2 Seismic Wave Propagation and Earth models

2. Seismic Wave Propagation and Earth models
A 0
A = n
x
ω t
−
2 Q
e
=
A
x
38
π x
−
0 Q T v
e n
=
A −
x
0 e n
ω x
2Q
v
, (2.31)
with A0/x n – the geometrical spreading term, exp(-x t/2Q) = exp(- π/Q T v) the attenuation
term, ω - angular frequency 2π /T, T – period of wave, t – travel time, v – propagation
velocity of wave, and n – exponential factor controlled by the kind of geometric spreading.
According to experimental data, n varies between about 0.3 and 3, depending also on the type
of seismic wave and distance range considered.
In ray theoretical methods, attenuation may be modeled through the use of the parameter t *
that is defined as the integrated value of the travel time divided by 1/Q
* dt
t = ∫ , (2.32)
path →
Q( r )
where →
r is the position vector. We can then write Eq. (2.31) as
A(ω) = A0(ω) e -ω t*/2 . (2.33)
Note, that P-wave attenuation Qα and S-wave attenuation Qβ differ. They are related to the
shear attenuation Qμ and the bulk attenuation Qκ by the relationships
Qβ = Qμ and 1/Qα = 4(β/α) 2 /3Qμ + [1 - 4(β/α) 2 /3]/Qκ. (2.34)
with P-wave velocity α = vp and S-wave velocity β = vs. In the Earth shear attenuation is
much stronger than bulk attenuation. While Qμ is smallest (and thus shear attenuation
strongest) in the upper mantle and the inner core, Qκ is generally assumed to be infinite,
except in the inner core. While the P- and S-wave velocities are rather well known and do not
differ much between different Earth models, the various model assumptions with respect to
Qα and Qβ as a function of depth still differ significantly (see Fig. 2.53). According to the
PREM model, Qμ is 600 for less than 80 km depth. It then drops between 80 and 220 km to
80, increases to 143 from 220 to 670 km, and is 312 for the lower mantle below 670 km
depth.
In practice, it is difficult to separate intrinsic attenuation and scattering Q. Particularly in local
earthquake records, which are strongly affected by scattering on crustal inhomogeneities,
scattering Q dominates. Scattering Q is usually determined from the decay of coda waves
following Sg (SmS) onsets (e.g., Fig. 2.40) and is called accordingly Qc. A full discussion on
these topics can be found,e.g., in Aki and Richards (1980; pp. 170-182).
In this context it should be mentioned that amplitudes of S waves are generally about five
times larger than those of P waves (see Fig. 2.3). This follows directly from Eq. (3.2) in
Chapter 3 or from the far-field term of the Green’s function when modeling earthquake shear
sources (see Equation (24) in the IS 3.1) taking into account that vP ≈ vS √3 (see this Chapter,
Eq. (2.9)). Also, the periods of S waves are longer than those of P waves, again by at least a
factor of √3, due to the differences in wave propagation velocity and the related differences in
the corner frequencies of the P- and S-wave source spectrum. Additionally, S waves are much
stronger attenuated than P waves (see following section), thus filtering out higher frequencies
more strongly. It should also be noted that S waves do not propagate in the fluid outer core