INTRODUCTION

In our last class we discussed the difference between forward modeling and inversion of data. Conceptually these are very different although they use the same mathematical formulae. We also derived formulae for travel-times for a single flat layer overlying a half-space. Today we’ll derive formulae for a set of two layers over a half-space and explore how that situation can lead to very misleading or ambiguous inversions.

FORMULAE FOR TRAVEL-TIMES

Our basic model has two layers overlying a half-space. Geologically they might correspond to dry sediment over saturated sediment over bedrock. In such a situation the velocity increases downward from layer to layer. We also know the layer thicknesses. In other words the model is described by five parameters.

The travel times for the direct ray and head wave from the water table are given by the same formula as for the single layer case. To derive a formula for the head wave from the basement, we use trigonometry, Snell’s Law of Refraction and our little wavefront trick: all legitimate rays paths linking the same two wavefronts have identical travel-times.

"Normally" the travel-time graph will have three straight segments with slopes equal to the inverses of the three velocities. As before the direct ray segment passes through the origin. The water table segment has a finite intercept and the bedrock intercept is even bigger. To invert field data we draw three straight line segments. From the slopes we calculate the velocities. Then, working from top to bottom, we use the intercepts to calculate the layer thicknesses and interface depths.

In many refraction surveys only the first arrival times are used as it’s difficult or impossible to pick later arrivals with any confidence because of overlapping and lengthy wave trains. This reliance on first arrival times can lead us astray in at least two situations.

THIN LAYER CASE

If the middle layer (second layer, "phreatic layer" in our example) is too thin there will be no first arrivals from the top of that layer ("water table"); only direct waves and head waves from the bedrock will be first arrivals. Thus when we invert the data we’ll probably use a model with only two components: the vadose zone and the bedrock. In addition to completely missing one layer, our inversion model will underestimate the depth to bedrock.

Local geologic information such as elevations of surface water bodies or water table depths in wells might suggest the presence of a hidden thin layer. In that case we might want to sketch in a fictitious water table arrival at some suitable velocity and reinterpret in terms of three velocities.

LOW-VELOCITY LAYER CASE

In some cases the second layer might have a lower velocity than the upper layer or the underlying half-space. Geologically an example would be permafrost over saturated sediment over bedrock.

We can use the same formulae to calculate the travel times but in this case the middle layer does not yield a head wave. All we see on our plots is a direct wave and a head wave from the half-space. The middle layer head wave isn’t hidden or obscured by earlier arrivals; it simply doesn’t exist! Thus we would invert the data using a single layer model. The middle layer just isn’t there and we overestimate the bedrock depth.

If we are lucky the missing layer may be located using reflected body waves or dispersive surface waves.

APPLICATIONS OF SEISMIC REFRACTION

This method is used on all scales for a wide variety of purposes. I’ll just mention a few here. You can read about others for your journal.

On a tiny scale refraction profiles have been used to obtain relative ages of boulders and hence of the moraines on which they are found. As a boulder weathers an outer rind of lower velocity grows. In effect we have a single layer over a half-space. If the boulder is small we need to use round-earth formulae, not the flat-earth ones derived in class. The rinds are thickest and velocities lowest on the oldest moraines.

Perhaps the most common uses are to find water table and bedrock in engineering and hydrologic studies. For example we might want to map out bedrock under a highway route to determine where blasting would be needed. Low bedrock velocities might indicate fractured and weathered rock that could be "ripped" by a bulldozer whereas high velocities might indicate great strength. Determination of sediment type is difficult or impossible using just P-wave velocities but we might profitably use S-waves or attenuation of P-wave amplitudes with distance.

Localized increases in the slopes of head wave travel-time curves often indicate the presence of faults or fractures zones in bedrock. These might be good places for bedrock water wells or bad ones to build dams.

Refraction data is commonly used in oil and gas exploration. The goal is usually not to find velocities and thicknesses of layers but rather to make "static" corrections to seismic reflection surveys. The static corrections are used to adjust travel times for passage through the thick, low-velocity "weathered zone" overlying solid rock. If these large time corrections are not done well the images of deeper layers are severely distorted.

On a larger scale seismic refraction is used to determine crustal layering, Moho depth and upper mantle velocities. Results show that average continental crust is about forty kilometers thick and the velocities indicate composition ranging from granitic near the top to gabbroic at the base. P-wave velocities of the uppermost mantle are about eight kilometers per second indicating an ultramafic composition. Oceanic crust is about seven kilometers thick with basalt flows at the top and gabbro plutons at the base. These seismic data form a large part of the evidence for isostasy.

Refraction of earthquake waves indicates that both P and S-wave velocity increase with depth. This effect is attributed mainly to increasing pressure. The gradual increase of velocity leads to concave upward ray paths in the mantle. At the core-mantle boundary P-waves refract strongly downward indicating a big decrease in velocity. S-waves don’t go through the core at all. This means that at least the outer core is liquid.

Although made of plasma and inaccessible to geophones it turns out that there are waves traveling through the sun. P-waves can travel though the sun as through any gas. A wave coming up shows a spectral shift to a bluer color according to the Doppler effect whereas waves going in are shifted towards red. In contrast to the sun, the moon has been the scene of conventional seismic exploration.