%A Wojciech Debski
%T Probabilistic Inverse Theory
%X Geophysical investigations which commenced thousands of years ago in China from observations of 
the Earth shaking caused by large earthquakes (Lee et al., 2003) have gone a long way in their 
development from an initial, intuitive stage to a modern science employing the newest technological 
and theoretical achievements. In spite of this enormous development, geophysical research still faces 
the same basic limitation. The only available information about the Earth comes from measurement 
at its surface or from space. Only very limited information can be acquired by direct measurements. 
It is not surprising, therefore, that geophysicists have contributed significantly to the development of 
the inverse theory — the theory of inference about sought parameters from indirect measurements. 
For  a  long  time  this  inference  was  understood  as  the  task  of  estimating  parameters  used  to 
describe the  Earth’s  structure  or  processes  within  it,  like  earthquake  ruptures.  The  problem was 
traditionally solved by using optimization techniques following the least absolute value and least 
squares criteria formulated by Laplace and Gauss. 
Today  the  inverse theory  faces  a  new challenge  in  its  development.  In  many geophysical  and 
related applications, obtaining the model  “best fitting”  a given set of data according to a selected 
optimization criterion is not sufficient any more. We need to know how plausible the obtained model 
is or, in other words, how large the uncertainties are in the final solutions. This task can hardly be 
addressed in the framework of the classical optimization approach. 
The probabilistic inverse theory incorporates a statistical point of view, according to which all 
available information, including observational data, theoretical predictions and a priori knowledge, 
can  be  represented  by  probability  distributions.  According  to  this  reasoning,  the  solution  of  the 
inverse problem is not a single, optimum model, but rather the a posteriori probability distribution 
over the model space which describes the probability of a given model being the true one. This path 
of development of the inverse theory follows a pragmatic need for a reliable and efficient method of 
interpreting observational data. 
The aim of this chapter is to bring together two elements of the probabilistic inverse theory. The 
first one is a presentation of the theoretical background of the theory enhanced by basic elements of 
the  Monte  Carlo computational  technique.  The  second  part  provides  a  review of  the  solid  earth 
applications of the probabilistic inverse theory.
%K Inverse theory, Inverse problems, Probabilistic inference, Bayesian inversion, Geophysical 
inversion, Parameter estimation, Nonparametric  inverse problems, Monte  Carlo technique, Markov 
Chain Monte Carlo, Reversible Jump Monte Carlo, Global optimization
%P 1-102
%B Advances in Geophysics
%E Renata Dmowska
%V 52
%C San Diego, USA
%D 2010
%I Elsevier
%L epos1304