One can study subsurface velocity and layer interface structure by

analyzing the first arrival times of P-waves (longitudinal or

compressional waves) at the surface of the earth. This technique is

termed seismic refraction. Applications of subsurface imaging include

locating buried archeological sites, assessing subsurface geological

hazards, defining aquifer geometry, and exploring for fossil fuel and

other natural resources.

P-waves traveling through rock are analogous to sound waves traveling

through air. The speed a P-wave propagates through a medium depends on

the physical properties (i.e. rigidity, density, saturation) and degree

of homogeneity of the rock. Spherical wave fronts emanate from a

source, as well as ray paths. Ray paths travel normal to the spherical

wave surface. For seismic refraction discussion, it is useful to

imagine seismic waves as ray paths.

When a ray encounters an inhomogeneity in its travels, for example a

lithological contact with another rock, the incident ray transforms into

several new rays. A reflected wave enters and exits at the same angle

measured to the normal of the boundary - angle of incidence equals angle

of reflection.

From Snell's Law, a ray path is dependent on the wave velocities through

different layers. For refraction seismology, the critical angle is the

most important angle value to understand. If angle (r) equals 90

degrees, then the refracted wave propagates along the boundary

interface. One can solve for the critical angle (ic) by calculating

inverse sine of (V1/V2). As the critically refracted wave propagates

along the boundary, according to Huygen's Theory of Wavelets, the

primary critically refracted wave acts as a source for new secondary

wave fronts and ray paths. These secondary ray paths exit at the

critical angle.

In the ideal world (of engineering), refraction seismology is most easily

understood through a horizontal two layer model.

Seismic waves are generated from a source (sledge hammer). Geophone

receivers record seismic signals received along the survey profile.

Since P-waves travel at the fastest speeds, the first seismic signal

received by a geophone represents the P-wave arrival. Five P-waves are

of interest in refraction seismology: direct, diving, reflected, head,

refracted. The direct wave propagates along the atmosphere-upper layer

1 boundary. A transmitted wave through layer 2 is termed a diving wave.

A reflected wave enters with the same angle of incidence as exit angle.

If the critical angle is achieved, the critically refracted head wave

travels along the layer 1-layer 2 interface. Refracted waves propagate

from the interface, with exit angles equal to the critical angle.

With arrival time data collected, arrival times for P-waves are noted or

computed from the seismographs. Arrival times can be represented on a

travel-time graph or T-X plot, that is P-wave arrival times (usually in

milliseconds) verses distance (geophone location).

What are we trying to calculate? Of interest are velocities of P-wave

propagation through layers 1 and 2, and also thickness of layer 1. To

obtain these values, a healthy combination of equations and

interpretation from the T-X plot is required.

Analysis of the direct wave yields V1. On the profile view, notice that

the wave arrives at a geophone located a known horizontal distance from

the source. Thus, V1 should equal geophone-source distance divided by

P-wave arrival time for a given geophone. On the T-X plot, the direct

wave is represented by an interpolated line for arrival time data

passing through the origin. The slope of this interpolated line is time

over distance, or the inverse of velocity. The slope of lines on the

T-X plot is termed slowness.

Another interpolated line can be observed on the T-X plot, a line

representing the refracted wave. The distance between the source and

first geophone to receive the refracted wave is termed critical

distance. Cross-over distance is defined as the position where the

refracted wave overtakes the direct wave.

A common analogy to the cross-over phenomenon are the travels of a

cyclist and motorist. Imagine a cyclist and motorist depart from the

Fall AGU meeting in San Francisco. Both are traveling to a field

excursion north of the Golden Gate Bridge in the Marin Headlands. The

cyclist decides to pedal along the bike path situated on the bay shore.

With regards to distance to the Marin Headlands, the bike path distance

is much less than highway distance. The cyclist can pedal at a constant

rate of 15 miles per hour. Think of the cyclist as a direct wave.

