Uniform weighting in processing GPS observations from 24h sessions could substantially reduce or even eliminate from GPS solutions the influence of changes of satellites configuration of diurnal and half-diurnal periods.

Significant differences in systematic and random errors in vector length, in the numbers of observations and ambiguities resolved, as well as in random errors in vertical component between two subsets examined are observed (Fig. 9). Bias in the vector components and their uncertainty estimated using GPS data from one subset exceeds maximum dispersion obtained. It results in discrepancy of constant sign between the corresponding parameters determined from two data sets considered. The reasons and mechanisms of generation of asymmetry in distribution of results require further investigation. Such differences may occur due to the errors generated in the process of ambiguity resolution. Those errors can be amplified by poor configuration of visible satellites. The “configurationally induced” problem of ambiguity resolution is in particular frequently faced when processing GPS data from the stations in high latitudes, in particular in Antarctic. Therefore, polar regions are considered suitable test areas for advanced analysis of GPS positioning.

Fig. 9. Time series for OHIG-MCM4 vector vertical component, its length, number of single differences used and number of ambiguites resolved

Similar, “ionospherically induced” problem of ambiguity resolution was already reported (Cisak et al., 2003a; 2003b). The problems addressed above can also affect GPS positioning at mid-latitude permanent GPS stations, but their effect is smaller and its separation becomes more difficult. Mutual analysis of time series of both fix and float GPS solutions is a powerful tool for studying such problems, using for example a wide range of data provided by EPN stations.

Fig. 10. Dispersion of lengths and vertical components of the vectors vs. internal accuracy of GPS solutions (fix - grey marks, and float - black marks)

Dispersion of the lengths and vertical components of BOGO-JOZE and BOGO-BOR1 vectors versus internal accuracy of GPS solutions (fix - grey marks, and float - black marks) are shown in Fig. 10. No essential difference in solutions and their error estimates for vectors of 42 and 250 km length is observed. On the other hand no significant biases were detected and the larger standard deviations estimated the lager are random errors.Strategy of GPS solution quality analysis
The strategy developed for detecting and modelling biases in time series of GPS solutions is given in Fig. 11.

Fig. 11. Flowchart of the strategy of GPS solutions quality analysis

Temporal resolution of a series of GPS output 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 than those based on longer ones. 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 with shortening the length of session processed. Shortening the sessions is, however, limited by the length of the vector determined. Therefore, in order to increase temporal resolution of time series the overlapped sessions need to be processed. Overlapping the sessions 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, efficiently detect and separate chaotic effects from biases. Power spectrum density at Nyquist frequency can be used as an indicator of need of a 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 series reflect very clearly periodic biases as well as chaotic terms. Therefore they are suitable to detect and estimate periodic biases. Similar strategy with four hours window stepped by 30 minutes, followed by a running average procedure, was used for investigation of the vertical shift caused by Earth tides (Neumeyer et al., 2002).

Long time series can be directly 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 to model biases. Finally the models could be correlated with the external data, e.g. troposphere or ionosphere parameters in order to separate biases and find their sources.

Conclusions

A dispersion of GPS-derived vector components that is considered as an external accuracy estimate does not coincide with processing software-provided estimated accuracy of vector components determination 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 the case investigated, by a scale factor of about 7 and 10 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 and precision of GPS solutions based on data from permanent stations or from long-term geodynamics campaigns, especially in polar regions, need to 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 their 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.

Acknowledgements

The research was supported by the Institute of Geodesy and Cartography in Warsaw and was partially financed by the Polish State Committee for Scientific Research (grant PBZ-KBN-081/T12/2002). 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.

Bibliography

Bruyninx C. (2001): Overview of the EUREF Permanent Network and the Network Coordination Activities, EUREF Publication, Eds. J. Torres, H. Hornik, Bayerischen Akademie der Wiessenschaf-ten, München, Germany, 2001, No 9, pp. 24-30.

