Specifications

University of Pretoria etd – Combrinck, M (2006)
The formulas for transforming the derivatives are straightforward, trivial to apply, and
analytically correct (i.e. no truncation error is induced in the transformation process). This
enables us to choose the best domain for performing numerical differentiation based on
the nature of data and available differentiation schemes.
Synthetic TEM data for a 10 ohm.m half space are used to illustrate this point. The
Lagrange 3-point formula was applied to the example data set in the three different
domains and transformed back to the linear domain using the appropriate formulas. The
results were compared to the analytical derivative of the data and the percentage errors are
shown in Figure 4-3. The advantage of differentiating in the logarithmic domain is very
clear. Also indicated on this graph are derivative values obtained from a two-point power-
law formula in the linear domain. These values are exactly equal to the Lagrange 3-point
formula applied in the logarithmic domain. This formula was derived by fitting a power-
law function to two points and taking the derivative of this interpolating function in exactly
the same way as the Lagrange formulas are derived from polynomials.
0
5
10
15
20
25
30
35
0.00001 0.0001 0.001 0.01
Time [s]
Percentage error
Three-point formula in linear domain
Three-point formula in semi-logarithmic domain
Three-point formula in logarithmic domain
Derivative of power-law function fitted in linear domain
Figure 4-3 Percentage errors for differentiation in different domains of TDEM data for a 10
Ohm.m half space.
4.4.2.2 Description of three polynomial based numerical differentiation techniques
The specific numerical differentiation implemented by Tartaras et. al. (2000) is described
only as “… a numerical differentiation scheme that computes a first order derivative for
the provided TDEM data using polynomial interpolation.” This is unfortunately not clear
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