INTRODUCTION

Today we’ll look carefully at three-layer soundings and some serious problems with nonuniqueness of inverting data. Before doing so it’s useful to introduce some new terminology.

THE GEOELECTRIC SECTION

A geoelectric section is a 1-m by 1-m stack of layers of different resistivities and thicknesses. The section has several important properties that are useful to us in discussing multi-layer cases, models and data (Mallet, 1947).

The total thickness of the section is just the sum of the individual thicknesses (m). The transverse resistance is the sum of the products of the layer resitivities times the layer thicknesses (ohm). For vertical currents the layers in the section act as resistors in series. We may assume that the section rests on an infinite half-space of very high resistivity.

For horizontal current flow the section acts as a set of resistors in parallel. The horizontal conductance is the sum of the quotients of the layer thicknesses divided by the layer resistivities (1/ohm). The horizontal resistance (ohm) is just the inverse of the horizontal conductance.

THREE-LAYER CASES

In these cases we have three resitivities and two thicknesses. Based on the relative resistivity values we can define four different kinds of sounding curves.

"H" curves have the lowest resistivity in the second (middle) layer. The sounding curve has a minimum at some intermediate a-spacing and higher values at either end.

"K" curves, in contrast, have maximum apparent resistivity at some intermediate spacing and lower values at each end because the section has maximum resistivity in the middle layer

For "A" curves the apparent resistivity increases with electrode spacing. The layer resistivities increase with depth.

Finally there are "Q" curves for which the apparent resistivity decreases with a-spacing. The layer resistivities decrease with depth.

NONUNIQUENESS OF MODELS

In theory all sounding curves have a unique solution. That is, only one layer model exactly fits an ideal data set. In practice our data is not ideal. For one thing the a-spacing is limited with a minimum and a maximum value. Between these limits the sounding curve is sampled, not known at every possible electrode space. In addition a typical data point has random error of up to about five per cent. Given these considerations it turns out that many models can fit the same data within experimental error. These models are said to be equivalent.

EQUIVALENCE OF H CURVES

The geoelectric section has lowest resistivity in the middle layer. Because of how current is refracted, much of it is channeled to flow more or less horizontally through the middle layer. In consequence models with the same resistivity/thickness quotient for the middle layer will have the same sounding curves (assuming that the other layer parameters are the same). In other words the equivalent models have the same horizontal conductance for the middle layer. As you’ll discover in a homework problem, this equivalence rule is only true within certain limits.

EQUIVALENCE OF K CURVES

In this case the middle layer has the highest resistivity. Refraction of current lines leads to roughly vertical flow in the middle layer. Thus the condition of equivalence is that the transverse resistance of the middle layer is the controlling factor, again assuming that the other parameters are the same. Within limits, all models with the same resistivity times thickness product for the middle layer will be equivalent.

SUPPRESSION OF THIN LAYERS

Layers that are "too thin" have no obvious expression in a sounding curve. Thus a three-layer geoelectric section may produce a sounding curve that looks just like a two-layer curve. In fact we can only discern the minimum number of layers from any given sounding curve. There can always be more layers.

To complicate matters it’s possible that the resistivity varies continuously with depth so that there are really no distinct layers at all! Clearly we need to use other data to constrain our models.

REFERENCES

Mallet, 1947, Geophysics, 12, 529-.