[cct1] HPF WP 3220

 

External calibration of GOCE SGG data with terrestrial gravity data

Report by

D. Arabelos* and C.C. Tscherning

University of Copenhagen, Department of Geophysics.

* Visiting from University of Thessaloniki

 

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, Australia and Scandinavia were used.

 

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 Australia (further on called region B)

 

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 Australia. The data set covering the continental Australia and the surrounding ocean (1,117,054 point values) was reduced to EGM96. The statistics of the original and reduced free-air gravity anomalies is shown in Table 2.

 

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 Australia and consequently, appropriate for the calibration requirements. For this reason, point gravity anomalies were selected from the area bounded by.

 

 

Figure 1. The free-air anomaly-EGM96 gravity field in Australia (simplified)

 

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 Scandinavia (simplified)

 

 

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.

 

(859 point values)

Component

Mean value

Std. dev.

Min. value

Max. value

Mean coll. err.

(0.004286)

0.0008

0.0041

-0.0057

0.0084

0.0031

(0.004948)

-0.0001

0.0027

-0.0073

0.0067

0.0040

(0.005080)

0.0003

0.0047

-0.0106

0.0122

0.0036

(0.004286)

0.0013

0.0026

-0.0034

0.0060

0.0030

(0.005080)

0.0005

0.0029

-0.0063

0.0063

0.0037

(0.007249)

-0.0021

0.0063

-0.0144

0.0090

0.0044

(1414 point values)

(0.004286)

0.0010

0.0044

-0.0067

0.0090

0.0026

(0.004948)

-0.0009

0.0034

-0.0091

0.0057

0.0036

(0.005080)

0.0005

0.0052

-0.0114

0.0128

0.0030

(0.004286)

0.0007

0.0033

-0.0055

0.0056

0.0026

(0.005080)

0.0006

0.0037

-0.0064

0.0074

0.0032

(0.007249)

-0.0017

0.0072

-0.0136

0.0115

0.0035

( 2385 point values)

(0.004286)

0.0011

0.0058

-0.0085

0.0113

0.0023

(0.004948)

-0.0019

0.0038

-0.0088

0.0067

0.0031

(0.005080)

0.0006

0.0066

-0.0132

0.0150

0.0024

(0.004286)

0.0006

0.0038

0.0070

-0.0072

0.0022

(0.005080)

0.0010

0.0041

0.0080

-0.0084

0.0027

(0.007249)

-0.0017

0.0091

-0.0158

0.0146

0.0029

 (3578 point values)

(0.004286)

0.0015

0.0058

-0.0088

0.0120

0.0020

(0.004948)

-0.0036

0.0046

-0.0119

0.0053

0.0026

(0.005080)

0.0006

0.0073

-0.0143

0.0157

0.0021

(0.004286)

0.0001

0.0045

-0.0092

0.0079

0.0020

(0.005080)

0.0011

0.0051

-0.0117

0.0095

0.0023

(0.007249)

-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: , Data-sampling 5’ (2252 point values)

Component

Mean value

Std. dev.

Min. value

Max. value

Mean coll. err.

(0.004286)

0.0010

0.0040

-0.0061

0.0085

0.0025

(0.004948)

-0.0011

0.0032

-0.0089

0.0053

0.0036

(0.005080)

0.0005

0.0048

-0.0110

0.0120

0.0029

(0.004286)

0.0008

0.0032

-0.0054

0.0057

0.0025

(0.005080)

0.0005

0.0037

-0.0066

0.0074

0.0031

(0.007249)

-0.0017

0.0067

-0.0128

0.0109

0.0032

Area: , Data-sampling 7.5’ (1814 point values)

(0.004286)

0.0010

0.0038

-0.0056

0.0083

0.0026

(0.004948)

-0.0011

0.0031

-0.0086

0.0051

0.0036

(0.005080)

0.0004

0.0046

-0.0111

0.0117

0.0030

(0.004286)

0.0009

0.0031

-0.0050

0.0057

0.0025

(0.005080)

