INTRODUCTION

Once we have completed a resistivity survey or a resistivity sounding we are faced with the fun of interpreting the data. First of all we need to find a distribution of underground resistivities that is consistent with the data. Then we have to make a geological interpretation; what are the materials underground? Today we’ll concentrate on the first step with examples of surveys and samples.

EMPIRICAL CORRELATION

This is the simplest method of all. Rather than seeking a model that quantitatively matches the data, we just look for clear correlations between aspects of the resistivity data and some other geological observables.

In the "Boulder Field" the data indicates that the Wenner array apparent resistivity (a=20 feet) is inversely related to bedrock depth determined by seismic refraction. Thus, to the extent that the correlation is valid, we can simply relabel the contours on the resistivity map as contours of bedrock depth.

The advantages of the empirical correlation method is that it is fast, cheap and, by definition, involves the use of at least one other kind of data. On the negative side the correlations are only of local value, the resistivity data is not fully utilized and that no real quantitative understanding is developed. I think one should always look for clear, explainable correlations but not stop there.

MASTER CURVES

Use of master curves for soundings and profiles is no longer common. They are useful for fast, cheap interpretations for two- or three-layer soundings. These can be used to monitor field work in progress or as starting models for some computer programs.

Often the master curves don’t match the field data very well and curves for more than three layer models are hard to find. It is difficult to evaluate the uniqueness of the model and hard to build in geological constraints.

TRIAL AND ERROR COMPUTER MODELLING

For a beginner this method is perhaps the best. Fitting data within experimental error can readily be done for models with only a few layers. In most programs it is easy to include geological constraints such as thickness of a layer. In addition one learns a lot about uniqueness, equivalence and suppression.

By its nature trial and error modeling is time consuming and thus expensive given the skill needed to it well. There’s a natural tendency to stop with the first good model and not explore the full range of possible models. There is a bias toward simple models with the minimum possible number of layers.

AUTOMATIC CURVE MATCHING

A common practice is to invert soundings automatically using some criterion of fit such as least-squares (Inman, 1975). We use a computer program to minimize some measure of misfit between our observed apparent resistivities and those of a model. A common measure of misfit ("objective function") is the sum of the squares of the misfits at each electrode spacing ("L-2 Norm"). Often the misfit is defined using the logarithms of the apparent resistivities rather than the values themselves.

In practice we specify an initial model. Then the computer automatically adjusts the model parameters (resistivities and thicknesses) until the objective function no longer decreases. At that point the program stops and the final "best"model is displayed.

The most serious problem with this procedure is that the final model depends on the initial model. In some cases the final model is unacceptable and we’ll have to try again with a new starting model. Another problem, related to equivalence and suppression, is that our model may have the wrong number of layers and thus the wrong thicknesses and resistivities. For any given sounding the least-squares solution is also the "roughest" model with the largest layer-to-layer jumps in resistivity.

In summary, the least-squares method is rapid, automatic and usually gives us acceptable models. On the other hand it gives roughest models, doesn’t explore equivalence and depends strongly on the initial model.

In the next class we’ll look at some other automatic modeling methods.

REFERENCES

Inman, 1975, Geophysics, 40, 798-817