Geology Reference
In-Depth Information
In Figures 8.9.1b and 8.9.1d, we indicated “static” three-dimensional color
plotting. This terminology refers to plots not being mouse-rotatable on screen.
The former is shown in “default” format, while the latter displays contour
plotting results. Additional results are also available in a “dynamic” rotatable
plotting mode, as shown at the left in Figure 8.9.3. This mode supports more
convenient viewing, but importantly, from a software user perspective, permits
direct display of the R
h
-R
v
plane along with bars of constant color voltage
amplitude as shown at the right of Figure 8.9.3.
The foregoing display is important for resistivity analysis, as we will now
discuss. We have replicated our Receiver 1 amplitude results in Figure 8.9.4,
where we have removed all resistivity headings in favor of their logarithms (to
the base 10, since resistivities were assumed to increase ten-fold run-to-run).
Closer examination of Figures 9.9.1b, 8.9.1d and 8.9.3 shows that we, in fact,
have plotted resistivity logarithms on both horizontal axes. The right side of
Figure 8.9.3 shows that lines of constant amplitude (that is, fixed color bands)
appear as straight lines in the rectangular log
10
R
h
- log
10
R
v
plane.
AMPLITUDE (VOLTS)
Frequency, 400,000 Hz; Transmitter-to-Receiver, 15.187 in.
R
v
= 0.1 m
log
10
R
v
= - 1
R
v
= 1 m
Log
10
R
v
= 0
R
v
= 10 m
log
10
R
v
= +1
R
h
= 0.1 m
log
10
R
h
= - 1
0.001225
0.004339
0.008836
R
h
= 1 m
log
10
R
h
= 0
0.0005528
0.001929
0.005180
R
h
= 10 m
log
10
R
h
= +1
0.0001075
0.0005670
0.001972
Figure 8.9.4.
Amplitudes versus logarithmic resistivity.
This foregoing observation is very significant since it allows us to employ
standard “y = mx + b” curve fitting, as suggested in Figure 8.9.5. In this figure,
(near) Receiver 1 results are given at the top, while (far) Receiver 2 results
appear at the bottom. Now, let us assume that 0.005 and 0.002 volts are
measured at near and far receivers, respectively. Parallel straight lines are
drawn within the required color bands. The corresponding slope and vertical
intercepts “m” and “b” are labeled directly on the plots. Then, simple algebra
shows that log
10
R
v
= ( b
2
m
1
-b
1
m
2
)/(m
1
-m
2
) and log
10
R
h
= ( b
2
-b
1
)/(m
1
-m
2
) so that
R
v
= 10^log
10
R
v
and R
h
= 10^log
10
R
h
where the logarithms are measured. The
results shown immediately after Figure 8.9.5 provide one typical estimate. A
short Fortran program gives R
v
, R
h
and R
v
/R
h
as follows -
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