122 Seismology and Earth Structure
sin i c = v 0 t/v 1 t
v 1 t
i c
i c
v 0 t
v 0
v 1
Fig. 3.2-3 Generation of an upgoing head wave by Huygens’ sources due
to a refracted pulse propagating along a boundary. The head wave travels
in the upper layer at a slower velocity (v 0 ) than the refracted wave creating
it, which travels in the layer below at velocity v 1 . (After Griffiths and King,
1981.)
Time (s)
250
200
150
100
50
0
500
Distance (km)
1000
S g
S n
P g
P n
Fig. 3.2-4 Schematic of Mohorovibia’s results showing the existence of a
distinct crust and mantle. The travel time curves are labeled using modern
nomenclature: the direct waves are P g and S g , and the head waves are P n
and S n . (After Bonini and Bonini, 1979. Eos, 60, 699–701, copyright by
the American Geophysical Union.)
Despite this solution’s elegance, the basic assumption about
the travel time of the head wave may seem unsatisfying, because
it is unclear why energy should follow this path. However,
the result conforms with observations — the experiment
diagrammed in Fig. 3.2-1 yields an arrival whose travel time is
given by Eqn 8. To understand why, we can view the head wave
in several ways. As shown in this chapter’s problems, it corresponds to a minimum time path between the source and the
receiver, so, by Fermat’s principle (Section 2.5.9), we expect
such a wave. Another approach, using Huygens’ principle
(Section 2.5.10), is to consider the refracted wave traveling
horizontally below the boundary at the velocity of the halfspace, generating spherical waves that propagate upward in the
lower-velocity layer (Fig. 3.2-3). The spherical waves interfere
to produce upgoing plane waves that leave the interface at the
critical angle. 2 However, our analysis of postcritical incidence
(Section 2.6.4), which showed that an evanescent wave propagates along the interface, does not fully describe the head
wave. A more sophisticated analysis than is appropriate here
shows that the geometry in Fig. 3.2-1 gives the head wave’s
travel time, but not its amplitude, because geometrical optics
are not applicable. Thus, although the energy propagation is
more complicated than along the geometric ray path, the travel
time predicted is correct.
Seismic refraction data led A. Mohorovibia 3 in 1909 to
one of the most important discoveries about earth structure.
Observing two P arrivals (Fig. 3.2-4), he identified the first as
having traveled in a deep high-velocity (7.7 km/s) layer, and
the second as a direct wave in a slower (5.6 km/s) shallow layer
about 50 km thick. These layers, now identified around the
world, are known as the crust and the mantle. The boundary
between them is known as the Mohorovibia discontinuity,
or Moho. We now denote the head wave as P n and the direct
wave as P g (“g” for “granitic”). Corresponding arrivals are also
observed for S waves. The Moho, which defines the boundary
2 This situation is analogous to a bow wave from a boat or a supersonic wave from a
jet airplane, in that the energy source travels faster than the wave it produces.
3 Andrija Mohorovibia (1857–1936), working in Zagreb, Croatia (then part of the
Austro-Hungarian Empire), studied travel times from earthquakes in the region using
recently invented pendulum seismographs.
between the crust and the mantle, has been observed around
the world. One of the first steps in studying the nature of the
crust is characterizing the depth to Moho, or crustal thickness,
and the variation in P n velocity from site to site.
Travel time plots for refraction experiments can be made by
displaying seismograms in record sections. Because seismograms are functions of time, aligning several as a function of
distance yields a travel time plot showing the different arrivals.
Figure 3.2-5 shows a record section of a profile of seismograms
recorded in England from explosive sources. In addition to
P n and P g , the reflection off the Moho, known as P m P, is well
recorded. As expected, the direct and head wave travel times
are linear with distance, whereas the reflection has a hyperbolic
curvature. The figure is plotted as a reduced travel time plot,
in which the time shown is the true time minus the distance
divided by a constant velocity. This reduces the size of the plot,
and makes waves arriving at the reducing velocity appear as a
line parallel to the distance axis.
The geometry discussed here can correspond to different
physical experiments. A single source can be recorded simultaneously at receivers at different distances. Alternatively,
multiple sources at different distances can be recorded by a single
receiver at different times. A single receiver can be moved away
from a fixed source, so the same source is recorded at different
distances. Similarly, a source can be moved away from a fixed
receiver. Results of various experiments can be combined,
using the principle of reciprocity, which states that the travel
time is unchanged if the source and the receiver are interchanged. As a result, we can use travel time measurements
without considering whether the source was at one position
and the receiver at another, or the reverse. Moreover, because
earth structure presumably is not changing during the experiment, data collected at different times can be combined.
sin i c = v 0 t/v 1 t
v 1 t
i c
i c
v 0 t
v 0
v 1
Fig. 3.2-3 Generation of an upgoing head wave by Huygens’ sources due
to a refracted pulse propagating along a boundary. The head wave travels
in the upper layer at a slower velocity (v 0 ) than the refracted wave creating
it, which travels in the layer below at velocity v 1 . (After Griffiths and King,
1981.)
