Outline of the talk

1. Approximate decorrelation and non-isotropic smoothing of time-variable GRACE gravity field modelsJürgen Kusche, Roland Schmidtwith input fromSusanna Werth, Roelof RietbroekGFZ PotsdamIUGG 2007, Perugia, GS002

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. “Stripes” in GRACE solutions

4. Stripes in GRACE solutions NS-oriented artefacts gravity field determination = ill-posed problem • Stochastic (noise) and deterministic (background model) errors cause unphysical oscillations Surface Mass 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

5. Decorrelation, “de-striping”

6. Filter Methods for GRACE-L2 Products • 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

7. This work • 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”

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

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

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

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

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

13. Scaling (bias) problem

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

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

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

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

18. Results

19. Gaussian Filter Geoid Surface Mass wrms=3.85cm 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

20. Decorrelation – Swenson and Wahr 2006 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) Surface Mass wrms=3.76cm

21. Decorrelation – This Work Geoid Surface Mass wrms=3.83cm 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

22. Decorrelation – This Work Surface Mass - WGHM Surface Mass – GRACE wrms=2.30cm wrms=3.85cm 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

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

24. Conclusions • 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

25. Thank you

26. Decorrelation – This Work Order/parity only Full W-matrix 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. 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. Decorrelation – This Work

29. Decorrelation – This Work

30. Decorrelation – This Work

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