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 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.