380 Seismograms as Signals
expected ground motion as a function of earthquake magnitude and distance.
Often the impulse response is defined in both space and time.
This is the basic approach used to find the response of the earth
to a seismic source (Chapter 4). The displacement at a point x
and time t is
u(x, t) =
ΎΎ
G(x − x ′; t − t′) f(x′, t ′)dt ′dV ′,
(14)
where G(x − x′; t − t ′) is the Green’s function, 1 the impulse
response to a source at position x′ and time t ′, and f(x′, t ′) is
the distribution of seismic sources. Thus the integral gives the
total response due to the distribution of sources. In most cases
the source is limited in space and time, so the integral is done
over the source region. Often the source is at a point in space or
time, so f(x′, t ′) contains delta functions and is easily integrated using the sifting property. A nice feature of this formulation is that the principle of reciprocity, which says that the
source and the receiver can be interchanged, emerges directly.
The Green’s function in Eqn 14 is for a laterally homogeneous medium, so the response depends only on the distance
between the source and the receiver. In a general medium
Eqn 14 becomes
u(x, t) =
ΎΎ
G(x, t; x′, t′) f(x′, t′) dt′dV ′.
(15)
When a system is described by a convolution, we can examine the effects of the different contributing factors using
deconvolution. We start with the output and one of the time
series that were convolved to form it, and then find the other.
For example, in Section 3.3.6 we discussed using seismic reflection data to obtain the sharpest resolution of reflectors in the
earth. We assumed that a seismogram s(t) results from convolution of a source pulse, or wavelet, w(t), and an earth structure operator, r(t). r(t), known as a reflector series, is presumed
to be a set of delta functions with positions corresponding to
the travel time for a reflection from an interface and amplitudes
corresponding to the amplitude of the reflected arrival. Thus
s(t) = w(t) * r(t) and S(ω) = W(ω )R(ω).
(16)
If the travel time differences between the arrivals corresponding to individual reflectors are shorter than the duration
of the wavelet, interference can occur, giving a complicated
signal. Hence it would be desirable to have a delta function
source wavelet whose Fourier transform is simply 1, so that
the seismogram would equal the reflector series. Although a
physical source wavelet is not a delta function, we simulate
such a wavelet by creating an inverse filter 2 w −1 (t), which, when
convolved with the wavelet, yields a delta function:
w −1 (t) * w(t) = δ(t).
(17)
As we saw in Section 3.3.6, the Fourier transform of the inverse
filter is just 1/W(ω), so deconvolution can be done by dividing
the Fourier transforms
S(ω)/W(ω) = R(ω).
(18)
This sometimes works well, but can be problematic at frequencies where the source wavelet spectrum W(ω) is small (causing
R(ω) to go to infinity), so a minimum amplitude threshold can
be set.
As an alternative, inverse filters can be designed in the time
domain to compress the source wavelet into a function as close
to a delta function as possible. This approach is a special case of
the general problem of finding a shaping filter that converts a
given input into a given output. We will shortly discuss another
approach, which relies not on the convolution, but on the
related cross-correlation operator.
Deconvolution is also used in other applications. A conceptually similar one is modeling seismograms from a distant
earthquake as a sum of secondary arrivals generated when the
upcoming wave encounters interfaces below the receiver
(Fig. 6.3-7). The vertical component is assumed to represent
the direct arrival, and is used as a Green’s function that is
deconvolved from a horizontal component to find a receiver
function characterizing the structure. The receiver function
corresponds to the reflector series in this geometry. Another
application of deconvolution is to take seismograms and deconvolve the effects of the seismometer to find the true ground
motion, or deconvolve a seismogram to try to find the source
pulse due to an earthquake (Section 4.3.3).
6.3.3 Finite length signals
We have seen that the Fourier transform describes a signal as
the sum of harmonic signals with different frequencies. One
important limitation is that the Fourier transform requires integration over all time. In reality, we only have data over a finite
interval of time.
To see how this affects our results, consider a window function b(t) which selects part of the data. Its effect on the data f(t)
is represented by multiplying f(t) by b(t). We then ask how the
Fourier transform of the function, including the effect of the
window
G(ω) =
Ύ
−∞
∞
b(t) f(t)e −iωt dt,
(19)
is related to the transform of the original function, F(ω).
1 The same entity is commonly termed a Green’s function in physical problems and
an impulse response in time series analysis. In seismology the terms are used essentially interchangeably.
