Geoscience Reference
In-Depth Information
time intervals Dt. N is the total number of the observations. The mean value of all
observations is computed as Y ; then the autocorrelation coefficient (R
h
) of the
observation series is computed as follows:
R
h
¼
C
h
=
C
o
;
ð
4
Þ
where C
h
is the autocovariance function:
C
h
¼
X
N
h
ð
Y
t
Y
Þð
Y
t
þ
h
Y
Þ=
N
;
ð
5
Þ
t¼1
and C
o
is the variance function:
ð
Y
t
Y
Þ
2
.
N
;
C
o
¼
X
N
ð
6
Þ
t
¼
1
h is time lag (h = 1, 2, 3, …).
The plot of R
h
for varying h is called the correlogram for the random process Y
k
.
The correlogram is used to check for serial dependency in an observed time series.
6 Results and Analysis
The plots of both filtered and unfiltered observations for the first 25 s are given in
Figs.
7
,
8
,
9
. It is shown that the implementation of the high pass filter has been
able to detect outliers in the observations. The observations of first 5 and 16 s are
outliers. The noises were due to the fact that the position solutions of first 5 s were
in float solution as the system was still in the initialization stage. The fixed solution
commenced from the sixth second observation.
The plot of the standard deviation values for the first 25 s is given in Fig.
10
.
The standard deviation values for the three components were generally high in
observations of first 5 and 16 s; Northing and Easting having the highest values of
about 1 m, and height more than 3 m. The standard deviation values for the
subsequent observations were about 1 mm for Northing and Easting and about
3 mm for Height.
The plot of the displacement vectors for Northing, Easting and Height com-
ponents, for the first 25 s is given in Fig.
11
. It is shown that the displacements are
affected by high standard deviation values. That is, large displacements have high
standard deviation values.
The autocorrelation functions of the GPS time series for Northing, Easting, and
Height, respectively are shown in Figs.
12
,
13
,
14
.
In the correlogram of Northing (Fig.
12
), the autocorrelation functions take the
value Ro = 0.003 and decrease exponentially until at time lag 300 s when the
autocorrelation of the observations is not so obvious. In the correlogram of Easting
