Improved Analysis of Time Series with Temporally Correlated Errors; An Algorithm that Reduces the Computation Time. Abstract


  • Most time series of geophysical phenomena are contaminated with temporally correlated errors that limit the precision of any derived parameters. For example, estimates of station velocity derived from a time series of daily geodetic position measurements will be biased and its uncertainty will be too small if temporally correlated errors are ignored [Langbein and Johnson, 1997, Williams, 2003]. In particular, the rate uncertainty can be underestimated by a factor 10 for many GNSS time series. Obtaining better estimates of uncertainties is limited by several factors, including selection of the correct model for the background noise and the computational requirements to estimate the parameters of the selected noise model when there are numerous observations. Here, I address the second problem of computational efficiency using maximum likelihood estimates (MLE). Most geophysical time series have background noise processes that can be represented as a combination of white and power-law noise, 1/f^n , with frequency, f. With missing data, standard spectral techniques involving Fourier transforms are not appropriate. Instead, time domain techniques involving construction and inversion of large data covariance matrices are employed. Bos et al. [2012] demonstrate one technique that substantially increases the efficiency of the MLE methods, but it provides only an approximate solution for power-law indices greater than 1.0 since they require the data covariance matrix to be Toeplitz. That restriction can be removed by simply forming a data-filter that adds noise processes rather than combining them in quadrature. Consequently, the inversion of the data covariance matrix is simplified and it provides robust results for a wide range of power-law indices. With the new formulation, the efficiency is typically improved by about a factor of 8 over previous MLE algorithms [Langbein, 2004]. The new algorithm can be downloaded at The main program, est_noise7.2x, provides a number of basic functions that can be used to model the time-dependent part of time series, (rate, rate change, offset, exponential, Omori-law, sinusoids, and other, user supplied functions) and a variety of models that describe the temporal covariance of the data. These models of background noise include white noise, power-law noise, Gauss-Markov noise, and band-passed filtered noise; these can be combined to provide a complex mixture of noise processes. In addition, the new code provides a choice between adding the noise in quadrature, which has been the standard method, and summing the filter functions representing each noise process, which is the newer, faster method. Furthermore, the main program is packaged with a variety of utilities that can remove outliers, and importantly, help assess the success of the noise model with respect to the observations. These components are combined into example scripts which can help users analyze their own data. References: Bos, M.S., R.M.S. Fernandes, S.P.D. Williams, and L. Bastos, Fast error analysis of continuous GNSS observations with missing data, J. Geod., doi:10.1007/s00190-012-0605-0 (2012). Langbein, J., and H. Johnson, Correlated error in geodetic time series: Implications for time-dependent deformation, J. Geophy. Res., 102, 591--604 (1997). Langbein, J., Noise in two-color electronic distance meter measurements revisited, J. Geophy. Res. doi:10.1029/ 2003JB002819. (2004) Williams, S.D.P, The effect of coloured noise on the uncertainties of rates estimated from geodetic time series., J. Geod, doi:10.1007/s00190-002-0283-4,(2003)


publication date

  • 2016

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