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

in Montillet, Jean-Philippe; Bos, Machiel (Eds.) Geodetic Time Series Analysis in Earth Sciences (2020)

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 ... [more ▼]

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. [less ▲]

The Combined Effect of Periodic Signals and Noise on the Dilution of Precision of GNSS Station Velocity Uncertainties

Poster (2016, April 05)

Station velocity uncertainties determined from a series of Global Navigation Satellite System (GNSS) position estimates depend on both the deterministic and stochastic models applied to the time series ... [more ▼]

Station velocity uncertainties determined from a series of Global Navigation Satellite System (GNSS) position estimates depend on both the deterministic and stochastic models applied to the time series. While the deterministic model generally includes parameters for a linear and several periodic terms, the stochastic model is a representation of the noise character of the time series in form of a power-law process. For both of these models the optimal model may vary from one time series to another while the models also depend, to some degree, on each other. In the past various power-law processes have been shown to fit the time series and the sources for the apparent temporally-correlated noise were attributed to, for example, mismodelling of satellites orbits, antenna phase centre variations, troposphere, Earth Orientation Parameters, mass loading effects and monument instabilities. [less ▲]

A Bayesian Monte Carlo Markov Chain Method for the Statistical Analysis of Geodetic Time Series

Poster (2013, September 06)

Geodetic time series provide information which help to constrain theoretical models of geophysical processes. It is well established that such time series, for example from GPS or gravity measurements ... [more ▼]

Geodetic time series provide information which help to constrain theoretical models of geophysical processes. It is well established that such time series, for example from GPS or gravity measurements, contain time-correlated noise which is usually assumed to be a combination of a long-term stochastic process (characterized by a power-law spectrum) and random noise. Therefore, when fitting a model to geodetic time series it is essential to also estimate the stochastic parameters beside the deterministic ones. In many cases the stochastic parameters have included the power amplitudes of both time-correlated and random noise as well as the spectral index of the power-law process. To date the most widely used method for obtaining these model parameter estimates is based on maximum likelihood estimation (MLE). We present a new Bayesian Monte Carlo Markov Chain (MCMC) method to estimate the deterministic and stochastic model parameters of geodetic time series. This method provides a sample of the likelihood function and thereby, using Monte Carlo integration, all parameters and their uncertainties are estimated simultaneously. One advantage of this method over MLE is that the computation time required increases linearly with the number of parameters, hence being very suitable for dealing with a large number of parameters. Another advantage is that the properties of the estimator used by the MCMC method do not depend on the stationarity of the time series. We assess the MCMC method through comparison with MLE, using a data set of 300 synthetic GPS-like time series and the JPL daily position time series for 90 GPS stations (the IGS core network). [less ▲]

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

in Journal of Geophysical Research. Solid Earth (2013), 118

The accuracy of Global Positioning System (GPS) time series is degraded by the presence of offsets. If these are not detected and adjusted correctly they bias velocities, and hence geophysical estimates ... [more ▼]

The accuracy of Global Positioning System (GPS) time series is degraded by the presence of offsets. If these are not detected and adjusted correctly they bias velocities, and hence geophysical estimates, and degrade the terrestrial reference frame. They also alter apparent time series noise characteristics as undetected offsets resemble a random walk process. As such, offsets are a substantial problem. A number of offset detection methods have been developed across a range of fields, and some of these are now being tested in geodetic time series. The DOGEx (Detection of Offsets in GPS Experiment) project aims to test the effectiveness of automated and manual offset detection approaches and the subsequent effect on GPS-derived velocities. To do this, simulated time series were first generated that mimicked realistic GPS data consisting of a velocity component, offsets, white and flicker noises (1/f spectrum noises) composed in an additive model. We focus on offset detection and together with velocity biases induced by incorrect offset detection. We show that, at present, manual methods (where offsets are hand -picked by GPS time series experts) 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 5th percentile ranges (5% to 95%) in velocity bias for automated approaches is equal to 4.2mm/year,whereas it is equal to 1.8mm/yr for the manual solutions. However the True Positive detection rate of automated solutions is significantly higher than those for the manual solutions, being around 37% for the best automated, and 42% for the best manual solution. The amplitude of offsets detectable by automated solutions is greater than for hand picked solutions, with the smallest detectable offset for the two best manual solutions equal to 5mm and 7mm and to 8mm and 10mm for the two best automated solutions. The best manual solutions yielded velocity biases from the truth commonly in the range ±0.2mm/yr, whereas the best automated solutions produced biases no better than double this range. Assuming the simulated time series noise levels continue to be representative of real GPS time series, robust geophysical interpretation of individual site velocities lower than these levels is therefore not robust. Further work is required before we can routinely interpret sub-mm/yr velocities for single GPS stations. [less ▲]