GPS coordinate time series measurements in Ontario and Quebec, Canada

Abstract

New precise network solutions for continuous GPS (cGPS) stations distributed in eastern Ontario and western Québec provide constraints on the regional three-dimensional crustal velocity field. Five years of continuous observations at fourteen cGPS sites were analyzed using Bernese GPS processing software. Several different sub-networks were chosen from these stations, and the data were processed and compared to in order to select the optimal configuration to accurately estimate the vertical and horizontal station velocities and minimize the associated errors. The coordinate time series were then compared to the crustal motions from global solutions and the optimized solution is presented here. A noise analysis model with power-law and white noise, which best describes the noise characteristics of all three components, was employed for the GPS time series analysis. The linear trend, associated uncertainties, and the spectral index of the power-law noise were calculated using a maximum likelihood estimation approach. The residual horizontal velocities, after removal of rigid plate motion, have a magnitude consistent with expected glacial isostatic adjustment (GIA). The vertical velocities increase from subsidence of almost 1.9 mm/year south of the Great Lakes to uplift near Hudson Bay, where the highest rate is approximately 10.9 mm/year. The residual horizontal velocities range from approximately 0.5 mm/year, oriented south–southeastward, at the Great Lakes to nearly 1.5 mm/year directed toward the interior of Hudson Bay at stations adjacent to its shoreline. Here, the velocity uncertainties are estimated at less than 0.6 mm/year for the horizontal component and 1.1 mm/year for the vertical component. A comparison between the observed velocities and GIA model predictions, for a limited range of Earth models, shows a better fit to the observations for the Earth model with the smallest upper mantle viscosity and the largest lower mantle viscosity. However, the pattern of horizontal deformation is not well explained in the north, along Hudson Bay, suggesting that revisions to the ice thickness history are needed to improve the fit to observations.

Appendices

Appendix 1: GPS data processing using Bernese version 5

In this paper, RINEX data for GPS stations spanning 2008–2012 (Table 6) are processed using the Bernese Processing Engine (BPE) (Beutler et al. 2007) and a double-differencing technique. Table 6 contains all information related to the GPS RINEX data in this analysis, including position of the sites, monuments, receiver and antenna types, the first date of data availability, length, and end date. The offset dates for the considered time interval are shown as well.

The updated precise orbit information and the Earth orientation parameter (EOP) spanning of our GPS campaign are introduced into the program to create the standard orbit files. In the preprocessing phase, the code observation files are used to synchronize the receiver clock with the GPS times, and then, the baselines are created based on the zero-difference observation files. In addition, at this stage, the cycle slips and outliers are detected and removed by considering the RMS value of the observations and multiple ambiguities are added for the phase observations using the triple-combination approach.

Table 6 GPS station information for stations from the POLARIS network and those operated by Canadian Geodetic Survey (CGS), Natural Resources Canada

The ionospheric delay signals are eliminated using ionosphere-free linear combination (L3) of carrier phase measurements. The effect of the higher-order ionosphere has not been considered as they have less than 1-mm effects (Hernández-Pajares et al. 2007; Petrie et al. 2010). The tropospheric effects are modeled by applying the (Niell 1996) mapping function for both the dry and the wet part which maps the zenith troposphere delay to the satellite-station direction. The benefit of using this model is that the calculation of the wet and dry mapping functions does not depend on the local surface meteorology and it gives an accurate positions for stations over latitude of 43$^{\circ }$N and below 75$^{\circ }$N for minimum elevation angle of three degrees.

Here, we considered the minimum elevation angle equal to five degrees that minimizes multipath errors. The ocean tidal loading effects that cause crustal deformation and therefore site displacement are corrected from the horizontal and vertical components by introducing a GOT00.2 model containing eleven coefficients for each particular site to the program. This file contains the magnitude of the ocean loading effect for a subset of IGS stations so that the amplitude and phase shifts values for other stations in our study were obtained from Bos and Scherneck (2011). These values were not corrected for the center of mass motion so that our frame origin is in the solid earth center (CM).

Next, all baselines are processed separately and the ambiguities are resolved by the quasi-ionosphere-free (QIF) ambiguity resolution strategy. New coordinates and troposphere parameters files are introduced, and the results consist of two parts. The first part refers to the solution where the ambiguities are estimated as real-valued measurement biases, whereas the second part reports the results after resolving the ambiguity parameters to integer values. It is important to know that the ambiguities larger than the specified RMS could not be resolved.

Then, the final network solution is implemented in which the correlation between the observations is considered and the ambiguities that have been resolved already are introduced as known parameters. In this way, the free network or the minimum constraint solution, in which no station is fixed to its a priori coordinates, is carried out in this paper. This approach is optimal for defining the geodetic datum with a minimum number of constraints where there are inconsistencies in the reference stations coordinates. Then, the coordinates obtained from the last solution are compared to the a priori coordinates for the IGS core sites so that the stations with the residual more than the considered threshold, 15 mm for north and east directions and 30 mm for up direction, would be rejected in the datum definition. In this phase, seven transformation parameters are calculated by comparing the two sets of coordinates, the output coordinates from the last step to the first input file containing stations coordinates in IGS08, in the Helmert transformation program. Then, the repeatability of the coordinate solution is checked to report the difference in each individual coordinate set to the mean value. In the final solution at each epoch, the troposphere parameters have to be pre-eliminated and to avoid singularities, all station coordinates have to be constrained. The station coordinates obtained in the last step are introduced here.

