1. 1 Approximate decorrelation and non-isotropic smoothing of time-variable GRACE gravity field models Jürgen Kusche, Roland Schmidt with input from Susanna Werth, Roelof Rietbroek GFZ Potsdam IUGG 2007, Perugia, GS002

2. 2 Outline of the talk GRACE fields exhibit artefacts (“stripes”) which may be seen as a realization of spatially correlated noise - smoothing and/or “de-striping” is required Theory: Discussion of ways to decorrelate (“de-stripe”) the noise in GRACE solutions (including method from Swenson-Wahr 2006 (SW06) and a new method) Theory: The scaling (bias) problem Results: De-striped GFZ GRACE RL4 fields, surface mass grids, and a time series of basin-averaged GRACE- derived OBP (  talk in JGS001) Conclusions

3. 3 “Stripes” in GRACE solutions

4. 4 Stripes in GRACE solutions NS-oriented artefacts gravity field determination = ill-posed problem  Stochastic (noise) and deterministic (background model) errors cause unphysical oscillations RMS variability of 40 GFZ RL04 monthly solutions in 2/03-12/06 relative to their mean (7-10/04 and 12/06 excluded), Gaussian 550km Surface Mass

5. 5 Decorrelation, “de-striping”

6. 6 degree-dependent (isotropic) Gauss (Jekeli 1981, Wahr & al 1998), Gauss-Weierstrass (Freeden 1998), Hanning (Jekeli 1981), Blackman (Schmidt & al 2006), CuP (Fengler & al 2006) degree- and order-dependent modified Gauss (Han 2005) removing single coefficients based on hypothesis testing (Sasgen & al 2005) full non-isotropic (general two-point kernel) constrained fields (Tikhonov) empirical signal decorrelation combined with Gaussian (Swenson & Wahr 2006) empirical error decorrelation and Tikhonov smoothing (Kusche 2007) Issues de-striping property amplitude damping (bias) and phase lags interpretability optimality criteria, multiresolution properties Filter Methods for GRACE-L2 Products

7. 7 combine approximate error decorrelation and Tikhonov smoothing (Kusche 2007) scaled dense synthetic, “smooth” normal matrix for 1 month synthetic, smooth signal variance model from Hydrology + Ocean circulation damping “on normal equation level” This work

8. 8 Construction of E and S GRACE orbits (coverage) Hydrology Model + Ocean Circulation Model

9. 9 LAT=60 o Cross-sections N-S direction (o) W-E direction (*) Impulse response Filter Properties This work LAT=0 o Distance from kernel center

10. 10 This work Swenson and Wahr (2006) black circle = Gaussian 500km Impulse response Filter Properties

11. 11 can be approximated as block-diagonal Decorrelation/Smooth. Filter W for L = 70

12. 12 C-Block (m+1) odd/even degrees Asymmetric order/parity weighting degree C-Block (m)

13. 13 Scaling (bias) problem

14. 14 ratio  between filtered and exact basin average depends on filter shape of basin signal within and outside basin All smoothed GRACE-based functionals, global maps or basin averages, are systematically biased low Scaling (bias) problem damping  of the global rms

15. 15 Relative bias from true and filtered signal, including hydrology apparent phase lag Scaling (bias) problem

16. 16 Relative bias from true and filtered signal, hydrology removed 400km: 56% Scaling (bias) problem year

17. 17 Comparison of filters based upon variance and standard scaling bias Comparison Gaussian – This Work

18. 18 Results

19. 19 Gaussian Filter RMS variability of 40 GFZ RL04 monthly solutions in 2/03-12/06 relative to their mean (7-10/04 and 12/06 excluded)  Further “de-striping” reduced amplitude (biased towards zero) Left: Gaussian 500km, Right: Gaussian 550km wrms=3.85cm Surface Mass Geoid

20. 20 Empirical signal decorrelation according to Swenson and Wahr (2006) Filter > l=10, Gaussian 400km RMS variability of 40 GFZ RL04 monthly Solutions in 2/03-12/06 relative to their Mean (7-10/04 and 12/06 excluded) wrms=3.76cm Decorrelation – Swenson and Wahr 2006 Surface Mass

21. 21 RMS variability of 40 GFZ RL04 monthly solutions in 2/03-12/06 relative to their mean (7-10/04 and 12/06 excluded) Left and right: approx. decorrelated using 8/03 orbits and LaD+ECCO for W-matrix (up to deg/ord = 70), a = 10E+14 wrms=3.83cm Decorrelation – This Work Surface Mass Geoid

22. 22 BOTH are decorrelated/smoothed using the SAME operator, i.e. 8/03 orbits and LaD+ECCO for W-matrix (up to deg/ord = 70), a = 10E+14  Approx. (GRACE) decorrelation does not distort hydrology model wrms=3.85cm wrms=2.30cm Decorrelation – This Work Surface Mass – GRACE Surface Mass - WGHM

23. 23  DFG-Mass Transport project STREMP  See talk by L. Fenoglio et al in JSG001 Regional Averaging GRACE “raw” time series of mass change over the Mediterranean by different methods

24. 24 Stripes in GRACE solutions still visible; although RL04 improvement over earlier releases Best strategy: remove during processing (but perfect de-aliasing impossible) Second-best strategy: post-processing using error correlation model (here: from an arbitrary GRACE- or GRACE-type orbit + a-priori model information) Proposed technique removed stripes much more effectively compared to Gaussian; simultaneously smoothing (“amplitude bias”) is comparable to Gaussian Use for mass transport studies (hydrology, ocean); higher resolution at comparable damping Conclusions

25. 25 Thank you

26. 26 Decorrelation – This Work Full W-matrix Order/parity only RMS variability of 40 GFZ RL04 monthly solutions in 2/03-12/06 relative to their mean (7-10/04 and 12/06 excluded) Approximately decorrelated using 8/03 orbits and LaD+ECCO for W-matrix (up to deg/ord = 70), a = 10E+14

27. 27 Decorrelation – This Work W-matrix based on synthetic normals from orbit 8/03 W-matrix based on covariance matrix for 8/03 GFZ-RL04 RMS variability of 40 GFZ RL04 monthly solutions in 2/03-12/06 relative to their mean (7-10/04 and 12/06 excluded) Apriori model information for W-matrix (70,70) from LaD+ECCO, a = 10E+14

28. 28 Decorrelation – This Work

29. 29 FilteringRegularization eq. interpretationestimate of x reduces variance yes introduces bias yes (“scaling factor”) yes alternative interpretation unbiased estimate of averaged x no required processing L2 dataL1 data (but…) Decorrelation – This Work

30. 30 FilteringRegularization parameter tuning S/N geophysical signals /GRACE (Wiener/”optimal”) S/N geophysical signals/ GRACE (LSC) or data-driven (GCV,VCE) latitude- dependence no (but…)automatically anisotropic decorrelation no (but…)automatically geometric interpretation yesno Decorrelation – This Work

31. 31 Spherical disc signal + Gaussian (can be analytically treated) Disc radius [km]  Gaussian smoothing radius [km] amplitude scaling error (relative bias) Mediterranean Amplitude Scaling Error - Gaussian