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

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).