416 Inverse Problems
the reference model does not improve significantly. When this
occurs, we are probably doing about as well as possible with
this type of model. For example, as discussed in Section 3.5,
laterally homogeneous global seismic velocity models have
become sufficiently accurate that more attention is now directed toward the lateral variations.
In this chapter, we discuss several inverse problems to introduce some of the methods used. Because such inverse problems are crucial to seismology and the earth sciences, and also
appear in other sciences, considerable attention has been directed toward them. It turns out that physically quite different
problems are often described in mathematically similar ways.
Our goal is to identity some common themes and approaches,
rather than discuss the details. Some more sophisticated treatments are listed in the suggested reading.
7.2 Earthquake location
We first consider the classic inverse problem of locating an
earthquake and finding its origin time using the arrival times of
seismic waves at various stations. The velocity structure, which
determines the ray paths and hence travel times, is crucial. We
first regard the velocity structure as known, and then explore
how it can also be estimated from the travel times.
7.2.1 Theory
Assume that an earthquake occurred at an unknown time t, at
an unknown position x = (x, y, z), known as the hypocenter,
or focus (Fig. 7.2-1). The point (x, y) on the surface above the
focus is called the epicenter. n seismic stations at locations
x i = (x i , y i , z i ) detect the earthquake at arrival times d i ′, which
depend on the origin time t and the travel time between the
source and the station T(x, x i ):
d i ′ = T(x, x i ) + t.
(1)
If the velocity structure is known, the forward problem can be
written using the formulation
Table 7.1-1 Some large-scale reference models.
Model for
Laterally homogeneous
earth structure
Relative plate motions
Thermal evolution of
oceanic lithosphere
Observables inverted and predicted
Travel times, eigenfrequencies
Rates and azimuths of plate motion
Variation with age in depth, heat
flow, and geoid
Parameters estimated
Average velocity and density
versus depth
Euler vectors
Plate thickness, asthenospheric
temperature, physical
properties (e.g., a, k, k)
Misfits (“anomalies”) indicate
Lateral velocity variation (subduction
zones, continental–ocean differences,
etc.)
Nonrigid plate behavior (plate
interiors and boundary zones)
Lateral thermomechanical variations
(swells, etc.)
significantly on the details of the location algorithm and velocity model, but have only limited resolution for where in the
slab the earthquakes occur. More detailed locations, which
are more useful for relating the earthquakes to the physics
of subduction, can be derived from sophisticated location
algorithms using a laterally variable velocity model that better
represents the slab. However, the improved resolution comes
at the price of stability, in that it depends on the specific velocity model used.
The results of inverse studies can be viewed in terms of
two end members. In one, we use an individual set of data to
characterize a specific phenomenon, such as the location of an
earthquake or the velocity structure in a specific area. In others,
we describe a set of data averaged over a region or the whole
earth with a simple physical model characterized by a relatively
small, or sparse, set of parameters. Such reference models a the
physical model with a specific set of parameters a are used
to characterize large sets of data in a simple way, predict data
where no observations exist, and thus identify misfits, or
“anomalies,” where the data deviate from the model predictions
and hence the global average. We then use reference models
to draw inferences about the processes that give rise to both the
average situation and deviations from it. For example, body
wave, surface wave, and normal mode data give average global
velocity structure. This structure is used to constrain models of
the average radial variations in composition and temperature,
and as a reference against which velocity perturbations due to
subducting slabs, continental roots, hot spots, ridges, etc. can
be identified and analyzed in terms of local processes that perturb the global model. As shown in Table 7.1-1, we can view
other reference models in a similar way. For example, the Euler
vectors describing a plate’s motion are a simple description
of its behavior, and places where earthquake mechanisms
differ from these predictions indicate deviations from rigid
plate behavior. Similarly, simple cooling models of the oceanic
lithosphere describe the average variations in depth, heat flow,
and the geoid, and so give a reference model for the temperature against which other effects can be identified and modeled.
