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ln
arg
Y
(ln
T
1
)]
2
2
Y
(ln
T
1
)
Y
(ln
T
1
)
Y
(
T
)
1
2
ln
2
R
Y
(
T
)
[arg
Y
(ln
T
1
)
−
=
+
−
ln
arg
Y
(ln
T
m
)]
2
2
M
−
1
Y
(ln
T
m
)
Y
(ln
T
m
)
[arg
Y
(ln
T
m
)
+
ln
+
−
m
=
2
ln
arg
Y
(ln
T
M
)]
2
2
Y
(ln
T
M
)
Y
(ln
T
M
)
1
2
ln
[arg
Y
(ln
T
M
)
+
+
−
.
(10
.
98)
Note that in the function space
R
the impedance and admittance metrics coincide:
Z
(
T
)
=
Y
(
T
)
2
R
2
R
Z
(
T
)
Y
(
T
)
−
−
.
(10
.
99)
Let us supplement the impedance and impedance metrics with the apparent-
resistivity metrics. By analogy with (10.91) and (10.92), we get
ln
2
dT
T
T
max
A
(ln
T
)
˜
2
R
A
(
T
)
˜
−
A
(
T
)
=
A
(ln
T
)
T
min
ln
2
ln
2
ln
2
M
−
1
1
2
A
(ln
T
1
)
A
(ln
T
1
)
˜
T
2
T
1
+
˜
A
(ln
T
m
)
A
(ln
T
m
)
T
m
+
1
T
m
−
1
+
1
2
A
(ln
T
M
)
A
(ln
T
M
)
˜
T
M
T
M
−
1
+
ln
ln
ln
,
m
=
2
.
(10
100)
where ˜
A
are apparent resistivities obtained from the field measurements and
model computations and
T
1
A
and
T
max
. On a logarithmic uniform grid
(10.93), the apparent-resistivity misfit takes the form
=
T
min
,
T
M
=
1
2
2
ln
2
ln
2
ln
M
−
1
A
(ln
T
1
)
A
(ln
T
1
)
˜
A
(ln
T
m
)
A
(ln
T
m
)
˜
1
2
˜
A
(ln
T
M
)
A
(ln
T
M
)
2
R
˜
A
(
T
)
−
A
(
T
)
=
ln
+
+
.
m
=
2
(10
.
101)
When ˜
A
comes close to
A
,wehave
ln
2
ln(1
)
2
˜
2
˜
˜
−
−
A
A
A
A
A
=
+
≈
.
(10
.
102)
A
A
A
Using such an approximation, we consider the apparent-resistivity misfit as a
quadratic sum of partial relative misfits with a factor ln
. The similar approximation
is valid for logarithmic misfit of the impedance or admittance.
Going over to the general case of the three-dimensional inversion, we introduce
coordinates
x
y
of the observation sites and define the impedance and admittance
tensors, [
Z
] and [
Y
], the tipper,
W
, and the apparent resistivity,
,
A
, in the function
space
R
.
