192 Seismology and Earth Structure
Energy decays faster than the amplitude, because the negative
exponent in Eqn 22 is twice as large as in Eqn 21.
3.7.6 Quality factor, Q
The solution for the damped harmonic oscillator incorporated
the damping through the quality factor, Q. Attenuation for
seismic waves and a variety of other physical phenomena are
often discussed in terms of Q or Q −1 . Although Q has more
convenient values, Q −1 has the advantage that it is directly,
rather than inversely, proportional to the damping. In some
cases, Q is used to describe the decay of an oscillation, whereas
in others it is used to describe the physical properties of the
system that cause a disturbance to attenuate. For example,
the Q of one of the earth’s normal modes, which is directly
analogous to a damped oscillator, describes how the mode
decays with time. This decay results from the distribution of
material in the earth that causes seismic energy to be lost
to heat. This distribution can be described in terms of a Q,
or anelastic attenuation, structure analogous to the elastic
velocity structure.
As a result, we speak of the Q of surface waves, body waves,
and crustal phases like Lg. We also speak of the variation
within the earth of Q α and Q β , which controls the attenuation
of P and S waves. The anelastic structure of the earth, given by
variations in Q α and Q β , is analogous to the elastic velocity
structure because Q can be viewed mathematically as an
imaginary part of the velocity. To see this, note that (9), which
we used to derive the decaying oscillation, can be viewed as an
oscillation with a complex frequency p
u(t) = A 0 e ipt = A 0 e i(a + ib)t
(23)
where the real and imaginary parts of the frequency are
a = ω
b = ω* = ω 0 /2Q ≈ ω/2Q
(24)
assuming that attenuation is small (Q large) enough that ω ≈
ω 0 . Hence we write
Q −1 = 2b/a = 2ω*/ω.
(25)
Treating the attenuation as an imaginary part of the
frequency and dividing by the wavenumber lets us treat the
corresponding velocity for a propagating wave as complex,
c + ic* = ω/k + iω*/k = ω/k + iωQ
−1 /2k
(26)
so
Q −1 = 2c*/c.
(27)
Thus we can express the attenuation of P- and S-waves by using
the quality factors Q α and Q β to give imaginary parts to the
velocities. If there is no attenuation (Q = ∞) the frequency and
the velocity have no imaginary parts. This formulation is useful
because it means that methods used to invert surface wave
velocities or normal mode eigenfrequencies to find velocity
in the earth can also be used to invert observations of their
attenuation to find the distribution of anelasticity.
We pose the complex parts of the velocities in terms of the
properties of the material causing attenuation by treating the
elastic moduli as having imaginary parts. For the shear velocity
β + iβ * = β(1 + iQ −1
β /2)
= ((µ + iµ*)/ρ) 1/2 = β(1 + iµ*/µ) 1/2
Ϸ β(1 + iµ*/2µ)
(28)
where the last step used the first term of the Taylor series,
because the attenuation and hence imaginary part is small.
Comparing terms shows that
Q −1
β = µ*/µ.
(29)
A similar analysis shows that the quality factor for P waves is
given by the imaginary parts of both the bulk and shear moduli
Q −1
α = (K* + 4/3µ*)/(K + 4/3µ).
(30)
Physically, it is useful to think of energy as being lost in either
compressional or shear deformation, so we express their
attenuation in terms of imaginary parts of the compressibility
and rigidity
Q −1
K = K*/K
Q −1
µ = µ*/µ = Q −1
β .
(31)
These quality factors are related to those for the velocities by
Q
−1
α = LQ
−1
µ + (1 − L)Q
−1
K
L = (4/3)( β/α)
2
.
(32)
In general little energy is lost in compression, so Q −1
K is very
small, and thus most of the attenuation for P waves occurs in
shear, making Q
−1
α ≈ (4/9)Q
−1
β .
Techniques for measuring Q in the earth follow from those
used to measure Q for the decay of an oscillation. From Eqn
20, taking the natural logarithm of the envelope shows that
ln A(t) = ln A 0 − ω 0 t/2Q,
(33)
so Q can be found from the slope of the logarithmic decay.
Alternatively, if successive peaks one full period T = 2π /ω 0
apart have amplitudes
A 1 (t 1 ) = A 0 exp (−ω 0 t 1 /2Q) and
A 2 (t 1 + T) = A 0 exp (−ω 0 (t 1 + T )/2Q),
(34)
their ratio is
A 1 /A 2 = exp [−ω 0 t 1 /2Q − ω 0 (t 1 + T )/2Q] = exp (π /Q),
(35)
Energy decays faster than the amplitude, because the negative
exponent in Eqn 22 is twice as large as in Eqn 21.
