APPLIED MATH SEMINAR

Title: Tutorial Lecture on Bayesian Methods and their use in Accounting for
Uncertainty in the Solution to Geophysical Inverse Problems (first of
a series of talks)

Speaker: Nick Bennett (Schlumberger)

When/where: Thurs., Oct. 13th, 4:15pm, AKW 200

Abstract:

The goal of this set of lectures will be to describe our experiences
with the use of Bayesian methods in solving some geophysical inverse
problems involving one spatial dimension and to raise a series of
issues with their wider application for geophysical imaging
applications involving two or three spatial dimensions. 

In particular, we plan to

(a)        give an short overview of two practical applications of
Bayesian methods in geophysical inverse problems that illustrate how
these methods allow one to quantify the uncertainties of a 1D Earth
model given measurement data. These examples will include the
inversion of vertical offset seismic profile data for imaging
structure ahead of the bit while drilling [1] and (2) the inversion of
dc electrical measurements to model a saturation front moving away
from an injection well [2]. 

(b)        describe in some detail the basic Bayesian machinery used in
solving 1D linear and nonlinear inverse problems ((a) the classical
linear Gaussian case and (b) the use of Monte Carlo and nonlinear
least squares methods for dealing with the nonlinear case.).  Issues
such as model parameterization, model likelihood, and computational
costs will be highlighted.  A good reference for this material is [3].

(c)        highlight some of the issues that one encounters when applying
Bayesian methods in 2D and 3D inverse problems.  These issues include
the problem of large model and data size as well as efficiently
representing natural geological structures.  I will give an overview
of the use of wavelets as a potential ingredient in dealing with the
model size issue [4].  I will show some early results from the use of
Bayesian methods in solving a 2D linear tomography inverse problem and
finally introduce the nonlinear versions of this problem that involve
ray-tracing and full-waveform tomography.

[1] A. Malinverno and S. Leaney, “A Monte-Carlo Method to Quantify
Uncertainty in the Inversion of Zero-Offset VSP Data”, 70th Ann.
Internat. Mtg: SEG, pp. 2393-2396, 2000.
[2] A. Malinverno and C. Torres-Verdin, “Bayesian Inversion of DC
Electrical Measurements with Uncertainties applied to Reservoir
Monitoring”, Inverse Problems 16, pp. 1343-1356, 2000.
[3] A. Tarantola, Inverse Problem Theory, SIAM, 2004.
[4] Bennett and Malinverno, “Fast Model Updates Using Wavelets”, SIAM
Multiscale Modeling and Simulation, Vol 3, pp. 106-138, 2005.