Gérard
C. Herman
(Department of Applied Mathematical Analysis, Delft University
of Technology) g.c.herman@its.tudelft.nl
Inverse
Scattering of Guided Waves
The
objective of seismic exploration is to determine rock properties
and geological structure of the subsurface of the earth
from seismic recordings. As such, it is an inverse scattering
problem with many challenges. One of those is the fact that
most of the seismic energy is trapped near the earth's surface
and never reaches the deeper layers of interest for oil
and gas exploration.
We
are developing methods for removing the effect of these
trapped (or guided) waves. By using asymptotic methods,
the propagation can be described accurately and efficiently.
An important step in the method is the estimation of global
properties of the near-surface and their local deviations.
These local deviations can be of the size of the seismic
wavelengths and, consequently, give rise to severe scattering.
By determining the parameters relevant for this scattering
process, we are able to reduce the effect of the guided
waves.
Suzhou
Huang (E.
Technology Research Department, Ford Research Lab.) shuang10@ford.com
An
Inverse Problem in Economics Slides:
html pdf powerpoint
A
consumer's behavior on a market should reflect his/her underlying
preference. Given a rational decision making process for
each consumer, extracting consumer preference from observed
market data is, therefore, a typical inverse problem. I
will first briefly describe how this kind of problems is
mathematically formulated in economics. Then, I will review
what are typically practiced in recent econometric studies.
There are two types of uncertainties that could impede the
accuracy of the inversion: measurement error and model misspecification.
The latter is a harder problem to deal with and can potentially
give rise to misleading results. Concrete illustrations
will be given to exemplify the problem and demonstrate its
seriousness. The purpose of this presentation is to stimulate
exchange of ideas and methodologies across various disciplines
and to solicit inputs from experts.
Alberto
Malinverno
(Schlumberger-Doll Research, 36 Old Quarry Road, Ridgefield,
CT 06877) alberto@ridgefield.sdr.slb.com
Using
Bayesian inference to quantify how measurements affect the
uncertainty of inversion results
In
many practical applications, parameters in an Earth model
are estimated by inverting geophysical measurements. The
inversion process is generally nonunique: many values of
the Earth model parameters may fit the measurements equally
well. In other words, the model parameters estimated by
inversion have an inherent uncertainty. Clearly, the more
informative and accurate the measurements are, the less
the uncertainty in the inversion.
Bayesian
inference is well suited to quantify how much measurements
reduce inversion uncertainty. The fundamental object of
Bayesian inference is a posterior distribution of the Earth
model parameters; this posterior distribution is proportional
to the product of a prior distribution (which contains information
on the overall variability of and correlations among the
model parameters) and of a likelihood function (which quantifies
how well an Earth model fits a set of measurements). More
information in the prior distribution or in the measurements
will result in a corresponding reduction in the spread of
the posterior distribution.
I
illustrate a Bayesian inference approach by showing how
to invert seismic data collected in a vertical seismic profile,
where seismic sources are at the surface and receivers are
in a well. The object is to predict elastic properties (compressional
and shear wave velocity and density) in a one-dimensional
layered Earth model beneath the deepest receiver. To quantify
posterior uncertainties, I use a Monte Carlo Markov chain
method that samples the posterior distribution of layered
models. These sampled layered models agree with prior information
and fit the seismic data, and their overall variability
defines the uncertainty in the predicted elastic properties.
It is then easy to show how much the uncertainty in the
predicted elastic properties is reduced by increasing the
seismic data coverage (in this case, increasing the interval
spanned by the seismic source positions at the surface).
References:
(1)
On parsimony in Bayesian inference: Malinverno, A., 2000,
A Bayesian criterion for simplicity in inverse problem parametrization,
Geophys. J. Int., v. 140, 267285.
(2)
On Markov chain Monte Carlo sampling: Malinverno, A. & Leaney,
S., 2000, A Monte Carlo method to quantify uncertainty in
the inversion of zero-offset VSP data, in SEG 70th Annual
Meeting, Calgary, Alberta, The Society of Exploration Geophysicists,
Tulsa, Oklahoma. Available at
http://seg.org/meetings/past/seg2000/techprog/pdf/papr0039.pdf
Susan
Minkoff (Department
of Mathematics and Statistics, University of Maryland, Baltimore
County) sminkoff@math.umbc.edu
Scaling:
the impact of coupled fluid flow and geomechanical deformation
modeling on 4D seismic
Co-authors:
Mike Stone (Sandia National
Labs), Steve Bryant, Malgo
Peszynska, Mary Wheeler
(Center for Subsurface Modeling, University of Texas at
Austin).
