Analysis of the time series of station coordinates - a comparison of the network and PPP approach

Acta Geodyn. Geomater., Vol. 12, No. 2 (178), 127–133, 2015 DOI: 10.13168/AGG.2015.0019 journal homepage: http://www.irsm.cas.cz/acta ORIGINAL PAPER ANALYSIS OF THE TIME SERIES OF STATION COORDINATES – A COMPARISON OF THE NETWORK AND PPP APPROACH Jan KOSTELECKÝ 1, 2)*, Jan DOUŠA 1), Jakub KOSTELECKÝ 1) and Pavel VÁCLAVOVIC 1, 3) 1) Research Institute of Geodesy, Topography and Cartography, p.r.i., GO Pecný, CZ-251 65 Ondřejov 2) Institute of Geodesy and Mine Surveying, Faculty of Mining and Geology, TU Ostrava, 17. listopadu 15, CZ-708 33 Ostrava Poruba 3) Department of Geomatics, Faculty of Civil Engineering, CTU in Prague, Thákurava 7, CZ-166 29 Praha 6 *Corresponding author‘s e-mail: mailto:[email protected] ABSTRACT Based on the example of fourteen selected permanent GNSS stations of the EPN (EUREF Permanent Network) the differences of time variations of coordinates – velocities – resulting from the network and PPP (Precise Point Positioning) solution are studied. In both approaches the coordinates are determinated using the Bernese software version 5.2. The time series analysis is made using the modified Vaníček´s method. The comparison shows that the results p rovided by the two methods cannot be accepted as statistically consistent. ARTICLE INFO A rticle history: Received 15 January 2015 Accepted 12 May 2015 Available online 22 May 2015 K eywords: GNSS Time series analysis Station velocities 2013/12. The entire EUREF-EPN (250+ stations) was included in the processing. The distribution of the stations can be seen from Figure 1. In processing the following parameters were used: parameters o f IERS2010 compliant models, I08 satellite antenna and p hase centers and ITRF2008 reference frame. Fo r initial coordinates of fiducial stations the NNT (No N et Translation) minimum constraints were applied. The datum was realized on a daily and weekly solution through a selected set of iteratively controlled fiducial stations. A multi-year combination was generated for station coordinates, velocities and discontinuities and residuals analysis. 2.2. PPP SOLUTION The Precise Point Positioning solution is based on a “direct non-differenced position determination”. The processing was carried out using the Bernese GPS Software Version 5.2 following processing steps recommended by software developers and provided with the software installation. This solution is consistent over the period 2000/01 – 2007/12 and was applied for selected stations only: BOGO, BOR1, DRES, GOPE, GRAZ, HERS, HFLK, KIRU, NICO, ONSA, POTS, REYK, TUBI, WTZR. As in the network solution, the precise products for satellite orbits, clocks, and Earth Rotation Parameters (ERP) were taken from the reprocessing repro1 provided by the International GNSS service (IGS). Such products were selected for their consistency along the p rocessing period. The Differential Code Biases (DCB) were taken from the CODE solution. The reference frame was taken from the orbits and the sampling interval 300 s was used for the solution. All 1. INTRODUCTION The goal is the analysis of time series o f coordinates of a selected set of 14 EPN (EUREF Permanent Network) stations over a long time interval. The coordinates were determined using two methods. First, the classical „network solution“ was used where the GNSS observations are processed by baseline vectors connecting two stations and tied to selected „fiducial stations“ using differenced observations. Second, the PPP (Precise Poin t Positioning) method was applied, where the single station coordinates are determined directly from all available GNSS observations carried out at this station only using precise final orbits and precise satellite clock corrections. In both cases the Bernese v. 5.2 software was used for the processing and the computations were carried out in the ITRF2008 terrestrial reference frame. The resulting coordinate time series were analysed using the Vaníček´s unharmonic analysis (Wells and Vaníček, 1978 or Kostelecký and Karský, 1987) slightly modified by Vondrák (1970, 1977). This analysis makes it possible to determine the trend as well as the most significant periods. The main goal is to compare the results obtained from the two p rocessing methods as to the time development o f coordinates and also the most significant detected periods. 