Specifications

University of Pretoria etd – Combrinck, M (2006)
4.4.2 Numerical differentiation of TEM data
In the late time, TDEM responses measured over a one-dimensional subsurface can be
described as a power-law function, equation 4.4, with k=2.5 for a half space and k=4 for a
thin conductive layer (S-layer). This creates the possibilities of using standard polynomial
approximations, but in the logarithmic and semi logarithmic domains. Consider the
following power-law decay:
constantdecay k
constant
=
=
−
=
A
k
AttV )4.4()(
If the natural logarithm of this function is taken on both sides, it becomes,
)5.4()ln(ln))(ln( tkAtV −=
Substituting ln(V(t)) with y(x), ln(t) with x and –k with m, we end up with the equation of a
straight line (and a polynomial function) in the logarithmic domain.
If there is a two- or three-dimensional conductor present it will contribute to the late-
time response in the form of an exponential function as shown in equation 4.6.
constantdecayk
constantA
=
=
=
−
)6.4()(
kt
AetV
If the natural logarithm of this function is taken on both sides, it becomes,
)7.4(
ln))(ln(
)ln())(ln(
⎪
⎭
⎪
⎬
⎫
−=
=
−
ktAtV
AetV
kt
Equation 4.7 is again a linear function in the semi-logarithmic domain. This behaviour
only becomes dominant on TDEM sounding data if the conductor is much more
conductive than the surrounding host rock or when the host rock response has been
removed from the data through subtraction or deconvolution. Raw field data (as on which
the S-layer differential transform is normally applied) would mostly exhibit predominantly
power-law decay, although it will be distorted slightly in the presence of conductive layers
or finite conductors. In the logarithmic domain TDEM data appear as functions deviating
from a general linear trend and can be locally approximated by second or third order
polynomials. Finding a relationship between the derivative of a function and the derivative
of the same function in the logarithmic or semi-logarithmic domains would enable us to
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