[cct1]HPF WP 3220
External calibration of GOCE SGG data with terrestrial gravity data
Report
by
D. Arabelos* and C.C. Tscherning
*
Visiting from
Draft
2004-09-24
Abstract
Terrestrial gravity anomalies
selected from three extended continental regions having a smooth gravity field
were used in order to determine the appropriate size of the area for gravity
data collection as well as the required data-sampling for calibration of the GOCE
satellite gravity gradient (SGG) data. Using Least Square Collocation (LSC),
prediction of gravity gradient components was carried out at points on a
realistic orbit of the IAG SC7 simulated data set. Based on the mean error
estimation it was shown that up to 80% of the signal of the gravity gradient
components, as it is expressed through the covariance function of the
terrestrial gravity data, can be recovered in the case of an optimal size of
the collection area and of the optimum resolution of the data. These optimal
conditions e.g. for the Australian gravity field, correspond to an 
area
extend and a 5’ data-sampling.
Key words: Satellite Gravity gradiometer data, External calibration, Systematic
error parameters
1. Introduction
In the last decade numerous
investigations were published, concerning the calibration of GOCE satellite
gravity gradiometer data, (e.g. Baumann et al, 2004). The use of Least Squares
Collocation (LSC) as a element of the space-wise approach methods has also been
discussed in a number of papers (e.g. Tscherning 2003). The aim of this work
was to determine the size of required areas with terrestrial gravity data, as
well as the required resolution and accuracy of the gravity data for
calibration when LSC is used. The aim is to detect possible systematic errors
in the GOCE SSG data.
The “simple” LSC method was used for the tests concerning the size of
the area and the resolution of the data, while the parametric LSC was used for
the tests concerning the detection of systematic errors.
The terrestrial gravity data sets used in this study are
described in details in section 2. As control data the realistic orbit of the IAG
SC7 simulated data set was used. This data set comprises one month “clean” gravity
gradient values from EGM96 to degree 300, referring to the instrument frame,
aligned with the velocity vector and the
z-axis lying in the plane formed by this vector and the position vector.
The precision of the calibration will be directly
proportional to the gravity field standard deviation for example expressed as
the standard deviation of gravity anomalies from which the contribution of a
reference field have been subtracted. The areas studied here are therefore
areas with a very smooth gravity field.
We have used EGM96 to degree 360 for the reduction of gravity
anomalies, in an order to smooth as far as possible the gravity anomalies used
in all test areas. Then, using the reduce gravity anomalies we have predicted
gradient values at points selected from the SC7 simulated data set, further on
called control values. In order to compare the predicted with the control
values, we have reduced also the control values using EGM96 to degree 360. In
this way, the differences between control and predicted values are actually due
to the spectral content of EGM96 between the degrees 301 and 360 and to the
spectral contents of the terrestrial gravity anomalies beyond the degree 360. This
procedure does not affect the calibration ability of our method, since dealing
with simulated control data instead of real data we are based rather on the
error estimation given by collocation instead of the statistics of the
differences between the predicted and the control values. More specifically, we
relate the mean collocation error, depending on the choice of the covariance
function, with the formal standard deviation of the gravity gradient
components, depending also on the covariance function used, in order to draw
conclusions about the optimal size of the area and the resolution of the data
needed for the calibration. Note, however, that the error in the middle of the
area typically is 90 % of the mean error.
It will be numerically shown in the next, that in all test
areas, using LSC and terrestrial gravity anomalies, up to 80% of the formal
standard deviation of the gravity gradient signal can be recovered, in the case
of a high data accuracy, size of the
area of collection of terrestrial gravity anomalies and of the data-sampling.
Furthermore, it will be also numerically shown that in this way it is possible
to calibrate the GOCE SGG data for systematic errors such as bias and tilt.
In the computations it was attempted, to keep the
data-sampling and the size of the terrestrial data collections areas constant
for the corresponding experiments from area to area.
2. Gravity data used
For reasons discussed in
earlier work (e.g. Arabelos & Tscherning, 1998), for the requirements of
the calibration, the terrestrial data have to be collected from areas with at
possible smooth free air gravity anomaly field. A further reason for this is to
avoid topographic reductions to smooth the gravity field, due to errors that
could be introduced to gravity anomalies from erroneous altitudes and density
hypotheses.
Another requirement was to collect data from extended regions
in different geographic latitudes due to the dependence of the distribution of
the GOCE data on the latitude.
For all these reasons, data
from the Canadian plains,
1.
Terrestrial free-air gravity anomalies from the Canadian plains (further
on called region A)
This data set was already
described in (Arabelos & Tscherning, 1998). In the present paper the
reduced values of free-air gravity anomalies to degree 360 were used. The corresponding
statistics is shown in Table 1.
Table 1.
Statistics of the Canadian plains free-air gravity anomalies (14177 point
values)
|
|
Observations |
EGM96 |
Difference |
|
Mean |
-10.768 |
-10.678 |
-0.090 |
|
Standard Deviation |
22.419 |
17.497 |
13.418 |
|
Max. Value |
133.000 |
51.044 |
114.168 |
|
Minimum value |
-81.100 |
-72.210 |
-124.857 |
2.
Surface gravity anomalies from
The
2004 edition of the Australian National Gravity Database contains over
1,200,000 point data values in the area bounded by
.
This data was made available by Geoscience
Table 2.
Statistics of the Australian free-air gravity anomalies
|
|
Observations |
EGM96 |
Difference |
|
Mean |
4.901 |
5.158 |
-0.258 |
|
Standard Deviation |
24.504 |
22.504 |
12.102 |
|
Max. Value |
248.602 |
94.223 |
219.875 |
|
Minimum value |
-211.327 |
-104.930 |
-194.732 |
As
it is shown in Fig. 1 the gravity field is very smooth in the central 
.
Figure
1. The free-air anomaly-EGM96
gravity field in
3.
Surface gravity anomalies from Scandinavian (further on called region C)
Terrestrial
as well as air-borne gravity anomaly data sets were made available by R. Forsberg,
P. Knudsen and G. Strykovski (private communication). The terrestrial data set
(62,126) cover the area
.
After the reduction to EGM96 we obtain the statistics given in Table 3.
Table 3.
Statistics of the terrestrial free-air gravity anomalies in Scandinavian
|
|
Observations |
EGM96 |
Difference |
|
Mean |
-8.466 |
-8.118 |
-0.349 |
|
Standard Deviation |
18.285 |
16.229 |
8.789 |
|
Max. Value |
71.740 |
36.066 |
76.805 |
|
Minimum value |
-84.021 |
-77.351 |
-47.250 |
The
air-borne data set cover the area 
.The
statistics of the reduced to EGM96 air-borne gravity anomalies (4778 point
values) is shown in Table 3.
Table 4.
Statistics of the terrestrial free-air gravity anomalies in Scandinavian
|
|
Observations |
EGM96 |
Difference |
|
Mean |
-18.443 |
-19.735 |
1.292 |
|
Standard Deviation |
19.834 |
18.209 |
10.024 |
|
Max. Value |
29.660 |
21.235 |
35.085 |
|
Minimum value |
-80.540 |
-68.107 |
-31.675 |
In the collocation experiments
both data sets were used jointly with common accuracy equal to 2 mGal. The
free-air gravity field reduced to EGM96 is shown in Fig. 2.
From Tables 1 to 4, it is shown that the reduced to EGM96
free-air gravity anomalies present very similar statistical characteristics.
This is more evident from the shape of the corresponding covariance functions
(see Figs.3, 4 and 6).
Figure 2. The free-air anomaly-EGM96 gravity field in
3. Numerical experiments
The experiments concern
recovery of 5 s sampling noise-free simulated GOCE data provided by IAG, along
1 month realistic orbit (250 km), using terrestrial gravity anomalies. For the
determination of the required size of the area for terrestrial data collection
in all cases 10 min data-sampling was used and prediction experiments were
carried out in four areas with different size. For the determination of the
required data-sampling, the prediction experiments were carried using data with
5, 7.5, 10, 15 and 20 min. sampling in areas with constant size.
The statistics of the gradient components used as control
points in each of the three test areas is given in order to be able to compare
with the statistics of the differences between control points and the
corresponding predicted ones. In all computations the GRAVSOFT programs
(Tscherning et al., 1992) EMPCOV, COVFIT and GEOCOL were used.
