Fredholm Equation of First Kind in Borehole Gamma Ray Measurements - Cubic Spline Weighing

Abstract

The problem in the processing of gamma-ray borehole data is to find out the radioactive zones down the bore depths which vary from 50m to 1000m in various geological formations and subsequent evaluations. The radiation field in the boreholes in cylindrical geometries follows the Fredholm Integral Equation of first kind. And in this work it has been dealt numerically using  cubic spline method to compute the second derivatives of the measured gamma ray intensity at various depths in the borehole. This is achieved by expressing this integral equation as a sum of products of the measured gamma ray intensities and some unknown coefficients which can be computed using Lagrange interpolation method. A finite window is selected for this summation which can be varied. In this paper the three windows have been selected. The logging probe is calibrated in primary borehole standard and then the real data for radiation field in the boreholes is acquired. Three such windows were selected and results for 3, 5, 7 point windows have been presented in this paper.

Keywords: Fredholm Equation, Borehole Gamma Ray Measurements, Cubic Spline Weighing, logging probe