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are the major and minor semi-axes of the ellipse. Here
AB
=
ad
bc
=
det [Re S τ ( r
|
r B )]
.
(4
.
109)
The slope of the major axis is
2( ac + bd )
tan
=
+ ( a 2
a 2
+
b 2
c 2
d 2
+
b 2
+
c 2
+
d 2 ) 2
4( ad
bc ) 2
( a 2
a 2
+
b 2
c 2
d 2
+
b 2
+
c 2
+
d 2 ) 2
4( ad
bc ) 2
(4
.
110)
=
2( ac + bd )
+ 2( ac + bd ) ( a 2
a 2
+ b 2
c 2
d 2
+ b 2
+ c 2
+ d 2 ) 2
4( ad bc ) 2
+ ( a 2
4( ad bc ) 2 .
=
a 2
+ b 2
c 2
d 2
2( ac + bd )
+ b 2
+ c 2
+ d 2 ) 2
Similarly we construct the perturbation ellipse for the Schmucker tensor [Im S τ ].
Substituting
a
=
Im S xx ( r
|
r B )
,
b
=
Im S xy ( r
|
r B )
,
c
=
Im S yx ( r
|
r B )
,
d
=
Im S yy ( r
|
r B )
into (4.108) and (4.110), we get the semi-axes and the slope of the major axis of the
imaginary perturbation ellipse associated with the imaginary perturbation vectors
Im p and Im q .
The same technique can be used for constructing the real and imaginary ellipses
of the horizontal magnetic tensor [ M ].
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