Region A
Simulated
GOCE data used as control data in the area 
,
(242
values). The statistics of these gravity gradient values is shown in Table 5.
Table 5.
The statistics of the gravity gradient components used as control data in the
region (A). (EU)
|
|
Mean value |
Std. dev. |
Min. value |
Max. value |
|
|
0.1389 |
0.0598 |
-0.0136 |
0.2356 |
|
|
0.0409 |
0.2321 |
-0.2521 |
0.3625 |
|
|
-0.0706 |
0.1071 |
-0.2527 |
0.1241 |
|
|
0.1215 |
0.0516 |
-0.0119 |
0.1959 |
|
|
-0.0151 |
0.2915 |
-0.3869 |
0.3737 |
|
|
-0.2605 |
0.0903 |
-0.4168 |
-0.0941 |
The covariance function used
(empirical and the corresponding analytical one is shown in Figure 3.
Fig.3.
Covariance function of free-air gravity anomalies in the region (A) (red=empirical,
green=model).
(a)
Experiments for the
determination of the size of the area for terrestrial data collection
The
experiments were carried out using terrestrial gravity data with constant 10
min sampling. Accuracy equal to 1 mGal was adopted for these data. Prediction
experiments were carried out collecting terrestrial data from four areas with
size
,
,
and
respectively. The first of them correspond to
size of the control points collection area. The results of these numerical
experiments are shown in Table 6.
The assessment of the prediction results
in collocation is based not only on the statistics of the differences between
observed (control) and predicted quantities, but also on the collocation error
estimation of the prediction. It is also well known that the actual standard
deviation of the observed quantities remains constant, while the formal
standard deviation and consequently the collocation error estimation depend on
the covariance function. For this reasons in the following results the formal
standard deviation of the control points as well as the mean error estimation
are given simultaneously with the statistics of the differences between control
and predicted quantities.
With the increase of collection area
from to a
continuous improvement of the mean error estimation concerning all gradient
components is shown in Table 6. This improvement is more significant in the
case of 
(37%).
The mean error of 0.0027 EU correspond to a 37% of the formal standard
deviation of ,
resulting from the covariance function of free-air gravity anomalies in this
region. This could be interpreted as the ability of the method to recover the
63% of the signal, in the case of real SGG data. On the other hand, the
increased value of the standard deviation of the differences between control
and predicted quantities represent only a 10.6% of the standard deviation of
the simulated control values (see Table 5), and as it was discussed in section
1 is due to the different resolution between the predicted and the simulated
SGG control data.
Table 6.
Results in terms of the statistics of the differences between
predicted-observed quantities in the region (A), concerning the determination
of the size of required area with terrestrial gravity data. (Unit is EU). In
the left column, in parenthesis, the formal standard deviation of the gravity
gradient components, and in the last column the mean collocation error are
shown.
|
|
|||||
|
Component |
Mean value |
Std. dev. |
Min. value |
Max. value |
Mean coll. err. |
|
|
0.0008 |
0.0041 |
-0.0057 |
0.0084 |
0.0031 |
|
|
-0.0001 |
0.0027 |
-0.0073 |
0.0067 |
0.0040 |
|
|
0.0003 |
0.0047 |
-0.0106 |
0.0122 |
