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Markov Chain Monte Carlo and the Application to Geodetic Time Series Analysis Olivares Pulido, German ; Teferle, Felix Norman ; Hunegnaw, Addisu 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 ▲] Detailed reference viewed: 111 (8 UL)On the combined effect of periodic signals and colored noise on velocity uncertainties ; Olivares Pulido, German ; Teferle, Felix Norman et al in GPS Solutions (2017) 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 125 (3 UL)The Combined Effect of Periodic Signals and Noise on the Dilution of Precision of GNSS Station Velocity Uncertainties ; Olivares Pulido, German ; Teferle, Felix Norman et al 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 ▲] Detailed reference viewed: 78 (3 UL)A Comparison of Bayesian Monte Carlo Markov Chain and Maximum Likelihood Estimation Methods for the Statistical Analysis of Geodetic Time Series Olivares Pulido, German ; Teferle, Felix Norman Scientific Conference (2013, December 10) 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 119 (6 UL)A Bayesian Monte Carlo Markov Chain Method for the Statistical Analysis of Geodetic Time Series Olivares Pulido, German ; Teferle, Felix Norman 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 ▲] Detailed reference viewed: 137 (3 UL)A Bayesian Monte Carlo Markov Chain Method for the Analysis of GPS Position Time Series Olivares Pulido, German ; Teferle, Felix Norman Poster (2013, April 12) 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 119 (1 UL)Detecting offsets in GPS time series: First results from the detection of offsets in GPS experiment ; ; et al 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 ▲] Detailed reference viewed: 211 (6 UL)A Bayesian Monte Carlo Markov Chain Method for Parameter Estimation of Fractional Differenced Gaussian Processes Olivares Pulido, German ; Teferle, Felix Norman in IEEE Transactions on Signal Processing (2013), 61(9), 2405-2412 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 190 (15 UL) |
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