Noise Analysis

In this thesis, the author assumes the reader to be familiar with the concepts and ter- minology of white and coloured noise. An introduction to this topic can be found in AppendixH.

It was initially expected that CGPS measurements would improve estimates of station coordinate velocities uncertainties by a factor of 1/√N , with N being the number of mea- surements, when compared to episodic GPS measurements. These expectations and the assumption that daily variations in the GPS coordinate solutions are purely random, are now widely accepted as unrealistic. Besides, random or white (time–independent) noise, errors have also been characterized as coloured (time–correlated) noise. Whereas white

Chapter 6. Analysis of the Continuous GPS Network 155

noise can be greatly reduced by increasing the number of measurements and averaging, coloured noise is not or to a far lesser degree reduced by these measures. Several analyses, e.g. Johnson and Agnew (1995); King et al. (1995); Zhang et al. (1997); Calais (1999);

Mao et al. (1999); Johnson and Agnew (2000); Lavall´ee (2000); Nikolaidis (2002) and

Williams et al. (2003), have shown that CGPS coordinate time series contain coloured noise. The general conclusion being that if coloured noise is not accounted for, station velocity uncertainties may be underestimated by an order of magnitude.

Two special cases of coloured noise with integer spectral indices have previously been discussed: random walk and flicker noise. Flicker noise, or 1/f noise, has been observed as fluctuations in e.g. the frequency of quartz crystal oscillators, average seasonal tem- perature, annual amount of rainfall or the rate of traffic flow (Keshner, 1982). Spurious motions of the geodetic monument, with respect to the underlying Earth’s crust, have been identified to follow a random walk process (Johnson and Agnew,1995;Langbein and Johnson,1997). Using 2.5 years of CGPS data from an 8 km baseline across the Hayward fault in California, King et al. (1995) were not able to detect such random walk noise in their baseline time series. However, the detectability of random walk noise depends on the length of the time series, the sampling frequency, and the relative amplitudes of the other noise components. By investigating the baseline time series between the two CGPS stations at the Pi˜non Flat Observatory separated by only 50 m, thus reducing the amount of white noise to a tenth of the amount normally observed in coordinate time series,

Johnson and Agnew (2000) did find that these well monumented sites do show a random walk process at low frequencies, but could not confirm this for the vertical component.

It is inherently important to understand the time–correlated noise content of coordinate time series in order to obtain realistic station velocity uncertainties. Zhang et al. (1997) analysed the CGPS measurements of 10 stations in southern California with only 1.6 years of data,Mao et al.(1999) assessed the noise characteristics in the time series of 23 globally distributed CGPS stations with 3 years of data and Calais (1999) performed a similar analysis on three CGPS stations in the Western Alps with 2.5 years of data. Lavall´ee

(2000) then compared their results with those obtained from 4 years of global weekly GPS solutions for 66 stations. In all these analyses, the stochastic model best describing the data was found to be a combination of white plus flicker noise. Following this, the most

Chapter 6. Analysis of the Continuous GPS Network 156

comprehensive noise analysis of CGPS data so far, was carried out for an 11 year period, for a total of 877 global and regional CGPS coordinate time series from 377 individual sites in seven different GPS solutions (Williams et al.,2003). They concluded that indeed for global GPS solutions the combination of white plus flicker noise is the most suitable stochastic model for all three coordinate components. For regional GPS solutions however, especially when spatial correlations were reduced using the regional filtering technique, it seemed that at different sites and networks different noise sources dominated, including residual common mode noise (white plus flicker noise), monument instabilities (random walk noise) and localized deformation due to changes in the groundwater table (unknown power–law noise plus annually repeating signals) (Williams et al., 2003). Furthermore, a latitude dependency of both white and flicker noise amplitudes was identified with amplitudes being largest close to the equator and amplitudes in general being noisier in the southern hemisphere than the northern hemisphere.

For CGPS coordinate time series in particular, Mao et al. (1999) concluded that mis– modelled satellite orbits, other reference frame effects, mis–modelled atmospheric effects, mis–modelled antenna phase centre variations, which may vary with satellite elevation, and local environmental factors can be regarded as sources for large amounts of coloured noise. It is further believed that the contribution of these effects to the noise spectrum of the CGPS coordinate time series dominates over the effect of monument motion. Calais

(1999) also suggested that temporal correlations observed in the CGPS coordinate time series are not due to site–specific noise such as monument motion, multipath, or antenna phase centre variations, but rather to noise sources common to all sites. This assumption is confirmed by the results shown in Williams et al. (2003) who stated that even the observed equatorial noise bulge may only partly be of tropospheric origin but also due to the reference frame via propagation.

Several methods for investigating the noise characteristics of the coordinate time series of UK CGPS stations were available to the author. These included the spectral analysis using the periodogram byScargle(1982) (Eq.H.2in AppendixH), two empirical methods byMao et al.(1999) andWilliams(2003a) (Eq.H.12, and Eqs.H.13andH.14respectively, both in AppendixH) and the most accurate and precise method using maximum likelihood estimation (MLE) (Eq. H.7in AppendixH).

