2 Seismic Wave Propagation and Earth models

2. Seismic Wave Propagation and Earth models
Different kinds of seismic waves follow different time paths, e.g., the reflected waves pP (see
Fig. 2.43) a true minimum path, the PP or the SKKS reflection (Fig. 2.42) a minimax path and
the reflected wave P'P' (PKPPKP) (Fig. 2.44) a true maximum path. Note that the character of
the stationary path influences the character (phase shift) of the reflected waveform. Whenever
a seismic ray travels in some parts of its raypath as a maximum time ray, it touches a caustic.
This caustic can be a focusing point (see 2.5.3.3 or 2.5.3.4) or a surface along which seismic
rays superimpose each other (see 2.5.4.3). In any case prominent phase distortion can be
observed and has to be taken into account during the analysis of seismograms.
2.5.2.1 Snell’s Law for a flat Earth
From Fermat’s Principle follows, with some simple geometry and mathematics, Snell’s Law
of wave refraction (e.g., Aki and Richards 1980 and 2002; Lay and Wallace, 1995; Shearer,
1999; Červeny, 2001; Kennett, 2001):
sin i/v = s sin i = sx = 1/vapp ≡ p = constant (2.12)
where i is the angle of incidence, measured between the ray and the vertical (see Fig. 2.24), v
is the velocity of wave propagation in the medium, s =1/v is called slowness, and p is the socalled
ray parameter, v/sin i = vapp is the apparent horizontal wave propagation velocity in xdirection
with vapp = ∞ for i = 0 (vertical incidence of the ray) and sx = 1/vapp is the horizontal
component of the slowness vector s. Note, however, that p is constant for laterally
homogeneous media only. In Fig. 2.24 the refraction of a seismic wavefront and of a related
seismic ray across the interface of two half spaces with different but constant seismic
velocities v1 and v2 is sketched. Such an instantaneous velocity jump is called first-order
discontinuity. Because the ray parameter must remain constant across the interface, the ray
angle has to change:
sin i1/v1 = sin i2/v2 = s1 sin i1 = s2 sin i2. (2.13)
Fig. 2.24 A plane wavefront with the associated ray crossing a medium boundary with v2>v1.
The ray in medium two is refracted away from the vertical, i.e., i2>i1.
2.5.2.2 Snell's Law for the spherical Earth
Above, a flat-layered case was considered. Yet the Earth is a sphere and curvature has to be
taken into account at distances greater than about 12°. In this case the ray parameter has to be
26