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carry similar amounts of information. Therefore, for a uniform comparison, it is best
to represent
T
and
.Atthe
same time, phases of impedances, arg
Z
, as well as real and imaginary parts of the
tipper, Re
W
and Im
W
, should be represented in a space with logarithmic abscissas
ln
T
and arithmetic ordinates arg
Z
and Re
W
,Im
W
.
Let us begin with the one-dimensional inversion. Consider the Tikhonov-
Cagniard impedance in bilogarithmic representation:
A
,
|
Z
|
in bilogarithmic coordinates ln
T
and ln
A
,ln
|
Z
|
e
i
arg
Z
(ln
T
)
ln
Z
(ln
T
)
=
ln
|
Z
(ln
T
)
|
=
ln
|
Z
(ln
T
)
| +
i
arg
Z
(ln
T
)
.
(10
.
89)
Assume that this function determining a relationship between ln
Z
and ln
T
belongs to the metric function
R
- norm. Thus, we introduce the
R
- norm for
the impedance
Z
(
T
):
2
R
2
L
2
2
L
2
2
L
2
Z
(
T
)
=
ln
Z
(ln
T
)
=
ln
|
Z
(ln
T
)
|
+
arg
Z
(ln
T
)
(10
.
90)
.
Compare impedances
Z
(
T
) and
Z
(
T
) obtained from the field measurements and
model computations. In the
R
- norm, the distance between these impedances, deter-
mining the impedance misfit, is expressed as
Z
(
T
)
2
R
Z
(
T
)
−
=
ln
Z
(ln
T
)
−
|
L
2
+
arg
Z
(ln
T
)
arg
Z
(ln
T
)
2
2
L
2
ln
|
Z
(ln
T
)
−
⎧
⎨
arg
Z
(ln
T
)
2
⎫
(10
.
91)
ln
2
Z
(ln
T
)
T
max
⎬
+
arg
Z
(ln
T
)
dT
T
,
=
−
⎩
⎭
|
Z
(ln
T
)
|
T
min
where
T
min
and
T
max
are minimal and maximal periods bounding the observation
interval. This equation defines the impedance metric in the
R
- norm.
In
practice,
the
impedance
is
measured
over
a
finite
number
of
periods
T
1
,
T
2
,...
T
M
−
1
,
T
M
, where
T
1
=
T
min
,
T
M
=
T
max
. On integrating (10.91) by trape-
zoid rule, we get
ln
+
[arg
Z
(ln
T
1
)
−
arg
Z
(ln
T
1
)]
2
ln
2
Z
(ln
T
1
)
Z
(ln
T
1
)
Z
(
T
)
−
Z
(
T
)
1
2
T
2
T
1
2
R
=
ln
+
[arg
Z
(ln
T
m
)
−
arg
Z
(ln
T
m
)]
2
ln
2
M
−
1
Z
(ln
T
m
)
Z
(ln
T
m
)
1
2
T
m
+
1
T
m
−
1
+
m
=
2
ln
+
[arg
Z
(ln
T
M
)
−
arg
Z
(ln
T
M
)]
2
ln
2
Z
(ln
T
M
)
Z
(ln
T
M
)
1
2
T
M
T
M
−
1
.
(10
+
.
92)
Let the impedances
Z
(
T
) and
Z
(
T
) be obtained on a logarithmic uniform grid
