11.1 Part 1
Part 1 of this thesis explained how we prepare the data by removing outliers and off-sets and filling gaps. The seasonal signals are extracted, so that we finally can get a better estimate of the linear trend. We are satisfied with the results generated by the different algorithms.
Our time series are modelled by an additive model Xi = Ti +Si+Ni, where the Ti represents the trend component, Si represents the seasonal component andNi represents the noise component. We are primarily interested in the trend component, which estimates the velocity of the site under investigation. We give a summary of our actions here.
• STEP1:
We implemented different algorithms to fix gaps, detect and correct offsets and handle outliers. We then extracted the linear trend from our data Yi =Xi−Ti = Si+Ni.
• STEP2:
We analyzed the spectral of Yi by Lomb periodogram. We were interested in the frequency of the peak in the Lomb periodogram. The next step is to estimate the coefficients Af andBf inSi by the least square method.
We use the normal equation from the Lomb method (Least Square, see ch. 8):
CC 0 0 SS
Af Bf
=
XC XS
105
0 200 400 600 800 1000 1200 1400
−50−30−1010
Time
East Direction
Abscis= −11.1 rate: 0.0105 [mm/year]
0 200 400 600 800 1000 1200 1400
−4002040
Time
East Direction
Abscis= −12.7 New rate: 0.0161 [mm/year]
Figure 11.1: Impact of jumps in TS
=⇒ Af
Bf
=
CC 0 0 SS
−1 XC
XS
• STEP3:
To get a better estimate of the site velocity, we need to extract the componentSifrom the original time seriesZi =Xi−Si =Ti0+Ni0. We then estimate the linear trendTi0 again, but this time we have to be more accurate than when we calculated from Ti. The main reason is that we have removed the seasonal component (improvement).
• STEP4:
After the extraction of the linear trend Ti0, the last step is to remove the noise component. This is done by first identifying the type of noise at the site; this will be introduced in chapter10.
The next table shows the slope for each direction without extracting the seasonal com-ponent, and the second one with extraction of the seasonal components.
We see that the abscissa and the slope are changed in the second table. This shows the improvement by extractingSi component.
11.2. PART 2 107 To show the impact of outliers, gaps and offsets on site velocity estima-tions, I created an artificial offset of size 20mm at the middle of a time serie . The site is randomly chosen. I calculated the abscissa and the slope in both cases. When looking at fig.11.1, we see that the presence of offsets in our time series influences the site velocity heavily.
11.2 Part 2
Part 2 of this thesis explained how we determine common fluctuations in our network.
The removal of this common variance is not included in this thesis. Two methods are im-plemented: Factor Analysis and Principal Component. We conclude that Factor Analysis is the preferred method.
• East Direction:
In the table for theEast Directionon the next pages, the values of the first loading component obtained from PFA are uniform. This means that the common global variation over the whole network is almost constant. The same information is ob-tained from MLE, with opposite sign.
The second loading component bring us interesting results: we have two groups;
the first group contains the sites ALES,KRIS,OSLO,STAV, and the second group BODO,NYA1,TRON,VAR. This component tries to explain the regional common variation, and is different for the southern and the northern Norway. The observation shows that Trondheim separates the two regions.
• North Direction:
In the table for the North Direction on the next pages, the values of the first loading component obtained from PFA, shows that we end up with two groups. The first group is the site of NYA1 and VARD, while the second group contains the rest of network. Each group has a common global variation. The same information is obtained from MLE, but with opposite sign of loading.
The second loading component bring the same results as for East Direction
• Height Direction:
In the table for the Height Direction on the next pages, the values of the first loading component obtained from PFA is explained as in East Direction. The
same information is obtained from MLE, with opposite sign.
The second loading component bring the same results as for East Direction and North Direction:.
• Choose between PFA or MLE:
We examined the residuals of Rb=S−ΛbbΛT −Ψ, and found that the MLE estimateb does a better job of producingS than principal components (PFA). The entries of Rb are almost zero compared to PFA.
