IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 10, NO. 3, MAY 2013 563 Extracting White Noise Statistics in GPS Coordinate Time Series Jean-Philippe Montillet, Member, IEEE, Paul Tregoning, Simon McClusky, and Kegen Yu, Member, IEEE Abstract—The noise in GPS coordinate time series is known to follow a power-law noise model with different components (white noise, flicker noise, and random walk). This work proposes an algorithm to estimate the white noise statistics, through the decomposition of the GPS coordinate time series into a sequence of sub time series using the empirical mode decomposition algorithm. The proposed algorithm estimates the Hurst parameter for each sub time series and then selects the sub time series related to the white noise based on the Hurst parameter criterion. Both simulated GPS coordinate time series and real data are employed to test this new method; the results are compared to those of the standard (CATS software) maximum-likelihood (ML) estimator approach. The results demonstrate that this proposed algorithm has very low computational complexity and can be more than 100 times faster than the CATS ML method, at the cost of a moderate increase of the uncertainty (∼5%) of the white noise amplitude. Reliable white noise statistics are useful for a range of applications including improving the filtering of GPS time series, checking the validity of estimated coseismic offsets, and estimating unbiased uncertainties of site velocities. The low complexity and computational efficiency of the algorithm can greatly speed up the processing of geodetic time series. Index Terms—Empirical mode decomposition (EMD), frac- tional Brownian motion (fBm), GPS coordinates, Hurst parame- ter, power-law noise, white noise amplitude. I. I NTRODUCTION T HE application of global navigation satellite system (GNSS) observations for monitoring geophysical phe- nomena such as earthquakes and tectonic movements requires also understanding the long-term coordinate time series error spectrum. This letter investigates different methods to best fit the noise contained within GPS station coordinate time series. The general method used is to fit and remove a linear trend to the coordinate time series and then model the noise characteristics of the residuals (e.g., [10] and [19]). It is crucial to know the statistics of the different noise components for applications such as checking the significance of estimated earthquake coseismic offsets and/or tectonic velocities from noisy GPS time series (e.g., [10] and [19]). Several studies have Manuscript received October 26, 2011; revised May 18, 2012 and August 3, 2012; accepted August 10, 2012. This work was supported by the Australian Research Council under Grant DP0877381. J.-P. Montillet, P. Tregoning, and S. McClusky are with the Research School of Earth Sciences, Australian National University, Canberra, A.C.T. 0200, Australia (e-mail: j.p.montillet@anu.edu.au; paul.tregoning@anu.edu.au; simon.mcclusky@anu.edu.au). K. Yu is with the Satellite Navigation and Positioning Laboratory, School of Surveying and Spatial Information Systems, The University of New South Wales, Sydney, N.S.W. 2052, Australia (e-mail: kegen.yu@unsw.edu.au). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LGRS.2012.2213576 shown that the error spectrum of geodetic GNSS time series is best characterized by a stochastic process following a power law (e.g., [19]) as S(f ) ≃ 1/f α , α ∈ [0, 2] (1) with S(f ) being the power spectrum and α being the spectral index. Following this model, researchers demonstrated that the noise is colored with mainly three components: white noise, flicker noise, and random walk [9]. White noise is independent of frequency and is generally associated with hardware noise or measurement errors. Clearly, white noise contains little or no geophysical information. However, it is useful to have good knowledge of the white noise statistics to enable efficient filtering. For example, GPS time series that require a priori knowledge of the noise statistics can be filtered using a Kalman filter [5]. Recently, the authors in [12] have developed a least mean squares adaptive filter to smooth GPS time series based on a complex noise model. In addition, knowledge of the noise statistics would help to exclude rather noisy GPS time series when processing measurements from a global network of sta- tions to estimate geophysical parameters such as the coseismic offsets associated with earthquakes [8] and the offsets arising from instrument upgrades and changes [17]. A power-law noise model means that S(f ) is not flat but is governed by long- range dependences. If the probability density function of the noise is Gaussian or has a different density function with a finite value of variance, its fractal properties can be described by the Hurst parameter (H). In 1968, Mandelbrod and Van Ness [11] defined the fractional Brownian motion (fBm) model where H is studied. In the case of H< 0.5, the process behaves as a Gaussian variable; if H> 0.5, the process exhibits long-range dependence, while the case of H =0.5 corresponds to a pure Brownian motion (white noise). From [16], H is directly connected with α by the relation α =2H − 1, α ≤ 2. (2) With this definition, flicker noise corresponds to α =1 or H = 1, random walk corresponds to α =2 or H =3/2, and white noise is related to α =0(H =0.5). Thus, random walk and flicker noise are classified as long-term dependence phenom- ena. It is, however, difficult to look directly into the GPS time series to characterize the various fractal properties because the different noise components are correlated. In this study, we apply the fBm model combined with the em- pirical mode decomposition (EMD) algorithm onto each GPS coordinate time series. First, a GPS time series is decomposed into sub time series using the EMD. Then, the Hurst parameter 1545-598X/$31.00 © 2012 IEEE