0.0005

0.0035

-0.0062

0.0067

0.0032

(0.007249)

-0.0019

0.0064

-0.128

0.0100

0.0033

Area: , Data-sampling 10’ (1414 point values)*

(0.004286)

0.0010

0.0044

-0.0067

0.0090

0.0026

(0.004948)

-0.0009

0.0034

-0.0091

0.0057

0.0036

(0.005080)

0.0005

0.0052

-0.0114

0.0128

0.0030

(0.004286)

0.0007

0.0033

-0.0055

0.0056

0.0026

(0.005080)

0.0006

0.0037

-0.0064

0.0074

0.0032

(0.007249)

-0.0017

0.0072

-0.0136

0.0115

0.0035

Area: , Data-sampling 15’ (743 point values)

(0.004286)

0.0009

0.0043

-0.0061

0.0090

0.0029

(0.004948)

-0.0009

0.0030

-0.0075

0.0060

0.0038

(0.005080)

0.0004

0.0051

-0.0117

0.0125

0.0033

(0.004286)

0.0010

0.0029

-0.0045

0.0058

0.0029

(0.005080)

0.0006

0.0032

-0.0061

0.0058

0.0035

(0.007249)

-0.0019

0.0069

-0.0136

0.0102

0.0041

Area: , Data-sampling 20’ (423 point values)

(0.004286)

0.0013

0.0034

-0.0044

0.0071

0.0032

(0.004948)

0.0004

0.0027

-0.0072

0.0059

0.0040

(0.005080)

0.0002

0.0043

-0.0105

0.0106

0.0037

(0.004286)

0.0013

0.0026

-0.0031

0.0065

0.0032

(0.005080)

0.0005

0.0028

-0.0063

0.0075

0.0038

(0.007249)

-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.

 

(1086 point values)

Component

Mean value

Std. dev.

Min. value

Max. value

Mean coll. err.

(0.003564)

-0.0013

0.0042

-0.0108

0.0063

0.0024

(0.004115)

-0.0012

0.00390

-0.0095

0.0067

0.0032

(0.004225)

-0.0000

0.0053

-0.0119

0.0115

0.0028

(0.003564)

-0.0009

0.0033

-0.0090

0.0075

0.0024

(0.004225)

0.0003

0.0045

-0.0113

0.0096

0.0028

(0.006028)

0.0021

0.0068

-0.0128

0.0188

0.0032

(1737 point values)

(0.003564)

-0.0016

0.0053

-0.0127

0.0084

0.00189

(0.004115)

-0.0017

0.0036

-0.0089

0.0057

0.0027

(0.004225)

-0.0000

0.0058

-0.0147

0.0136

0.0022

(0.003564)

-0.0014

0.0033

-0.0081

0.0059

0.0019

(0.004225)

0.0004

0.0048

-0.0106

0.0087

0.0022

(0.006028)

0.0031

0.0077

-0.0141

0.0205

0.0023

(2831 point values)

(0.003564)

-0.0015

0.0068

-0.0170

0.0101

0.0016

(0.004115)

-0.0030

0.0045

-0.0119

0.0066

0.0021

(0.004225)

0.0000

0.0074

-0.0187

0.0182

0.0176

(0.003564)

-0.0019

0.0042

-0.0108

0.0072

0.0016

(0.004225)

0.0007

0.0061

-0.0136

0.0125

0.0017

(0.006028)

0.0034

0.0099

-0.0165

0.0274

0.0017

 (3879 point values)

(0.003564)

-0.0006

0.0069

-0.0174

0.0118

0.0015

(0.004115)

-0.0044

0.0055

-0.0144

0.0066

0.0018

(0.004225)

0.0001

0.0088

-0.0210

0.0209

0.0016

(0.003564)

-0.0026

0.0046

-0.0129

0.0065

0.0015

(0.004225)

0.0009

0.0067

-0.0152

0.0151

0.0015

(0.006028)

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:, Data-sampling 5’ (6090 point values)

Component

Mean value

Std. dev.