• Principal Component or Factor Analysis :
Since our purpose is to identify the latent variables that are contributing to the common fluctuations in our network, the Factor Analysis meets our needs better than Principal Component, which does not distinguish between the common variance and specific variance.
• Including more Loading Components:
Including more sites into our network and including more factor loadings, allows us to discover more regarding the variations due to the location of the site.
• Loading generated by PC:
The next table represents the factor loading generated by principal components from the previous section, and provide us with almost the same information.
• Peaks from Lomb Periodogram - Part1 :
The number of peaks generated by frequency analysis (Lomb periodogram) from Part 1 (see Appendix A), must comply with the number of factor loading explain-ing the fluctuations.
I believe it is amazing that we can determine how many factor loading components we need by counting the number of peeks from the spectral analysis.
11.3 Part 3
Part 3focused on the implementation of a CGPS Filter. The reason why I chose to proceed with this part, was that all the building blocks needed to construct the filter were already implemented in Part 1.
This chapter is NOT an obligatory part of my thesis. I included it because this information is useful when working with CGPS time series and because I find this theory
11.3. PART 3 109 very interesting. I did not, however, have the time to implement the algorithms needed to identify the dominant type of noise in our network.
appendix A, the graphs generated from each steps are presented.
110CHAPTER11.SUMMARYAND
Site Name East Direction North Direction Height Direction
Intercept Corrected Slope Corrected Intercept Corrected Slope Corrected Intercept Corrected Slope Corrected
ales.res -50.08 -51.53 0.0092 0.0112 -56.64 -58.77 0.0108 0.0133 -16.237 -16.805 0.0032 0.00381
berg.res -77.64 -80.96 0.0147 0.0181 -52.75 -43.08 0.01001 0.0078 -22.426 -21.205 0.00456 0.0046
bodo.res -53.56 -55.18 0.0094 0.0114 -53.21 -55.13 0.01014 0.0124 -23.953 -24.976 0.0045 0.0056
hers.res -52.34 -57.08 0.0116 0.0134 -49.7 -51.97 0.01055 0.0125 -8.973 -8.617 0.0021 0.0022
hofn.res -40.38 -42.6 0.0088 0.0107 -51.8 -53.85 0.01039 0.0123 -51.5 -53.19 0.01083 0.01263
kris.res -56.35 -54.19 0.012 0.0182 -51.72 -15.91 0.0117 0.0013 -13.584 -5.154 0.00320 0.00096
nya1.res -42.78 -43.66 0.007 0.0085 -53.26 -54.81 0.00961 0.0116 -49.19 -51.23 0.0087 0.010827
oslo.res -56.87 -57.92 0.01063 0.0129 -51.39 -51.18 0.00999 0.0115 -25.984 -26.705 0.00512 0.00614
pots.res -68.69 -72.78 0.01221 0.0144 -50.03 -59.97 0.00974 0.013096 -4.566 -6.026 0.0012252 0.0017
stav.res -53.84 -54.96 0.01005 0.0120 -54.81 -46.03 0.01036 0.00857 -11.451 -11.705 0.0022 0.0026132
tro1.res -14.07 -22.794 0.0082 0.0118 -48.95 -51.31 0.00850 0.01015 -19.778 -20.629 0.00402 0.0046
tron.res -52.2 -53.6 0.009411 0.0113 -55.08 -57.14 0.01045 0.0128 -23.071 -23.733 0.00444 0.00535
vard.res -69.69 -71.7 0.012425 0.0155 -38.23 -39.85 0.00809 0.0101 -18.918 -19.699 0.00368 0.00466
wsrt.res -60.6 -62.14 0.01168 0.01276 -55.07 -57.67 0.01039 0.012112 -0.6819 0.14106 0.0004642 0.00014
Table 11.1: Site Velocity Improvement