3.5 GPS Time Series Analysis
3.5.3 Analysis Strategy
A simple analysis strategy has been adopted for the investigations presented in this section. Data between 1998-2004 were utilised (where available), to standardise the available data across the four solutions. Daily topocentric (north, east and up) coordinates for each time series solution formed the fundamental dataset for analyses. Outlier detection was done using a simple sliding window with a length of 1 year run over de-trended data. Within the 1 year period, data were eliminated if its absolute value was greater than 2.5 times the inter-quartile range of the windowed data. This technique allows for improvements in the variability of GPS time series over time (see Herring, 1999, for a historic perspective).
Following the outlier detection, standard weighted least squares techniques were used to fit a simple model defining the offset, linear trend, and annual and semi- annual terms according to Eqn 3-10:
( )
( )
( )
( )
( )
(
1 2 3 4 5 6 sin 2 cos 2 ... ... sin 4 cos 4 y t x x t x t x t)
x t x tπ
π
π
π
= + + + + + Eqn 3-10where y(t) defines the modelled site coordinate at time, t, (in decimal years), and x1-x6 are model parameters. No attempt was made to model discontinuities or
seismic deformation for this analysis, as it was not required for the sites of interest.
3.5.3.1
Spectral Analysis
Spectral analysis was undertaken using the Welch method of spectral estimation (Welch, 1967), as implemented in Matlab®. Any gaps within the time series
(arising from outages or outlier removal), were filled with white noise (5 mm for north and east components and 8 mm for the vertical components) prior to estimating the spectral density estimates. A standard Hanning window was used (with the length defined as the maximum power of two less than the length of the input time series).
Two separate spectral analyses were undertaken on each coordinate component: 1) Raw data with just the offset and linear trend removed; and
2) Raw data with offset, linear trend and periodic components removed.
The approximate spectral index (k) was estimated using a linear regression computed in log-log space, according to Eqn 3-11:
( )
( )
10 10 log x P f k f = Eqn 3-11where Px is the power spectrum of the input signal, x, estimated over the set of
frequencies f. To avoid biasing the regression from spectral peaks present at the 13-14 day period in both the SOP and JPL solutions, the spectral index was computed over frequencies lower than 1/(15 days). See §3.5.4.4 for further discussion.
3.5.3.2
Wavelet Analysis
Wavelet analysis represents a powerful technique for the analysis of non stationary periodic signals. For the discrete analysis, coefficients were computed using a discrete approximation of the Meyer wavelet, “dmey”, in the Matlab® Wavelet
Toolbox. The “dmey” wavelet was well suited to this particular analysis due to its properties of symmetry, infinite regularity and suitability for single level decomposition. For the continuous analysis, the Morlet wavelet (Figure 3-16), was used due to its suitability for time series applications (see Daubechies, 1992 and Percival and Walden, 2000 for further discussion).
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 -1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1.0
Mortlet Wavelet (blue) and Centre Frequency Based Approximation (red)
Period: 1.2308; Cent. Freq: 0.8125
Amplitude
Figure 3-16 Mortlet wavelet and the centre frequency based approximation. For the continuous analysis, the wavelet scale coefficients were converted to estimates of power and displayed on a two dimensional grid with colour scaled across blue, cyan, green, yellow and red according to wavelet power (with time on the x-axis and pseudo-frequency derived from wavelet scale, converted to period plotted on the y-axis). An analysis of a test dataset is shown in Figure 3-17. The synthetic signal consists of random white noise (standard deviation of 10 mm), an annual signal for the first half of the signal (amplitude 20 mm), a semi-annual signal for the last half of the signal (amplitude 20 mm) and a fortnightly signal throughout the entire signal (amplitude 10 mm). Each signal component is clearly visible in the continuous wavelet analysis, highlighting its usefulness for the analysis of non-stationary periodic time series.
Wavelet Power (High Frequency Components)
Wavelet Power (Low Frequency Components)
1998.5 1999 1999.5 2000 2000.5 2001 2001.5 2002 2002.5 2003 2003.5 -50 0 50 Time (years) U (mm)
Continuous Wavelet Analysis: Test Data
Approximate Signal Period (Days)
Time (years) 1998 1999 2000 2001 2002 2003 2004 2 4 6 8 10 12 14 16 18 20 22 24 26
Approximate Signal Period (Months)
Time (years) 1998.5 1999 1999.5 2000 2000.5 2001 2001.5 2002 2002.5 2003 2003.5 2 4 6 8 10 12 14 16
Figure 3-17 Continuous wavelet analysis of the test dataset.