Min. value

Max. value

Mean coll. err.

(0.003564)

-0.0009

0.0050

-0.0119

0.0088

0.0017

(0.004115)

-0.0015

0.0036

-0.0089

0.0060

0.0026

(0.004225)

0.0000

0.0055

-0.0146

0.0142

0.0021

(0.003564)

-0.0011

0.0034

-0.0075

0.0069

0.0018

(0.004225)

0.0004

0.0047

-0.0091

0.0080

0.0020

(0.006028)

0.0020

0.0074

-0.0155

0.0183

0.0018

Area:, Data-sampling 7.5’ (3088 point values)

(0.003564)

-0.0009

0.0053

-0.0124

0.0096

0.0018

(0.004115)

-0.0017

0.0038

-0.0095

0.0066

0.0026

(0.004225)

0.0000

0.0058

-0.0152

0.0146

0.0021

(0.003564)

-0.0010

0.0038

-0.0080

0.0080

0.0018

(0.004225)

0.0005

0.0050

-0.0093

0.0088

0.0021

(0.006028)

0.0019

0.0081

-0.0171

0.0195

0.0020

Area:, Data-sampling 10’ (1737 point values)**

(0.003564)

-0.0016

0.0053

-0.0127

0.0084

0.0019

(0.004115)

-0.0017

0.0036

-0.0089

0.0057

0.0027

(0.004225)

-0.0000

0.0058

-0.0147

0.0136

0.0022

(0.003564)

-0.0014

0.0033

-0.0081

0.0059

0.0019

(0.004225)

0.0004

0.0048

-0.0106

0.0087

0.0022

(0.006028)

0.0031

0.0077

-0.0141

0.0205

0.0023

Area:, Data-sampling 15’ (773 point values)

(0.003564)

-0.0007

0.0045

-0.0094

0.0091

0.0023

(0.004115)

-0.0019

0.0037

-0.0090

0.0058

0.0030

(0.004225)

0.0000

0.0052

-0.0130

0.0119

0.0027

(0.003564)

-0.0008

0.0037

-0.0068

0.0083

0.0023

(0.004225)

0.0003

0.0048

-0.0110

0.0090

0.0026

(0.006028)

0.0014

0.0073

-0.0171

0.0162

0.0032

Area:, Data-sampling 20’ (436 point values)

(0.003564)

-0.008

0.0027

-0.0079

0.0049

0.0028

(0.004115)

-0.0015

0.0021

-0.0061

0.0038

0.0034

(0.004225)

0.0000

0.0036

-0.0095

0.0098

0.0032

(0.003564)

-0.0008

0.0020

-0.0053

0.0041

0.0028

(0.004225)

0.0002

0.0024

-0.0060

0.0047

0.0032

(0.006028)

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:, Data-sampling 20’, (982 point values)

Component

Mean value

Std. dev.

Min. value

Max. value

Mean coll. err.

(0.006028)

0.0009

0.0050

-0.0100

0.0173

0.0041

Area:, Data-sampling 20’, (2163 point values)

(0.006028)

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:, Data-sampling 7.5’, (6834 point values)

Component

Mean value

Std. dev.

Min. value

Max. value

Mean coll. err.

(0.006028)

0.0021

0.0111

-0.0206

0.0302

0.0014

Area: , Data-sampling 20’, (13476 point values)

(0.006028)

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

(EU/s)

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.

 

(1072 point values)

Component

Mean value

Std. dev.

Min. value

Max. value

Mean coll. err.

