INTRODUCTION

Up till now all the interfaces in our refraction models have been planar, either horizontal or inclined. Our reading and everyday observations on campus tell us that at some sites we'll have to account for interface irregularities in some way. Before deriving formulae let’s think a little about irregularities or roughness of different scales.

SCALES OF ROUGHNESS

We might expect the bedrock surface to be the roughest refracting interface in New Hampshire and the water table to be the smoothest. Although the same ideas are true for any interface I'll talk about bedrock for simplicity.

At one extreme we have bedrock bumps and depressions with spacings and widths much less than the profile length. Head waves ascending from the bumps will have shorter than average travel-times whereas waves from the hollows will arrive later than average. The travel-time graph will show a lot of scatter about a roughly straight line. Probably we won’t try to resolve and model each individual bump. We'll try to calculate the average bedrock position using planar layer formulae and perhaps estimate the typical relief of the bedrock surface using the amplitude of the scattering..

At the other extreme the bedrock bumps and depressions are so low and broad that the interface has a roughly constant slope from one end of our profile to the other. In such cases we can use our planar layer formulae.

The real trouble starts when the bumps and depressions are of intermediate size. Perhaps a few bumps on a long profile. In such cases we can devise simple procedures to get good models.

ELEVATION CORRECTIONS

The simplest irregular interface case is irregular topography over a horizontal high-speed refracting layer; perhaps water table or bedrock. In this case the head waves will be delayed or "late" for geophones on the bumps and early for geophones in the topographic depressions.

Our basic strategy is to calculate what the head wave times would be if the sources and geophones were all on a datum plane located underground. The datum plane should lie above the refracting interface and be parallel to it. The travel-time corrections or adjustments are made using a formula based on our "wavefront trick". For "gentle" bumps we can use the direct travel-times without change. Now we can interpret the adjusted data as a single flat layer case.

DELAY TIME METHOD

The classic delay time method and its variations can be used for refracting interface slopes of ten degrees or less with respect to the (planar) ground surface. Using the wavefront trick we derive formulae for head wave travel-times from each end of the profile to a phone at an arbitrary location on the profile. This formula includes the bedrock depth at that phone as an unknown parameter. We also find an expression for the travel-time of a head wave all across the profile.

We now have three linear equations for three unknowns: the bedrock depths at each end and at the particular geophone location. Given the velocities in each layer (see below) we can easily solve the three equations for the three unknown depths. We repeat the procedure for each geophone location for which we have observed head waves times from each end of the profile.

To get a good value for the refractor velocity we use the difference of the head wave times from the left and the right ends of the profile for each of the previous ‘phone locations. A plot of these difference times versus geophone location yields a straight line with a slope of two divided by the refractor velocity. Now we can go ahead and calculate the refractor depths.

As mentioned this method is only accurate for gentle slopes. The basic reason is that the head waves reaching a given geophone from the left or the right do not "take off" from the same points on the refracting interface. This introduces a fundamental false assumption in derivation of the procedure. How can we deal with steeper slopes?

SCOTT METHOD

This computer method was used by the United States Geologic Survey in many projects. The basic assumption is the refracting surface is planar between neighboring geophones but can "bend" at each phone location. The computer automatically shifts the bedrock inflections up and down, calculates travel-times from all sources to all phones and repeats the process until the observed and calculated times agree to some given extent. The program can cope with multiple layers.

Although used with success for a long time the method is fundamentally illogical. If we shifted our phones the locations of the bumps and hollows would also move! This problem can be minimized by spacing the phones close together.

GENERALIZED RECIPROCAL METHOD

This method, also called the Palmer method, tries to overcome the weakness of the classical delay time method by using pairs of head waves that take off from the same refractor location and end up at different geophones. This contrasts with the delay time method which uses one phone and two take off points.

This method can give accurate results for refractor slopes of over twenty degrees. The down side is that it requires substantial operator intervention and very detailed data. That is, the geophones must be very close together.

REFERENCES

Hagedoorn, 1959, Geophysical prospecting, 7, 158-.

Hawkins, 1961, Geophysics, 26, 806-.

Cummings, 1979, Geophysicsa,, 44, 1987-

Scott, 1973, Geophysics, 38, 271-284.

Palmer, 1981, Geophysics, 46, 1508-