7 Inverse Problems
Most people, if you describe a train of events to them, will tell you what the result would be. There are few people, however, who,
if you told them a result, would be able to evolve from their own inner consciousness what the steps were which led up to that result.
This power is what I mean when I talk of reasoning backwards.
Sherlock Holmes, in A Study in Scarlet by Arthur Conan Doyle
7.1 Introduction
Throughout this book we have noted that seismology is largely
directed at solving inverse problems dealing with earthquake
sources and earth structure. We start with the end result,
seismograms, and work backwards to characterize the earthquakes that generated the seismic waves and the medium
through which the waves passed. To do this, we first addressed
the forward problems of how features of seismic waves that are
observable from seismograms, such as travel times, amplitudes,
waveforms, eigenfrequencies, dispersion, and attenuation,
depend on the seismic source and the medium. We have also
discussed how the properties of the medium and the source,
such as velocity structure and earthquake mechanisms, reflect
tectonic processes within the earth. These are specific examples
of the fundamental question of what we can say about the earth
from seismological and other observations at its surface.
We now end our discussions by addressing some issues in
solving inverse problems. Inverse problems can be posed by
assuming that we understand the physics of a process which,
for a set of model parameters described by a vector m, gives rise
to a set of observed data described by the vector d. The data
can thus be considered the result of a function, or operator, A
acting on the model parameters,
d = A(m).
(1)
The forward problem, predicting the data d that would result
from a given model described by m, is tractable if we understand the process. The corresponding inverse problem, finding
what gave rise to a specific set of observed data, is more difficult. We assume that some physical model describes the process, and then use the data to estimate a set of model parameters
that are consistent with the data. We solve the inverse problem
using either mathematical inverse techniques to find m directly
from d, or trial-and-error techniques that solve the forward
problem repeatedly and look for the best solution. Each
approach has advantages in some applications.
We have already mentioned solving inverse problems in
contexts including studying the cooling of oceanic lithosphere
using surface wave dispersion (Section 2.8.3), inverting travel
time and amplitude data to find earth structure (Chapter 3),
inverting polarity, waveform, and geodetic data to study
earthquake mechanisms (Chapter 4), and using earthquake
mechanisms to study plate motions and regional tectonics
(Chapter 5). We have noted (Section 1.1.2) that although forward problems typically can be solved in a straightforward
way, giving a unique solution, inverse problems often have
no unique, exact, or “correct” solutions. Because the data are
generally somewhat inconsistent due to errors, and our models
simplify complex reality, no model exactly describes the data.
Similarly, a range of parameters can describe the data equally
well for a given model, and we have various models to choose
from based on various criteria and preconceptions. Moreover,
the data are often insufficient to resolve aspects of the model.
We can thus only recognize and accept these limitations on the
solutions. 1
A consequence of these limitations is a trade-off between
the model’s resolution, how detailed it is, and its stability, or
robustness. For example, inverting travel times with simple
earthquake location algorithms using a laterally homogeneous
velocity model shows the Wadati–Benioff zones of dipping
seismicity. These results are stable, in that they do not depend
1 This situation is summarized by the title of a paper “Interpretation of inaccurate,
insufficient, and inconsistent data” (Jackson, 1972).
Most people, if you describe a train of events to them, will tell you what the result would be. There are few people, however, who,
if you told them a result, would be able to evolve from their own inner consciousness what the steps were which led up to that result.
This power is what I mean when I talk of reasoning backwards.
Sherlock Holmes, in A Study in Scarlet by Arthur Conan Doyle
7.1 Introduction
Throughout this book we have noted that seismology is largely
directed at solving inverse problems dealing with earthquake
sources and earth structure. We start with the end result,
seismograms, and work backwards to characterize the earthquakes that generated the seismic waves and the medium
through which the waves passed. To do this, we first addressed
the forward problems of how features of seismic waves that are
observable from seismograms, such as travel times, amplitudes,
waveforms, eigenfrequencies, dispersion, and attenuation,
depend on the seismic source and the medium. We have also
discussed how the properties of the medium and the source,
such as velocity structure and earthquake mechanisms, reflect
tectonic processes within the earth. These are specific examples
of the fundamental question of what we can say about the earth
from seismological and other observations at its surface.
We now end our discussions by addressing some issues in
solving inverse problems. Inverse problems can be posed by
assuming that we understand the physics of a process which,
for a set of model parameters described by a vector m, gives rise
to a set of observed data described by the vector d. The data
can thus be considered the result of a function, or operator, A
acting on the model parameters,
d = A(m).
(1)
The forward problem, predicting the data d that would result
from a given model described by m, is tractable if we understand the process. The corresponding inverse problem, finding
what gave rise to a specific set of observed data, is more difficult. We assume that some physical model describes the process, and then use the data to estimate a set of model parameters
that are consistent with the data. We solve the inverse problem
using either mathematical inverse techniques to find m directly
from d, or trial-and-error techniques that solve the forward
problem repeatedly and look for the best solution. Each
approach has advantages in some applications.
We have already mentioned solving inverse problems in
contexts including studying the cooling of oceanic lithosphere
using surface wave dispersion (Section 2.8.3), inverting travel
time and amplitude data to find earth structure (Chapter 3),
inverting polarity, waveform, and geodetic data to study
earthquake mechanisms (Chapter 4), and using earthquake
mechanisms to study plate motions and regional tectonics
(Chapter 5). We have noted (Section 1.1.2) that although forward problems typically can be solved in a straightforward
way, giving a unique solution, inverse problems often have
no unique, exact, or “correct” solutions. Because the data are
generally somewhat inconsistent due to errors, and our models
simplify complex reality, no model exactly describes the data.
Similarly, a range of parameters can describe the data equally
well for a given model, and we have various models to choose
from based on various criteria and preconceptions. Moreover,
the data are often insufficient to resolve aspects of the model.
We can thus only recognize and accept these limitations on the
solutions. 1
A consequence of these limitations is a trade-off between
the model’s resolution, how detailed it is, and its stability, or
robustness. For example, inverting travel times with simple
earthquake location algorithms using a laterally homogeneous
velocity model shows the Wadati–Benioff zones of dipping
seismicity. These results are stable, in that they do not depend
1 This situation is summarized by the title of a paper “Interpretation of inaccurate,
insufficient, and inconsistent data” (Jackson, 1972).