Seismograms as Signals
We shall introduce the concepts of signal and noise. We define the signal as the desired part of the data and the noise as the unwanted
part. Our definition of signal and noise is subjective in the sense that a given part of the data is “signal” for those who know how to
analyze and interpret the data, but it is “noise” for those who do not. For example, for many years the times of the first arrivals of
P- and S-waves were the only signals conveyed by an earthquake, and the rest of the seismogram, such as surface waves and coda
waves, had to be considered as useless until appropriate methods of interpretations were found.
Thus, through the application of a new technique to old data, an analyst can experience a moment of discovery as joyful as a data
gatherer does using a new observational device.
Aki and Richards, Quantitative Seismology, 1980
6.1 Introduction
Seismology uses various techniques to study the displacement
field as a function of position and time associated with elastic
waves in the earth, and to draw inferences from it about the
nature of seismic sources and the earth. Although some techniques depend on specific aspects of seismic waves in the earth,
others rely on general properties of functions of space and time.
We thus often use a class of techniques known as signal
processing or time series analysis. Signal processing considers
functions of time or space, also called series or signals, in general terms without regard to the specific physics involved. As a
result, many wave propagation subjects, including seismology,
radar, sonar, and optics, can be treated in similar ways. The
signals can have different forms. For example, in seismology,
we can treat either a continuous (analog) record of ground
motion or the digital data that result from representing the
ground motion as being sampled at discrete intervals, providing numbers that can be manipulated using a computer.
In general terms, we can think of filtering a signal, or applying some operation that modifies the signal. We have already
discussed several examples. A seismometer is a filter, in that it
yields a record of ground motion that differs from the actual
ground motion. Similarly, processes in the earth such as dispersion or attenuation have effects that can be described as a
filter acting on the wave field. We can also consciously apply
filters to enhance parts of a seismogram or seismic wave field
and suppress others. In this chapter we extend these ideas by
considering mathematical approaches that are common to
such applications and then seeing how these approaches give
additional insight into the physical processes. We discuss some
basic concepts and provide references at the end of the chapter
for more extensive treatments.
6.2 Fourier analysis
6.2.1 Fourier series
In many applications, we use an approach based on the idea
that any time series can be decomposed into the sum or integral
of harmonic waves of different frequencies, using methods
known as Fourier analysis. We derived the properties of seismic
waves using a harmonic wave, a sinusoid of a single frequency,
and noted that any wave could be treated as the sum of
harmonic waves. Thus we showed that waves on a string could
be viewed as the sum of the string’s normal modes, or standing
waves (Section 2.2.5), and that waves in a spherical earth can
be written as the sum of the earth’s normal modes (Section 2.9).
This concept is especially useful when the components with
various frequencies behave differently. For example, surface
waves of different frequencies have different apparent velocities (Section 2.8) and seismic wave attenuation varies with
frequency (Section 3.7). Similarly, we will see shortly that
seismometers respond differently to ground motion of different
frequencies. Fourier analysis lets us decompose the signal
into harmonic waves, consider each harmonic wave separately,
and then recombine the harmonic waves. Thus we use this
approach to analyze situations where the effect of the earth or a
seismometer can be described by a filter. We also use Fourier
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We shall introduce the concepts of signal and noise. We define the signal as the desired part of the data and the noise as the unwanted
part. Our definition of signal and noise is subjective in the sense that a given part of the data is “signal” for those who know how to
analyze and interpret the data, but it is “noise” for those who do not. For example, for many years the times of the first arrivals of
P- and S-waves were the only signals conveyed by an earthquake, and the rest of the seismogram, such as surface waves and coda
waves, had to be considered as useless until appropriate methods of interpretations were found.
Thus, through the application of a new technique to old data, an analyst can experience a moment of discovery as joyful as a data
gatherer does using a new observational device.
Aki and Richards, Quantitative Seismology, 1980
6.1 Introduction
Seismology uses various techniques to study the displacement
field as a function of position and time associated with elastic
waves in the earth, and to draw inferences from it about the
nature of seismic sources and the earth. Although some techniques depend on specific aspects of seismic waves in the earth,
others rely on general properties of functions of space and time.
We thus often use a class of techniques known as signal
processing or time series analysis. Signal processing considers
functions of time or space, also called series or signals, in general terms without regard to the specific physics involved. As a
result, many wave propagation subjects, including seismology,
radar, sonar, and optics, can be treated in similar ways. The
signals can have different forms. For example, in seismology,
we can treat either a continuous (analog) record of ground
motion or the digital data that result from representing the
ground motion as being sampled at discrete intervals, providing numbers that can be manipulated using a computer.
In general terms, we can think of filtering a signal, or applying some operation that modifies the signal. We have already
discussed several examples. A seismometer is a filter, in that it
yields a record of ground motion that differs from the actual
ground motion. Similarly, processes in the earth such as dispersion or attenuation have effects that can be described as a
filter acting on the wave field. We can also consciously apply
filters to enhance parts of a seismogram or seismic wave field
and suppress others. In this chapter we extend these ideas by
considering mathematical approaches that are common to
such applications and then seeing how these approaches give
additional insight into the physical processes. We discuss some
basic concepts and provide references at the end of the chapter
for more extensive treatments.
6.2 Fourier analysis
6.2.1 Fourier series
In many applications, we use an approach based on the idea
that any time series can be decomposed into the sum or integral
of harmonic waves of different frequencies, using methods
known as Fourier analysis. We derived the properties of seismic
waves using a harmonic wave, a sinusoid of a single frequency,
and noted that any wave could be treated as the sum of
harmonic waves. Thus we showed that waves on a string could
be viewed as the sum of the string’s normal modes, or standing
waves (Section 2.2.5), and that waves in a spherical earth can
be written as the sum of the earth’s normal modes (Section 2.9).
This concept is especially useful when the components with
various frequencies behave differently. For example, surface
waves of different frequencies have different apparent velocities (Section 2.8) and seismic wave attenuation varies with
frequency (Section 3.7). Similarly, we will see shortly that
seismometers respond differently to ground motion of different
frequencies. Fourier analysis lets us decompose the signal
into harmonic waves, consider each harmonic wave separately,
and then recombine the harmonic waves. Thus we use this
approach to analyze situations where the effect of the earth or a
seismometer can be described by a filter. We also use Fourier
6