2. DATA ANALYSIS 2.1. NETWORK SOLUTION The network solution is based on the doubledifferenced solution on the pre-defined baselines. The solution is consistent over the period 1996/01 – J. Kostelecký et al. 128 Fig. 1 Distribution of the stations used for the network solution. Fig. 2 Distribution of permanent stations used in PPP solution. 3. METHOD OF ANALYSIS Let have the set of homogenized quantities H are considered the only realization of random process. Drawing on the physical characteristics of the process of determining quantity H, our realization of the deterministic process can be expressed in terms o f a function of time t.   01 cos 2 KJ k ikijjiji kj H tAtR ft st        (1) models in the processing were applied according to the IERS2010 conventions. The type antenna calibrations I08 were applied for satellite and receive r p hase center offsets and variations. Zenith total delays were estimated in PPP as well as in the network solution. Moreover, ambiguities were solved as float values in the PPP because phase biases were not applied. The solution is based only on GPS observations obtaining one solution per day. The distribution of selected GNSS permanent stations used in the PPP solution is displayed in Figure 2. ANALYSIS OF THE TIME SERIES OF STATION COORDINATES – A COMPARISON OF THE … . 129 of function     ik x tQt. Conditions (6) and (7) lead to classical method of least squares, Kbeing determined in (1) and (2) from the apriori known character of the function, or from experimental computations. Vondrák’s (1977) modification of the method is in the following: (a) The optimum normalized spectrum is expressed as the function     2 g ff   (9) (b) Consequently, the values of the normalized spectrum are defined as follows:    , 1, ikJj j ikJ xQQ gf xQ                      (10) where 01 cos 2 KJ k kJ k j j j kj QAtR ft         . (11) Function   g f is realized by a set of discrete values   min max ,, j j gf f f f, where fmin and max fare empirically optional. The values   i g fcan be calculated from (10) in each case for commonly adjusted parameters A, R, 0Φ from (11) unde r conditions similar to (6) and (7) with values o f 1 j j fP  set in advance. Moreover, J = J0 + J1, J0 being the number of frequencies j f imposed in advance, and J1the number of frequencies determined earlier. The spectral analysis is applied gradually in each case with a kJ Q newly formed from (11) for j f from the whole interval. For the next iteration step Jis increased by 1 and the frequencies for which   j g f was maximum is added to the set of frequencies earlier. After the last iteration step one should have       ikJi x tQt st , (12) and the set     , i s ttT should converge to the normal distribution. In the computer program this is effected by the condition ,  0 max j g fg, 0 g being selected empirically by deliberating the physical nature of the phenomenon being studied, of by means of model test. (c) Last modification resulted in speeding up the computation of the coefficients of polynomials (11) by omitting the quite rigorous (with a view to the discrete realization of the set with a generally non-uniform data distribution) calculation o f where si(t) has a normal distribution with dispersion 2  and mean value  0 st  , Ak are polynomial coefficients, 1 j j fP  is the frequency, Pj the period (in days), Rj amplitude, and j   the phase. We shall seek the optimum estimate of these parameters, p reserving the ergodicity condition and the properties of realization (1) of the process. We shall also introduce the notations 0 ,cos2 Kk ki k j ji j j k A tQ R ft Q      (2) In seeking the hidden periods, one usually uses periodograms of the type      2 1 0 2exp T I fTxtitfdt     (3) as a criterion, in which f is the frequency, xthe continuous realization of the random process in interval T, 1i. However, we have used a slightly different procedure, Vaníček (Wells and Vaníček, 1978) unharmonic analysis, modified by Vondrák (1977). According to Vaníček, the optimum normalized spectrum, using (2) reads    , 1, ikj i ik xQQ xxQ         (4) where   2 , tT F GFtGt     (5) defines the metric of the two function F and G define d on T. The coefficients Ak of polynomials