Theory of second order stationary random processes applied to GPS coordinate time-series

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.

[1]  K. W. Clark,et al.  Continuous GPS observations across the Hayward Fault, California, 1991-1994 , 1995 .

[2]  Jessica R. Murray,et al.  Coseismic and initial postseismic deformation from the 2004 Parkfield, California, earthquake, observed by global positioning system, electronic distance meter, creepmeters, and borehole strainmeters , 2006 .

[3]  John Langbein,et al.  Correlated errors in geodetic time series: Implications for time‐dependent deformation , 1997 .

[4]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[5]  T. Dixon,et al.  Noise in GPS coordinate time series , 1999 .

[6]  Per Knudsen,et al.  Simultaneous Estimation of the Gravity Field and Sea Surface Topography From Satellite Altimeter Data By Least-Squares Collocation , 1991 .

[7]  M. Tamisiea,et al.  On seasonal signals in geodetic time series , 2012 .

[8]  Simon D. P. Williams,et al.  Fast error analysis of continuous GPS observations , 2008 .

[9]  Y. Bock,et al.  Anatomy of apparent seasonal variations from GPS‐derived site position time series , 2001 .

[10]  R. Forsberg,et al.  Airborne Gravity Field Determination , 2010 .

[11]  J. Ray,et al.  Anomalous harmonics in the spectra of GPS position estimates , 2008 .

[12]  L. Stenseng,et al.  Airborne gravity tests in the Italian area to improve the geoid model of Italy , 2009 .

[13]  Simon D. P. Williams,et al.  Fast error analysis of continuous GNSS observations with missing data , 2013, Journal of Geodesy.

[14]  Helmut Moritz,et al.  Least‐squares collocation , 1978 .

[15]  Robert B. Gramacy,et al.  A Brief History of Long Memory: Hurst, Mandelbrot and the Road to ARFIMA, 1951-1980 , 2014, Entropy.

[16]  S. Williams.  The effect of coloured noise on the uncertainties of rates estimated from geodetic time series , 2003 .

[17]  Walter H. F. Smith,et al.  New, improved version of generic mapping tools released , 1998 .

[18]  F. Wyatt.  Displacement of surface monuments: Vertical motion , 1989 .

[19]  P. Teunissen,et al.  Assessment of noise in GPS coordinate time series : Methodology and results , 2007 .

[20]  Roland Klees,et al.  An approach for estimating time-variable rates from geodetic time series , 2016, Journal of Geodesy.

[21]  H. Akaike.  A new look at the statistical model identification , 1974 .

[22]  Alessandro Caporali,et al.  Statistical analysis of the time series of permanent GPS stations , 2003 .

[23]  Geoffrey Blewitt,et al.  Effect of annual signals on geodetic velocity , 2002 .

[24]  H. Moritz.  Advanced Physical Geodesy , 1980 .

[25]  P. Phillips,et al.  Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? , 1992 .

[26]  Riccardo E. M. Riva,et al.  GPS monitoring and earthquake prediction: A success story towards a useful integration , 2009 .

[27]  A. Amiri-Simkooei,et al.  Non-negative least-squares variance component estimation with application to GPS time series , 2016, Journal of Geodesy.

[28]  R. Barzaghi,et al.  The SISMA prototype system: integrating Geophysical Modeling and Earth Observation for time-dependent seismic hazard assessment , 2013, Natural Hazards.

[29]  Yehuda Bock,et al.  Southern California permanent GPS geodetic array: Error analysis of daily position estimates and site velocities , 1997 .

[30]  F. Wyatt.  Displacement of surface monuments: Horizontal motion , 1982 .

[31]  Manfredo Massironi,et al.  Average Strain Rate In The Italian Crust Inferred From A Permanent GPS Network , 2001 .

[32]  R. Dach,et al.  Bernese GNSS Software Version 5.2 , 2015 .

[33]  Knudsen.  Estimation and modelling of the local empirical covariance function using gravity and satellite altimeter data , 1987 .

[34]  Simon D. P. Williams,et al.  CATS: GPS coordinate time series analysis software , 2008 .

[35]  A. Aoudia,et al.  Precursory slow-slip loaded the 2009 L'Aquila earthquake sequence , 2016 .

[36]  Philip Hans Franses,et al.  Generalizations of the KPSS‐test for stationarity , 2004 .

[37]  Anthony L Bertapelle.  Spectral Analysis of Time Series. , 1979 .

[38]  Benoit B. Mandelbrot,et al.  Some noises with I/f spectrum, a bridge between direct current and white noise , 1967, IEEE Trans. Inf. Theory.

[39]  D. Agnew,et al.  The time-domain behavior of power-law noises. [of many geophysical phenomena] , 1992 .

[40]  Alireza Amiri-Simkooei,et al.  On the nature of GPS draconitic year periodic pattern in multivariate position time series , 2013 .

[41]  Yehuda Bock,et al.  Error analysis of continuous GPS position time series , 2004 .

[42]  W. Förstner,et al.  A Metric for Covariance Matrices , 2003 .

[43]  P. Teunissen,et al.  Least-squares variance component estimation , 2008 .

[44]  A. I. McLeod,et al.  Preservation of the rescaled adjusted range: 1. A reassessment of the Hurst Phenomenon , 1978 .

[45]  J. C. Savage,et al.  Postseismic relaxation and transient creep , 2005 .

[46]  C. C. Tscherning,et al.  Collocation and least squares methods as a tool for handling gravity field dependent data obtained through space research techniques , 1978 .

[47]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[48]  Thomas A. Herring,et al.  A method for detecting transient signals in GPS position time-series: smoothing and principal component analysis , 2013 .

[49]  Jan Beran,et al.  Statistics for long-memory processes , 1994 .

[50]  Gwilym M. Jenkins,et al.  Time series analysis, forecasting and control , 1971 .