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Authors: Barzaghi, R.* 
Borghi, Alessandra* 
Title: Theory of second order stationary random processes applied to GPS coordinate time-series
Issue Date: Jul-2018
Series/Report no.: /22(2018)
DOI: 10.1007/s10291-018-0748-4
Keywords: GPS
time series
Stationary stochastic process
Auto-covariance function
Power law spectrum
Subject Classification04.03. Geodesy 
Abstract: The analysis of Global Positioning System (GPS) coordinates time series is a valuable tool in quantifying crustal deformations. The longer continuous GPS time series allow estimation of nonlinear signatures. As a matter of fact, besides the linear and periodic behaviors, other relevant signals are present in such time series as the so-called transient deformations. They can be related to, e.g., slow slip events, which play a crucial role in studying fault mechanisms. To give reliable estimates of these signals, an appropriate and rigorous approach for defining the deterministic and the stochastic models of the data is needed. We prove that the theory of the second order stationary random process (SOSRP) can be used to describe the stochastic behavior of the daily GPS time series. In particular, the second order stationarity condition has to be verified for the daily GPS coordinate time series to be described as a SOSRP. This method has been already used for modeling the gravity field of the earth and in predicting/filtering problems, and this work shows that it can also be useful for characterizing the colored noise in the GPS time series.
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