Time-lapse
seismic feasibility studies for compactible reservoirs such
as Ekofisk in the North Sea require coupled flow simulation
and geomechanical deformation modeling. After presenting
an algorithm for staggered-in-time, 2-way coupling of flow
and geomechanics, we describe an interesting numerical experiment
based on the Belridge Field, California, in which microscale
permeability changes appear to have a large-scale impact
on seismic velocities.
Douglas
W. Oldenburg
(Department of Earth and Ocean Sciences, University of British
Columbia, Canada) doug@eos.ubc.ca
Geophysical
Inversion for Mineral Exploration Slides:
pdf
Mineral
deposits are characterized by a variety of physical properties.
For each property, a surface geophysical survey is carried
out and a first goal of the analysis is to find a 3D distribution
that reproduces the data and provides a geologically interpretable
image. The inverse problem is large, and ill-posed, and
is typically solved as an unconstrained optimization problem.
The critical elements are the specification of measures
of misfit and model norm, and deciding upon the appropriate
relative weighting of these parameters in the final optimization.
Considerable effort is required to design appropriate measures
and I will outline our approaches in this talk.
Solution
of the inverse problem provides a first image from which
questions can be posed pertaining to existence, or detail,
of structure. Answering these questions often requires further
inversions of the data. Unfortunately, the cost of carrying
out a single inversion is high, and practicality dictates
that we answer our questions by carrying out a limited number
of subsequent inversions. I illustrate our progress in this
area with examples, and put forth some avenues for future
research.
George
C. Papanicolaou
(Mathematics Department, Stanford University) papanico@math.stanford.edu
http://georgep.stanford.edu
Array
imaging, time reversal and communications in random media
Slides: pdf
postscript
I will present an exposition of the mathematical problems
that arise in using arrays of transducers for imaging and
communications in random media. The key to understanding
their performance capabilities is the phenomenon of statistically
stable super-resolution in time reversal, which I will explain
carefully. Signals that are recorded, time reversed and
re-emitted by the array into the medium tend to focus on
their source location with much tighter resolution when
there is multipathing because of random inhomogeneities.
I will explain how this super-resolution enters into array
imaging and communications when there is multipathing.
Robert
L. Parker
(Institute of Geophysics and Planetary Physics, Scripps
Institution of Oceanography, UCSD) rlparker@ucsd.edu
Estimation
of the magnetizations of geologic bodies: the poles of uncertainty
Slides
Paleomagnetists
usually determine the magnetization of geological units
by sampling the material. In some environments this approach
may be prohibitively expensive or even impossible. Then
we must resort to inferences based the external magnetic
fields of the bodies. The mathematical problem of inversion
of a potential field for its sources does not have a unique
solution.
The
ambiguity can be reduced by: (1) introducing assumptions
about the sources -- but this is often overdone; (2) estimating
specific properties, rather than a complete source distribution;
(3) both of the preceding.
The
magnetization of seamounts, which are ancient submarine
volcanos, is an important geological question. Paleomagnetists
are primarily concerned with the direction of the magnetization
vector, and so placing bounds on the direction of the vector
is a useful exercise. The solution of this optimization
problem will be discussed. Another setting, the planet Mars,
provides different challenges. The very large size of the
observed anomalies raises the question of a the magnitude
of the magnetizations, which have been reported as ten times
larger than those found anywhere on Earth. We describe how
a strict lower bound can be calculated on the intensity,
without assumptions about magnetization direction.
Gerhard
Pratt
(Department of Geological Sciences, Queen's University Kingston,
Ontario) pratt@geol.queensu.ca
http://geol.queensu.ca/people/pratt/
Local
minima in seismic waveform tomography
Seismic
traveltime tomography has proven to be an effective imaging
tool in many applications. More recently, the technique
of waveform tomography has emerged, in which we use wave-theoretical
methods and the direct arrival waveform.
Traveltime
methods, while reasonably robust, are known to have a reduced
resolution that scales with the Fresnel zone, while the
superior resolution of waveform methods scales with the
wavelength (Williamson, 1991; Schuster, 1996). Waveform
methods, however, are far more likely to fail due to a lack
of robustness.
Resolution is usually estimated by examining the "impulse
response" of a given algorithm: This procedure can be carried
out analytically; numerical studies generally confirm the
predictions (Williamson and Worthington, 1993). However,
studies based on spatial delta functions fail to account
for non-linear effects created by distortions of the wavefield,
or by high order scattering. In this paper we show that
non-linear effects can dramatically affect the performance
of high resolution of waveform tomography. We use "chequerboard"
models in which the anomaly sizes vary from the dominant
wavelength to the approximate size of the first Fresnel
zone.