Chapter 6. Analysis of the Continuous GPS Network 157

As outlined in Appendix H, the use of the periodogram (Eq. H.2) rather than the commonly applied Fast Fourier Transform (FFT) ensures the correct computation of the power spectrum for unevenly sampled data. Although, for most stations analysed, data gaps only extend from one to several days, as indicated by the high percentages of over 98% for the total number of days cleaned for most stations in Table 5.3, there are a number of sites with longer periods of missing data. Hence, the use of a FFT, which requires evenly spaced data to estimate the power spectrum of the coordinate time series may lead to spurious bulges of power at low frequencies (Press et al.,1992), thus affecting the estimation of coloured noise and in severe cases, making it even impossible to detect any. Filling large gaps has been shown to be problematic, as many interpolation methods perform poorly (Press et al.,1992).

By fitting a line in the form of Eq.H.5 to the power spectrum in the log–log space, an estimate of the spectral index κ of a power–law process (Eq. H.1 in AppendixH) can be computed. This has previously been carried out in a simple manner byZhang et al.(1997), who concluded that this method would underestimate the spectral index, especially if the higher frequency range is dominated by white noise. Therefore, Mao et al. (1999) fitted a more sophisticated curve to the log–log power spectra, modelling both coloured noise in the low frequency range and white noise in the higher frequency band. Mao et al.

(1999), however, reported of convergence problems of this model fit when an annual signal was removed from the coordinate time series prior to estimating the power spectra and had to resort to not removing the annual term before computing the spectral indices. In this thesis, the author follows a different approach in that a line is only fitted to the low frequency band of the power spectrum rather than the whole frequency range investigated (Calais, 1999; Lavall´ee, 2000; Nikolaidis, 2002). This has the advantage that there are no convergence problems if a known annual signal has been removed, and also that the estimated spectral index is not affected by the power in the higher frequency band.

BothMao et al.(1999) andWilliams(2003a) derived simple methods to determine the amplitudes of white and flicker noise. Mao et al.(1999) based their method on simple linear correlations of the WRMS statistic and the noise amplitudes (Eq. H.12 in Appendix H).

Williams(2003a) derived a comparable method from the equivalence of the RMS statistic and the power spectrum (Eq.H.13 in Appendix H). Although not exact, these empirical

Chapter 6. Analysis of the Continuous GPS Network 158

methods allow the computation of more realistic station velocity uncertainties, Eq. H.19

and H.20in AppendixH, without carrying out a spectral analysis or the time consuming MLE.

As mentioned, the MLE can be regarded as the most accurate and precise method to analyse the noise characteristics of coordinate time series (Langbein and Johnson, 1997;

Zhang et al., 1997; Mao et al., 1999; Williams, 2003a). Using the MLE, it is possible to simultaneously estimate the noise amplitudes for several stochastic models and the parameters of, e.g. a linear trend, periodic signals and coordinate offsets. The initial algorithm (Langbein and Johnson,1997;Zhang et al.,1997;Mao et al.,1999) was based on the assumption that the noise content is either classical white (WN) or a combination of white plus flicker (WN+FN) or white plus random walk (WN+RWN) noise, i.e. both cases with integer spectral indices (see Appendix H). The main reason for choosing these specific models was that at the time of these studies, the general form for the covariance matrix, describing the noise properties based on power–law noise (Eq.H.1in AppendixH), was not known (Williams, 2003b). The exact covariance matrix for random walk noise was shown in Langbein and Johnson (1997) while an approximation for the flicker noise covariance matrix was shown inZhang et al.(1997). Recently,Williams(2003a) derived a general form for a power–law covariance matrix (Eqs.H.8andH.9in AppendixH), which now allows the MLE to also estimate the spectral index. As no assumption on the a–priori noise model is made, the MLE determines the fractional spectral index best describing the noise characteristics of the coordinate time series.

Although, the MLE with fractional spectral indices, i.e. fitting a white plus power– law (WN+PLN) noise model, has the advantage that no a–priori noise model selection is required, the MLE with integer spectral indices is much faster as either a combination of WN+FN or of WN+RWN is assumed (Williams,2003a). However, this is achieved by disregarding the fact that the spectral index of the coordinate time series is most likely not an integer value.

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To test which of the noise models best fits the data, Zhang et al. (1997) applied the maximum–log–likelihood ratio test statistic (Kendall and Stuart,1979)

Λ = max likelihood 1_{max likelihood 2}

= exp[(max−log−likelihood 1) − (max−log−likelihood 2)]

(6.4)

Assuming the null hypothesis that the white noise only model best fits the data, it is possible to test whether the null hypothesis can be rejected in favour of the alternative hypothesis that any of the combined white and coloured noise models describe the data more appropriately. From tests on synthetic coordinate time series, Zhang et al. (1997) concluded that the maximum–log–likelihood test statistic was a reliable and powerful test to identify the correct model. They stated that if the Λ–statistic equals 1.0, then this would imply that the log–likelihood values from the null and alternative hypotheses were identical and in case Λ << 1 that the alternative hypothesis was to be favoured.

By evaluating the above mentioned methods of spectral analysis, the MLE and the two empirical methods presented by Mao et al.(1999) and Williams (2003a), to estimate the spectral index and the amplitudes of white and coloured noise, the author aimed to get an insight into the stochastic characteristics of the coordinate time series of the UK CGPS stations and the comparability between the different methods described.