Markov Chain Monte Carlo and the Application to Geodetic Time Series Analysis

The time evolution of geophysical phenomena can be characterised by stochastic time series. The stochastic nature of the signal stems from the geophysical phenomena involved and any noise, which may be due to, e.g., un-modelled effects or measurement errors. Until the 1990's, it was usually assumed that white noise could fully characterise this noise. However, this was demonstrated to be not the case and it was proven that this assumption leads to underestimated uncertainties of the geophysical parameters inferred from the geodetic time series. Therefore, in order to fully quantify all the uncertainties as robustly as possible, it is imperative to estimate not only the deterministic but also the stochastic parameters of the time series. In this regard, the Markov Chain Monte Carlo (MCMC) method can provide a sample of the distribution function of all parameters, including those regarding the noise, e.g., spectral index and amplitudes. After presenting the MCMC method and its implementation in our MCMC software we apply it to synthetic and real time series and perform a cross-evaluation using Maximum Likelihood Estimation (MLE) as implemented in the CATS software. Several examples as to how the MCMC method performs as a parameter estimation method for geodetic time series are given in this chapter. These include the applications to GPS position time series, superconducting gravity time series and monthly mean sea level (MSL) records, which all show very different stochastic properties. The impact of the estimated parameter uncertainties on sub-sequentially derived products is briefly demonstrated for the case of plate motion models. Finally, the MCMC results for weekly downsampled versions of the benchmark synthetic GNSS time series as provided in Chapter 2 are presented separately in an appendix

New Estimates of Present-Day Crustal/Land Motions in the British Isles Based on the BIGF Network

n this study we present results from a recent reprocessing effort that included data from more than 120 continuous Global Positioning System (CGPS) stations in the British Isles for the period from 1997 to 2008. Not only was the CGPS network dramatically densified from previous investigations by the authors, it now also includes, for the first time, stations in Northern Ireland, providing new constraints on glacio-isostatic processes active in the region. In our processing strategy we apply a combination of re-analysed satellite orbit and Earth rotation products together with updated models for absolute satellite and receiver antenna phase centers, and for the computation of atmospheric delays. Our reference frame implementation uses a semi-global network of 37 stations, to align our daily position estimates, using a minimal constraints approach, to ITRF2005. This network uses a combination of current IGS reference frame stations plus additional IGS stations in order to provide similar network geometries throughout the complete time span. The derived horizontal and vertical station velocities are used to investigate present-day crustal/land motions in the British Isles. This first solution provides the basis for our contri- bution to the Working Group on Regional Dense Velocity Fields, 2007 - 2011 of the International Asso- ciation of Geodesy Subcommission 1.3 on Regional Reference Frames

Ambiguity resolution in precise point positioning with hourly data

Precise Point Positioning (PPP) has become a recognized and powerful tool for scientific analysis of Global Positioning System (GPS) measurements. Until recently, ambiguity resolution at a single station has been considered difficult, due to the non-integer uncalibrated hardware delays (UHD) originating in receivers and satellites. Fortunately, recent studies show that if these UHD can be determined precisely with a network in advance, then ambiguity resolution at a single station is possible. In this study, the method proposed by Ge et al (2007) is adopted with a refinement in which the fractional parts of single-difference narrow-lane UHD for a satellite pair are determined within each full pass over a regional network. This study uses the European Reference Frame Permanent Network (EPN) to determine these UHD from Day 245 to 251 in 2007, and 27 IGS stations inside and outside the EPN are used to conduct ambiguity resolution in hourly PPP. It is found that the total hourly position accuracy is improved from 3.8 cm, 1.5 cm and 2.8 cm to 0.5 cm, 0.5 cm and 1.4 cm in East, North and Up, respectively, for the stations inside the EPN. For the stations outside the EPN, some of which are even over 2000 km away from the EPN, their total hourly East, North and Up position accuracies still achieve 0.6 cm, 0.6 cm and 2.0 cm, respectively, when the EPN-based UHD are applied to the ambiguity resolution at these stations. Therefore, it is feasible and beneficial for the operators of GPS networks, such as the providers of PPP-based online services, to provide these UHD estimates as an additional product to allow users to conduct ambiguity resolution in PPP

Integer ambiguity resolution in precise point poistioning: method comparison

Integer ambiguity resolution at a single receiver can be implemented by applying improved satellite products where the fractional-cycle biases (FCBs) have been separated from the integer ambiguities in a network solution. One method to achieve these products is to estimate the FCBs by averaging the fractional parts of the float ambiguity estimates, and the other is to estimate the integer-recovery clocks by fixing the undifferenced ambiguities to integers in advance. In this paper, we theoretically prove the equivalence of the ambiguity-fixed position estimates derived from these two methods by assuming that the FCBs are hardware-dependent and only they are assimilated into the clocks and ambiguities. To verify this equivalence, we implement both methods in the Position and Navigation Data Analyst software to process 1 year of GPS data from a global network of about 350 stations. The mean biases between all daily position estimates derived from these two methods are only 0.2, 0.1 and 0.0 mm, whereas the standard deviations of all position differences are only 1.3, 0.8 and 2.0 mm for the East, North and Up components, respectively. Moreover, the differences of the position repeatabilities are below 0.2 mm on average for all three components. The RMS of the position estimates minus those from the International GNSS Service weekly solutions for the former method differs by below 0.1 mm on average for each component from that for the latter method. Therefore, considering the recognized millimeter-level precision of current GPS-derived daily positions, these statistics empirically demonstrate the theoretical equivalence of the ambiguity-fixed position estimates derived from these two methods. In practice, we note that the former method is compatible with current official clock-generation methods, whereas the latter method is not, but can potentially lead to slightly better positioning quality

Detecting offsets in GPS time series: first results from the detection of offsets in GPS experiment

The accuracy of Global Positioning System (GPS) time series is degraded by the presence of offsets. To assess the effectiveness of methods that detect and remove these offsets, we designed and managed the Detection of Offsets in GPS Experiment. We simulated time series that mimicked realistic GPS data consisting of a velocity component, offsets, white and flicker noises (1/f spectrum noises) composed in an additive model. The data set was made available to the GPS analysis community without revealing the offsets, and several groups conducted blind tests with a range of detection approaches. The results show that, at present, manual methods (where offsets are hand picked) almost always give better results than automated or semi-automated methods (two automated methods give quite similar velocity bias as the best manual solutions). For instance, the fifth percentile range (5% to 95%) in velocity bias for automated approaches is equal to 4.2 mm/year (most commonly ±0.4 mm/yr from the truth), whereas it is equal to 1.8 mm/yr for the manual solutions (most commonly 0.2 mm/yr from the truth). The magnitude of offsets detectable by manual solutions is smaller than for automated solutions, with the smallest detectable offset for the best manual and automatic solutions equal to 5 mm and 8 mm, respectively. Assuming the simulated time series noise levels are representative of real GPS time series, robust geophysical interpretation of individual site velocities lower than 0.2-0.4 mm/yr is therefore certainly not robust, although a limit of nearer 1 mm/yr would be a more conservative choice. Further work to improve offset detection in GPS coordinates time series is required before we can routinely interpret sub-mm/yr velocities for single GPS stations