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Magnetotelluric Tensor Impedance Analysis

Processing:Taking the time series data and determining the MT impedance tensor response estimatesAnalysis:Taking the MT impedance tensor and analysing it for dimensionality and directionality, deriving the optimum responses, checking for internal consistency, and studying and removing static shi

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Magnetotelluric Tensor Impedance Analysis

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    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 Parnaíba

    60. Response derivation in 2-D

    61. Testing internal consistency

    62. Static shift

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