Qk are determined under assumption of independence, so that  ,min. ik xQ   (6) And, similarly, coefficients j R and j   from Qj are determined for every frequency f , where  is a set of reasonably optional frequencies, so that  ,min. ikj xQQ   (7) Coefficients j R, j = 1, 2, ….., J, are thus functions o f frequency fj , and, as regards   i x  , it is easy to prove that for the given realization of the process  :0,1 i x   . (8) The separate maxima of function   , i x t  , the spectrum peaks, determine the frequencies which make the principal contribution to the overall variation J. Kostelecký et al. 130 Table 1 List of analyzed stations. acronym localization character of locality results are in Table # BOGO Borowa Góra, Poland stable part of Eurasian Plate 2 BOR1 Borowiec, Poland stable part of Eurasian Plate 3 DRES Dresden, Germany stable part of Eurasian Plate 4 GOPE Pecny, Ondrejov, Czech Rep. stable part of Eurasian Plate 5 GRAZ Graz, Austria stable part of Eurasian Plate 6 HERS Hailsham, United Kingdom British Isles 7 HFLK Hafelekar, Innsbruck, Austria high mountain station 8 KIRU Kiruna, Sweden Scandinavian plate 9 NICO Nicosia, Cyprus border zone of Euro-Asian plate 10 ONSA Onsala, Sweden Scandinavian plate 11 POTS Potsdam, Germany, stable part of Eurasian Plate 12 REYK Reykjavik, Iceland, North-American plate 13 TUBI Gebze, Turkey area of intensive seismicity 14 WTZR Wettzell, Germany stable part of Eurasian Plate 15  rms dispersion σ0 of analyzed time series for the individual component N (south-north), E (west – east), U (up)  time interval of network (NET) solution, taken for the analysis  value of the linear trend for N, E, U for the described interval of PPP solution  amplitude of the annual (one year) period  “long periodic terms” means period greater than one year and also the differences between the network (NET) and PPP solutions. RMS error of the trend in all coordinates is b etween 0.01 mm/year and 0.04 mm/year and RMS error in determined amplitudes is between 0.2 mm and 0.3 mm. the summations of the type cos , , i i ttT    which is replaced with analytically compute d integrals 0cos Ttdt  . 4. RESULTS OF ANALYSIS As already mentioned above, the time series analysis was performed for 14 selected permanen t stations the coordinates of which were determined by two methods – the network solution based on differenced observations and the precise point p ositioning. All the analysed stations are located in the areas that are interesting from the point of view o f geodynamics – see Table 1 The results of analysis are given in the following ta b les. Each table contains for both processing methods Table 2 - result of analysis. σ0 N, E, U linear trend [mm/y] NET solution1996.7 – 2012.1 amplitude of annual period [mm] BOGO [mm] N E U N E U NET 3.9,2.5,6.7 14.6 21.0 -0.2 - - - PPP 3.5,4.5,7.4 14.6 20.4 -0.4 - - 5.1 diference 0 0.6 0.2 long periodic changes Table 3 - result of analysis. σ0 N, E, U linear trend [mm/y] NET solution 1996.1 - 2012.1 amplitude of annual period [mm] BOR1 [mm] N E U N E U NET 3.8,2.9,6.3 14.5 20.3 -0.1 - - - PPP 2.1,3.4,5.5 14.8 19.7 -0.1 - - 5.4 diference -0.3 0.6 0.0 long periodic changes ANALYSIS OF THE TIME SERIES OF STATION COORDINATES – A COMPARISON OF THE … . 131 Table 4 result of analysis. σ0 N, E, U linear trend [mm/y] NET solution 2003.1 – 2012.1 amplitude of annual period [mm] DRES [mm] N E U N E U NET 2.5,2.2,6.5 15.6 19.8 -0.8 1.3 1.8 2.6 PPP 1.9,3.4,7.3 15.7 20.4 -0.4 1.1 - 6.6 diference -0.1 -0.6 -0.4 --- Table 5 result of analysis. σ0 N, E, U linear trend [mm/y] NET solution 2000.0 – 2008.0 amplitude of annual period [mm] GOPE [mm] N E U N E U NET 2.6,2.4,5.8 15.5 19.8 1.7 - - 2.1 PPP 1.8,2.8,5.8 16.4 19.0 0.8 - - 5.2 diference 0.9 0.8 0.9 long periodic changes Table 6 - result of analysis. σ0 N, E, U linear trend [mm/y] NET solution 1996.1 – 2012.1 amplitude of annual period [mm] GRAZ [mm] N E U N E U NET 4.0,2.9,6.5 15.8 21.8 0.5 - - 2.3 PPP 2.5,3.6,6.2 15.8 21.8 0.9 - - 6.5 diference 0.0 0.0 -0.4 long periodic changes Table 7 -result of analysis. σ0 N, E, U linear trend [mm/y] NET solution 2001.7 – 2010.7 amplitude of annual period [mm] HERS [mm] N E U N E U NET 3.1,2.8,9.0 16.6 16.8 0.6 0.9 0.9 1.8 PPP 2.0,2.7,4.8 16.4 16.5 0.5 - - 4.8 diference 0.2 0.3 0.1 long periodic changes Table 8 - result of analysis. σ0 N, E, U linear trend [mm/y] NET solution 1996.1 – 2003.2 amplitude of annual period [mm] HFLK [mm] N E U N E U NET 4.3,3.7,7.2 16.8 21.5 4.0 4.0 2.4 3.1 PPP 2.5,5.2,6.2 15.8 21.3 2.1 5.3 3.1 6.3 diference 1.0 0.2 1.9 --- Table 9 - result of analysis. σ0 N, E, U linear trend [mm/y] NET solution 1996.1 – 2011.1 amplitude of annual period [mm] KIRU [mm] N E U N E U NET 6.8,3.9,13.1 15.0 15.8 6.9 1.7 - 6.9 semi 4.0 PPP 4.1,4.3,13.4 15.1 15.5 7.3 - - 7.2 semi 7.4 diference -0.1 0.3 -0.4 long periodic changes J. Kostelecký et al. 132 Table 10 - result of analysis. σ0 N, E, U linear trend [mm/y] NET solution 1999.7 – 2006.5 amplitude of annual period [mm] NICO [mm] N E U N E U NET 4.6,4.8,9.3 15.4 19.9 -0.5 2.8 - 3.9 PPP 2.5,3.9,7.2 15.5 19.0 0.2 1.7 - 5.8 diference -0.1 0.9 -0.7 --- Table 11 - result of analysis. σ0 N, E, U linear trend [mm/y] NET solution 1996.1 – 2012.1 amplitude of annual period [mm] ONSA [mm] N E U N E U NET 3.4,2.7,5.6 14.5 17.0 2.3 - - 1.5 PPP 2.3,3.9,6.4 14.5 16.8 2.8 - - 4.3 diference 0.0 0.2 -0.5 long periodic changes Table 12 - result of analysis. σ0 N, E, U linear trend [mm/y] NET solution 1996.1 – 2012.1 amplitude of annual period [mm] POTS [mm] N E U N E U NET 3.3,2.8,6.1 15.2 19.0 -0.2 - - 2.0 PPP 3.9,4.5,6.1 15.2 18.9 0.2 - - 5.2 diference 0.0 0.1 -0.4 long periodic changes Table 13 - result of analysis. σ0 N, E, U linear trend [mm/y] NET solution 1996.1 – 2012.1 amplitude of annual period [mm] REYK [mm] N E U N E U NET 4.4,5.4,9.0 20.7 -10.8 -1.3 - 2.6 7.9 PPP 2.9,4.6,10.1 20.8 -11.5 -2.6 - 2.8 3.4 diference -0.1 -0.3 1.3 long periodic changes Table 14 - result of analysis. σ0 N, E, U linear trend [mm/y] NET solution 1999.8 – 2012.1 amplitude of annual period [mm] TUBI [mm] N E U N E U NET 4.3,4.1,7.7 10.0 23.7 -1.9 1.7 semi 1.0 2.7 4.7 PPP 2.1,3.2,6.2 9.1 23.6 -1.5 - - 5.4 diference 0.9 0.1 -0.4 long periodic changes Table 15 - result of analysis. σ0 N, E, U linear trend [mm/y] NET solution 2000.0 – 2008.0 amplitude of annual period [mm] WTZR [mm] N E U N E U NET 3.2,2.7,6.1 15.4 20.4 0.0 - - 2.0 PPP 2.1,3.1,5.7 15.7 20.2 -0.1 - - 5.2 diference -0.3 0.2 0.1 long periodic changes ANALYSIS OF THE TIME SERIES OF STATION COORDINATES – A COMPARISON OF THE … . 133 ACKNOWLEDGEMENT This contribution was made possible thanks to the project TB01CUZK006 of TA ČR. Data for the GOPE station was taken via project LM2010008 – CzechGeo EPOS. REFERENCES Kalvoda, J.: 2015, private communication. Kostelecký, J. and Karský, G.: 1987, Analysis of 1970 – 1983 circumzenithal measurements. Bull. Astron. Inst. Czechosl., 38, 16–23. Vondrák, J.: 1970, The new circumzenithal of the Research Institute for Geodesy, Topography and Cartography in Prague. Bull. Astron. Inst. Czechosl., 21, No. 4, 264. Vondrák, J.: 1977, The rotation of the Earth between 1955.5 and 1976.5 Studia geophys. geod., 21, No. 2, 107– 117. Wells, D.E. and Vaníček, P.: 1978, Least Squares Spectral Analysis. Bedford Institute of Oceanography, Dartmouth, Canada, 43 pp. 5. CONCLUSIONS Summarizing the obtained results we get the values given in Table 16. Table 16 Summarized values. value mm/y difference mean min max dN 0.43 -0.3 0.9 dE 0.39 -0.6 0.9 dU 0.74 -0.7 1.3 It is possible to state that for the PPP solution the statistically significant amplitude of the annual term in the height component U occurs at each station. For the network solution the corresponding amplitude is smaller which can be explained by phase-consonant p eriodical changes of heights of the fiducial stations which are used in the network solution. The majority of stations also display significant long-periodic (greater than one year) changes of some of coordinates which may indicate that the linea r approximation of the trend might be insufficient. From the obtained results it can be inferred that a) from statistical point of view (dispersion of value with respect to value) the differences of the results are in many cases statistically significant, b) the annual variations of the vertical component, determined by PPP method have real basis according meaning o f geomorfologists – see i.e. (Kalvoda, 2015) and c) the differences in the trend of coordinate changes resulting from two processing methods cannot be neglected and might, in some way, influence geodynamical interpretations.