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Evaluating the utility of gravity gradient tensor components Mark Pilkington

Evaluating the utility of gravity gradient tensor components Mark Pilkington Geological Survey of Canada. Tensor component choice. Txx. Txy. Txz. Which to use?. Single components Combinations Concatenations. Tyz. Tyy. Qualitative interpretation Quantitative interpretation. Tzz.

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Evaluating the utility of gravity gradient tensor components Mark Pilkington

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  1. Evaluating the utility of gravity gradient tensor components Mark Pilkington Geological Survey of Canada

  2. Tensor component choice Txx Txy Txz Which to use? • Single components • Combinations • Concatenations Tyz Tyy • Qualitative interpretation • Quantitative interpretation Tzz

  3. Tensor component choice Quantitative interpretation [Inversions] (Txx, Txy, Txz, Tyy, Tyz) Li, 2001 (Tuv, Txy), Tzz Zhdanov et al., 2004 (Txz, Tyz, Tzz, Tuv) Droujinine et al., 2007 (Tuv, Txy) Li, 2010 (Tuv, Txy), Tzz, (Tzz, Tuv, Txy) Martinez & Li, 2011 Tzz, (Txz, Tyz, Tzz), (Txz, Tyz, Txz, Tyy, Txx) Martinez et al., 2013 • Rating the solutions: • goodness of fit • sharp/smooth • close to geology

  4. Inversion versus component combinations Components inverted: Tzz Txz, Tyz, Tzz Txz, Tyz, Txz, Tyy, Txx Txz, Tyz, Txz, Tzz, Tyy, Txx RMS error TxxTxyTxzTyyTyzTzz 1-C 23.9 23.2 31.8 23.1 26.1 16.5 3-C 17.5 16.0 15.9 16.0 12.4 22.5 5-C 16.6 12.6 16.3 15.8 12.2 24.3 6-C 15.7 13.0 17.9 13.8 13.8 21.4 Martinez et al., 2013

  5. Outline Aim: quantitative rating of component/combinations Approach: inversion using a simple model – estimate parameter errors Method: linear inverse theory – analyse model/data relations

  6. Inversion method used Inversion Parametric [underdetermined inversion problem] n data m parameters m >> n m << n Model 3-D volume Specified shape quantity Solution Physical property Parameters (density …) (depth, dip…) Methodology Regularized inversion Overdetermined least – squares Solution Resolution, covariance Parameter errors appraisal

  7. Prism model xc yc z t r b w

  8. Inverse theory Forward problem: b = f (x) b = data x = parameters (linearized)db = Adx A = Jacobian [model dependent] aij = dbi/dxj Inverse problem : dx = A+db A = ULVTsingular value decomposition

  9. Inverse theory A = ULVTsingular value decomposition U = data eigenvectors V = parameter eigenvectors L = singular values R = VVT Resolution matrix (=I) S = UUTData information matrix C = CdVL-2VTCovariance matrix

  10. Model parameter errors C = CdVL-2VTParameter covariance matrix Cd = Data covariance L =singular values small L large C large L small C Cd = e2I Equal data error Cd = D Variable data error

  11. Variable component errors • Components have different error levels: e.g., e(Txx) = e(Txz) • only relative levels required • estimate based on FFT or equivalent source method • ratio Tzz : Txz, Tyz : Txy : Txx, Tyy = 1 : 0.70 : 0.37 : 0.59 • Component quantities are combined: e.g., H1 = sqrt(Txz2+Tyz2) • combine errors: e(Tuv) = [0.5 (e(Txx)2+e(Tyy)2)]1/2

  12. Component quantities tested Single components: Txx Tyy Tzz Txy Tyz Txz Tuv Invariants: I1 = TxxTyy+TyyTzz+TxxTzz-Txy2-Tyz2-Txz2 I2 = Txx(TyyTzz-Tyz2)+Txy(TyzTxz-TxyTzz)+Txz(TxyTyz-TxzTyy) H1 = sqrt(Txz2+Tyz2) H2 = sqrt[Txy2+0.25(Tyy-Txx)2] Concatenations: (Tuv, Txy) (Txz, Tyz, Tzz) (Txy, Tyz, Txz) (Txx, Tyy, Txy) (Txz, Tyz, Txz, Txy, Txx) (Tyy, Tyz, Txz, Txy, Txx)

  13. Inversion tests • Procedure: • Specify model and evaluate matrix A [db=Adx] • Calculate covariance matrix C • Get parameter standard deviations (p.s.d.) • Rank p.s.d. for each parameter versus component quantity Models tested: xc yc z t w b r

  14. Eigenvector matrix V

  15. Eigenvector matrix V Invariants: I1 = TxxTyy+TyyTzz+TxxTzz-Txy2-Tyz2-Txz2 I2 = Txx(TyyTzz-Tyz2)+Txy(TyzTxz-TxyTzz) +Txz(TxyTyz-TxzTyy) H1 = sqrt(Txz2+Tyz2) H2 = sqrt[Txy2+0.25(Tyy-Txx)2]

  16. Eigenvector matrix V

  17. Correlation matrix corrij = covij [ covii covjj ]1/2

  18. Parameter errors xc,yc = location z = depth t = thickness w = width b = breadth r = density

  19. Parameter errors xc,yc = location z = depth t = thickness w = width b = breadth r = density

  20. Parameter errors xc,yc = location z = depth t = thickness w = width b = breadth r = density

  21. Parameter error ranking [29 models] high error low

  22. Parameter errors versus averaging With averaging correction No averaging correction

  23. Conclusions • Concatenated components produce smallest parameter errors • Invariants I1, I2 best performers in combined component category • Purely horizontal components poor performers • Tzz best single component

  24. Parameter rankings Txz I1 higher error higher error

  25. Width error versus coordinate rotation body axis b coordinate axis

  26. Information density matrix

  27. Information density versus eigenvector

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