Magnetotelluric Tensor Impedance Analysis

1. Magnetotelluric Tensor Impedance Analysis

2. Definitions

3. MT impedance tensor

4. MT impedance tensor

5. Analysis

6. Expected MT impedance tensors

7. 1-D

8. 1-D: Example

9. 2-D

10. 2-D: strike direction

11. 2-D: in strike direction

12. 2-D: Example

13. 3-D

14. 3-D/2-D: Special case

15. 3-D/2-D: Special case

16. 3-D/2-D: Special case

17. 3-D/2-D: Special case

18. 3-D/2-D: Special case

19. 3-D/2-D: In strike direction

20. Rotational properties Let�s examine some rotational properties of the MT impedance tensor:

21. Rotational properties Multiplying out the elements:

22. Rotational invariants Combining some of these, it is possible to show that there exist a number of combinations that are rotationally invariant (are not a function of rotation angle), e.g., the sum of the diagonal elements

23. Rotational invariants Combining some of these, it is possible to show that there exist a number of combinations that are rotationally invariant (are not a function of rotation angle), e.g., the sum of the diagonal elements

24. Rotational invariants Combining some of these, it is possible to show that there exist a number of combinations that are rotationally invariant (are not a function of rotation angle), e.g., the sum of the diagonal elements

25. Rotational invariants Other rotational invariants are: Zxy � Zyx difference of the off-diagonal terms:

26. Examples: 1-D

27. Examples: 1-D

28. Examples: 2-D

29. Examples: 2-D

30. Examples: 3-D

31. Examples: 3-D

32. Examples: 3-D/2-D

33. Examples: 3-D/2-D

34. Examples of polar diagrams - 1

35. Examples of polar diagrams - 2

36. Directionality indicators

37. Dimensionality indicators

38. Problem to solve Determine the dimensionality If 2-D, how to determine the �best� rotation angle??? Amplitude-based methods can be highly erroneous in the presence of distorting features Need to use phase-based methods How to do this within a statistical framework??? Measures of dimensionality and directionality need to be with reference to the errors in the data

39. Groom-Bailey Decomposition-1 The Groom-Bailey method (first presented by Bailey and Groom, 1987) undertakes a tensor decomposition of Z. Nothing new about tensor decomposition. Many had been proposed in the past, e.g., Eggers (1982) � classical eigenstates of Z Spitz (1985) � another eigenstates approach LaTorraca et al. (1986) � SVD analysis Yee and Paulson (1987) � canonical decomposition These are all mathematically-based decompositions

40. Groom-Bailey Decomposition-2 All these prior tensor decomposition methods however result in mixed amplitude and phase-based parameters Bailey and Groom were the first to realize that what is required is a separation of the determinable and undeterminable parts of Z. The determinable parts relate to phase rotational sensitivity, whilst the undeterminable parts relate to the amplitudes.

41. Groom-Bailey Decomposition-3 G-B distortion decomposition uses a factorization of the impedance tensor into matrices that resemble Pauli spin matrices

42. General case In the general case:

43. Site gain - g The site gain, g, merely scales both true impedances by a single real number

44. Twist - T The twist tensor, T, �twists� the E-field in a similar manner to a rotation.

45. Shear - S The shear tensor, S, �shears� the E-field.

46. Anisotropy - A The site anisotropy, A, multiplies one off-diagonal term by the same amount that it reduces the other off-diagonal term

47. Effects of gain and anisotropy Site gain and anisotropy together scale the true 2-D impedances by multiplicative factors. Thus, this affects only the amplitudes, not the phases. There is no way to differentiate in MT between Z2D and a scaled version of it Z�2D, where

48. Replace Z2D by Z�2D We cannot distinguish between Z2D and Z�2D (= gAZ2D) Recast equation in terms of Z�2D

49. Model fitting Equations recast for numerical stability reasons (see GB89 and MJ01). At a single site and a single frequency, fitting a 7 parameter model to 8 data. For multiple frequencies, then solving for the same ?, t and s, but a different Z2D at each frequency. For n freqs, have 8n data and 3 + 4n unknowns

50. Model fitting For multiple sites, then solving for the same ?, but different t and s, and different Z2D at each frequency. For m sites, have 8m data and 1 + 2m + 4m unknowns For n freqs from m sites, then have 8mn data and 1 + 2m + 4mn unknows.

51. Methodology: Step 1 Must build up an understanding of a dataset in a stepwise process. Should never just use strike as a black box and throw everything in. Step 1: Totally unconstrained Undertake a GB analysis using strike setting the bandwidth such that each frequency is in its own band (no averaging). Inspect output and look for frequency stable parameter

52. Far-md2

53. Far-md2

54. Far-md2

55. Far-md2

56. Far-md2

57. Two sites: lit007 & lit008

58. Dados bacia do Parna�ba

60. Response derivation in 2-D

61. Testing internal consistency

62. Static shift