INTRODUCTION
SINGLE DIPPING LAYER
Using Snell’s Law, our "wavefront trick" and a little trigonometry we get formulae for head wave travel-times in the "down-dip" and "up-dip" directions. For a down-dip profile the successive ray paths extend farther and farther down-dip. In other words the origin of the horizontal coordinate is located at the shallow end of the profile. The down- dip travel-time of the head wave has a smaller time intercept (at x=0) and a steeper slope than the up-dip head wave. The head wave time intercepts at maximum source to receiver separation are the same for the up-dip and down-dip travel times. This is termed the "condition of reciprocity".
Interpreted in isolation as flat-layer cases, both the down-dip and up-dip profiles give erroneous results. For the down-dip case we'll get too low estimates of bedrock velocity and depth. In contrast, the up-dip profile will give excessive estimates. However, as shown in class, we can interpret both profiles together and get the correct results. Of course the usual caveats about hidden layers and three-dimensional effects still hold true.
INTERPRETATION OF FIELD DATA
Now we lightly draw in the direct wave times from either end. For a constant velocity top layer all across the profile, these lines must meet exactly in the middle of the profile. If they don't (allowing for a little observational error and "fudging") you can't continue with today’s formulae. You have a more complex situation and need other formulae to make an interpretation.
Next we draw in the two head wave lines being sure that the far intercepts are identical for both. This "condition of reciprocity" must be true for all cases regardless of how complex the distribution of velocities. The reason is that at the far ends of the profile the ray path from left to right is the same as that in the opposite direction.
A third shot in the middle of the profile gives very useful control on how we draw in our lines. For one thing it gives us two more estimates of the direct wave velocity. Secondly the head wave slopes must correspond to those of the full profile. If they don't the actual geometry is too complex for our formulae. In effect we have three profiles that must all tell the same story.
MULTI LAYER CASES
Although we could carry out the calculations with a calculator it’s much easier to use a computer program such as "Bison*", a True BASIC program available on the class web site. I'll give you a handout on how to enter the data and run the program.
REFERENCES
So far we have considered refraction cases where all the layers or interfaces are horizontal or, at least, parallel to the ground surface. As geologists we know that in many places strata may dip or interfaces such as the buried bedrock surface may slope with respect to the ground surface. How will head wave travel-times be influenced by these sloping interfaces?
The simplest dipping case has one layer overlying a dipping surface of a higher velocity material. For example we might consider dry regolith lying over bedrock. The bedrock surface slopes at an angle "alpha" with respect to ground level.
As with the flat layer case, we begin by plotting the travel-time data (first arrivals) in the reversed profile format. That means that the distance origin for the forward profile is at the left side of the graph and the distance origin for the reverse profile is to the right.
The travel-time formulae for multiple dipping layers are similar to the ones we have just derived (Ewing et al., 1939). Of course they are more complex. Draw in the direct and head wave lines in the same way as before. Be sure to honor the requirement of reciprocity!
Ewing et al., 1939, Geological Society of America Bulletin, 50, 257-296