Specifications

University of Pretoria etd – Combrinck, M (2006)

However, low order polynomials will result in loss of high frequency information and high

degree polynomials still suffer from unwanted oscillations.

TEM is based on diffusion of electrical currents into the earth and this is a smooth process,

never oscillating in nature, and therefore not suited to this type of interpolation.

Furthermore, although TEM data locally approximate polynomial functions in the

logarithmic domain they cannot in general be represented by the same order polynomial on

all time channels. A better strategy to follow is piecewise polynomial approximation. One

method to interpolate data under these conditions is known as the cubic spline method

that involves fitting third order polynomials to each consecutive pair of data points,

requiring the function values to be equal to the measured data values as well as continuity

of the first and second order derivatives ensuring smoothness of the function. As with the

Lagrange three-point formula, an approximation to the derivative of the sampled function

is obtained by taking the analytical derivative of the third order cubic spline polynomials at

every point. The cubic spline method is computationally more intensive than the Lagrange

three-point formula but it can also be used to resample data to equally spaced intervals

(which simplifies smoothing of data) if required. The free (or natural) boundary conditions

were invoked in this study (assume that the second derivatives of the end points are zero)

as the information required for clamped boundary points are not available.

4.4.2.2.3 Differentiation as the inverse of integration

In contrast to numerical differentiation, numerical integration (its inverse) is a stable

method and not particularly sensitive to noise. The same relationship holds for the

downward (unstable) and upward (stable) continuation filters used in processing of

potential field data. Cooper (2004) introduced the “inverse of upward continuation” as a

more stable downward continuation filter. The same principle can now be extended to the

problem at hand and differentiation can be substituted by a stable “inverse of integration”

filter. This can be formulated as follows. Define

as the n data points to be

differentiated,

as the required derivative values and A an operator of integration.

Then

)(

)(xf

′

(4.12)....0),( nixf)(xf

′

⋅A