As illustrated in Fig. 1.1-8, the models are refined over time
using new data and model parameterizations. Eventually,
the reference model does not improve significantly. When this
occurs, we are probably doing about as well as possible with
this type of model. For example, as discussed in Section 3.5,
laterally homogeneous global seismic velocity models have
become sufficiently accurate that more attention is now directed toward the lateral variations.
In this chapter, we discuss several inverse problems to introduce some of the methods used. Because such inverse problems are crucial to seismology and the earth sciences, and also
appear in other sciences, considerable attention has been directed toward them. It turns out that physically quite different
problems are often described in mathematically similar ways.
Our goal is to identity some common themes and approaches,
rather than discuss the details. Some more sophisticated treatments are listed in the suggested reading.
7.2 Earthquake location
We first consider the classic inverse problem of locating an
earthquake and finding its origin time using the arrival times of
seismic waves at various stations. The velocity structure, which
determines the ray paths and hence travel times, is crucial. We
first regard the velocity structure as known, and then explore
how it can also be estimated from the travel times.
7.2.1 Theory
Assume that an earthquake occurred at an unknown time t, at
an unknown position x = (x, y, z), known as the hypocenter,
or focus (Fig. 7.2-1). The point (x, y) on the surface above the
focus is called the epicenter. n seismic stations at locations
x i = (x i , y i , z i ) detect the earthquake at arrival times d i ′, which
depend on the origin time t and the travel time between the
source and the station T(x, x i ):
d i ′ = T(x, x i ) + t.
(1)
If the velocity structure is known, the forward problem can be
written using the formulation
Table 7.1-1 Some large-scale reference models.
Model for
Laterally homogeneous
earth structure
Relative plate motions
Thermal evolution of
oceanic lithosphere
Observables inverted and predicted
Travel times, eigenfrequencies
Rates and azimuths of plate motion
Variation with age in depth, heat
flow, and geoid
Parameters estimated
Average velocity and density
versus depth
Euler vectors
Plate thickness, asthenospheric
temperature, physical
properties (e.g., a, k, k)
Misfits (“anomalies”) indicate
Lateral velocity variation (subduction
zones, continental–ocean differences,
etc.)
Nonrigid plate behavior (plate
interiors and boundary zones)
Lateral thermomechanical variations
(swells, etc.)
significantly on the details of the location algorithm and velocity model, but have only limited resolution for where in the
slab the earthquakes occur. More detailed locations, which
are more useful for relating the earthquakes to the physics
of subduction, can be derived from sophisticated location
algorithms using a laterally variable velocity model that better
represents the slab. However, the improved resolution comes
at the price of stability, in that it depends on the specific velocity model used.
The results of inverse studies can be viewed in terms of
two end members. In one, we use an individual set of data to
characterize a specific phenomenon, such as the location of an
earthquake or the velocity structure in a specific area. In others,
we describe a set of data averaged over a region or the whole
earth with a simple physical model characterized by a relatively
small, or sparse, set of parameters. Such reference models a the
physical model with a specific set of parameters a are used
to characterize large sets of data in a simple way, predict data
where no observations exist, and thus identify misfits, or
“anomalies,” where the data deviate from the model predictions
and hence the global average. We then use reference models
to draw inferences about the processes that give rise to both the
average situation and deviations from it. For example, body
wave, surface wave, and normal mode data give average global
velocity structure. This structure is used to constrain models of
the average radial variations in composition and temperature,
and as a reference against which velocity perturbations due to
subducting slabs, continental roots, hot spots, ridges, etc. can
be identified and analyzed in terms of local processes that perturb the global model. As shown in Table 7.1-1, we can view
other reference models in a similar way. For example, the Euler
vectors describing a plate’s motion are a simple description
of its behavior, and places where earthquake mechanisms
differ from these predictions indicate deviations from rigid
plate behavior. Similarly, simple cooling models of the oceanic
lithosphere describe the average variations in depth, heat flow,
and the geoid, and so give a reference model for the temperature against which other effects can be identified and modeled.
As illustrated in Fig. 1.1-8, the models are refined over time
using new data and model parameterizations. Eventually,