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

2. Tensor component choice Txx TxyTxz Tyz Tyy Tzz  Single components  Combinations  Concatenations Which to use?  Qualitative interpretation  Quantitative interpretation

3. Tensor component choice Quantitative interpretation [Inversions] (T xx, T xy, T xz, T yy, T yz )Li, 2001 (T uv, T xy ), T zz Zhdanov et al., 2004 (T xz, T yz, T zz, T uv )Droujinine et al., 2007 (T uv, T xy )Li, 2010 (T uv, T xy ), T zz, (T zz, T uv, T xy )Martinez & Li, 2011 T zz, (T xz, T yz, T zz), (T xz, T yz, T xz, T yy, T xx )Martinez et al., 2013 Rating the solutions:  goodness of fit  sharp/smooth  close to geology

4. Inversion versus component combinations Martinez et al., 2013 T zz T xz, T yz, T zz T xz, T yz, T xz, T yy, T xx T xz, T yz, T xz, T zz, T yy, T xx Components inverted: RMS error Txx Txy Txz Tyy Tyz Tzz 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

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 parametersm >> nm << n Model3-D volumeSpecified shape quantity SolutionPhysical propertyParameters (density …) (depth, dip…) MethodologyRegularized inversionOverdetermined least – squares SolutionResolution, covarianceParameter errors appraisal

7. Prism model z t b w xc yc 

8. Inverse theory Forward problem: b = f (x) b = data x = parameters (linearized)db = Adx A = Jacobian [ model dependent ] a ij = db i /dx j Inverse problem : dx = A + db A = U  V T singular value decomposition

9. Inverse theory A = U  V T singular value decomposition U = data eigenvectors V = parameter eigenvectors  = singular values R = VV T Resolution matrix (=I) S = UU T Data information matrix C = C d V  -2 V T Covariance matrix

10. Model parameter errors C = C d V  -2 V T Parameter covariance matrix C d = Data covariance  =  singular values small  large C large  small C C d = e 2 I Equal data error C d = DVariable data error

11. Variable component errors Components have different error levels: e.g., e(T xx ) = e(T xz )  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 (T xz 2 +T yz 2 )  combine errors: e(Tuv) = [0.5 (e(T xx ) 2 +e(T yy ) 2 )] 1/2

12. Component quantities tested Single components: T xx T yy T zz T xy T yz T xz T uv Invariants: I1 = T xx T yy +T yy T zz +T xx T zz -T xy 2 -T yz 2 -T xz 2 I2 = T xx (T yy T zz -T yz 2 )+T xy (T yz T xz -T xy T zz )+T xz (T xy T yz -T xz T yy ) H1 = sqrt (T xz 2 +T yz 2 ) H2 = sqrt [T xy 2 +0.25(T yy -T xx ) 2 ] Concatenations: (T uv, T xy ) (T xz, T yz, T zz) (T xy, T yz, T xz) (T xx, T yy, T xy) (T xz, T yz, T xz, T xy, T xx ) (T yy, T yz, T xz, T xy, T xx )

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 

14. Eigenvector matrix V

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

16. Eigenvector matrix V

17. Correlation matrix corr ij = cov ij [ cov ii cov jj ] 1/2

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

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

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

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

22. Parameter errors versus averaging No averaging correction With 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

25. Parameter rankings I1T xz higher error

26. Width error versus coordinate rotation  coordinate axis body axis

27. Information density matrix

28. Information density versus eigenvector