Specifications

University of Pretoria etd – Combrinck, M (2006)
(4.20)
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(4.21)
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The matrix
A is square and symmetrical with the highest values on the diagonal as pivot
elements except for the first and last rows. The inverse of this matrix,
A
-1
, depends on the
number of data points, n, and can be calculated using any standard algorithm. In the
general case where data points are unequally spaced the weights (a
ii
) are different for every
data set and both the matrix and its inverse have to be recalculated for every sounding.
The derivation of Simpson’s rule for unequally spaced points is given in APPENDIX A.
4.4.2.3 Smoothing and re-sampling of data points
Before testing of the three integration methods discussed above, one need to consider the
process and effect of smoothing of the data at various stages of the differentiation process.
Smoothing of data before and after numerical differentiation is an accepted method of
reducing the noise in data (Tartaras et. al., 2000). However, any smoothing applied to the
data, will change the data and will influence the final interpretation. In the S-layer
differential transform algorithm there are four possible opportunities to smooth data and
the question arises which of these will optimise the algorithm with minimum alteration of
the data.
The last factor taken into consideration is the spacing of data points. The time channels at
which data are measured are almost equally spaced in the logarithmic domain. The
calculated depths (to which the second differentiation has to be performed) will have
intervals dependent on the conductivity of the subsurface. The advantage of re-sampling
44