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

Abstract

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 1990s, it was usually assumed that white noise could fully characterise this noise. However, this was demonstrated to not be 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 model 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, an evaluation of the MCMC method against the Hector method using weekly downsampled versions of the Benchmark Synthetic GNSS (BSG) time series as provided in Chap. 2 is presented separately in an appendix.

Appendix

The appendix presents a cross-evaluation of the MCMC and Hector methods using the Benchmark Synthetic GNSS (BSG) time series (Chap. 2 of this book).

3.1.1 An Evaluation of MCMC Using Hector and the Benchmark Synthetic GNSS Time Series

Here we provide an evaluation of the MCMC method as implemented by Olivares and Teferle (2013) using Hector and the Benchmark Synthetic GNSS (BSG) time series. While providing the results from MCMC by themselves, we also carry out a basic comparison with the results provided and computed in-house with Hector. We have computed our own Hector parameter estimates since MCMC is a computationally intensive method and we have down–sampled the daily time series into weekly ones for all BSG series, see Fig. 3.36 for an example showing both original and down–sampled time series. Figure 3.37 shows the differences in the trend estimates $\Delta v$ between the two methods. Overall the parameter estimates (trend, amplitude of the annual term and phase-lag, white and power-low noise amplitudes as well as spectral index) are in good agreement between MCMC and Hector. A detailed comparison is shown in Tables 3.17, 3.18, and 3.19 for the deterministic parameter estimates, and in Tables 3.20, 3.21 and 3.22 for the stochastic parameter estimates.

Fig. 3.36

A weekly sampled time series (green line) superimposed on the daily time series (red line) for one of the time series for North, East, and Up components

Fig. 3.37

Trend differences between Hector and MCMC for the 20 weekly BSG time series. North, East, and Up components are displayed in green, red, and blue, respectively. A box whisker plot showing minimum, 25th percentile, median, 75th percentile and maximum values is to the right

3.1.2 Gaussian Properties of Parameters Estimates from MCMC

The parameters estimated from MCMC follow in general a Gaussian distribution. While these histograms can provide valuable additional information it is clear that in several cases the MCMC method has failed to provide converged results. This is most likely due to instabilities in the variance/covariance matrix within the MCMC method. Figure 3.38 shows the histograms for the Up trend components for the 20 time series.

Table 3.17 Hector and MCMC parameter estimates for North component time series compared: trend v, amplitude of the annual term $A_{1yr}$ and phase–lag $\phi$

Table 3.18 Hector and MCMC parameter estimates for East component time series compared: trend v, amplitude of the annual term $A_{1yr}$ and phase–lag $\phi$

Table 3.19 Hector and MCMC parameter estimates for Up component time series compared: trend v, amplitude of the annual term $A_{1yr}$ and phase–lag $\phi$

Further we have tested the Gaussian properties of the parameter estimates from the MCMC method by constructing the histograms for the amplitude estimates of the annual terms as shown in Figs. 3.39, 3.40, and 3.41 for the North, East and Up components, respectively.

We have also compared the mean and median trend estimates for all the time series considered in this analysis. The mean and median values show a similar magnitude. An indication that the estimates from MCMC exhibit unbiased and uncorrelated properties. A further test for the Skewness and the Kurtosis (“tailedness” of the probability distribution) for the trends again show the majority of the estimates indeed follow a Gaussian distribution, see Tables 3.23, 3.24, and 3.25. Formally, the Skewness of a Gaussian distribution is 0 while the Kurtossis is 3.

Table 3.20 Hector and MCMC stochastic parameter estimates for North component time series compared: spectral index $\kappa$, power–law $\sigma _{pl}$ and white $\sigma _{wn}$ noise amplitude

Table 3.21 Hector and MCMC stochastic parameter estimates for East component time series compared: spectral index $\kappa$, power–law $\sigma _{pl}$ and white $\sigma _{wn}$ noise amplitude

Table 3.22 Hector and MCMC stochastic parameter estimates for Up component time series compared: spectral index $\kappa$, power–law $\sigma _{pl}$ and white $\sigma _{wn}$ noise amplitude

Fig. 3.38

Histograms of the trend estimates of the Up component, continued on next page

Fig. 3.39

Histograms for the annual term amplitude of the North component, continued on next page

Fig. 3.40

Histograms for the annual term amplitude of the East component, continued on next page

Fig. 3.41

Histograms for the annual term amplitude of the Up component, continued on next page

Table 3.23 Statistics for the Gaussian distributions of the North component time series MCMC results: trend v, amplitude of the annual term $A_{1yr}$, phase–lag $\phi$, Skewness and Kurtosis

Table 3.24 Statistics for the Gaussian distributions of the East component time series MCMC results: trend v, amplitude of the annual term $A_{1yr}$, phase–lag $\phi$, Skewness and Kurtosis

Table 3.25 Statistics for the Gaussian distributions of the Up component time series MCMC results: trend v, amplitude of the annual term $A_{1yr}$, phase–lag $\phi$, Skewness and Kurtosis