0.0036 |
|
|
0.0013 |
0.0026 |
-0.0034 |
0.0060 |
0.0030 |
|
|
0.0005 |
0.0029 |
-0.0063 |
0.0063 |
0.0037 |
|
|
-0.0021 |
0.0063 |
-0.0144 |
0.0090 |
0.0044 |
|
|
|||||
|
|
0.0010 |
0.0044 |
-0.0067 |
0.0090 |
0.0026 |
|
|
-0.0009 |
0.0034 |
-0.0091 |
0.0057 |
0.0036 |
|
|
0.0005 |
0.0052 |
-0.0114 |
0.0128 |
0.0030 |
|
|
0.0007 |
0.0033 |
-0.0055 |
0.0056 |
0.0026 |
|
|
0.0006 |
0.0037 |
-0.0064 |
0.0074 |
0.0032 |
|
|
-0.0017 |
0.0072 |
-0.0136 |
0.0115 |
0.0035 |
|
|
|||||
|
|
0.0011 |
0.0058 |
-0.0085 |
0.0113 |
0.0023 |
|
|
-0.0019 |
0.0038 |
-0.0088 |
0.0067 |
0.0031 |
|
|
0.0006 |
0.0066 |
-0.0132 |
0.0150 |
0.0024 |
|
|
0.0006 |
0.0038 |
0.0070 |
-0.0072 |
0.0022 |
|
|
0.0010 |
0.0041 |
0.0080 |
-0.0084 |
0.0027 |
|
|
-0.0017 |
0.0091 |
-0.0158 |
0.0146 |
0.0029 |
|
|
|||||
|
|
0.0015 |
0.0058 |
-0.0088 |
0.0120 |
0.0020 |
|
|
-0.0036 |
0.0046 |
-0.0119 |
0.0053 |
0.0026 |
|
|
0.0006 |
0.0073 |
-0.0143 |
0.0157 |
0.0021 |
|
|
0.0001 |
0.0045 |
-0.0092 |
0.0079 |
0.0020 |
|
|
0.0011 |
0.0051 |
-0.0117 |
0.0095 |
0.0023 |
|
|
-0.0016 |
0.0096 |
-0.0159 |
0.0161 |
0.0027 |
(b) Experiments for the determination of data-sampling
Concerning
the determination of the data-sampling, experiments were carried out with
terrestrial data collected from the same area
(
)
but with different resolution (5, 7.5, 10, 15 and 20 min). The results of these
experiments are shown in Table 7.
Table 7.
Results in terms of the statistics of the differences between predicted-observed
quantities from the experiments in the region (A), concerning the influence of
the terrestrial gravity data-sampling. (Unit is EU)
|
Area: |
|||||
|
Component |
Mean value |
Std. dev. |
Min. value |
Max. value |
Mean coll. err. |
|
|
0.0010 |
0.0040 |
-0.0061 |
0.0085 |
0.0025 |
|
|
-0.0011 |
0.0032 |
-0.0089 |
0.0053 |
0.0036 |
|
|
0.0005 |
0.0048 |
-0.0110 |
0.0120 |
0.0029 |
|
|
0.0008 |
0.0032 |
-0.0054 |
0.0057 |
0.0025 |
|
|
0.0005 |
0.0037 |
-0.0066 |
0.0074 |
0.0031 |
|
|
-0.0017 |
0.0067 |
-0.0128 |
0.0109 |
0.0032 |
|
Area: |
|||||
|
|
0.0010 |
0.0038 |
-0.0056 |
0.0083 |
0.0026 |
|
|
-0.0011 |
0.0031 |
-0.0086 |
0.0051 |
0.0036 |
|
|
0.0004 |
0.0046 |
-0.0111 |
0.0117 |
0.0030 |
|
|
0.0009 |
0.0031 |
-0.0050 |
0.0057 |
0.0025 |
|
|
0.0005 |
0.0035 |
-0.0062 |
0.0067 |
0.0032 |
|
|
-0.0019 |
0.0064 |
-0.128 |
0.0100 |
0.0033 |
|
Area: |
|||||
|
|
0.0010 |
0.0044 |
-0.0067 |
0.0090 |
0.0026 |
|
|
-0.0009 |
0.0034 |
-0.0091 |
0.0057 |
0.0036 |
|
|
0.0005 |
0.0052 |
-0.0114 |
0.0128 |
0.0030 |
|
|
0.0007 |
0.0033 |
-0.0055 |
0.0056 |
0.0026 |
|
|
0.0006 |
0.0037 |
-0.0064 |
0.0074 |
0.0032 |
|
|
-0.0017 |
0.0072 |
-0.0136 |
0.0115 |
0.0035 |
|
Area: |
|||||
|
|
0.0009 |
0.0043 |
-0.0061 |
0.0090 |
0.0029 |
|
|
-0.0009 |
0.0030 |
-0.0075 |
0.0060 |
0.0038 |
|
|
0.0004 |
0.0051 |
-0.0117 |
0.0125 |
0.0033 |
|
|
0.0010 |
0.0029 |
-0.0045 |
0.0058 |
0.0029 |
|
|
0.0006 |
0.0032 |
-0.0061 |
0.0058 |
0.0035 |
|
|
-0.0019 |
0.0069 |
-0.0136 |
0.0102 |
0.0041 |
|
Area: |
|||||
|
|
0.0013 |
0.0034 |
-0.0044 |
0.0071 |
0.0032 |
|
|
0.0004 |
0.0027 |
-0.0072 |
0.0059 |
0.0040 |
|
|
0.0002 |
0.0043 |
-0.0105 |
0.0106 |
0.0037 |
|
|
0.0013 |
0.0026 |
-0.0031 |
0.0065 |
0.0032 |
|
|
0.0005 |
0.0028 |
-0.0063 |
0.0075 |
0.0038 |
|
|
-0.0026 |
0.0057 |
-0.0136 |
0.0068 |
0.0050 |
* The results for this area and data sampling are the
same with the corresponding results for the same area and data-sampling of
Table 6.
