Geology Reference
In-Depth Information
quence. A useful additional parameter is the resistivity
transform
T
(
l
) defined by
I
T
(
lr l
)
=
K
(
)
(8.18)
i
i
i
θ
r
where
T
i
(
l
) is the resistivity transform of the
i
th layer
which has a resistivity
r
i
and a kernel function
K
i
(
l
).
T
(
l
) can similarly be constructed using recurrence
relationships.
By methods analogous to those used to construct
equation (8.16), a relationship between the apparent
resistivity and resistivity transform can be derived. For
example, this relationship for a Wenner spread with elec-
trode spacing
a
is
z
(
r
,
θ
,
z
)
Fig. 8.13
Cylindrical polar coordinates.
•
Ú
=
2
aT
J a J
)
-
2
a
d
r
(
l
)
[
(
l
(
l
)
]
l
(8.19)
The curve-matching methods have been almost
completely superseded by more sophisticated interpre-
tational techniques described below. Curve-matching
methods might still be used, however, to obtain interpre-
tations in the field in the absence of computing facilities,
or to derive an approximate model that is to be used as a
starting point for one of the more complex routines.
Equation (8.13) represents the potential at the surface
resulting from a single point of current injection over
two horizontal layers as predicted by the method of im-
ages. In general, however, the potential arising from any
number of horizontal layers is derived by solution of
Laplace's equation (see Section 6.11). The equation in
this case is normally represented in cylindrical coordi-
nates as electrical fields have cylindrical symmetry with
respect to the vertical line through the current source
(Fig. 8.13). The solution and application of the relevant
boundary conditions are complex (e.g. Koefoed 1979),
but show that the potential
V
at the surface over a series
of horizontal layers, the uppermost of which has a resis-
tivity
r
1
, at a distance
r
from a current source of strength
I
is given by
a
0
0
The resistivity transform function has the dimensions
of resistivity and the variable
l
has the dimensions of in-
verse length. It has been found that if
T
(
l
) is plotted as a
function of
l
-1
the relationship is similar to the variation
of apparent resistivity with electrode spacing for the
same sequence of horizontal layers. Indeed only a simple
filtering operation is required to transform the
T
(
l
) vs.
l
-1
relationship (resistivity transform) into the
r
a
vs.
a
relationship (apparent resistivity function). Such a filter
is known as an indirect filter.The inverse operation, that
is, the determination of the resistivity transform from the
apparent resistivity function, can be performed using a
direct filter.
Apparent resistivity curves over multilayered models
can be computed relatively easily by determining the re-
sistivity transform from the layer parameters using a re-
currence relationship and then filtering the transform
to derive the apparent resistivity function. Such a tech-
nique is considerably more efficient than the method
used in the derivation of equation (8.13).
This method leads to a form of interpretation similar
to the indirect interpretation of gravity and magnetic
anomalies, in which field data are compared with data
calculated for a model whose parameters are varied in
order to simulate the field observations.This comparison
can be made between either observed and calculated
apparent resistivity profiles or the equivalent resistivity
transforms, the latter method requiring the derivation of
the resistivity transform from the field resistivity data by
direct filtering. Such techniques lend themselves well to
automatic iterative processes of interpretation in which
a computer performs the adjustments necessary to a
I
r
•
1
Ú
V
=
KJ
r
d
(8.17)
(
lll
)
(
)
0
2
p
0
l
is the variable of integration.
J
0
(
l
r
) is a specialized
function known as a Bessel function of order zero whose
behaviour is known completely.
K
(
l
) is known as a ker-
nel function and is controlled by the thicknesses and re-
sistivities of the underlying layers. The kernel function
can be built up relatively simply for any number of layers
using
recurrence relationships
(Koefoed 1979) which pro-
gressively add the effects of successive layers in the se-
