2 Seismic Wave Propagation and Earth models

2. Seismic Wave Propagation and Earth models
Eqs. (2.7) and (2.8) are solutions of the wave equation for the propagation of two independent
types of seismic body waves, namely longitudinal (compressional - dilatational) P waves and
transverse (shear) S waves. Their velocities are
and
vp =
λ + 2μ
=
ρ
6
κ + 4μ
/ 3
ρ
(2.9)
μ
vs = . (2.10)
ρ
Accordingly, for a Poisson solid with λ = μ vp/vs = 3 . This comes close to the vp/vs ratio of
consolidated sedimentary and igneous rocks in the Earth’s crust (see Tab. 2.1). Eqs. (2.9) and
2.10) also mean that P (primary) waves travel significantly faster than S (secondary) waves
and thus arrive ahead of S in a seismic record (see Fig. 2.3). The Poisson’s ratio is often used
as a measure of the vp/vs ratio, namely
σ = (vp 2 /vs 2 – 2)/2(vp 2 /vs 2 – 1) (2.11)
Fig. 2.3 The three components of ground-velocity proportional digital records of the P and S
waves from a local event, an aftershock of the Killari-Latur earthquake, India (18.10.1993), at
a hypocentral distance of about 5.3 km.
Note the simple transient waveform (wavelet) of P in the Z-component of Fig. 2.3. The
waveform and duration of the primary body wave is related to the shape and duration of the
source-time function. It is for an earthquake shear rupture usually a more or less complex
displacement step (see Figs. 2.4 and 3.4) which can be described by the moment-release
function M(t) (see 3.5). In the far-field, i.e., at distances larger than the source dimension and
several wavelengths of the considered signal, the related displacement u(t) looks, in the
idealized case, bell-shaped and identical with the moment-rate M (t)
& (or velocity source-time)
function (see Fig. 2.4 middle). The base-width of this far-field displacement source pulse u(t)
corresponds to the duration of displacement at the source (for examples see Fig. 3.7).
However, usually broadband seismometers record ground velocity u& (t)
instead of ground
displacement. The recorded waveform then looks similar to the ones seen in Fig. 2.3 and Fig.