Frits
H. Ruymgaart
(Department of Mathematics and Statistics, Texas Tech University,
Lubbock, TX 79409) ruymg@koch.math.ttu.edu
On
Weak Convergence of the Integrated Squared Error of Inverse
Estimators
In this paper various limit theorems for the integrated
squared error of inverse estimators are derived when spectral-cut-off
and Moore-Penrose regularization schemes are used. Conditions
for the generalized Fourier transform are given to obtain
a normal limit law. As a particular example, the sinc kernel
is investigated in a direct regression problem, which extends
some results that can be found in the literature for kernels
of finite order.The proofs are quite different , however,
and are partially based on results from approximation theory.
John
A. Scales (Department of Geophysics, Colorado
School of Mines) jscales@ruelle.mines.edu
There
is Plenty of Signal in the Noise
The
concept of noise is a subjective one. A practical definition
of noise might be that it consists of that portion of the
data which one has no interest in explaining or which one
is unable to explain. In geophysics, by far the largest
component of uncertainty in the data is unmodeled physics.
In exploration seismology, for example, surface waves, mode-converted
waves, multiply-scattered waves, are all frequently treated
as noise. Indeed concerted effort to suppress multiply-reflected
waves is one of the most important practical problems in
seismic exploration. Whereas in global seismology, surface
waves and mode-converted waves are prime tools for imaging
the earth's interior. Using examples from a variety of scientific
disciplines, we will see how data once regarded as noise,
can be exploited as signal to make useful inferences. Indeed
there is a lot of signal lurking in our "noise.''
Gerard
T. Schuster
(Department of Geology and Geophysics, University of Utah)
schuster@mines.utah.edu
Resolution
Limits of Traveltime Tomograms based on Diffraction Physics
Material
from Talk
Uncertainty
estimates typically assume that the correct physics is used
to predict the data, thereby ignoring uncertainty due to
incorrect physics. This can be a major problem for inversion
algorithms, particularly traveltime tomography. Ray-based
traveltime tomography assumes a high-frequency approximation
to the wave equation, yet many types of seismic measurements
violate this approximation. Consequently, the uncertainty
estimates of traveltime tomograms based strictly on statistical
arguments are significantly flawed. To account for uncertainty
based on diffraction physics, we apply the Generalized Radon
Transform to the Rytov equation and show how to construct
traveltime wavepaths that connect each source-receiver pair.
The intersection of these wavepaths define the spatial limits
of resolution about a scattering point. I illustrate how
to use these wavepaths for several traveltime tomography
examples, including surface-based refraction tomography
and crosswell tomography.
Delphine
Sinoquet
(Institut Francais du Petrole) Delphine.Sinoquet@ifp.fr
Uncertainty
analysis of the solution model of 3D seismic reflection
tomography
Joint
work with Carole Duffet.
Reflection
tomography aims to determine the velocity model that best
fits the travel time data associated with reflections of
seismic waves in the subsurface. This solution model is
only one model among many possible models. Indeed, the uncertainties
on the observed travel times (resulting from an interpretative
event picking on seismic sections) and more generally the
underdetermination of the inverse problem lead to uncertainties
on the solution model. An a posteriori uncertainty analysis
is then crucial to delimit the range of possible solution
models that fit, with the expected accuracy, the data and
the a priori information. The large size non linear least-square
problem is solved classicaly by a Gauss-Newton method based
on successive linearizations of the forward operator. A
linearized a posteriori analysis is then possible by an
analysis of the a posteriori covariance matrix (inverse
of the Hessian matrix). The computation of this matrix is
generally expensive (the matrix is huge for 3D problems)
and the physical interpretation of the results is difficult.
A formalism based on macro-parameters (linear combinations
of model parameters) allows to compute uncertainties on
relevant geological quantities for a reduced computational
time (the matrices to be manipulated are reduced to the
macro-parameter space). A first application on a synthetic
example with basic macro-parameters shows their potentialities.
The generality of the formalism allows a wide flexibility
for the construction of the macro-parameters. Nevertheless,
this approach is only valid in the vicinity of the solution
model (linearized framework) and complex cases may require
a non linear approach.
Roel
K. Snieder
(Department of Geophysics, Colorado School of Mines) rsnieder@mines.edu
http://www.mines.edu/~rsnieder
The
role of nolinearity in inverse problems Paper.pdf
Inverse
problems are often formulated as a minimization problem
of a quantity that mesures the misfit between the recorded
data and synthetic data for a given model. One normally
assumes that the main effect of nonlinearity of the forward
problem is to create secondary minima of the mistfit function.
Several examples are shown that this is an oversimplification
of the real situation and that nonlinearity can have much
more pathological effects. The related instability of that
can occur in nonlinear inverse problems is illustrated using
perturbation theory. Modern optimization techniques generate
not single models that are compatible with the data, but
a large class of models that is compatible with the data.