Region B
Experiments
were carried out with constant 10 min. data-sampling. An empirical covariance
function was computed from the 10 min gravity data covering the area 
(Fig. 4).
Fig. 4.
Covariance function of free-air gravity anomalies in the region (B)
(red=empirical, green=model)
The
simulated GOCE data were collected from the area: 
(245
values). The statistics of the gravity gradient components used as control data
is shown in Table 8.
Table 8.
The statistics of the gravity gradient components used as control data in the
region (B). (EU)
|
|
Mean value |
Std. dev. |
Min. value |
Max. value |
|
|
0.1206 |
0.1058 |
-0.1137 |
0.2958 |
|
2 |
0.0204 |
0.4166 |
-0.5133 |
0.5621 |
|
|
0.0152 |
0.02404 |
-0.4630 |
0.4862 |
|
|
0.1585 |
0.0748 |
0.0063 |
0.3120 |
|
|
0.0286 |
0.1754 |
-0.3482 |
0.3390 |
|
|
-0.2792 |
0.1675 |
-0.5499 |
0.0750 |
The terrestrial data were
assumed to be accurate to 1 mGal. The simulated GOCE data were assumed to have
accuracy equal to 0.001 EU.
(a) Experiments concerning the size of required area with terrestrial
gravity data.
The
experiments were carried out using terrestrial gravity data with constant 10
min sampling. Prediction experiments were carried out collecting terrestrial
data from four areas with size,
,
and
respectively. The first of them correspond to
size of the control points collection area. The results of the prediction are
shown in Table 9.
In the region B the formal standard
deviation of is 0.0060 EU when the covariance function is
computed from terrestrial gravity anomalies in the area ,
while is increased to 0.0089 EU when the covariance function is computed from
terrestrial data in the area 
.
From this point of view, the mean collocation error estimation should be
considered as a measure of the part of the real signal that could be recovered,
if of course, the covariance function used reflects the statistical
characteristics of the real signal.
Table 9.
Results in terms of the statistics of the differences between
predicted-observed quantities in the region (B), concerning the determination
of the size of required area with terrestrial gravity data. (Unit is EU). In
the left column, in parenthesis, the formal standard deviation of the gravity
gradient component, and in the last column the mean collocation error are
shown.