Subsequently, the obtained daily GPS position time series spanning approximately 5 years are analyzed to estimate the velocities of the GPS sites. The outliers are eliminated as follows: First coordinates with jumps of more than 300 mm are removed manually, and then, the Hampel filter (Hampel 1974) is employed. In this method, if a point differs from the mean by more than three times the standard deviation, it is replaced by the median of the window containing the six surrounding points.

We employed the sigma averaging (SIGAVG) method presented by Goudarzi et al. (2013) in the GPS interactive time series analysis (GITSA) software (Goudarzi et al. 2013) to detect jumps and discontinuities in the position time series. This approach divides the time series into different segments based on the introduced threshold, here set at 3 mm, and detects discontinuities at the border of adjacent segments without jumps.

In order to analyze the GPS time series more adequately, the rate uncertainties and the linear trend are determined by employing Hector software (Bos et al. 2013). A maximum likelihood estimation (MLE) approach is used to calculate the noise in the time series (Williams et al. 2004) by computing the parameters of the noise model, including the amplitude and spectral index. This software also computes the constant velocity, offsets which may occur due to GPS equipment changes, annual and semiannual variations, and velocity uncertainties. Here, the combination of the power-law noise (colored noise) and white noise is taken into account as source of noises in the time series. In addition, the AmmarGrag method is employed for the likelihood computation because the percentage of missing data is less than 50% of the total time series length. The covariance matrix which represents the time-dependent positions is computed from (Williams 2008) (Eq. 4):

$$\begin{aligned} C=a_{w}^{2}I + b_{k}^{2}J_{k} \end{aligned}$$

(4)

where $a_{w}$ and $b_{k}$ are the white and power-law amplitudes, respectively. These depend on the variance of the noise, innovation noise, $\sigma ^2$. I is the unit matrix with n $\times$ n dimension, and $J_{k}$ is the covariance matrix for the power-law noise with spectral index k.

Figure 15 illustrates the daily vertical position time series for station TYNO and its corresponding fitted sine function.

Fig. 15

Detrended position changes time series at station TYNO in eastern Ontario. The pink line corresponds to the best fit annual sinusoidal function

To verify the correctness of choice of the noise model, the power spectra analysis is carried out by fitting the combination of white plus power-law noise model to the computed spectrum for the GPS observations. This analysis represents the difference between observations minus the estimated linear trend and additional offsets and periodic signals. Figure 16 shows the power spectral density (PSD) plot for the vertical position time series of station TYNO. At high frequencies, the fitted model is flat, which is representative of the white noise. At lower frequencies, the fitted model obeys a power-law noise with a slope of approximately one, which implies the presence of flicker noise in the time series.

Fig. 16

Computed PSD of the residuals for up component for station TYNO. The fitted white plus power-law noise model is shown by the solid green line. The red $''$x$''$ is the calculated spectrum for the observations

Appendix 2: GPS data processing using GIPSY/OASIS-II software

The time series of GPS stations on the NGL Web site (http://geodesy.unr.edu/) were processed using GIPSY/OASIS-II software (Webb and Zumberge 1997) and made available by Jet Propulsion Laboratory (JPL). The precise point positioning (PPP) technique (Zumberge et al. 1997) was applied to the ionosphere-free carrier phase and pseudorange data. The daily GPS coordinate time series are produced using GPS satellite orbit, GPS satellite clock, and satellite antenna calibration models. The elevation cutoff angle was set at seven degrees. Troposphere effects were modeled using the global mapping function (GMF) proposed by Boehm et al. (2006), and horizontal gradients were estimated using a random walk stochastic process at every five minutes (Bar-Sever et al. 1998).

The first order of ionosphere effect was removed with the ionosphere-free carrier phase and pseudorange data combination. As the higher-order ionosphere effect has very low amplitude, less than 1mm, it was not considered in the process (Hernández-Pajares et al. 2007). Non-tidal atmospheric loading model was not applied, and only the effects of ocean loading were corrected by using the FES2004 tidal model (Lyard et al. 2006) which was provided by http://holt.oso.chalmers.se/loading (Scherneck 1991). The ocean loading effect was modeled in the CM frame (Blewitt 2003; Fu et al. 2012). In addition, the integer ambiguities for every station were solved using the wide lane and phase bias (WLPB) approach (Bertiger et al. 2010). The resulted coordinates were obtained in the frame of JPL’s fiducial-free orbit so that they were transformed into reference frame IGS08 employing a seven-parameter transformation computed with JPL’s orbit products (Blewitt 2014).

A similar time series analysis was performed for the time series obtained from this solution to estimate the velocities, velocity uncertainties, and spectral index associated with the power-law noise.