%From doser@garrett.geo.ep.utexas.edu Mon Dec 26 17:44:45 1994
%From: Dr. Diane Doser <doser@garrett.geo.ep.utexas.edu>

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\title{Estimating Uncertainties for Geophysical Tomography}
\author{D. I. Doser, K. D. Crain, M. R. Baker, V. Kreinovich, \\M. C. 
Gerstenberger, and J. L. Williams}

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\markright{APIC'95, El Paso,
Extended Abstracts,
A Supplement to the international journal of {\rm Reliable
Computing}\ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ }

\maketitle

  
\auffil{The authors are with the 
Department of Geological Sciences (DID, KDC, MRB, MCG, JLW)
and Department of Computer Science (VK),
University of Texas at El Paso,
El Paso, TX 79968. This work was partially 
supported by NSF grant No. CDA-9015006 and NASA Research Grant No.
NAG 9-757.}

	We present new techniques for evaluating the uncertainties 
associated with geophysical tomographic inversion problems 
including estimation of data errors, model errors, the 
information density matrix and total solution uncertainty.    
Our inversion method uses the conjugate gradient technique 
\cite{Scales et al 1988}, incorporating a priori estimates of 
data and model uncertainties to stabilize the solution.   We 
apply this inversion method to travel time data collected in 
a cross well seismic experiment \cite{Gerstenberger 1994} and to 
data collected during the experimental freezing and thawing 
of a soil filled tank \cite{Williams 1995}.

	Seismic tomography involves using the travel times of 
seismic (sonic) waves to reconstruct the velocity or inverse 
velocity (slowness) distribution of the material between 
sources and receivers.  The velocities or slownesses provide 
estimates of the possible geologic formations or rock 
compositions that may exist between the sources and 
receivers; factors important to groundwater, mining, 
petroleum, and geotechnical studies.

	The ``standard'' estimate of image quality from travel times 
test only limited contributions to uncertainty and are 
qualitative (i.e. \cite{Hearn and Clayton 1986}).  The ``standard'' 
techniques estimate the solution variance by calculating 
artificial travel times for a synthetic model.  Gaussian 
noise is added to the travel times.   Next, the noisy travel 
times are inverted using the same ray paths as the original 
(observed) data set.  A comparison of the tomographic image 
from noisy data with the image obtained from the observed 
data provides a qualitative measure of the image quality.  
Resolution is estimated by constructing a slowness model, 
such as a checkerboard pattern, calculating travel times 
through the model using the same ray paths as the observed 
data set, and inverting the travel times to see if the 
slowness image matches the initial slowness model.   These 
estimates of resolution and variance are generally done for 
only one test case (one set of noisy artificial data, one 
checkerboard slowness model), and assume the travel times 
and slownesses are known perfectly in the inversion of the 
original data.

	Our approach differs from the ``standard'' technique in the 
following ways:
\begin{itemize}

\item[1)] We allow for model and data error by assigning a priori 
uncertainties to the travel times and  using an initial 
starting model of slownesses that also has associated 
uncertainty values.  These uncertainties are based on other 
geological and geophysical information.  The inversion 
approach uses the conjugate gradient technique where a 
priori information can be included to increase stability and 
speed of convergence.

\item[2)] We obtain a quantitative estimate of the information 
density matrix \cite{Kreinovich et al 1991} by repeatedly 
adding noise to the observed travel times and inverting for 
slowness images.  The variations between the slowness images 
gives the covariance of each velocity model block.

\item[3)] We estimate the total uncertainty of the solution by 
adding noise to both the data and the starting slowness 
model.  We then conduct multiple inversions of the data and 
slowness models with different noise sets.

\item[4)] We have examined the effects of error in picking raw 
travel time data by inverting subsets of data that have bad 
data points removed.  In addition, we examined the 
distribution of data errors (difference between every 
observed travel time and its corresponding calculated travel 
time) to test whether the $l^2$ norm was the appropriate data 
norm for our inversion process.

