Specifications

University of Pretoria etd – Combrinck, M (2006)
data to equal intervals is that the weights (and inverse matrices) used for differentiation and
smoothing only has to be calculated once. Applying the transform to unequally spaced
data points means less tampering with the original data but it is much more time intensive.
The method use for smoothing is the three-point moving average filter for equally spaced
points and a weighted extension for unequally spaced points (equation 4.22).
(
)
()
(
)
()
(4.22))
3
(
13
12
2
)
2
()
1
(
13
23
2
)
2
( xf
xx
xx
xfxf
xx
xx
AverageMovingWeighted
xf ⋅
−
−⋅
++⋅
−
−
⋅
=
4.4.3 Comparison of differentiation methods applied to synthetic data
4.4.3.1 First order numerical differentiations compared to the analytical derivative of a function
A TDEM decay curve is simulated by an analytical function (equation 4.23), defined as the
sum of exponential and power law terms. (There are no analytical formulations of
complete layered earth TEM responses in closed form, therefore an “equivalent” analytical
function is used.)
(4.23)
t-t-t-
--/-
eee
t.t.t.f(x)
40080002500
3425
10120002000
0000000101000000000000000010
⋅+⋅+⋅+
⋅+⋅+⋅=
The first order analytical derivative of this function is compared to the following numerical
scenarios:
• All three numerical differentiation algorithms are applied to unequally spaced data
points and without smoothing of the data. The results are shown in figure 4.5(a).
• The same as (1) but with smoothing of the function before the derivatives were
calculated. The results are shown in figure 4.5(b).
• The same as (1), but with the unequally spaced data points re-sampled to equally
spaced data points using the cubic spline interpolation method. Figure 4.5(c)
shows the results without smoothing of the data, as in Figure 4.5(a).
• The results in Figure 4.5(d) is the equally spaced data points equivalent of 4.5(b).
• A summary of the average errors (absolute values) for every alternative is given in
Figure 4.6.
From these comparisons the following are concluded:
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