|
|
|||||
|
Component |
Mean value |
Std. dev. |
Min. value |
Max. value |
Mean coll. err. |
|
|
-0.0013 |
0.0042 |
-0.0108 |
0.0063 |
0.0024 |
|
|
-0.0012 |
0.00390 |
-0.0095 |
0.0067 |
0.0032 |
|
|
-0.0000 |
0.0053 |
-0.0119 |
0.0115 |
0.0028 |
|
|
-0.0009 |
0.0033 |
-0.0090 |
0.0075 |
0.0024 |
|
|
0.0003 |
0.0045 |
-0.0113 |
0.0096 |
0.0028 |
|
|
0.0021 |
0.0068 |
-0.0128 |
0.0188 |
0.0032 |
|
|
|||||
|
|
-0.0016 |
0.0053 |
-0.0127 |
0.0084 |
0.00189 |
|
|
-0.0017 |
0.0036 |
-0.0089 |
0.0057 |
0.0027 |
|
|
-0.0000 |
0.0058 |
-0.0147 |
0.0136 |
0.0022 |
|
|
-0.0014 |
0.0033 |
-0.0081 |
0.0059 |
0.0019 |
|
|
0.0004 |
0.0048 |
-0.0106 |
0.0087 |
0.0022 |
|
|
0.0031 |
0.0077 |
-0.0141 |
0.0205 |
0.0023 |
|
|
|||||
|
|
-0.0015 |
0.0068 |
-0.0170 |
0.0101 |
0.0016 |
|
|
-0.0030 |
0.0045 |
-0.0119 |
0.0066 |
0.0021 |
|
|
0.0000 |
0.0074 |
-0.0187 |
0.0182 |
0.0176 |
|
|
-0.0019 |
0.0042 |
-0.0108 |
0.0072 |
0.0016 |
|
|
0.0007 |
0.0061 |
-0.0136 |
0.0125 |
0.0017 |
|
|
0.0034 |
0.0099 |
-0.0165 |
0.0274 |
0.0017 |
|
|
|||||
|
|
-0.0006 |
0.0069 |
-0.0174 |
0.0118 |
0.0015 |
|
|
-0.0044 |
0.0055 |
-0.0144 |
0.0066 |
0.0018 |
|
|
0.0001 |
0.0088 |
-0.0210 |
0.0209 |
0.0016 |
|
|
-0.0026 |
0.0046 |
-0.0129 |
0.0065 |
0.0015 |
|
|
0.0009 |
0.0067 |
-0.0152 |
0.0151 |
0.0015 |
|
|
0.0032 |
0.0104 |
-0.0172 |
0.0294 |
0.0018 |
From Table 9 it is shown that e.g. in
the case of ,
the mean collocation error is rapidly changing from 54% to 31% of the formal
standard deviation of ,
when the size of the collection of terrestrial data was increased from 50´60 to 100´120. On the other hand, the standard
deviation between observed and predicted was
increased from 2.5% to 6.2% of the standard deviation of the control data.
These numbers, even in the worst case of 6.2%, could be considered as
fulfilling the requirements of the calibration procedure. So the decision
concerning the size of the area for the collection of terrestrial data it is
reasonable to be based on the mean collocation error rather than on the
standard deviation of the differences between observer-predicted quantities.
This means that with an area extent of 100´120 about 65% of the signal of the real
data could be recovered, while the standard deviation of the differences is
still acceptable, for the calibration requirements.
(b). Experiments concerning the data-sampling.
Terrestrial gravity anomalies
with data-sampling 5, 7.5, 10, 15 and 20 min were collected from the area
bounded by
.
The prediction results are shown in Table 10.
Table 10. Results in terms of the statistics of the differences between
predicted-observed quantities from the experiments in the region (B),
concerning the influence of the terrestrial gravity data-sampling. (Unit is EU)
|
Area: |
|||||
|
Component |
Mean value |
Std. dev. |
Min. value |
Max. value |
Mean coll. err. |
|
|
-0.0009 |
0.0050 |
-0.0119 |
0.0088 |
0.0017 |
|
|
-0.0015 |
0.0036 |
-0.0089 |
0.0060 |
0.0026 |
|
|
0.0000 |
0.0055 |
-0.0146 |
0.0142 |
0.0021 |
|
|
-0.0011 |
0.0034 |
-0.0075 |
0.0069 |
0.0018 |
|
|
0.0004 |
0.0047 |
-0.0091 |
0.0080 |
0.0020 |
|
|
0.0020 |
0.0074 |
-0.0155 |
0.0183 |
0.0018 |
|
Area: |
|||||
|
|
-0.0009 |
0.0053 |
-0.0124 |
0.0096 |
0.0018 |
|
|
-0.0017 |
0.0038 |
-0.0095 |
0.0066 |
0.0026 |
|
|
0.0000 |
0.0058 |
-0.0152 |
0.0146 |
0.0021 |
|
|
-0.0010 |
0.0038 |
-0.0080 |
0.0080 |
0.0018 |
|
|
0.0005 |
0.0050 |
-0.0093 |
0.0088 |
0.0021 |
|
|
0.0019 |
0.0081 |