\item[5)] Model validity was evaluated by using a variety of 
different starting models that assumed isotropic or 
anisotropic velocity distributions.  The distribution of 
model errors (differences between each slowness value in the 
starting model and its corresponding slowness valued 
calculated from the inversion) was also determined.
\end{itemize}

	Our most exhaustive studies of error estimation used a data 
set of travel times collected from a cross well experiment 
run in an oil field near Rifle, Colorado \cite{Albright et al 1988}.  
The two wells were separated by 34 m and we 
concentrated on a depth interval between 1800 and 2100 m.  A 
2.2 kHz source transmitted signals at a rate of 1.5 
signals/m.  The receiver was held stationary for each 
transit of the source and then moved 1.5 m between transits.   
The velocity model between wells was constrained by velocity 
logs run in the source and receiver wells.  We constructed 
both anisotropic and isotropic a priori starting models 
consistent with the velocity logs.

	We found that between 10 and 15 inversions sets were 
adequate to accurately estimate the information density 
matrix and total uncertainty.  Best fit to the data came 
from a starting model of uniform isotropic layers, although 
we expected an anisotropic model to fit better.  It appeared 
that incorrect travel time picks altered much of the 
anisotropic information, especially in the lower part of the 
well where the data indicated anisotropy would be the 
greatest.  The data norm appears to be near 2, but the model 
norm is close to 1, so that using an $l^2$ norm for the
inversion process may not be appropriate.  No clear 
relationship between the total uncertainty and known geology 
was observed, but vertical slowness uncertainties were much 
less than horizontal uncertainties.

	We are still completing analysis of travel time data 
collected during a laboratory experiment where a tank of 
soil underwent freeze/thaw cycles.  Exhaustive estimates of 
travel time errors were made during data collection.  These 
observation errors were found to underestimate by a factor 
of 4 the level of errors required for consistency with a 
reasonable slowness model.  We are still searching for the 
source of this inconsistency and initial results indicate 
ASTM procedures specifying the seismic source are 
inappropriate for the scale of the experiment.

	The explicit use of a priori model estimates and data 
uncertainties in tomographic estimation problems permits 
identification of the source of data errors as well as 
providing solution uncertainty estimates based on 
information density and resolution.  These techniques can be 
applied to any problem where a useful solution estimation 
algorithm exists.

\begin{thebibliography}{99}
    
\bibitem{Albright et al 1988}
J. N. Albright, P. A. Johnson, W. S. Phillips, C. R. Bradley, 
and J. T. Rutledge, {\bf The crosswell, acoustic 
survey project},  Los Alamos National Laboratory Rept. 
LA-11157-MS, 1988, 121 pp.

\bibitem{Hearn and Clayton 1986}
T. M. Hearn and R. W. Clayton, ``Lateral velocity 
variations in southern California.  I.  Results for the 
upper crust from Pg waves'',  {\it Bull. Seismol. Soc. Amer.}, 1986, Vol. 76, 
pp. 495--509.

\bibitem{Gerstenberger 1994}
M. C. Gerstenberger, {\bf Development of new techniques in 
crosswell seismic travel time tomography}, M.S. Thesis, 
University of Texas at El Paso, 1994, 44 pp.

\bibitem{Kreinovich et al 1991}
V. Kreinovich, A. Bernat, E. Villa, and Y. Mariscal, 
``Parallel computers estimate errors caused by imprecise 
data'', {\it Interval Computations}, 1991, No. 2, pp. 31--46.

\bibitem{Scales et al 1988}
J. A. Scales, A. Gerstenkorn, and S. Treitel, ``Fast 
$l^p$ solutions of large, sparse, linear systems:  Application 
to seismic travel time tomography'', {\it J. Comput. Phys.}, 1988, Vol. 75, 
pp. 314--333.

\bibitem{Williams 1995}
J. L. Williams,  {\bf Seismic tomography of freezing and 
thawing soil}, M.S. Thesis, University of Texas at El Paso, 1995, 141 pp.

\end{thebibliography}


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