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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 WORKSHOP6-8 November 2006

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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.

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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

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Data The data sets used for the numerical experiments were: EGM96 generated T rr ESA/SGG data In both cases the data points for the experiments were selected using different criteria

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Numerical experiments Experiments with EGM96 generated T rr : 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

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6-8 November 2006 Experiments with EGM96 generated data T rr were computed from EGM96 and statistically homogenized by removing the EGM96 to degree 24. (Further homogenization is necessary). Global covariance function used: where

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6-8 November 2006 Experiments with EGM96 generated data With the following values of the parameters: R=R E -1.561 km, A =540.9 mgal 2, σ l from 2 to 24 from EGM9 with scale factor equal to 1.03 common standard deviation of observations equal to 0.01 E

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Table 1. Statistics of the differences between computed and true EGM96 harmonic coefficients resulted from the radial component T rr (A) with data in the polar caps (10,448 data points) and (B) without (10,428). All values are multiplied by10 9. In the last column the degree standard deviation is shown. (Only values every second degree are shown) EGM96 data on a 2 o equal-area grid DegreeStandard deviation computed-true Mean collocation error estimates Standard deviation computed-true Mean collocation error estimates Degree standard deviation of the EGM96 AB 253.692.793.742.82134.69 263.292.703.322.73124.57 283.252.553.222.57107.47 303.392.433.402.4593.66 322.962.332.952.3682.36 343.362.263.362.2972.98 362.692.202.652.2365.11 382.862.162.862.1958.45 402.652.142.682.1752.76

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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 T rr values on a 2 o ×2 o equiangular grid were used in this experiment. All numbers are multiplied by 10 9. EGM96 generated data on a 2 o x2 o grid

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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 10 9. 41,522 T rr from EGM96 on 1 0 equal-area grid

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Table 2. Statistics of the differences between ESA/SGG (A) and T rr from EGM96 as well as between ESA/SGG and T rr from the computed coefficients to degree 150, using the 41,522 data points (B). The number of observations is 509,222. Unit is E. PredictionsDifferencePredictionsDifference ObservationsAB Mean value-0.0049-0.0048-0.000096-0.004826-0.000093 Standard deviation0.23540.23300.0161870.2371340.015766 Maximum value1.78471.83060.0900121.9179030.107806 Minimum value-1.5811-1.5639-0.134011-1.659430-0.134011 EGM96 data on a 1 o equal-area grid

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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 10 9. (Only values every second degree are shown) Degree10,448 points on the nodes of a 2 o equal-area grid 8,500 points on real GOCE orbit 9,686 points on real GOCE orbit ababab 253.692.7911.227.665.154.71 263.292.7010.667.384.914.54 283.252.5511.537.335.664.32 303.392.4310.457.025.604.14 322.962.338.856.754.884.01 343.362.269.926.505.353. 90 362.692.208.396.324.263.82 382.862.168.026.204.293.76 402.652.148.186.104.083.73 EGM96 data on real GOCE orbit

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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 T rr

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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).

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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: σ l from EGM96 with scale factor = 0.2689 R=R E – 1.061 km Common 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 E 2 and correlation distance equal to 7 degrees.

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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 T rr values selected as closest to the nodes of a 2 o equal-area grid. All numbers are multiplied by 10 9. ESA/SGG data close to the nodes of a 2 o grid

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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 T rr values selected as closest to the nodes of a 2 o ×2 o equiangular grid. All numbers are multiplied by 10 9. ESA/SGG data close to the nodes of a 2x2 grid

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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 T rr values selected using the max|T rr | criterion. Green: The coefficients were computed using 15,340 T rr values selected using the max|T rr | and the var(T rr )> 0.01 E criteria. Blue: The coefficients were computed using 16,601T rr values selected as closest to the nodes of 2 o equal-area grid and the var(T rr )> 0.01 E criterion. All numbers are multiplied by 10 9. ESA/SGG T rr selected using different criteria

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Table 4. Comparison between ESA/SGG and T rr computed from EGM96 (A) as well as between ESA/SGG and T rr computed from the model resulted from the experiment 3 (B) to degree 90. The number of observations is 509,222. Unit is E. ObservationsPredictionsDifferencePredictionsDifference AB Mean value-0.0049-0.004828-0.000092-0.004812-0.000107 Standard deviation0.23560.2319070.0308710.2358430.032015 Maximum value1.78471.9461740.3280242.0038180.308217 Minimum value-1.5811-1.516500-0.348907-0.004812-0.000107

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

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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

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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

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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 T rr values from the noisy ESA/SGG, reduced to EGM96 and to predicted coefficients up to degree 90. Summary and Conclusion

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