On a strategy for the use of GOCE gradiometer data for the development of a

geopotential model by LSC

D.N. Arabelos C.C.Tscherning

University of Thessaloniki University of Copenhagen

3rd INTERNATIONAL GOCE USER WORKSHOP 6-8 November 2006

Aim and method (I)

Use of Least-Squares Collocation (LSC) requires as many equations as observations to be solved.

Thus not all GOCE data can be used: Use gridded data with uncorrelated noise.

But:

LSC gives the possibility for using data at position of satellite and with correlated noise modelled.

Aim and method (II)

Find a strategy for the collection of point data on the real GOCE orbit, leading to optimal LSC determination of spherical harmonic coefficients, with the minimum number of data points

The strategy which will be suggested was based on:

•The comparison between true and computed coefficients using different data distributions

•The collocation error estimates of the coefficients

•The comparison of the original data with data generated from the computed coefficients

Data

The data sets used for the numerical experiments were:

•EGM96 generated Trr

•ESA/SGG data

In both cases the data points for the experiments were selected using different criteria

Numerical experiments

Experiments with EGM96 generated Trr:

On the nodes of an equal-area grid

On the nodes of an equiangular grid

On the positions of real GOCE orbit and selected closest to the nodes of an equal-area grid, using different collection radius

6-8 November 2006

Experiments with EGM96 generated data

Trr were computed from EGM96 and statistically homogenized by removing the EGM96 to degree 24. (Further homogenization is necessary).

Global covariance function used:

∑2=

2+2 )()( )(=)(k

ll

lQPlQP tPrr/RgΔ,gΔσgΔ,gΔcov

∑∞

1+=

2+2 )())((+kl

ll

QPl tPrr/RgΔ,gΔc

))(2(

)1(

Bll

lAcl

where

6-8 November 2006

Experiments with EGM96 generated data

With the following values of the parameters:

R=RE -1.561 km,

A =540.9 mgal2,

σl from 2 to 24 from EGM9 with scale factor equal to 1.03

common standard deviation of observations equal to

0.01 E

Table 1. Statistics of the differences between computed and true EGM96 harmonic coefficients resulted from the radial component Trr (A) with data in the polar caps (10,448 data points) and (B) without (10,428). All values are multiplied by109. In the last column the degree standard deviation is shown. (Only values every second degree are shown)

EGM96 data on a 2o equal-area grid

Degree

Standard deviation

computed-true

Mean collocation

error estimates

Standard deviation

computed-true

Mean collocation

error estimates

Degree standard

deviation of the EGM96

A B

25 3.69 2.79 3.74 2.82 134.69

26 3.29 2.70 3.32 2.73 124.57

28 3.25 2.55 3.22 2.57 107.47

30 3.39 2.43 3.40 2.45 93.66

32 2.96 2.33 2.95 2.36 82.36

34 3.36 2.26 3.36 2.29 72.98

36 2.69 2.20 2.65 2.23 65.11

38 2.86 2.16 2.86 2.19 58.45

40 2.65 2.14 2.68 2.17 52.76

Fig. 1 Standard deviation per degree of the differences between computed and true coefficients from degree 25 to 90 (blue line) and mean collocation error estimates for the same degrees (red). 15,300 Trr values on a 2o×2o equiangular grid were used in this experiment. All numbers are multiplied by 109.

EGM96 generated data on a 2ox2o grid

Fig. 2 Standard deviation per degree of the differences between computed and true coefficients from degree 25 to 75 (blue line) and mean collocation error estimate for the same degrees (red). The standard deviation of the differences computed-true at degree 150 is 3.369. All numbers are multiplied by 109.

41,522 Trr from EGM96 on 10 equal-area grid

Table 2. Statistics of the differences between ESA/SGG (A) and Trr from EGM96 aswell as between ESA/SGG and Trr from the computed coefficients to degree 150,using the 41,522 data points (B). The number of observations is 509,222. Unit is E.

Predictions

Difference

Predictions

Difference

Observations

A B

Mean value -0.0049 -0.0048 -0.000096

-0.004826

-0.000093

Standard deviation

0.2354 0.2330 0.016187 0.237134 0.015766

Maximum value

1.7847 1.8306 0.090012 1.917903 0.107806

Minimum value -1.5811 -1.5639 -0.134011

-1.659430

-0.134011

EGM96 data on a 1o equal-area grid

Table 3. Standard deviation of the differences computed – true (a) and the corresponding mean collocation error (b) of the harmonic coefficients determined using the data sets of 8,500 (collection radius = 30 km) and 9,686 point values (collection radius=50 km) with GOCE distribution (columns 4 – 7). In columns 2 and 3 the corresponding results for the complete grid of Table 1 are shown. All numbers are multiplied by 109. (Only values every second degree are shown)

Degree

10,448 points on the nodes of a 2o

equal-area grid

8,500 pointson real GOCE orbit

9,686 pointson real GOCE

orbit

a b a b a b

25 3.69 2.79 11.22 7.66 5.15 4.71

26 3.29 2.70 10.66 7.38 4.91 4.54

28 3.25 2.55 11.53 7.33 5.66 4.32

30 3.39 2.43 10.45 7.02 5.60 4.14

32 2.96 2.33 8.85 6.75 4.88 4.01

34 3.36 2.26 9.92 6.50 5.35 3. 90

36 2.69 2.20 8.39 6.32 4.26 3.82

38 2.86 2.16 8.02 6.20 4.29 3.76

40 2.65 2.14 8.18 6.10 4.08 3.73

EGM96 data on real GOCE orbit

•The quality of the computed coefficients is better when the data are distributed on a grid.