Time (s)
250
200
150
100
50
0
500
Distance (km)
1000
S g
S n
P g
P n
Fig. 3.2-4 Schematic of Mohorovibia’s results showing the existence of a
distinct crust and mantle. The travel time curves are labeled using modern
nomenclature: the direct waves are P g and S g , and the head waves are P n
and S n . (After Bonini and Bonini, 1979. Eos, 60, 699–701, copyright by
the American Geophysical Union.)
Despite this solution’s elegance, the basic assumption about
the travel time of the head wave may seem unsatisfying, because
it is unclear why energy should follow this path. However,
the result conforms with observations — the experiment
diagrammed in Fig. 3.2-1 yields an arrival whose travel time is
given by Eqn 8. To understand why, we can view the head wave
in several ways. As shown in this chapter’s problems, it corresponds to a minimum time path between the source and the
receiver, so, by Fermat’s principle (Section 2.5.9), we expect
such a wave. Another approach, using Huygens’ principle
(Section 2.5.10), is to consider the refracted wave traveling
horizontally below the boundary at the velocity of the halfspace, generating spherical waves that propagate upward in the
lower-velocity layer (Fig. 3.2-3). The spherical waves interfere
to produce upgoing plane waves that leave the interface at the
critical angle. 2 However, our analysis of postcritical incidence
(Section 2.6.4), which showed that an evanescent wave propagates along the interface, does not fully describe the head
wave. A more sophisticated analysis than is appropriate here
shows that the geometry in Fig. 3.2-1 gives the head wave’s
travel time, but not its amplitude, because geometrical optics
are not applicable. Thus, although the energy propagation is
more complicated than along the geometric ray path, the travel
time predicted is correct.
Seismic refraction data led A. Mohorovibia 3 in 1909 to
one of the most important discoveries about earth structure.
Observing two P arrivals (Fig. 3.2-4), he identified the first as
having traveled in a deep high-velocity (7.7 km/s) layer, and
the second as a direct wave in a slower (5.6 km/s) shallow layer
about 50 km thick. These layers, now identified around the
world, are known as the crust and the mantle. The boundary
between them is known as the Mohorovibia discontinuity,
or Moho. We now denote the head wave as P n and the direct
wave as P g (“g” for “granitic”). Corresponding arrivals are also
observed for S waves. The Moho, which defines the boundary
2 This situation is analogous to a bow wave from a boat or a supersonic wave from a
jet airplane, in that the energy source travels faster than the wave it produces.
3 Andrija Mohorovibia (1857–1936), working in Zagreb, Croatia (then part of the
Austro-Hungarian Empire), studied travel times from earthquakes in the region using
recently invented pendulum seismographs.
between the crust and the mantle, has been observed around
the world. One of the first steps in studying the nature of the
crust is characterizing the depth to Moho, or crustal thickness,
and the variation in P n velocity from site to site.
Travel time plots for refraction experiments can be made by
displaying seismograms in record sections. Because seismograms are functions of time, aligning several as a function of
distance yields a travel time plot showing the different arrivals.
Figure 3.2-5 shows a record section of a profile of seismograms
recorded in England from explosive sources. In addition to
P n and P g , the reflection off the Moho, known as P m P, is well
recorded. As expected, the direct and head wave travel times
are linear with distance, whereas the reflection has a hyperbolic
curvature. The figure is plotted as a reduced travel time plot,
in which the time shown is the true time minus the distance
divided by a constant velocity. This reduces the size of the plot,
and makes waves arriving at the reducing velocity appear as a
line parallel to the distance axis.
The geometry discussed here can correspond to different
physical experiments. A single source can be recorded simultaneously at receivers at different distances. Alternatively,
multiple sources at different distances can be recorded by a single
receiver at different times. A single receiver can be moved away
from a fixed source, so the same source is recorded at different
distances. Similarly, a source can be moved away from a fixed
receiver. Results of various experiments can be combined,
using the principle of reciprocity, which states that the travel
time is unchanged if the source and the receiver are interchanged. As a result, we can use travel time measurements
without considering whether the source was at one position
and the receiver at another, or the reverse. Moreover, because
earth structure presumably is not changing during the experiment, data collected at different times can be combined.