\magnification=1200
\baselineskip=20pt
\nopagenumbers
\font\big=cmr12 scaled \magstep2

\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENTAL SEMINAR}
\bigskip

\baselineskip=12pt
\centerline{4:15 p.m., Tuesday, November 30, 1999}
\centerline{Sequoia Hall Rm. 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

\bigskip
\baselineskip=15pt
\centerline{\sl Jef Caers}
\centerline{\sl Petroleum Engineering department}
\centerline{\sl Stanford University}
\bigskip
\centerline{\bf Stochastic inversion using Markov chain Monte Carlo simulation}
\bigskip


Stochastic inversion is an increasingly important topic in the modeling and
analysis of Earth science data or spatially varying phenomena in general.
The geologist/engineer/statistician is charged with the difficult task of
constructing a set of "equi-probable" million cell three dimensional Earth
models (of rock properties) based on a large variety of usually remotely
sensed data (or indirect measurements) such as geophysical surveys.
Furthermore, these 3D models should depict realistic geological scenarios.

Most current inverse methods either neglect the stochastic part (the
ill-posedness of the problem), do not account for realistic spatial
variation of the phenomenon or their computational aspect prohibits them
from being implemented for actual day-to-day application. The latter is
caused by the fact that the forward model, relating the indirect
measurements to the rock properties, is usually a finite element/difference
formulation. 

In this seminar, I present a Markov chain Monte Carlo simulation approach
to sampling such 3D models conditioned to indirect measurements. First, a
new MCMC method is presented for estimating and sampling from a
Gauss-Markov random field and its is shown how models can be conditioned to
the indirect measurements. The CPU-limitation due to computationally
demanding finite element/difference formulations is circumvented by using
neural networks. Next, it is shown how the Markov chain formulation can be
extended to sample more realistic spatial variation than the geologically
unrealistic maximum entropy Gaussian fields. I formulate this problem as an
image analysis (pattern recognition) and image construction (pattern
reproduction or sampling) problem.  Several case studies are presented to
illustrate the application of these methods. 

\bye