INTRODUCTION

The basic purpose of a resistivity sounding is to determine the distribution of resistivity with depth at a given site. The basic concept is that for a small a-spacing the injected current flows mainly in a small volume near the surface. Thus the apparent resistivity would mainly be determined by the actual resistivity of shallow material. For larger a-spacings the current flows more deeply and the apparent resistivity is progressively influenced by deep material. By convention we plot apparent resistivity versus a-spacing on log-log graph paper. The resulting curves depend on array geometry and on true resistivities underground.

RESISTIVITY ARRAYS

The Wenner array dates to the early years of the twentieth century when it was developed at the United States Bureau of Standards. On the plus side it is easy to lay out with little room for blunders. Because the voltage electrodes are relatively far apart this array does not require a very sensitive or expensive instrument. On the negative side all four electrodes must be moved as the a-spacing is increased. For large spacings this means that larger field crews are needed compared to some other arrays. Another negative is that moving the inner electrodes each time leads to geologic noise from non-uniform surface materials.

The Schlumberger array was developed in France in the 1920’s. The Schlumberger brothers were wealthy textile magnates who did geophysics as a hobby. Eventually this led to a huge world-wide well-logging company (some well logs are just resistivity profiles down a well). With this array the inner (potential) electrodes are not moved until it’s absolutely necessary to get an adequate voltage difference. With fewer electrodes to move the field crews can be smaller and less geologic noise is introduced. One negative feature is that the sounding curves must be adjusted to join smoothly when the inner electrodes are moved. To do this well requires repeat, overlapping measurements, thus losing some of the time that otherwise would be saved.

The double-dipole array uses a small current array separated from a small potential array. In this way large field crews are not needed although a very sensitive instrument is. For large spacing we normally need long cables and high currents. Both of these features lead to serious safety issues. Guarding a 100-meter long dipole cable is much easier than a 10-km high-voltage cable.

Many other arrays for special purposes have been used.

FORWARD MODELLING

For a single layer over an infinite half space ("two layer case") we can easily calculate apparent resistivity versus electrode spacing. In class we’ll look at formulae for the Wenner and Schlumberger arrays. In both cases the formulae involve infinite series; probably 25 terms or so are enough for our purposes.

TWO LAYER SOUNDING CURVES FOR WENNER ARRAY

In the olden days before computers geophysicists used printed graphs showing sounding curves for many specific cases. Typically these "master curves" showed the log of apparent resistivity divided by top layer resistivity versus the log of a-spacing divided by top layer thickness. In this way the same curves could be used for any parameter values expressed in any set of units.

Each curve is labelled by "k", the resistivity contrast. This non-dimensional parameter equals the difference of bottom layer and top layer resistivities divided by the sum of the two resistivities.

For positive "k" the lower layer is more resistive and the apparent resistivity increases along with the electrode spacing. If the lower layer has essentially infinite resistivity the curve rises at a 45 degree angle. Otherwise it eventually levels off at the lower layer value regardless of a-spacing.

For negative values of "k" the sounding curve decreases. For a perfect conductor the curve descends at an ever steeper angle. Other wise the curve flattens out at the lower layer resistivity.

In practice the field data would be plotted on tracing paper using the same log-log scale. Then, using a light table, the field plot would be slid around over the master curves until an approximate match was found. Then it’s a simple matter to read off top layer resistivity and thickness. The the lower layer resistivity would be calculated using the "k" value of the matched curve.

In the next class we’ll look at the three-layer case.