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GOCE data analysis: the space-wise approach and the first space-wise gravity field model

GOCE data analysis: the space-wise approach and the first space-wise gravity field model. F. Migliaccio, M. Reguzzoni, F. Sansò Politecnico di Milano. C.C. Tscherning, M. Veicherts University of Copenhagen. The space-wise approach.

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GOCE data analysis: the space-wise approach and the first space-wise gravity field model

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  1. GOCE data analysis: the space-wise approach and the first space-wise gravity field model F. Migliaccio, M. Reguzzoni, F. Sansò Politecnico di Milano C.C. Tscherning, M. Veicherts University of Copenhagen

  2. The space-wise approach The main idea behind the space-wiseapproach is to estimate the spherical harmonic coefficients of the geo-potential model by exploiting the spatial correlation of the Earth gravitational field. [ E2] [degrees] spatial dependent signalcovariance time dependent noise covariances (spectra) COLLOCATION data model coeffs

  3. 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. spherical harmonics spherical grid with local patches local gridding harmonic analysis data model coeffs space-wise solver

  4. 0 10 -2 10 -4 10 -6 -4 -2 0 10 10 10 10 The space-wise approach 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] local gridding harmonic analysis Wiener filter data model coeffs - + prior model prior model space-wise solver

  5. The space-wise approach 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 Wiener filter and GRF/LORF corrections along track synthesis local gridding harmonic analysis Wiener filter data model coeffs - + + prior model prior model space-wise solver

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

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

  8. 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 termsof 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. Preprocessing philosophy Outliers and data gaps are replaced with values estimated by collocation. The idea is to preserve the stochastic characteristics of the observations time empirical cov. function empirical cov. function mean collocation time

  10. 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. SST solution philosophy energy conservation collocation gridding numerical integration SST model SST data + prior model space-wise solver prior model 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.

  12. Estimated potential along track Absolute differences w.r.t. EGM08 Predicted error standard deviation empirical error rms(w.r.t. EGM08) predicted error rms (from MC) 1.704 m2/s2 1.523 m2/s2 Non-stationary noise covariance is used in the gridding

  13. 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 Remarks: • 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 2 cpr period Error calibration introducing prior information (EGM2008)

  14. Choice of prior model • 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 full signal degree variances (estimated from EGM08) quick-look predicted error degree variances rescaled signal degree variances scale factor = 0.975 predicted residual degree variances if a rescaled quick-look model is used DEGRADED QUICK-LOOK QUICK-LOOK Used for signal covariance modelling in the gridding

  15. Estimated potential on the grid Predicted error [m2/s2] Estimated signal [m2/s2] latitude interval empirical (w.r.t. EGM08) error rms predicted error rms 0.026 m2/s2 0.041 m2/s2 -83° <  < 83° 0.106 m2/s2 0.135 m2/s2 -90° <  < 90°

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

  17. SST+SGG solution philosophy SST SGG Final model Data synthesis along orbit test Energy conservation Harmonic analysis - - FFT Space-wise solver complementary Wiener filter LORF/GRF correction Data gridding FFT -1 Wiener filter + + FFT Space-wise solver Low degree model Data gridding Harmonic analysis

  18. Estimation error along the orbit Error rms of the Wiener filtered observations along the orbit Empirical values from differences w.r.t. EGM08 Error rms of the Wiener filtered observations along the orbit Predicted values from Monte Carlo simulations

  19. Signal covariance modelling residualsignal 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. Estimation error on the grid T predicted error [m2/s2] Trr predicted error [mE] latitude interval empirical error rms predicted error rms latitude interval empirical error rms predicted error rms || < 83° || < 83° 2.64 mE 1.44mE 0.016m2/s2 0.020 m2/s2 || < 90° || < 90° 3.92 mE 1.71 mE 0.048 m2/s2 0.026 m2/s2 empirical error computed w.r.t. EGM08

  21. Error degree variances Estimated space-wise model EGM08 degree variances GOCE vs GRACE Error degree variances w.r.t. EGM08 GOCE vs EGM08 Error degree variances w.r.t. ITG-GRACE GOCE Predicted error degree variances Geoid error [cm] Geoid error [cm] Differences w.r.t. EGM08 (d/o 150)  = 8.4 cm Differences w.r.t. ITG-GRACE (d/o 150)  = 5.1 cm

  22. Estimated space-wise model Predicted error variances of the GOCE space-wise model Log10 scale

  23. Estimated space-wise model Predicted cumulative geoid error [cm] Geoid error [cm] 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 Predicted geoid error Predicted gravity anomaly error 10.86 cm 3.03 mgal for || < 83° and up to d/o 200

  24. 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.

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