1. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) GOCE data analysis: the space-wise approach and the space-wise approach and the first space-wise gravity field model the first space-wise gravity field model C.C. Tscherning, M. Veicherts University of Copenhagen F. Migliaccio, M. Reguzzoni, F. Sansò Politecnico di Milano

2. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) COLLOCATION The space-wise approach data The main idea behind the space-wise approach is to estimate the spherical harmonic coefficients of the geo-potential model by exploiting the spatial correlation of the Earth gravitational field. model coeffs time dependent noise covariances (spectra) spatial dependent signal covariance [ E 2 ] [degrees]

3. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) The space-wise approach A unique collocation solution is computationally unfeasible due to the huge amount of data downloaded from the GOCE satellite. A two-step collocation solution is implemented. data local gridding local gridding harmonic analysis harmonic analysis model coeffs space-wise solver spherical harmonics spherical grid with local patches

4. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) The space-wise approach Wiener filter Wiener filter data local gridding local gridding harmonic analysis harmonic analysis - prior model model coeffs space-wise solver In order to implement the local gridding: - a prior model is used to reduce the spatial correlation of the signal - a Wiener orbital filter is used to reduce the highly time correlated noise of the gradiometer + [Hz] 10 -6 10 -4 10 -2 10 0 -4 10 -2 10 0

5. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) The space-wise approach Wiener filter Wiener filter data local gridding local gridding harmonic analysis harmonic analysis along track synthesis along track synthesis Wiener filter and GRF/LORF corrections Wiener filter and GRF/LORF corrections + prior model The procedure is iterated to: - recover the signal frequencies cancelled by the Wiener orbital filter - improve the rotation from gradiometer to local orbital reference frame model coeffs space-wise solver - prior model +

6. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) The space-wise approach Wiener filter Wiener filter data local gridding local gridding harmonic analysis harmonic analysis prior model Intermediate results that can be used for local applications: - filtered data (potential and gravity gradients) along the orbit - grid values at mean satellite altitude model coeffs space-wise solver filtered data gridded data + - prior model + along track synthesis along track synthesis Wiener filter and GRF/LORF corrections Wiener filter and GRF/LORF corrections

7. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) The processed data common mode accelerations satellite attitude quaternions gravity gradients reduced dynamic orbits (for geo-locating gravity gradients) kinematic orbits with their error estimates (for low-degree gravity field recovery) from the gradiometer: from the GPS receiver: external information such as Sun and Moon ephemerides or ocean tides for modelling tidal effects Output data spherical harmonic coefficients and their error covariance matrix GOCE quick look as prior model other geopotential models as reference and to compute signal degree variances geopotential models: Input data (from November 2009 to mid January 2010)

8. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) The data processing The data analysis basically consisted of three steps: Data preprocessing: outlier detection, data gap filling, unexpected behaviours tagging, etc. SST solution: to estimate the low degrees of the field (that are then removed from the SGG data) SST+SGG solution: to estimate the final model in terms of spherical harmonic expansion Error estimates are computed by Monte Carlo methods. In particular, few samples are used to control the evolution of the solution, while the final error covariance matrix is based on a larger set of samples.

9. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) Preprocessing philosophy empirical cov. function empirical cov. function mean collocation time Outliers and data gaps are replaced with values estimated by collocation. The idea is to preserve the stochastic characteristics of the observations time

10. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) Preprocessing example An example of data gap filling applied to the difference between kinematic and reduced dynamic orbits. Cubic spline interpolation around the data gap to recover the long period behaviour Collocation interpolation inside the gap to recover the stochastic behaviour of the signal

11. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) space-wise solver SST solution philosophy The energy conservation approach requires to: detect outliers and data gaps in the kinematic orbits; derive velocities from positions by least-squares interpolation; calibrate biases in the common mode accelerations; correct potential estimates from non-gravitational and tidal effects. energy conservation collocation gridding numerical integration SST data SST model + prior model prior model

12. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) Estimated potential along track Absolute differences w.r.t. EGM08 Predicted error standard deviation 1.704 m 2 /s 2 predicted error rms (from MC) 1.523 m 2 /s 2 empirical error rms (w.r.t. EGM08) Non-stationary noise covariance is used in the gridding

13. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) Error calibration of the estimated potential Error spectrum w.r.t. EGM08 Predicted error spectrum If we do not remove spikes we get this error pattern on the grid low frequency zoom some periodical behaviours are not modelled (the highest with 2 cpr period) at very low frequency, the predicted spectrum is lower than the empirical one Error calibration introducing prior information (EGM2008) Remarks: 2 cpr period

14. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) Choice of prior model full signal degree variances (estimated from EGM08) quick-look predicted error degree variances predicted residual degree variances if a rescaled quick-look model is used Used for signal covariance modelling in the gridding rescaled signal degree variances scale factor = 0.975 A degraded version of the GOCE quick-look is used as prior model to reduce the influence on the final solution Predicted (residual) degree variances QUICK-LOOK DEGRADED QUICK-LOOK

15. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) Estimated potential on the grid Estimated signal [m 2 /s 2 ] -83° <  < 83° empirical (w.r.t. EGM08) error rms 0.041 m 2 /s 2 0.106 m 2 /s 2 -90° <  < 90° latitude interval predicted error rms 0.026 m 2 /s 2 0.135 m 2 /s 2 Predicted error [m 2 /s 2 ]

16. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) Estimated SST model EGM08 degree variances ITG-GRACE predicted error degree variances Error degree variances w.r.t. ITG-GRACE SST model predicted error degree variances Above degree 60 the estimated model is the (degraded) quick look model, as corrections are negligible Gravity gradients are needed for further improvements GRACE SST GOCE Error degree variances

17. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) SST+SGG solution philosophy Data gridding FFT complementary Wiener filter Data synthesis along orbit Wiener filter FFT + LORF/GRF correction test Harmonic analysis Space-wise solver Final model SSTSGG + Energy conservation Harmonic analysis Data gridding Space-wise solver -- Low degree model

18. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) Estimation error along the orbit T [m 2 /s 2 ] T XX [mE] T XZ [mE] T YY [mE] T ZZ [mE] 0.0912.44.44.66.0 Error rms of the Wiener filtered observations along the orbit Empirical values from differences w.r.t. EGM08 T [m 2 /s 2 ] T XX [mE] T XZ [mE] T YY [mE] T ZZ [mE] 0.0892.54.24.65.9 Error rms of the Wiener filtered observations along the orbit Predicted values from Monte Carlo simulations

19. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) Signal covariance modelling residual signal variances after removing SST-model zoom variances of degree 30 log10 scale Approximate degree variances are used for collocation Single coefficient variances are used for error modelling by Monte Carlo

20. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) Estimation error on the grid |  | < 83° empirical error rms 0.020 m 2 /s 2 0.048 m 2 /s 2 |  | < 90° latitude interval predicted error rms 0.016 m 2 /s 2 0.026 m 2 /s 2 |  | < 83° empirical error rms 2.64 mE 3.92 mE |  | < 90° latitude interval predicted error rms 1.44 mE 1.71 mE empirical error computed w.r.t. EGM08 T predicted error [m 2 /s 2 ]T rr predicted error [mE]

21. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) Estimated space-wise model EGM08 degree variances Error degree variances w.r.t. EGM08 Error degree variances w.r.t. ITG-GRACE Predicted error degree variances Differences w.r.t. EGM08 (d/o 150)   = 8.4 cm Differences w.r.t. ITG-GRACE (d/o 150)   = 5.1 cm GOCE GOCE vs EGM08 GOCE vs GRACE Error degree variances Geoid error [cm]

22. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) Estimated space-wise model Log10 scale Predicted error variances of the GOCE space-wise model

23. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) Estimated space-wise model Assuming a mission length of 18 months, (9 sets of two months + some refinement) one can expect an improvement of factor 3 in terms of accuracy, with the same spatial error distribution 10.86 cm Predicted gravity anomaly error 3.03 mgal Predicted geoid error for |  | < 83° and up to d/o 200 Geoid error [cm] Predicted cumulative geoid error [cm]

24. ESA Living Planet Symposium, 29 June 2010, Bergen (Norway) Conclusions and future work The analysis of the first two months of GOCE data shows that the space-wise approach is able to provide good results. The main characteristic of the space-wise solution is to be a solution fully computed by collocation, with its pros and cons. Furthermore, intermediate results such as filtered data along track and grid values at satellite level can be used for local applications. At medium-high degrees the solution is driven by GOCE data, while at very low degrees a dependence from prior models can be seen. This dependence will be removed in the next solutions. A new solution will be computed for a longer data period, that implies to optimally combine grids at mean satellite altitude based on different data subsets.