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)

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

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

On the combined effect of periodic signals and colored noise on velocity uncertainties

The velocity estimates and their uncertainties derived from position time series of Global Navigation Satellite System stations are affected by seasonal signals and their harmonics, and the statistical properties, i.e., the stochastic noise, contained in the series. If the deterministic model in the form of linear trend and periodic terms is not accurate enough to describe the time series, it will alter the stochastic model, and the resulting effect on the velocity uncertainties can be perceived as a result of a misfit of the deterministic model. The effects of insufficiently modeled seasonal signals will propagate into the stochastic model and falsify the results of the noise analysis, in addition to velocity estimates and their uncertainties. We provide the general dilution of precision (GDP) of velocity uncertainties as the ratio of uncertainties of velocities determined from to two different deterministic models while accounting for stochastic noise at the same time. In this newly defined GDP, the first deterministic model includes a linear trend, while the second one includes a linear trend and seasonal signals. These two are tested with the assumption of white noise only as well as the combinations of power-law and white noise in the data. The more seasonal terms are added to the series, the more biased the velocity uncertainties become. With increasing time span of observations, the assumption of seasonal signals becomes less important, and the power-law character of the residuals starts to play a crucial role in the determined velocity uncertainties. With reference frame and sea level applications in mind, we argue that 7 and 9 years of continuous observations is the threshold for white and flicker noise, respectively, while 17 years are required for random-walk to decrease GDP below 5% and to omit periodic oscillations in the GNSS-derived time series taking only the noise model into consideration

A Bayesian Monte Carlo Markov Chain Method for Parameter Estimation of Fractional Differenced Gaussian Processes

We present a Bayesian Monte Carlo Markov Chain method to simultaneously estimate the spectral index and power amplitude of a fractional differenced Gaussian process at low frequency, in presence of white noise, and a linear trend and periodic signals. This method provides a sample of the likelihood function and thereby, using Monte Carlo integration, all parameters and their uncertainties are estimated simultaneously. We test this method with simulated and real Global Positioning System height time series and propose it as an alternative to optimization methods currently in use. Furthermore, without any mathematical proof, the results from the simulations suggest that this method is unaffected by the stationary regime and hence, can be used to check whether or not a time series is stationary

A Comparison of Bayesian Monte Carlo Markov Chain and Maximum Likelihood Estimation Methods for the Statistical Analysis of Geodetic Time Series

One of the objectives of TIGA is to compute precise station coordinates and velocities for GPS stations of interest. Consequently, a comprehensive knowledge of the stochastic features of the GPS time series noise is crucial, as it affects the velocity estimation for each GPS station. For that, we present a Monte Carlo Markov Chain (MCMC) method to simultaneously estimate the velocities and the stochastic parameters of the noise in GPS time series. This method allows to get a sample of the likelihood function and thereby, using Monte Carlo integration, all parameters and their uncertainties are estimated simultaneously. We propose this method as an alternative to optimization methods, such as the Maximum Likelihood Estimator (MLE) method implemented in the widely used CATS software, whenever the likelihood and the parameters of the noise are to be estimated in order to obtain more robust uncertainties for all parameters involved. Furthermore, we assess the MCMC method through comparison with the widely used CATS software using daily height time series from the Jet Propulsion Laboratory

A Bayesian Monte Carlo Markov Chain Method for the Analysis of GPS Position Time Series

Position time series from continuous GPS are an essential tool in many areas of the geosciences and are, for example, used to quantify long-term movements due to processes such as plate tectonics or glacial isostatic adjustment. It is now widely established that the stochastic properties of the time series do not follow a random behavior and this affects parameter estimates and associated uncertainties. Consequently, a comprehensive knowledge of the stochastic character of the position time series is crucial in order to obtain realistic error bounds and for this a number of methods have already been applied successfully. We present a new Bayesian Monte Carlo Markov Chain (MCMC) method to simultaneously estimate the model and the stochastic parameters of the noise in GPS position 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 the MCMC method is that the computational time increases linearly with the number of parameters, hence being very suitable for dealing with a high number of parameters. A second advantage is that the properties of the estimator used in this method do not depend on the stationarity of the time series. At least on a theoretical level, no other estimator has been shown to have this feature. Furthermore, the MCMC method provides a means to detect multi-modality of the parameter estimates. We present an evaluation of the new MCMC method through comparison with widely used optimization and empirical methods for the analysis of GPS position time series

A linear scale height Chapman model supported by GNSS occultation measurements

Global Navigation Satellite Systems (GNSS) radio occultations allow the vertical sounding of the Earth’s atmosphere, in particular, the ionosphere. The physical observables estimated with this technique permit to test theoretical models of the electron density such as, for example, the Chapman and the Vary-Chap models. The former is characterized by a constant scale height, whereas the latter considers a more general function of the scale height with respect to height. We propose to investigate the feasibility of the Vary-Chap model where the scale height varies linearly with respect to height. In order to test this hypothesis, the scale height data provided by radio occultations from a receiver on board a low Earth orbit (LEO) satellite, obtained by iterating with a local Chapman model at every point of the topside F2 layer provided by the GNSS satellite occultation, are fitted to height data by means of a linear least squares fit (LLS). Results, based on FORMOSAT-3/COSMIC GPS occultation data inverted by means of the Improved Abel transform inversion technique (which takes into account the horizontal electron content gradients) show that the scale height presents a more clear linear trend above the F2 layer peak height, hm, which is in good agreement with the expected linear temperature dependence. Moreover, the parameters of the linear fit obtained during four representative days for all seasons, depend significantly on local time and latitude, strongly suggesting that this approach can significantly contribute to build realistic models of the electron density directly derived from GNSS occultation data.JRC.E.2-Technology Innovation in Securit

A New Method of Electron Density Retrieval from MetOp-A’s Truncated Radio Occultation Measurements

The radio occultation (RO) measurements of the Global Navigation Satellite System’s (GNSS’s) signals onboard a Low Earth Orbiting (LEO) satellite enable the computation of the vertical electron density profile from the LEO satellite’s orbit height down to the Earth’s surface. The ionospheric extension experiment performed by the GNSS Receiver for Atmospheric Sounding (GRAS) receiver on board MetOp-A provides opportunities for ionospheric sounding but with the RO measurements only taken with an impact parameter height below 600 and 300 km within two different experiments, although MetOp-A was flying at an orbit height of about 800 km. Here, we present a model-assisted RO inversion technique for electron density retrieval from such kind of truncated data. The topside ionosphere and plasmasphere above the LEO orbit height are modelled by a Chapman layer function superposed with an exponential decay function representing the plasmasphere. Our investigation shows that the model-assisted technique is stable and robust and can successfully be used to retrieve the electron density values up to the LEO height from the truncated MetOp-A data, in particular when observations are available until 600 km. Moreover, this model-assisted technique is also successful with the availability of a small number of observations of the topside above the peak density height. For observations available only up to 300 km, the accuracy of the retrieved profile is comparable to the one obtained by the data truncated at a 600 km height only when the peak electron density lies below the 250 km altitude level