The effect of coloured noise on the uncertainties of rates estimated from geodetic time series

Williams, Simon

doi:10.1007/s00190-002-0283-4

Cited by 377 publications

(337 citation statements)

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“…The values of the estimated trend and its associated error for these timeseries were estimated using the Maximum Likelihood Estimation method M a n u s c r i p t (Mao et al, 1999;Williams, 2003b;Langbein, 2004) and are presented in Table 1. One can see that the East component is always less accurate than the North component.…”

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confidence: 99%

The influence of seasonal signals on the estimation of the tectonic motion in short continuous GPS time-series

Bos

Bastos

Fernandes

2010

Journal of Geodynamics

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Please cite this article as: Bos, M.S., Bastos, L., Fernandes, R.M.S., The influence of seasonal signals on the estimation of the tectonic motion in short continuous GPS time-series, Journal of Geodynamics (2008Geodynamics ( ), doi:10.1016Geodynamics ( /j.jog.2009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Page 1 of 19A c c e p t e d M a n u s c r i p t However, Blewitt and Lavallée (2002) showed that including an annual signal in the estimation process still can give a larger linear trend error than the trend error estimated from data from which the annual signal has been removed by other means. They assumed that the GPS data only contained white noise and we extend their result to the case of power-law plus white noise which is known to exist in most GPS observations. For the GPS stations CASC, LAGO, PDEL and TETN the difference in trend error between having or not having an annual signal in the data is around ten times larger when a power-law plus white noise model is used instead of a pure white noise model. A c c e p t e d M a n u s c r i p t Next, our methodology can be used to estimate for any station how much the accuracy of the linear trend will improve when one tries to subtract the annual signal from the GPS time-series by using a physical model.Finally, we demonstrate that for short time-series the trend error is more influenced by the fact that the noise properties also need to be estimated from the data. This causes on average an underestimation of the trend error.

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confidence: 99%

The influence of seasonal signals on the estimation of the tectonic motion in short continuous GPS time-series

Bos

Bastos

Fernandes

2010

Journal of Geodynamics

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“…The spectral indices of the power-law noise estimated for GPS position time series ranging within fractional Gaussian part with the flicker noise being dominated. In both cases the covariance matrix of the unknown parameters of the deterministic model became more complex in comparison to the one of white noise (Williams 2003). We found, than the spectral indices are consistent within a particular technique, unlike the amplitudes of power-law noise, quite regular in case of GPS and very scattered in SG, which also may be an evidence of local influences.…”

Section: Discussionmentioning

confidence: 68%

“…The second task is to concern and unify the time series sampling rate. It is widely acknowledged, that noise characteristics may depend on sampling rate (Williams 2003). GPS-derived coordinates are rated every 24 h, while superconducting gravimeters record once every second, but are decimated to a 1-min sampling interval after a low-pass filtering using either a digital filter provided by the GWR manufacturer or a home-made one.…”

Section: Data and Pre-processingmentioning

confidence: 99%

“…Non-integer spectral indices include fractional Brownian motion with the spectral index varying between random-walk and flicker noise or fractional Gaussian noises for spectral indices falling between flicker and white noise (Mandelbrot and Van Ness 1968). Any assumed noise model affects the errors of rates (changes in time) estimated from any data (Williams 2003). Noises in GPS observations have already been widely discussed in the papers by Zhang et al (1997), Mao et al (1999), Williams et al (2004), Amiri-Simkooei et al (2007), Teferle et al (2008), Bos et al (2008), Acta Geod Geophys Santamaria-Gomez et al (2011) or Klos et al (2015).…”

Section: Introductionmentioning

confidence: 99%

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On the noise characteristics of time series recorded with nearby located GPS receivers and superconducting gravity meters

et al. 2018

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The inter-comparison of ground gravity measurements and vertical surface displacements enables to better understand the structure, dynamics and evolution of the Earth's system. Within this research we analyzed the Global Positioning System vertical position time series acquired in the vicinity of the superconducting gravimeters. We estimated of noise character of GPS and SG by comparison of the satellite and terrestrial measurements collected at 18 globally distributed neighboring sites. The comparable results were provided by applying the appropriate and corresponding models of geophysical phenomena to obtain residual time series, and by unifying the sampling rate since the noise characteristics may depend on it. The deterministic part of the series was assumed to follow the Polynomial Trend Model and was subtracted prior to noise analysis. Then, a combination of power-law and white noise was presumed and the Maximum Likelihood Estimation implemented in the Hector software to investigate the stochastic part was applied. Within the paper, we show that the spectral indices for all SG time series fall in the area of fractional Brownian motion (-2 \ j \ -1), while GPS data are best characterized by fractional Gaussian noises (-1 \ j \ 0). The estimated ratio between spectral indices of GPS and SG is stable worldwide with a global median value of about 0.5. Concerning the power-law amplitudes, these are very consistent worldwide for the GPS position time series and fluctuate around 15 mm/year -j/4 , while in the case of SG records they spread between 60 and 300 nm/ s 2 /year -j/4 . The fraction of power-law noise employed in the assumed combination is equal to 100% for almost all SG stations, while in case of GPS it varies between 26.1 and 99.9%. The main finding of this research is that the assumption of power-law noise is much more preferred for SG data than the assumption of a pure white noise being used until now.

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“…The methods of estimating a linear trend from continuous GNSS observations have received considerable attention in the last decades (Johnson and Agnew, 1995;Zhang et al, 1997;Williams, 2003). This interest stems mostly from the power spectral density of the noise at the low frequencies where it can be described by a power-law model (Williams et al, 2004;Santamaría-Gómez et al, 2011).…”

Section: Introductionmentioning

confidence: 99%

Fast error analysis of continuous GNSS observations with missing data

Bos

Fernandes

Williams

et al. 2012

J Geod

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One of the most widely used method for the time-series analysis of continuous Global Navigation Satellite System (GNSS) observations is Maximum Likelihood Estimation (MLE) which in most implementations requires O(n 3 ) operations for n observations. Previous research by the authors has shown that this amount of operations can be reduced to O(n 2 ) for observations without missing data. In the current research we present a reformulation of the equations that preserves this low amount of operations, even in the common situation of having some missing data.Our reformulation assumes that the noise is stationary to ensure a Toeplitz covariance matrix. However, most GNSS time-series exhibit power-law noise which is weakly non-stationary. To overcome this problem, we present an Toeplitz covariance matrix that provides an approximation for power-law noise that is accurate for most GNSS time-series. * mbos@ciimar.up.pt † rmanuel@di.ubi.pt ‡ sdwil@noc.ac.uk § lcbastos@fc.up.pt 1 Numerical results are given for a set of synthetic data and a set of International GNSS Service (IGS) stations, demonstrating a reduction in computation time of a factor of 10-100 compared to the standard MLE method, depending on the length of the time-series and the amount of missing data.

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The effect of coloured noise on the uncertainties of rates estimated from geodetic time series

Cited by 377 publications

References 16 publications

The influence of seasonal signals on the estimation of the tectonic motion in short continuous GPS time-series

The influence of seasonal signals on the estimation of the tectonic motion in short continuous GPS time-series

On the noise characteristics of time series recorded with nearby located GPS receivers and superconducting gravity meters

Fast error analysis of continuous GNSS observations with missing data

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