GATE-2018 (24) :Wyllie – Time average equation

101) An MT Survey was conducted over a fault. If one of the axis of measurement coincides with the fault extension, then the impedance elements Z_{xx, Zyy}, Z_xy and Z_yx are 

(Thanks to Chandrasekhar, ANU)

 Solution:

As we all know that the impedance can be defines as the ratio of electric field intensity to the magnetic field intensity.

 

                                                      $Z=\frac{E}{H}$

 

 In the presence of the 2-D or 3-D structures the impedance is not invariant but it depends on the distance of the point of measurement from the structure and the angle between the strike and the co-ordinate axis.

 

After getting  the four tensor components from the processing the data they align the components parallel and perpendicular to the structure. This procedure is known as ‘”location of Principle axes” and the relations are as follows

 

$Z_{xx}=\frac{1}{2}[Z_{xy}^{'}+Z_{yx}^{'}]\sin2\theta$

 

$Z_{yy}=-\frac{1}{2}[Z_{xy}^{'}+Z_{yx}^{'}]\sin2\theta$

 

$Z_{xy}=Z_{xy}^{'}-[Z_{xy}^{'}+Z_{yx}^{'}]\sin^{2}\theta$

 

$Z_{yx}=Z_{yx}^{'}-[Z_{xy}^{'}+Z_{yx}^{'}]\sin^{2}\theta$

 

 

 Where $\theta$ is the clockwise rotation angle between the survey and the principle axes.

 

·      During the rotation of 180⁰ the off diagonal elements $Z_{xy}$ and $Z_{yx}$ have two maximas(0⁰, 180⁰) and one minimum(90⁰).

 

·      Where as the diagonal elements $Z_{xx}$ has two minima(-45⁰,135⁰) with a maximum at 45⁰ where as $Z_{yy}$ is the exact convers

 

 

·      If the ground is isotropic $Z_{xx}=Z_{yy}=0$  and $Z_{xy} = -Z_{yx}$

 

·      If the ground is anisotropic $Z_{xx}=-Z_{yy}$  and $Z_{xy}≠Z_{yx}$

 

 

·      If the axes of measurements are parallel and normal to strike then the condition is

$Z_{xx}=Z_{yy}=0$ 

 

 

In the given problem the axes of measurements are parallel then the relation between the impedance tensor is as follows

     $Z_{xx}=Z_{yy}=0$    and $Z_{xy}=-Z_{yx}$

 

 Reference: Applied Geophysics by W.M.Telford, L.P.Geldart and R.E.Sheriff.