A technique is presented to extract the robust features
from these populations of models.
Biographical
sketch:
Roel Snieder holds the Keck
Foundation Endowed Chair of Basic Exploration Science at
the Colorado School of Mines. He received in 1984 a Masters
degree in Geophysical Fluid Dynamics from Princeton University,
and in 1987 a Ph.D. in seismology from Utrecht University.
In 1993 he was appointed as professor of seismology at Utrecht
University, where from 1997-2000 he was appointed as Dean
of the Faculty of Earth Sciences. In 1997 he was a visiting
professor at the Center for Wave Phenomena. Roel served
on the editorial boards of Geophysical Journal International,
Inverse Problems, and Reviews of Geophysics. In 2000 he
was elected as Fellow of the American Geophysical Union
for important contributions to geophysical inverse theory,
seismic tomography, and the theory of surface waves.
Philip
B. Stark
( Department of Statistics University of California, Berkeley) pbstark@uclink.berkeley.edu
http://www.stat.berkeley.edu/~stark
Statistical
measures of uncertainty in inverse problems Slides:
html
pdf
powerpoint
Inverse problems can be viewed as special cases of statistical
estimation problems. From that perspective, one often can
study inverse problems using standard statistical measures
of uncertainty, such as bias, variance, mean squared error
and other measures of risk, confidence sets, and so on.
It is useful to distinguish between the intrinsic uncertainty
of an inverse problem and the uncertainty of applying any
particular technique for "solving" the inverse problem.
The intrinsic uncertainty depends crucially on the prior
constraints on the unknown (including prior probability
distributions in the case of Bayesian analyses), on the
forward operator, on the statistics of the observational
errors, and on the nature of the properties of the unknown
one wishes to estimate. I will try to convey some geometrical
intuition for uncertainty, and the relation between the
intrinsic uncertainty of linear inverse problems and the
uncertainty of some common techniques applied to them.
Albert
Tarantola
(Institut de Physique du Globe de Paris) tarantola@ipgp.org
http://www.ccr.jussieu.fr/tarantola/
Strong nonlinearity and large dimensionality: should
we go to work or to sleep? slides.pdf
paper.pdf
This
speaker believes that any serious formulation of an inverse
problem leads to the definition of a probability distribution
in the parameter space. and that solving the inverse problem
means sampling this probability distribution. When using
realistic parameterizations, many inverse problems lead
to terribly multimodal probability distributions, that are
trivial to sample in low dimension but difficult (or impossible?)
to sample in the typical situation where the parameter space
has thousands or millions of dimensions. As an introduction
to this serious issue, the speaker will explain how an inverse
problem should be formulated, pointing his finger to some
of the most common mistakes.
Luis
Tenorio
(Mathematical & Computer Sciences, Colorado School of Mines)
ltenorio@Mines.EDU
Adapting
to nonstationary behavior. Examples from geophysics and
cosmology Slides:
pdf
postscript
Bias
in inverse problem estimates can be reduced by selecting
models that can adapt to the true nature of the unknown
signals. We present some examples of applications of mixture
models that can adapt to nonstationary behavior in deconvolution
problems in geophysics and in point source modeling problems
in cosmology. The deconvolution method generalizes the classical
Wiener-Levinson approach while the point source application
can be used to model coupled homogeneous random fields with
prescribed cross-spectrum and marginal structure.
Benjamin
S. White
(ExxonMobil Corporate Strategic Research) benjamin.s.white@exxonmobil.com
Random
Scattering and Uncertainty in Magnetotellurics
In
magnetotelluric (MT) surveys, surface measurements of the
earth's electrical impedance over a broad frequency range
at a number of different sites are analyzed to produce maps
of electrical resistivity in the subsurface. Naturally occurring
ambient electromagnetic (EM) radiation is used as a source.
In this work, we examine the effects of the earth's fine
scale microstructure, which is well documented from well
log resistivity measurements, on scattering of the EM waves.
Using a locally plane stratified earth model, we show how
MT data may be viewed as arising statistically from a smoothed
"effective medium'' version of the resistivity vs.
depth profile. The difference between the data produced
by the true medium and that produced by the effective medium
is due to random scattering noise. This noise is fundamental
to MT and other diffusive-wave EM exploration methods, since
it arises from the very small spatial scales that are usually
ignored. The noise has unique statistical properties, which
we characterize from first principles, using a limit theorem
for stochastic differential equations with a small parameter.
We show that when scattering is the dominant noise source,
a thin layer of increased resistivity at depth can be reliably
detected only if the noise statistics are incorporated properly
into the detection algorithm. This sets a new fundamental
limit on the vertical detection capability of MT data. The
theory agrees well with Monte Carlo simulations of MT responses
from random resistivity microlayers.