-0.0171 |
0.0195 |
0.0020 |
|
Area: |
|||||
|
|
-0.0016 |
0.0053 |
-0.0127 |
0.0084 |
0.0019 |
|
|
-0.0017 |
0.0036 |
-0.0089 |
0.0057 |
0.0027 |
|
|
-0.0000 |
0.0058 |
-0.0147 |
0.0136 |
0.0022 |
|
|
-0.0014 |
0.0033 |
-0.0081 |
0.0059 |
0.0019 |
|
|
0.0004 |
0.0048 |
-0.0106 |
0.0087 |
0.0022 |
|
|
0.0031 |
0.0077 |
-0.0141 |
0.0205 |
0.0023 |
|
Area: |
|||||
|
|
-0.0007 |
0.0045 |
-0.0094 |
0.0091 |
0.0023 |
|
|
-0.0019 |
0.0037 |
-0.0090 |
0.0058 |
0.0030 |
|
|
0.0000 |
0.0052 |
-0.0130 |
0.0119 |
0.0027 |
|
|
-0.0008 |
0.0037 |
-0.0068 |
0.0083 |
0.0023 |
|
|
0.0003 |
0.0048 |
-0.0110 |
0.0090 |
0.0026 |
|
|
0.0014 |
0.0073 |
-0.0171 |
0.0162 |
0.0032 |
|
Area: |
|||||
|
|
-0.008 |
0.0027 |
-0.0079 |
0.0049 |
0.0028 |
|
|
-0.0015 |
0.0021 |
-0.0061 |
0.0038 |
0.0034 |
|
|
0.0000 |
0.0036 |
-0.0095 |
0.0098 |
0.0032 |
|
|
-0.0008 |
0.0020 |
-0.0053 |
0.0041 |
0.0028 |
|
|
0.0002 |
0.0024 |
-0.0060 |
0.0047 |
0.0032 |
|
|
0.0015 |
0.0043 |
-0.0080 |
0.0132 |
0.0043 |
** The results for this area and data sampling are the
same with the corresponding results for the same area and data-sampling of
Table 9.
From Table 10 it is shown that the mean error of estimation for the same
size of the area of collection of terrestrial gravity anomalies was decreased when
the density of the data was increased, for all the gravity gradient components.
In terms of percentage of the formal standard deviation of the various
components this decrease is more significant in the case of ,
since it drops from 71% to 30% as the data-sampling changes from 20 to 5 min.
As it is shown in another experiment
(see Table 11), a further increase of the size of the area in the case of 20’
data-sampling, has no significant effect on the prediction results e.g. of .
Combining the results of Tables 9 and 10
it is reasonable to expect that using data in the larger area (
)
of Table 9 with the more dense data-sampling of Table 10 we will get the better
results in terms of the formal mean error estimation. This was verified with a relevant experiment (see
Table 12)
Table 11. Results of experiments in the
region (B), showing that in the case of a coarse data-sampling the increase of
the size of data collection area has no significant effect.(Unit is EU)
|
Area: |
|||||
|
Component |
Mean value |
Std. dev. |
Min. value |
Max. value |
Mean coll. err. |
|
|
0.0009 |
0.0050 |
-0.0100 |
0.0173 |
0.0041 |
|
Area: |
|||||
|
|
0.0007 |
0.0050 |
-0.0110 |
0.0168 |
0.0041 |
Table 12. Results of prediction of (EU) in the area (B), showing the improvement of the mean
collocation error when terrestrial gravity anomalies with 7.5 and 5min sampling
are used in the area 
|
Area: |
|||||
|
Component |
Mean value |
Std. dev. |
Min. value |
Max. value |
Mean coll. err. |
|
|
0.0021 |
0.0111 |
-0.0206 |
0.0302 |
0.0014 |
|
Area: |
|||||
|
|
0.0025 |
0.011 |
-0.0194 |
0.0296 |
0.0012 |
Finally
the ability of the method was evaluated with respect to the detection of
systematic errors in the GOCE data. For this reason, a new data set was added
to the 7.5 min sampling terrestrial gravity data in the area
.
This data set was the same as the control values
used in the previous experiments but assuming that they are affected by
systematic errors, as it is expected for the real SGG measurements. After the
exception of two very short tracks including 6 and 8 points respectively, the
data set consist of 233 values distributed in16 tracks (see Fig.5).