(0.004726)

0.0012

0.0032

-0.0036

0.0100

0.0032

(0.005455)

-0.0003

0.0035

-0.0090

0.0060

0.0043

(0.005601)

0.0004

0.0044

-0.0117

0.0107

0.0037

(0.004726)

0.0010

0.0034

-0.0045

0.0094

0.0031

(0.005601)

0.0000

0.0037

-0.0087

0.0101

0.0039

(0.007992)

-0.0022

0.0057

-0.0170

0.0077

0.0043

(1984 point values)

(0.004726)

0.0022

0.0064

-0.0069

0.0164

0.0026

(0.005455)

0.0009

0.0062

-0.0120

0.0111

0.0037

(0.005601)

0.0013

0.0075

-0.0161

0.0184

0.0028

(0.004726)

0.0023

0.0066

-0.0075

0.0187

0.0023

(0.005601)

0.0005

0.0073

-0.0129

0.0169

0.0031

(0.007992)

-0.0040

0.0111

-0.0306

0.0109

0.0029

(3167 point values)

(0.004726)

0.0019

0.0071

-0.0088

0.0164

0.0021

(0.005455)

0.0016

0.0066

-0.0115

0.0119

0.0031

(0.005601)

0.0016

0.0083

-0.0169

0.0193

0.0022

(0.004726)

0.0020

0.0068

-0.0088

0.0187

0.0020

(0.005601)

0.0007

0.0079

-0.0136

0.0183

0.0026

(0.007992)

-0.0039

0.0122

-0.0323

0.0142

0.0023

 (4165 point values)

(0.004726)

0.0016

0.0071

-0.0089

0.0157

0.0019

(0.005455)

0.0010

0.0066

-0.0122

0.0114

0.0026

(0.005601)

0.0016

0.0086

-0.0164

0.0189

0.0019

(0.004726)

0.0022

0.0068

-0.0091

0.0189

0.0018

(0.005601)

0.0008

0.0079

-0.0130

0.0181

0.0022

(0.007992)

-0.0038

0.0122

-0.0309

0.0149

0.0022

 

        These results are very similar to the previous ones in Tables 6 and 9. Increasing the collection area from to , the mean error estimation, e.g. in the case was decreased up to 51% of its original value. In terms of percentage on the formal standard deviation of  the mean estimation error drops from 53.8% to 27.5%.

 

(b). Experiments concerning the requirements in data-sampling.

 

For this purpose 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 16.

 

Table 16. Results in terms of the statistics of the differences between predicted-observed quantities from the experiments in the region (C), concerning the influence of the terrestrial gravity data-sampling. (Unit 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.

 

Area: , Data-sampling 5’ (7404 point values)

Component

Mean value

Std. dev.

Min. value

Max. value

Mean coll. err.

(0.004726)

0.0021

0.0072

-0.0078

0.0175

0.0024

(0.005455)

0.0008

0.0065

-0.0121

0.0107

0.0036

(0.005601)

0.0014

0.0082

-0.0172

0.0199

0.0026

(0.004726)

0.0021

0.0069

-0.0083

0.0192

0.0022

(0.005601)

0.0006

0.0078

-0.0136

0.0184

0.0030

(0.007992)

-0.0042

0.0125

-0.0335

0.0125

0.0025

Area: , Data-sampling 7.5’ (3488 point values)

(0.004726)

0.0022

0.0070

-0.0078

0.172

0.0025

(0.005455)

0.0007

0.0064

-0.0119

0.0101

0.0036

(0.005601)

0.0014

0.0082

-0.0171

0.0197

0.0027

(0.004726)

0.0024

0.0067

-0.0080

0.0188

0.0022

(0.005601)

0.0006

0.0077

-0.0128

0.0176

0.0030

(0.007992)

-0.0046

0.0123

-0.0330

0.0113

0.0026

Area: , Data-sampling 10’ (1984 point values)***

(0.004726)

0.0022

0.0064

-0.0069

0.0164

0.0026

(0.005455)

0.0009

0.0062

-0.0120

0.0111

0.0037

(0.005601)

0.0013

0.0075

-0.0161

0.0184

0.0028

(0.004726)

0.0023

0.0066

-0.0075

0.0187

0.0023

(0.005601)

0.0005

0.0073

-0.0129

0.0169

0.0031

(0.007992)