•Between equal-area or equiangular grids the first is beneficial because using considerably less number of data better quality of the computed coefficients was achieved.

•In the case of the equal-area grid, the transition from the grid nodes to the actual positions of the data on a real GOCE orbit results generally in a 50% decrease of the quality of the coefficients.

•The standard deviation of the differences between computed and true coefficients remains very close to their mean errors estimates.

Experience gained with EGM96 generated Trr

Experiments with ESA/SGG data

Realistic 5 second end-to-end simulated data with colored

noise was made available by the POLIMI GOCE-HPF group.

The noise based on the a-priori Fourier spectrum

characteristics of GOCE, primarily added by ESA, was

(partly) removed using Wiener orbital filtering (Reguzzoni

and Tselfes, 2006).

Experiments with ESA/SGG data

The data were reduced to EGM96 to degree 50 and the contribution above degree 150 was removed

Global covariance function was used

with the following parameters: σl from EGM96 with scale factor = 0.2689R=RE – 1.061 kmCommon standard deviation of the observations = 0.0134 E.

The correlated errors affecting the data were represented by a finite error covariance function (Sansò & Schuh, 1987) wit noise variance equal to (0.0134 E)2 = 0.00018 E2 and correlation distance equal to 7 degrees.

∑150

51=

2+2 )()( )(=)(l

ll

QPlQP tPrr/RgΔ,gΔσgΔ,gΔcov

Fig. 3. Standard deviation per degree of the differences between computed and true coefficients from degree 51 to 90 (blue) and mean collocation error for the corresponding degrees (red). The coefficients were computed using 9,696 Trr

values selected as closest to the nodes of a 2o equal-area grid. All numbers are multiplied by 109.

ESA/SGG data close to the nodes of a 2o grid

Fig. 4. Standard deviation per degree of the differences between computed and true coefficients from degree 51 to 90 (blue) and mean collocation error estimation for the corresponding degrees (red). The coefficients were computed using 14,131 Trr values selected as closest to the nodes of a 2o×2o equiangular grid. All numbers are multiplied by 109.

ESA/SGG data close to the nodes of a 2x2 grid

Fig. 5. Standard deviation per degree of differences between computed and true coefficients from degree 51 to 90. Red: The coefficients were computed using 10,218 Trr values selected using the max|Trr| criterion. Green: The coefficients were computed using 15,340 Trr values selected using the max|Trr| and the var(Trr)> 0.01 E criteria. Blue: The coefficients were computed using 16,601Trr values selected as closest to the nodes of 2o equal-area grid and the var(Trr)> 0.01 E criterion. All numbers are multiplied by 109.

ESA/SGG Trr selected using different criteria

Table 4. Comparison between ESA/SGG and Trr computed from EGM96 (A) as well as between ESA/SGG and Trr computed from the model resulted from the experiment 3 (B) to degree 90. The number of observations is 509,222. Unit is E.

Observations

Predictions

Difference

Predictions

Difference

A B

Mean value -0.0049 -0.004828 -0.00009

2

-0.00481

2

-0.000107

Standard deviation

0.2356 0.231907 0.030871

0.235843

0.032015

Maximum value 1.7847 1.946174 0.328024

2.003818

0.308217

Minimum value -1.5811 -1.516500 -0.34890

7

-0.00481

2

-0.000107

Distribution of the differences between Trr from ESA/SGG and Trr computed from EGM96 (upper) and Trr computed from the coefficients determined from the experiment 3 (lower)

Numerical experiments for the determination of a geopotential model by LSC using simulated GOCE data indicate that a distribution of the data based on an equal-area grid is superior compared to a distribution based on an equiangular grid, because using considerably fewer data points it results in better quality of the coefficients.

Since a global covariance function is to be used, a homogenization of the data is necessary.

Summary and Conclusion

In order to take advantage of the possibility of LSC to use the data in their original positions on the real GOCE orbit, it is essential to have a distribution providing a homogeneous quality of the computed coefficients.

The numerical experiments indicate that for the lower degrees an evenly distribution of the data closest to the nodes of the equal-area grid is necessary, while for the higher degrees the amount of the data in blocks with rough gravity field is critical.

Summary and Conclusion

This issue does not support selection of data points with other criteria, such as to have the maximum absolute value within the equal-area block, because the distribution of such points could be not even.

The excellent agreement between EGM96 and the coefficients computed according to the suggested strategy was reflected also in the agreement of the residuals of Trr values from the noisy ESA/SGG, reduced to EGM96 and to predicted coefficients up to degree 90.

Summary and Conclusion