Meanwhile, the motorist (e.g. refracted wave) must deal with numerous

one-way streets and earthquake-retrofit construction, traveling at a

snail's pace of 5 miles per hour. Eventually the motorist finds the on

ramp onto Highway 101 and heads north for the Golden Gate Bridge at a

brisk 55 mph. Since the motorist is traveling at a significant speed,

the cyclist can only wave to the motorist as the car speeds past the

cyclist on the bridge. The motorist must wait several tens of minutes

at the outcrop of Franciscan melange before the cyclist arrives at the

designated field trip meeting place.

In this example, the cross-over distance occurred at the southern end of

the Golden Gate Bridge. The speed of the cyclist (15 mph) represents

the P-wave velocity of layer 1. Highway speed for the motorist (55 mph)

would represent the P-wave velocity for layer 2. Back to the T-X

plot...

From the slowness of the direct and refracted wave, velocities in layers

1 and 2 can be calculated. To determine depth of layer 1 (Z1), the time

intercept (ti) of the refracted wave must be noted. Kearey & Brooks

(1984) derive and summarize the equations necessary to calculate V1, V2,

and Z1.

When discussing dipping layers, one wants to quantify the amount of dip.

For a simple case of two dipping layers, seismic refraction can be

utilized to calculated dip of the layers. For a given survey profile,

sources must be located at the beginning of the profile (forward shot)

and at the end of the profile (reverse shot).

P-wave arrival times for both forward and reverse shots can be plotted on

a T-X plot. From the Principle of Reciprocity, time required for a ray

to travel along the forward and reverse shot should be the same, since

the ray pathways are the same. From the T-X plot, V1 and V2 velocities

for forward and reverse shots can be calculated, as well as the

time-intercepts for forward and reverse refracted waves.

From Kearey & Brooks (1984), the following equations yield layer 1

thicknesses normal to the interface at the forward source (Zforward) and

reverse source (Zreverse). Small delta represents the dip of layer-1

layer-2 boundary.

Why only stop with interpretation of two horizontal layers?

Calculation of layer velocities and thicknesses for multi-layers requires

patience with many equations chock full of algebra and trigonometry.

Please refer to Kearey & Brooks(1984), Fowler (1990), or Burger (1996)

for these equations. Interpretation of T-X plots remains the same.

Each layer yields an interpolated refracted wave slowness, and time

intercept used to calculate layer thickness.

The preceding models assume planar boundary interfaces. Conformable

sequences of sedimentary rock may form planar boundaries. However,

erosion and uplift easily produce irregular boundary contacts. More

sophisticated algorithms can process refraction surveys where irregular

interfaces might be expected.

Profile length and source energy limit the depth penetration of the

refraction method. Typically, a profile can only detect features at a

depth of one-fifth survey length. Thus, refraction imaging of the Moho

would require profile lengths of over one hundred kilometers; an

unreasonable experiment. Larger sources could be utilized for greater

depth detection, but certain sources (e.g. explosives) may cause

problems in urban areas.

Refraction depends on layers to increase in velocity with depth. In the

hidden slow layer senario, a buried layer is overlain by a faster

layer. No critical refraction will occur along the boundary interface.

Thus, refraction will not detect the slow layer. All is not lost since

reflection seismology could detect the slower layer.

Seismograms require careful analysis to pick first arrival times for

layers. If a thin layer produces first arrivals which cannot easily be

identified on a seismogram, the layer may never be identified. Thus,

another layer may be misinterpreted as incorporating the hidden layer.

As a result, layer thicknesses may increase.

These sources offer excellent discussion of theory, derivation of

formulas, and practical examples of refraction seismology.

Burger, H.R. 1996. Exploration Geophysics of the Shallow Subsurface.

Prentice Hall. 489 pp.

Fowler, C.M.R. 1990. The Solid Earth - An Introduction to Global

Geophysics. Cambridge University Press. 472 pp.

Kearey, P. & Brooks, M. 1984. An Introduction to Geophysical

Exploration. Blackwell Scientific Publications. 296 pp.

For a more rigorous discussion of refraction and reflection seismology,

visit An Introduction to Geophysical Exploration

Check out equipment and seismic refraction applications in industry at

Geosphere Inc.

Lithoprobe is a Canadian program to study North American continental crust using

refraction seismology as one of many exploration methods.

If you have any comments or questions, please email me, Eric Cannon, at cannon@ymir.ucdavis.edu