The Magnetovariational Response Functions - Models and Methods of Magnetotellurics - page 147

S yy sin 2 S yy sin 2 + S xx =

S yy sin cos − S yy sin cos

S xy = S yx =− (4 . 95)

S yy = S yy cos 2

+

S yy cos 2

.

These relations form the equations for S zy , S zy and S yy , S yy .

=

With the solution of the equations system (4.94) is given by

+ +

S zx cos S zy sin S zx cos S zy sin

S zy = S zy =−

, . (4 . 96)

sin( − ) sin( − )

The equations system (4.95) is overdetermined and compatible. Its solution can
be given by

2

{

tr [ S τ ] } tr [ S τ ]

2

det [ S τ ]
sin 2 (

S yy =

+ −

4 − )

(4 . 97)

2

tr [ S τ ]
2

{ tr [ S τ ] } det [ S τ ]

sin 2 (

S yy =

− −

− ) ,

4

where tr [ S τ ] and det [ S τ ] are the trace and determinant of the matrix [ S τ ]:

tr [ S τ ] = S xx + S yy det [ S τ ] = S xx S yy − S xy S yx .

matrices S z , S τ and Using (4.96) and (4.97), we determine the partial

S z , S τ :

S yy sin 2 − S yy sin cos

S z = − S zy sin S zy cos , S τ =

,

− S yy sin cos S yy cos 2

S yy sin 2 − S yy sin cos

S z = − S zy sin S zy cos , S τ =

.

− S yy sin cos S yy cos 2

(4 . 98)

The decomposition of the matrices [ M ] and [ W ] is easily derived from the
decomposition of the matrices [ S τ ] and [ S z ]. By virtue of (4.84), (4.85), (4.86),
and (4.87)

[ S z ][ M ] − 1 [ M ] = [ S τ ] + [ I ] , [ W ] = , (4 . 99)

where [ I ] is the identity matrix: