These equations allow us to find the the second nodal plane
and the slip vector on it (φ f2 , δ 2 , λ 2 ) from the first nodal plane
and the slip on it (φ f1 , δ 1 , λ 1 ). The hard part, getting the angles
in the appropriate quadrants, can be done by first finding δ 2
from Eqn 18, and then finding sin λ 2 from Eqn 19 and cos λ 2
by combining Eqns 16 and 17. Given both sine and cosine,
λ 2 can be placed in the correct quadrant. We then find φ f 2 from
Eqns 22 and 16. Finally, if 90° < δ 2 < 180°, we change (φ f2 , δ 2 ,
λ 2 ) to (180° + φ f2 , 180° − δ 2 , 360° − λ 2 ).
If the nodal planes have been found from first motions using
a stereonet, the situation differs because the strike and dip of
both planes are known, but the slip angles are not. We then
choose one nodal plane and find the slip angle on it. This can be
done using Eqns 16 and 18 to find cos λ 1 and sin λ 1 , and then
placing λ 1 in the correct quadrant.
4.3 Waveform modeling
As noted in the previous section, P-wave first motions are often
inadequate to constrain focal mechanisms. Additional information is obtained by comparing the observed body and surface waves to theoretical, or synthetic, waveforms computed
for various source parameters, and finding a model that best
fits the data, either by forward modeling or by inversion. Waveform analysis also gives information about the earthquake
depths and rupture processes which cannot be extracted from
the first motions. We discuss such analysis first for body waves
and then for surface waves.
4.3.1 Basic model
To generate synthetic waveforms, we regard the ground motion
recorded on a seismogram as a combination of factors: the
earthquake source, the earth structure through which the waves
propagated, and the seismometer. Each factor can be thought
of as an operation whose effects depend on the frequency of
the seismic waves. Hence it is often useful to think of the
seismogram u(t) in terms of its Fourier transform U(ω), which
represents the contribution of the different frequencies:
u t
U e d
i t
( )
( )
=
−∞
∞
1
2π
ω
ω
ω
Ύ
U
ute d t
i t
( )
( )
ω
ω
=
−∞
∞
−
Ύ
(1)
As as in earlier discussions (Sections 2.8, 3.3, 3.7), we use
the Fourier transform and related concepts while deferring
more general treatment of Fourier analysis to Chapter 6. The
essence of this approach is that we represent a seismogram or
individual factors that make it up either as a time series or by
its Fourier transform, depending on which is more convenient,
and switch back and forth using the transform and inverse
transform relations.
This approach to generating synthetic seismograms from
earthquakes is conceptually the same as that discussed in Section 3.3.6 for reflection seismograms. There, we described the
combined effect of various factors as the convolution of time
series representing each factor. Recall that the convolution of
two time series w(t) and r(t) is written
s(t) = w(t) * r(t) = Ύ
−∞
∞
w(t − τ)r(τ)dτ.
(2)
Thus a seismogram u(t) can be written
u(t) = x(t) * e(t) * q(t) * i(t),
(3)
where x(t) is the source time function, the “signal” the earthquake puts into the ground, e(t) and q(t) represent the effects
of earth structure, and i(t) describes the instrument response
of the seismometer. We also noted (and will prove in Section
6.3.1) that convolution in the time domain is equivalent to
multiplication in the frequency domain, so Eqn 3 can be
written as the product of Fourier transforms of the four factors
U(ω) = X(ω)E(ω)Q(ω)I(ω).
(4)
Each factor can be described in the time domain or the frequency domain. For example, the seismogram depends on how
the seismometer responds to ground motion of different frequencies. Figure 4.3-1 (top) shows the instrument response,
the amplification of a signal as a function of period, for a longperiod seismometer. Ground motion with periods around the
peak response (T = 15 s) is enhanced relative to that at longer
or shorter periods. As discussed in Section 6.6, seismometer
responses differ; some have peak response at short (e.g., 1 s)
periods, whereas others have better response at longer periods.
The seismometer response can also be described in the time
domain by taking its inverse Fourier transform (Fig. 4.3-1,
bottom). The resulting time series, i(t), is the impulse response,
describing how the seismometer responds to a sharp impulse.
For the seismometer illustrated in Fig. 4.3-1, the impulse
response has a sharp initial peak, followed by a smaller
“backswing.”
In this formulation, the effects of earth structure are divided
into two factors. One, e(t), gives the effect of reflections and
conversions of seismic waves at different interfaces along the
ray path and the effect of geometric spreading of the rays due
to the velocity structure (Section 3.4.2). All these effects are
elastic wave phenomena. There is also anelastic attenuation
described by q(t), whereby some of the seismic waves’ mechanical energy is lost by conversion into heat. Attenuation, discussed in Section 3.7, is illustrated by the decay with time of a
damped harmonic oscillation with frequency ω :
f(t) = Ae iωt e −ωt /2Q .