Cisak J., Kry_ski J., Zanimonskiy Y.M., (2002): Variations of Ionosphere Versus Variations of Vector Components Determined from Data of IGS Antarctic GPS Stations – New Contribution to the GIANT Project “Atmospheric Impact on GPS Observations in Antarctic”, Paper presented at the AGS-02, Wellington, New Zealand, http://www.scar-ggi.org.au/geodesy/ags02/cisak_ionosphere_gps.pdf

Cisak J., Kry_ski J., Zanimonskiy Y.M., Wielgosz P., (2003a): Variations of Vector Components Determined from GPS Data in Antarctic. Ionospheric Aspect in New Results Obtained within the Project “Atmospheric Impact on GPS Observations in Antarctic”, Paper presented at the scientific conference, 22-23 January 2003, Lviv, Ukraine.

Cisak J., Krynski J., Zanimonskiy Y., Wielgosz P., (2003b): Study on the Influence of Ionosphere on GPS Solutions for Antarctic Stations in the Framework of the SCAR Project „Atmospheric Impact on GPS Observations in Antarctic", Paper presented at the 7-th Bilateral Meeting Italy-Poland, Bressanone, Italy, 22-23 May 2003

Dubbini M., Gandolfi S., Gusella L., Perfetti N., (2003): Accuracy and precision vs software and different conditions using Italian GPS fiducial network (IGFN) data, Presented at 7th Bilateral Geodetic Meeting Italy-Poland, Bressanone, Italy, 22-23 May 2003.
Harris F.J., (1978): On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform, Proc. IEEE, Vol. 66, January 1978, pp. 51-83.

King M., Coleman R., Nguyen L.N., (2002): Spurious Periodic Horizontal Signals in Sub-Daily GPS Position Estimates, Journal of Geodesy, Vol. 77, No 1-2, May 2003, pp. 15-21.Krynski J., Zanimonskiy Y., Wielgosz P. (2000): Short Time Series Analysis of Precise GPS Positioning, Presented at the XXV General Assembly of European Geophysical Society, Millenium Conference on Earth, Planetary and Solar System Sciences, Nice, France, 25-29 April 2000.

Krynski J., Zanimonskiy Y., (2001): Contribution of Data from Polar Regions to the Investigation of Short Term Geodynamics. First Results and Perspectives, SCAR Report, No 21, January 2002, Publication of the Scientific Committee on Antarctic Research, Scott Polar Research Institute, Cambridge, UK.

Krynski J., Cisak J., Zanimonskiy Y.M., Wielgosz P., (2002a): Variations of GPS Solutions for Positions of Permanent Stations – Reality or Artefact, Symposium of the IAG Subcommission for Europe (EUREF) held in Ponta Delgada, Portugal, 5-7 June 2002, EUREF Publication No 8/2, Mitteilungen des Bundesamtes für Kartographie und Geodäsie, Frankfurt am Main (in press).

Krynski J., Zanimonskiy Y., Wielgosz P. (2002b): Modelling biases in GPS Positioning Based on Short Observing Sessions, Presented at the XXVII General Assembly of European Geophysical Society 2002, Nice, France, 21-26 April 2002.

Krynski J., Zanimonskiy Y., (2002): Quality Structure of Time Series of GPS Solutions, Proceedings of the NKG General Assembly, 1-5 September 2002, Espoo, Finland, pp. 259-264.

Neumeyer J., Barthelmes F., Combrink L., Dierks O., Fourie P., (2002): Analysis Results from the SG Registration with the Dual Sphere Supercon-ducting Gravimeter at SAGOS (South Africa), BIM 135, 15 July 2002, Paris, pp. 10607-10616.

Poutanen M., Koivula H., Ollikainen M. (2001): On the Periodicity of GPS Time Series, Proceedings of IAG 2001 Scientific Assembly, 2-7 September 2001, Budapest, Hungary.

Teunissen P.J.G. (2002): The Parameter Distributions of the Integer GPS Model, Journal of Geodesy (2002), Vol. 76, No 1, pp. 41-48.