Figure 5. The additional input data (16 GOCE tracks)
used in the experiment for the estimation of systematic errors
For each of them two parameters, a bias
and a tilt were assumed. The results of this experiment in terms of the
parameters and their error estimates are shown in Table 13. Concerning the
tilt, almost zero values were resulted with very high accuracy, while the error
estimates for the bias does not exceed 33% of the formal standard deviation of .
Table 13. Results of the estimation of systematic errors using parametric
collocation
|
Track No. |
Bias (EU) |
Error of estimation |
Tilt ( |
Error of Estimation |
|
1 |
-0.0020 |
0.0021 |
0.12 |
0.66 |
|
2 |
0.0050 |
0.0016 |
-2.64 |
6.55 |
|
3 |
0.0046 |
0.0016 |
3.69 |
6.09 |
|
4 |
0.0017 |
0.0016 |
-0.42 |
6.55 |
|
5 |
-0.0006 |
0.0016 |
3.74 |
6.19 |
|
6 |
-0.0001 |
0.0020 |
-0.13 |
0.43 |
|
7 |
0.0011 |
0.0017 |
2.69 |
6.07 |
|
8 |
0.0061 |
0.0016 |
-4.34 |
6.64 |
|
9 |
0.0040 |
0.0015 |
1.71 |
6.00 |
|
10 |
-0.0015 |
0.0016 |
-3.33 |
6.69 |
|
11 |
-0.0016 |
0.0017 |
5.58 |
6.19 |
|
12 |
0.0017 |
0.0017 |
-0.15 |
0.42 |
|
13 |
0.0025 |
0.0016 |
3.11 |
6.00 |
|
14 |
0.0060 |
0.0016 |
-7.56 |
6.74 |
|
15 |
0.0028 |
0.0015 |
0.14 |
5.98 |
|
16 |
-0.0042 |
0.0018 |
-5.45 |
6.93 |
Region C
The empirical
covariance function was computed from gravity data covering the area 
.
This is shown in Fig. 6 together with its analytical fitting.
Simulated GOCE data were collected from
the area bounded by 
(250 values) in order to be used as control
data. For these data accuracy equal to 0.001 EU was adopted. The statistics of
the simulated GOCE data used as control values is shown in Table 14.
Fig. 6.
Covariance function of free-air gravity anomalies in the region (C) (red=empirical,
green=model).
Table 14. The statistics of the gravity gradient components used as control data
in the region (C). (EU)
|
|
Mean value |
Std. dev. |
Min. value |
Max. value |
|
|
0.1340 |
0.0303 |
0.0525 |
0.2076 |
|
|
0.0358 |
0.0574 |
-0.0979 |
0.2291 |
|
|
0.0019 |
0.0842 |
-0.1919 |
0.1921 |
|
|
0.1249 |
0.0724 |
-0.0080 |
0.3200 |
|
|
-0.0137 |
0.0525 |
-0.1076 |
0.1200 |
|
|
-0.2590 |
0.0826 |
-0.4653 |
-0.0836 |
(a). Experiments concerning
the size of required area with terrestrial gravity data.
The
experiments were carried out using terrestrial gravity data with constant 10
min sampling. Accuracy equal to 2 mGal was adopted for these data. Prediction
experiments were carried out collecting terrestrial data from four areas with
size,
,
and
respectively. The first of them correspond to
size of the control points collection area. The results of these numerical
experiments are shown in Table 15.
Table 15.
Results in terms of the statistics of the differences between
predicted-observed quantities in the region (C), concerning the determination
of the size of required area with terrestrial gravity data. (Unit is EU). In
the first column (in parenthesis) the formal standard deviation of the
corresponding gravity gradient component is shown. In the last column the mean
collocation error is shown.
|
|
|||||
|
Component |
Mean value |
Std. dev. |
Min. value |
Max. value |
Mean coll. err. |
|
|
0.0012 |
0.0032 |
-0.0036 |
0.0100 |
0.0032 |
|
|
-0.0003 |
0.0035 |
-0.0090 |
0.0060 |
0.0043 |
|
|
0.0004 |
0.0044 |
-0.0117 |
0.0107 |
0.0037 |
|
| |||||