(5)
The quality factor Q characterizes the attenuation: the amplitude decays by e −1 in a time 2Q/ω (Fig. 3.7-11), so the higher
4.3 Waveform modeling 229
and the slip vector on it (φ f2 , δ 2 , λ 2 ) from the first nodal plane
and the slip on it (φ f1 , δ 1 , λ 1 ). The hard part, getting the angles
in the appropriate quadrants, can be done by first finding δ 2
from Eqn 18, and then finding sin λ 2 from Eqn 19 and cos λ 2
by combining Eqns 16 and 17. Given both sine and cosine,
λ 2 can be placed in the correct quadrant. We then find φ f 2 from
Eqns 22 and 16. Finally, if 90° < δ 2 < 180°, we change (φ f2 , δ 2 ,
λ 2 ) to (180° + φ f2 , 180° − δ 2 , 360° − λ 2 ).
If the nodal planes have been found from first motions using
a stereonet, the situation differs because the strike and dip of
both planes are known, but the slip angles are not. We then
choose one nodal plane and find the slip angle on it. This can be
done using Eqns 16 and 18 to find cos λ 1 and sin λ 1 , and then
placing λ 1 in the correct quadrant.
4.3 Waveform modeling
As noted in the previous section, P-wave first motions are often
inadequate to constrain focal mechanisms. Additional information is obtained by comparing the observed body and surface waves to theoretical, or synthetic, waveforms computed
for various source parameters, and finding a model that best
fits the data, either by forward modeling or by inversion. Waveform analysis also gives information about the earthquake
depths and rupture processes which cannot be extracted from
the first motions. We discuss such analysis first for body waves
and then for surface waves.
4.3.1 Basic model
To generate synthetic waveforms, we regard the ground motion
recorded on a seismogram as a combination of factors: the
earthquake source, the earth structure through which the waves
propagated, and the seismometer. Each factor can be thought
of as an operation whose effects depend on the frequency of
the seismic waves. Hence it is often useful to think of the
seismogram u(t) in terms of its Fourier transform U(ω), which
represents the contribution of the different frequencies:
u t
U e d
i t
( )
( )
=
−∞
∞
1
2π
ω
ω
ω
Ύ
U
ute d t
i t
( )
( )
ω
ω
=
−∞
∞
−
Ύ
(1)
As as in earlier discussions (Sections 2.8, 3.3, 3.7), we use
the Fourier transform and related concepts while deferring
more general treatment of Fourier analysis to Chapter 6. The
essence of this approach is that we represent a seismogram or
individual factors that make it up either as a time series or by
its Fourier transform, depending on which is more convenient,
and switch back and forth using the transform and inverse
transform relations.
This approach to generating synthetic seismograms from
earthquakes is conceptually the same as that discussed in Section 3.3.6 for reflection seismograms. There, we described the
combined effect of various factors as the convolution of time
series representing each factor. Recall that the convolution of
two time series w(t) and r(t) is written
s(t) = w(t) * r(t) = Ύ
−∞
∞
w(t − τ)r(τ)dτ.
(2)
Thus a seismogram u(t) can be written
u(t) = x(t) * e(t) * q(t) * i(t),
(3)
where x(t) is the source time function, the “signal” the earthquake puts into the ground, e(t) and q(t) represent the effects
of earth structure, and i(t) describes the instrument response
of the seismometer. We also noted (and will prove in Section
6.3.1) that convolution in the time domain is equivalent to
multiplication in the frequency domain, so Eqn 3 can be
written as the product of Fourier transforms of the four factors
U(ω) = X(ω)E(ω)Q(ω)I(ω).
(4)
Each factor can be described in the time domain or the frequency domain. For example, the seismogram depends on how
the seismometer responds to ground motion of different frequencies. Figure 4.3-1 (top) shows the instrument response,
the amplification of a signal as a function of period, for a longperiod seismometer. Ground motion with periods around the
peak response (T = 15 s) is enhanced relative to that at longer
or shorter periods. As discussed in Section 6.6, seismometer
responses differ; some have peak response at short (e.g., 1 s)
periods, whereas others have better response at longer periods.
The seismometer response can also be described in the time
domain by taking its inverse Fourier transform (Fig. 4.3-1,
bottom). The resulting time series, i(t), is the impulse response,
describing how the seismometer responds to a sharp impulse.
For the seismometer illustrated in Fig. 4.3-1, the impulse
response has a sharp initial peak, followed by a smaller
“backswing.”
In this formulation, the effects of earth structure are divided
into two factors. One, e(t), gives the effect of reflections and
conversions of seismic waves at different interfaces along the
ray path and the effect of geometric spreading of the rays due
to the velocity structure (Section 3.4.2). All these effects are
elastic wave phenomena. There is also anelastic attenuation
described by q(t), whereby some of the seismic waves’ mechanical energy is lost by conversion into heat. Attenuation, discussed in Section 3.7, is illustrated by the decay with time of a
damped harmonic oscillation with frequency ω :
f(t) = Ae iωt e −ωt /2Q .
(5)
The quality factor Q characterizes the attenuation: the amplitude decays by e −1 in a time 2Q/ω (Fig. 3.7-11), so the higher
4.3 Waveform modeling 229