A Volumetric Probability and Probability Density 38 B Conditional and Marginal Probability Densities 38

In ‘inverse problems’ data from indirect measurements are used to estimate unknown parameters of physical systems. Uncertain data, (possibly vague) prior information on model parameters, and a physical theory relating the model parameters to the observations are the fundamental elements of any inverse problem. Using concepts from probability theory, a consistent formulation of inverse problems can be made, and, while the most general solution of the inverse problem requires extensive use of Monte Carlo methods, special hypotheses (e.g., Gaussian uncertainties) allow, in some cases, an analytical solution to part of the problem (e.g., using the method of least squares). ∗This text has been published as a chapter of the International Handbook of Earthquake & Engineering Seismology (Part A), Academic Press, 2002, pages 237–265. It is here complete, with its appendixes. †Niels Bohr Institute; Juliane Maries Vej 30; 2100 Copenhagen OE; Denmark; mailto:[email protected] ‡Institut de Physique du Globe; 4, place Jussieu; 75005 Paris; France; mailto:[email protected]