Statement of Inverse Problem - Models and Methods of Magnetotellurics - page 371

We have

ln Z ( x , y , ln T ) = ln | Z ( x , y , ln T ) | e i arg Z ( x , y , ln T ) = ln | Z ( x , y , ln T ) | + i arg Z ( x , y , ln T )

ln Y ( x , y , ln T ) = ln | Y ( x , y , ln T ) | e i arg Y ( x , y , ln T )

= ln | Y ( x , y , ln T ) | + i arg Y ( x , y , ln T )

W ( x , y ,

=

Re W ( x , y ,

+ i Im W ( x , y ,

ln T ) ln T ) ln T )

ln A ( x , y , ln T ) ,

103)
where Z is a component of t he tensor [ Z ] (for instance, Z xy ) or its scalar invari-
ant (for instance, Z eff

(10 .

Z xx Z yy −

Z xy Z yx ), Y is a component of the tensor
[ Y ] (for instance, Y xy ) or its scalar invariant that re duces to the Tikh onov-Cagnard

=

admittance in the 1D-model (for instance, Y eff = Y xx Y yy −

Y xy Y yx ), W is a com-
ponent o f the tippe r W (for instance, W zx ) or its scalar invariant (for instance,
W

W zx +

A is a component of the apparent resistivity (for instance,
xy ) or one of its scalar invariant (for instance,

= W zy ),

2

/ o ).
The impedance, admittance, tipper and apparent-resistivities metrics (misfits) are
defined as

eff = | Z eff |

Z ( x , y , T ) − Z ( x , y , T )

2
R

⎧
⎨
⎩

+ [arg Z ( x , y , ln T ) − arg Z ( x , y , ln T )] 2 ⎫

ln Z ( x , y , ln T )
| Z ( x , y , ln T ) |

2

⎬
⎭

= X Y

T max

dT
T

dydx

T min

(10 . 104)

Y ( x , y , T ) − Y ( x , y , T )

2
R

⎨
⎩

ln T )] 2 ⎬
⎭

ln Y ( x , y , 2

ln T )
| Y ( x , y , ln T ) |

= X Y

T max

dT
T

[arg Y ( x + , y , ln T ) − arg Y ( x , y , dydx

T min

(10 . 105)

T )

2
R

W ( x

, , − , ,

y T ) W ( x y

=
X

T max

ln T )] 2 dT
T

[Re W ( x , y , ln T ) − Re W ( x , y , dydx

(10 . 106)

Y T min

+ X Y

T min

T max

ln T )] 2 dT
T

[Im W ( x , y , ln T ) − Im W ( x , y , dydx