Analysis of the moments of the sensitivity function for resistivity over a homogeneous half-space: Rules of thumb for pseudoposition, offline sensitivity and resolution

作者：Butler S L^*

来源：Journal of Applied Geophysics, 2017, 143: 149-155.

DOI：10.1016/j.jappgeo.2017.06.005

It is instructive to consider the sensitivity function for a homogeneous half space for resistivity since it has a simple mathematical formula and it does not require a priori knowledge of the resistivity of the ground. Past analyses of this function have allowed visualization of the regions that contribute most to apparent resistivity measurements with given array configurations. The horizontally integrated form of this equation gives the sensitivity function for an infinitesimally thick horizontal slab with a small resistivity contrast and analysis of this function has admitted estimates of the depth of investigation for a given electrode array. Recently, it has been shown that the average of the vertical coordinate over this function yields a simple formula that can be used to estimate the depth of investigation. The sensitivity function for a vertical inline slab has also been previously calculated. In this contribution, I show that the sensitivity function for a homogeneous half-space can also be integrated so as to give sensitivity functions to semi-infinite vertical slabs that are perpendicular to the array axis. These horizontal sensitivity functions can, in turn, be integrated over the spatial coordinates to give the mean horizontal positions of the sensitivity functions. The mean horizontal positions give estimates for the centres of the regions that affect apparent resistivity measurements for arbitrary array configuration and can be used as horizontal positions when plotting pseudosections even for non-collinear arrays. The mean of the horizontal coordinate that is perpendicular to a collinear array also gives a simple formula for estimating the distance over which offline resistivity anomalies will have a significant effect. The root mean square (rms) widths of the sensitivity functions are also calculated in each of the coordinate directions as an estimate of the inverse of the resolution of a given array. For depth and in the direction perpendicular to the array, the rms thickness is shown to be very similar to the mean distance. For the direction parallel to the array, the rms thickness is shown to be proportional to the array length and similar to the array length divided by 2 for many arrays. I expect that these formulas will provide useful rules of thumb for estimating the centres and extents of regions influencing apparent resistivity measurements for survey planning and for education.

• 出版日期2017-8
• 单位Saskatchewan; Saskatoon