1Introduction
1.1 General Comments
Given a physical system, the ‘forward’ or ‘direct’ problem consists, by definition, in using a physical theory to
predict the outcome of possible experiments. In classical physics this problem has a unique solution. For instance,
given a seismic model of the whole Earth (elastic constants, attenuation, etc. at every point inside the Earth) and
given a model of a seismic source, we can use current seismological theories to predict which seismograms should
be observed at given locations at the Earth’s surface.
The ‘inverse problem’ arises when we do not have a good model of the Earth, or a good model of the seismic
source, but we have a set of seismograms, and we wish to use these observations to infer the internal Earth structure
or a model of the source (typically we try to infer both).
Many factors make the inverse problem underdetermined (non-unique). In the seismic example, two different
Earth models may predict the same seismograms
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, the finite bandwidth of our data will never allow us to resolve
very small features of the Earth model, and there are always experimental uncertainties that allow different models
to be ‘acceptable’.
The term ‘inverse problem’ is widely used. The authors of this text only like this name moderately, as we
see the problem more as a problem of ‘conjunction of states of information’ (theoretical, experimental and prior
information). In fact, the equations used below have a range of applicability well beyond ‘inverse problems’:
they can be used, for instance, to predict the values of observations in a realistic situation where the parameters
describing the Earth model are not ‘given’, but only known approximately.
We take here a probabilistic point of view. The axioms of probability theory apply to different situations. One is
the traditional statistical analysis of random phenomena, another one is the description of (more or less) subjective
states of information on a system. For instance, estimation of the uncertainties attached to any measurement
usually involves both uses of probability theory: some uncertainties contributing to the total uncertainty are
estimated using statistics, while some other uncertainties are estimated using informed scientific judgement about
the quality of an instrument, about effects not explicitly taken into account, etc. The International Organization
for Standardization (ISO) in Guide to the Expression of Uncertainty in Measurement (1993), recommends that
the uncertainties evaluated by statistical methods are named ‘type A’ uncertainties, and those evaluated by other
means (for instance, using Bayesian arguments) are named ‘type B’ uncertainties. It also recommends that former
classifications, for instance into ‘random’ and ‘systematic uncertainties’, should be avoided. In the present text, we
accept ISO’s basic point of view, and extend it by downplaying the role assigned by ISO to the particular Gaussian
model for uncertainties (see section 4.3) and by not assuming that the uncertainties are ‘small’.
In fact, we like to think of an ‘inverse’ problem as merely a ‘measurement’. A measurement that can be quite
complex, but the basic principles and the basic equations to be used are the same for a relatively complex ‘inverse
problem’ as for a relatively simple ‘measurement’.
We do not normally use, in this text, the term ‘random variable’, as we assume that we have probability
distributions over ‘physical quantities’. This a small shift in terminology that we hope will not disorient the reader.
An important theme of this paper is invariant formulation of inverse problems, in the sense that solutions
obtained using different, equivalent, sets of parameters should be consistent, i.e., probability densities obtained as
the solution of an inverse problem, using two different set of parameters, should be related through the well known
rule of multiplication by the Jacobian of the transformation.
This paper is organized as follows. After a brief historical review of inverse problem theory, with special
emphasis on seismology, we give a small introduction to probability theory. In addition to being a tutorial, this
introduction also aims at fixing a serious problem of classical probability, namely the non-invariant definition of
conditional probability. This problem, which materializes in the so-called Borel paradox, has profound consequences
for inverse problem theory.
A probabilistic formulation of inverse theory for general inverse problems (usually called ‘nonlinear inverse
problems’) is not complete without the use of Monte Carlo methods. Section 3 is an introduction to the most
versatile of these methods, the Metropolis sampler. Apart from being versatile, it also turns out to be the most
natural method for implementing our probabilistic approach.
In sections 4, 5 and 6 time has come for applying probability theory and Monte Carlo methods to inverse
problems. All the steps of a careful probabilistic formulations are described, including parameterization, prior
information over the parameters, and experimental uncertainties. The hitherto overlooked problem of uncertain
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For instance, we could fit our observations with a heterogeneous but isotropic Earth model or, alternatively, with an homogeneous
but anisotropic Earth.
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