3.7.6 Quality factor, Q
The solution for the damped harmonic oscillator incorporated
the damping through the quality factor, Q. Attenuation for
seismic waves and a variety of other physical phenomena are
often discussed in terms of Q or Q −1 . Although Q has more
convenient values, Q −1 has the advantage that it is directly,
rather than inversely, proportional to the damping. In some
cases, Q is used to describe the decay of an oscillation, whereas
in others it is used to describe the physical properties of the
system that cause a disturbance to attenuate. For example,
the Q of one of the earth’s normal modes, which is directly
analogous to a damped oscillator, describes how the mode
decays with time. This decay results from the distribution of
material in the earth that causes seismic energy to be lost
to heat. This distribution can be described in terms of a Q,
or anelastic attenuation, structure analogous to the elastic
velocity structure.
As a result, we speak of the Q of surface waves, body waves,
and crustal phases like Lg. We also speak of the variation
within the earth of Q α and Q β , which controls the attenuation
of P and S waves. The anelastic structure of the earth, given by
variations in Q α and Q β , is analogous to the elastic velocity
structure because Q can be viewed mathematically as an
imaginary part of the velocity. To see this, note that (9), which
we used to derive the decaying oscillation, can be viewed as an
oscillation with a complex frequency p
u(t) = A 0 e ipt = A 0 e i(a + ib)t
(23)
where the real and imaginary parts of the frequency are
a = ω
b = ω* = ω 0 /2Q ≈ ω/2Q
(24)
assuming that attenuation is small (Q large) enough that ω ≈
ω 0 . Hence we write
Q −1 = 2b/a = 2ω*/ω.
(25)
Treating the attenuation as an imaginary part of the
frequency and dividing by the wavenumber lets us treat the
corresponding velocity for a propagating wave as complex,
c + ic* = ω/k + iω*/k = ω/k + iωQ
−1 /2k
(26)
so
Q −1 = 2c*/c.
(27)
Thus we can express the attenuation of P- and S-waves by using
the quality factors Q α and Q β to give imaginary parts to the
velocities. If there is no attenuation (Q = ∞) the frequency and
the velocity have no imaginary parts. This formulation is useful
because it means that methods used to invert surface wave
velocities or normal mode eigenfrequencies to find velocity
in the earth can also be used to invert observations of their
attenuation to find the distribution of anelasticity.
We pose the complex parts of the velocities in terms of the
properties of the material causing attenuation by treating the
elastic moduli as having imaginary parts. For the shear velocity
β + iβ * = β(1 + iQ −1
β /2)
= ((µ + iµ*)/ρ) 1/2 = β(1 + iµ*/µ) 1/2
Ϸ β(1 + iµ*/2µ)
(28)
where the last step used the first term of the Taylor series,
because the attenuation and hence imaginary part is small.
Comparing terms shows that
Q −1
β = µ*/µ.
(29)
A similar analysis shows that the quality factor for P waves is
given by the imaginary parts of both the bulk and shear moduli
Q −1
α = (K* + 4/3µ*)/(K + 4/3µ).
(30)
Physically, it is useful to think of energy as being lost in either
compressional or shear deformation, so we express their
attenuation in terms of imaginary parts of the compressibility
and rigidity
Q −1
K = K*/K
Q −1
µ = µ*/µ = Q −1
β .
(31)
These quality factors are related to those for the velocities by
Q
−1
α = LQ
−1
µ + (1 − L)Q
−1
K
L = (4/3)( β/α)
2
.
(32)
In general little energy is lost in compression, so Q −1
K is very
small, and thus most of the attenuation for P waves occurs in
shear, making Q
−1
α ≈ (4/9)Q
−1
β .
Techniques for measuring Q in the earth follow from those
used to measure Q for the decay of an oscillation. From Eqn
20, taking the natural logarithm of the envelope shows that
ln A(t) = ln A 0 − ω 0 t/2Q,
(33)
so Q can be found from the slope of the logarithmic decay.
Alternatively, if successive peaks one full period T = 2π /ω 0
apart have amplitudes
A 1 (t 1 ) = A 0 exp (−ω 0 t 1 /2Q) and
A 2 (t 1 + T) = A 0 exp (−ω 0 (t 1 + T )/2Q),
(34)
their ratio is
A 1 /A 2 = exp [−ω 0 t 1 /2Q − ω 0 (t 1 + T )/2Q] = exp (π /Q),
(35)