INTRODUCTION

World-wide the seismic reflection method dominates applied geophysics because it’s central to the discovery and efficient exploitation of oil and gas fields. It is the principal shipboard method for studying sedimentary strata below the ocean floor. Historically seismic reflection has not been used much in engineering and environmental investigations but that situation is beginning to change with better field equipment and computerized data processing.

The refraction method is based primarily on rays that travel along interfaces between materials of contrasting velocities. Classically analysis has focused on first-arrival travel-times with less attention to amplitudes. The reflection method, in contrast, exploits rays that bounce off interfaces between materials of different acoustic impedances (velocity time density). The significant rays are more or less vertical. Analysis routinely uses many travel times and amplitudes. Given these contrasts we are not surprised to find big differences in the field operations.

FIELD PROCEDURES

The simplest field technique is small-scale profiling done by boats in shallow waters of the shelf or in lakes. In many cases a single source is towed near the surface as is a single receiving array of hydrophones; the received signals are added to form a single data channel. The signals are displayed as intensity variations on a running plot of travel-time versus time of day (obviously related to distance along the profile). This method has been used to map surmarine tunnel routes, to study Quaternary stratigraphy and to locate bedrock depressions favorable for placer deposits of diamonds and tin. Depths are approximately equal to half the travel-time multiplied by an appropriate average velocity.

On land the equivalent method is single-channel common offset profiling. Here a single source and geophone are moved along a profile, always keeping them the same distance apart. Thus the rays are not exactly vertical and we use a hyperbolic approximation to convert travel-times to depth. Usually this method is used only if the goal is to trace a single strong reflector such as the bedrock.

The traditional method for oil and gas exploration used the split-spread. In this technique a single shot (typically an explosion) was located in the center of a line of perhaps a dozen equally spaced geophones. Analog recording was initially on photographic film and processing done using graphs, mechanical drawing instruments and desk calculators ("adding machines"). Reflecting interfaces would plot as hyperbolae (more or less) owing to the increasing lengths of ray paths to the distant geophones. In contrast to seismic refraction, which uses first breaks, reflection times from all parts of the record can be used. It’s a common convention to pick the troughs of the arrivals, not to search hopelessly for arrival times.

With the advent of digital recording and digital computers the common-mid-point (CMP,CDP) method has almost completely taken over the field. Here we use shot points and geophones symmetrically arranged on either side of a central point (mid point). The traces from the gingival geophones are adjusted for the slant of the raypaths and then added together ("stacked"). This "dynamic correction" for "normal moveout" brings corresponding echoes from the different geophones into alignment in time. We can then stack the traces to improve signal to noise ratio, to reduce the amplitude of multiple echoes and even to determine average velocities to the different reflectors. We’ll investigate these procedures after deriving travel-time formulae for some important cases.

TRAVEL-TIME FORMULAE

The simplest case involves a point diffractor that scatters incident energy in all directions. This occurs for incident wavelengths equal to or larger than the diffracting object. For a constant velocity medium the travel-time can be found using only the Pythagorean Theorem. In terms of the source-receiver separation, the time plots as a hyperbola.

Using Snell’s Law of Reflection we see that the same formula applies to an echo off a half-space (single layer case). A plot of time squared versus separation squared yields a straight line from which we can easily determine the seismic velocity of the layer. The velocity in the half-space influences the amplitude of the echo but not the travel-time.

For multiple flat layers we can use Snell’s Law of Refraction to derive exact expressions for travel-time as a function of angle of incidence at the surface. We can also derive an exact expression for distance (x, source-receiver separation) as a function of the incidence angle. WE CANNOT DERIVE AN EXACT EXPRESSION FOR TIME VERSUS DISTANCE! In this case the echoes are almost , but not exactly, hyperbolae. On an t-squared versus x-squared plot the echoes plot as approximately straight lines from which we can estimate average velocities down to the various reflectors.

Using the average velocities and travel-times at zero separation we can work out the depths, thicknesses and layer velocities of all the layers. The trick is to work from the top down and account for the layers one-by-one.

HARMONIC MEAN VELOCITY

We’ll soon discover that there are many definitions of average velocity. Some are basically defined as properties of idealized, forward modeling cases. Others are derived from field data. Most books and articles are very unclear and inconsistent in terminology and usage. In the future we’ll study some of these average velocities. One, however, is clearly defined and useful.

The harmonic mean velocity strictly is defined as the total vertical distance through a stack of flat layers (each of constant velocity) divided by total travel-time. You’ll get the correct result using one-way travel for both or round-trip (two-way) travel for numerator and denominator. Just don’t mix one-way and two-way!