-0.0040

0.0111

-0.0306

0.0109

0.0029

Area: , Data-sampling 15’ (894 point values)

(0.004726)

0.0012

0.0066

-0.0078

0.0160

0.0028

(0.005455)

0.0006

0.0057

-0.0111

0.0091

0.0038

(0.005601)

0.0011

0.0075

-0.0144

0.0184

0.0031

(0.004726)

0.0008

0.0058

-0.0081

0.0160

0.0027

(0.005601)

0.0008

0.0065

-0.0119

0.0165

0.0034

(0.007992)

-0.0021

0.0112

-0.0272

0.0134

0.0037

Area: , Data-sampling 20’ (506 point values)

(0.004726)

0.0008

0.0043

-0.0064

0.0090

0.0032

(0.005455)

0.0014

0.0042

-0.0082

0.0080

0.0041

(0.005601)

0.0007

0.0043

-0.0061

0.0118

0.0031

(0.004726)

0.0007

0.0043

-0.0061

0.0118

0.0031

(0.005601)

0.0006

0.0047

-0.0092

0.0118

0.0038

(0.007992)

-0.0015

0.0038

-0.0186

0.0094

0.0046

*** The results for this area and data sampling are the same with the corresponding results for the same area and data-sampling of Table 15.

 

Also these results are similar to the results of corresponding experiments in the regions A and B.

 

Conclusion

 

Extended prediction experiments were carried out using terrestrial gravity anomalies from the Canadian plains, from Australia and from Scandinavian, 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 SGG data.

        The experiments concerning the size requirements showed that in all regions the mean error of estimation was continuously decreaing up to 35% (in average) when the size of the area was continuously increased from  to . This  size is considered satisfactory since according to the mean error of estimation a 70% of the signal of the real data can be recovered by the method used.

        In the same way, the results of the experiments for the determination of the required resolution of the terrestrial gravity data showed that for constant size of the collection area, when the data-sampling was changed from 20 to 5 min, the mean error of estimation dropped up to 50% (in average) of its original value. In this case, the mean error estimation of  correspond to 30% of its formal standard deviation.

        However, the combination of 5 min data-sampling in an area results in a mean error estimation that drops further up to 20% of the formal standard deviation of .

        Finally, using parametric LSC it was shown that the estimation of systematic errors such as bias and tilt in the SGG data is possible using a combination of terrestrial with satellite data.

        It is also possible to determine a scale factor for the total gravity gradients but, of course, the implementation of this possubility requires intricate programming changes if it is to be implemented in the GRAVSOFT program GEOCOL. An off-line estimation of scale-factors, which has been used for CHAMP accelerometer data (Howe et al., 2003) may be a good alternative..

 

Acknowledgments: This is a contribution to the ESA funded GOCE HPF development project.

 

References

 

Arabelos, D. and C.C. Tscherning, 1998, Calibration of satellite gradiometer data aided by ground gravity data. JoG, 72, 617-625.

 

Bouman, J., R. Koop, C.C.Tscherning and P.Visser (2004): Calibration of GOCE SGG Data Using High-Low SST, Terrestrial Gravity data, and Global Gravity Field Models. Journal of Geodesy, Vol. 78, no. 1 -2. DOI 10.1007/s00190-004-383-5.
Howe, E., L.Stenseng and C.C.Tscherning: Analysis of one month of CHAMP state vector and accelerometer data for the recovery of the gravity potential. Advances in Geoscience, (2003), 1, p. 1-4, 2003.
Tscherning, C.C.: Testing frame transformation, gridding and filtering of GOCE gradiometer data by Least-Squares Collocation using simulated data. In print proceedings IAG General Assembly, Sapporo July 2003.
 
Tscherning, C.C., R.Forsberg and P.Knudsen: The GRAVSOFT package for geoid determination. Proc. 1. Continental Workshop on the Geoid in Europe, Prague, May 1992, pp. 327-334, Research Institute 
of Geodesy, Topography and Cartography, Prague, 1992.

 

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