2 The notation w −1 (t) does not mean 1/w(t).
expected ground motion as a function of earthquake magnitude and distance.
Often the impulse response is defined in both space and time.
This is the basic approach used to find the response of the earth
to a seismic source (Chapter 4). The displacement at a point x
and time t is
u(x, t) =
ΎΎ
G(x − x ′; t − t′) f(x′, t ′)dt ′dV ′,
(14)
where G(x − x′; t − t ′) is the Green’s function, 1 the impulse
response to a source at position x′ and time t ′, and f(x′, t ′) is
the distribution of seismic sources. Thus the integral gives the
total response due to the distribution of sources. In most cases
the source is limited in space and time, so the integral is done
over the source region. Often the source is at a point in space or
time, so f(x′, t ′) contains delta functions and is easily integrated using the sifting property. A nice feature of this formulation is that the principle of reciprocity, which says that the
source and the receiver can be interchanged, emerges directly.
The Green’s function in Eqn 14 is for a laterally homogeneous medium, so the response depends only on the distance
between the source and the receiver. In a general medium
Eqn 14 becomes
u(x, t) =
ΎΎ
G(x, t; x′, t′) f(x′, t′) dt′dV ′.
(15)
When a system is described by a convolution, we can examine the effects of the different contributing factors using
deconvolution. We start with the output and one of the time
series that were convolved to form it, and then find the other.
For example, in Section 3.3.6 we discussed using seismic reflection data to obtain the sharpest resolution of reflectors in the
earth. We assumed that a seismogram s(t) results from convolution of a source pulse, or wavelet, w(t), and an earth structure operator, r(t). r(t), known as a reflector series, is presumed
to be a set of delta functions with positions corresponding to
the travel time for a reflection from an interface and amplitudes
corresponding to the amplitude of the reflected arrival. Thus
s(t) = w(t) * r(t) and S(ω) = W(ω )R(ω).
(16)
If the travel time differences between the arrivals corresponding to individual reflectors are shorter than the duration
of the wavelet, interference can occur, giving a complicated
signal. Hence it would be desirable to have a delta function
source wavelet whose Fourier transform is simply 1, so that
the seismogram would equal the reflector series. Although a
physical source wavelet is not a delta function, we simulate
such a wavelet by creating an inverse filter 2 w −1 (t), which, when
convolved with the wavelet, yields a delta function:
w −1 (t) * w(t) = δ(t).
(17)
As we saw in Section 3.3.6, the Fourier transform of the inverse
filter is just 1/W(ω), so deconvolution can be done by dividing
the Fourier transforms
S(ω)/W(ω) = R(ω).
(18)
This sometimes works well, but can be problematic at frequencies where the source wavelet spectrum W(ω) is small (causing
R(ω) to go to infinity), so a minimum amplitude threshold can
be set.
As an alternative, inverse filters can be designed in the time
domain to compress the source wavelet into a function as close
to a delta function as possible. This approach is a special case of
the general problem of finding a shaping filter that converts a
given input into a given output. We will shortly discuss another
approach, which relies not on the convolution, but on the
related cross-correlation operator.
Deconvolution is also used in other applications. A conceptually similar one is modeling seismograms from a distant
earthquake as a sum of secondary arrivals generated when the
upcoming wave encounters interfaces below the receiver
(Fig. 6.3-7). The vertical component is assumed to represent
the direct arrival, and is used as a Green’s function that is
deconvolved from a horizontal component to find a receiver
function characterizing the structure. The receiver function
corresponds to the reflector series in this geometry. Another
application of deconvolution is to take seismograms and deconvolve the effects of the seismometer to find the true ground
motion, or deconvolve a seismogram to try to find the source
pulse due to an earthquake (Section 4.3.3).
6.3.3 Finite length signals
We have seen that the Fourier transform describes a signal as
the sum of harmonic signals with different frequencies. One
important limitation is that the Fourier transform requires integration over all time. In reality, we only have data over a finite
interval of time.
To see how this affects our results, consider a window function b(t) which selects part of the data. Its effect on the data f(t)
is represented by multiplying f(t) by b(t). We then ask how the
Fourier transform of the function, including the effect of the
window
G(ω) =
Ύ
−∞
∞
b(t) f(t)e −iωt dt,
(19)
is related to the transform of the original function, F(ω).
1 The same entity is commonly termed a Green’s function in physical problems and
an impulse response in time series analysis. In seismology the terms are used essentially interchangeably.
2 The notation w −1 (t) does not mean 1/w(t).