INTRODUCTION

In the last class we discussed Snell’s Law of Refraction, the concept of critical angle and the generation of head waves in the context of a low-velocity layer overlying a higher velocity material. Before examining this situation in detail I want to say a little about "forward modeling" and "inversion", topics that we’ll deal with in all the geophysical methods.

FORWARD MODELLING

This phrase means calculating synthetic data for some known or hypothetical situation. For example we might want to predict the first arrival travel times for a field experiment we plan to do. Let’s suppose we’re dealing with a "dry" sediment layer (vadose zone) overlying a saturated sediment layer (phreatic zone) overlying bedrock. Perhaps we have a general idea of the geology based on a well but want more detailed estimates of the materials and thicknesses in the same general vicinity. To plan our measurements we need estimates of the travel times. This is essential so that we place our geophones at appropriate places.

Based on the geology, we might set up a mathematical model of the stratigraphy. That might consist of a set of layers of uniform velocities with depths given by the well data and velocities from a textbook or local experience. This forward model will be a gross oversimplification of the true geology. It might not differentiate between various sediment types (clay, sand, etc.) and might omit bedrock irregularities of various kinds.

The next step is to calculate the travel times for significant ray paths using formulae based on principles of physics such as Snell’s law of refraction. In other words we are calculating synthetic data for an idealized approximation to a geological situation.

Forward modeling is essentially a one-way, unique process once we’ve set up the mathematical model. The travel times are unique and accurate within the accuracy of our arithmetic.

INVERSION

By inversion we mean the process of finding a mathematical model that is consistent with actual field data. That is to say, what are the parameters (thicknesses, velocities) of a model that fits the data within experimental error.

Unlike forward modeling, which is unique for a given mathematical model, inversion is always non-unique. We can find an infinite number of models that fit the data. One reason is that our data are always incomplete. For example we only have a finite number of geophones so can not totally sample the wave field. In addition the selected travel-times will have errors for many different reasons. A big source of non-uniqueness lies in the basic model that we select. Should it have two, three, four or more layers? Are the layers flat or dipping? Are they planar or irregular? Are we really dealing with a two-dimensional situation?

For these and other reasons our final mathematical model is not unique. We can always find alternative models that fit our data.

GEOLOGICAL INTERPRETATION

Once we have a satisfactory mathematical model it’s our obligation to make a geological interpretation. At a minimum this should be to give our best estimates of the geological materials. Ideally we should also justify the model in terms of geologic history, discuss any significant features and point out plausible alternative interpretations.

In our hypothetical situation we might easily misinterpret the sediment types. P-wave velocities of clay, mud, sand and some glacial tills typically overlap a great deal. Only with other data, perhaps S-wave velocities or electrical resistivities, can we narrow our estimate of sediment type. Local knowledge and experience can be a big help or may lock us into misleading traditionalism.

Rough, weathered and fractured bedrock may have seismic velocities well below those listed in text and reference books. Local knowledge may help but we should be alert for unsuspected discoveries.

SINGLE HORIZONTAL LAYER OVER A HALF-SPACE

Your lecture notes and textbook show how we can derive travel-time curves for this important case. Using trigonometry and Snell’s Law of Refraction we can derive the times as functions of the two velocities and one thickness that define the model. We discover that both phases plot as straight lines on a time versus distance graph. The velocities are simply the inverse of the slopes of the straight travel-time curves and the thickness is found from the time intercept of the head wave line at zero distance.

An alternative derivation of the head wave times is based on the principle that any legitimate ray path between the same two wavefronts represents the exact same travel time. This isn’t a big help for us in this simple case but we’ll use it to greatly simplify several more complicated ones.

To invert field data we first fit straight lines to segments of the travel-time data. The direct wave line segment must pass through the origin of the plot. The head wave line segment must have a positive time intercept and a lower slope than the direct wave line. Then we determine the upper layer velocity from the slope of the direct wave line. Next we determine the half-space velocity from the slope of the head wave line. Finally we calculate the layer thickness from the time intercept of the head wave line. It’s easier to do that to read about!