Simple formulas for pseudoposition for electrical resistivity and IP in vertical boreholes based on mean positions of the sensitivity

Abstract

The electrical conductivity method in boreholes has been applied for exploration as well as engineering and environmental investigations. The simplest deployment involves placing electrodes at varying heights within a single borehole. Borehole surveys differ from surface surveys using colinear arrays in that the ground surface is in the line of the electrodes and so it influences the measured potential in the ground differently. Multiple electrodes can be deployed on a single multichannel cable resulting in measurements from nonstandard array configurations. The choice of the plot point for pseudosections can be difficult for these nonstandard arrays. The mean of the sensitivity function of a constant resistivity half space has been shown to yield simple and useful formulas for pseudopositions for four electrode surface arrays. In this contribution, I first derive the sensitivity function for electrodes in a vertical borehole and then calculate the vertical and horizontal sensitivity functions. I then derive simple formulas for the vertical and horizontal positions of the mean of the sensitivity function for electrodes in a vertical borehole. Pseudosections for synthetic data are shown to be more easily interpretable than pseudosections plotted using averages of the electrode positions. The simple formulas will be useful for plotting pseudosections for initial data visualization and for survey planning.