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On Optimization Techniques for the One-Dimensional Seismic Problem
M. Argaez¹ J. Gomez¹ J. Islas¹ V. Kreinovich³ C. Quintero ¹
L. Salayandia³ M.C. Villamarin¹ A.A. Velasco² L. Velazquez¹
¹ Numerical Optimization Group
Department of Mathematical Sciences
² Department of Geological Sciences
³ Department of Computer Science
Future Goal
Data Measurements
Abstract
An optimization code is being developed by the Numerical Optimization Group
at UTEP that is to be used with the Hole's Algorithm for solving onedimensional seismic travel time tomography problem. The new code will offer
the use of restrictions in material properties and parameters by applying
Interior-Point Methodology. The current Hole's algorithm does not incorporate
such restrictions explicitly in the formulation of the nonlinear least squares
problem. Our goal is to incorporate our optimization algorithms into the Hole's
algorithm. This work is being funded by NSF CyberShare Crest Center, Grant
No. HRD-0734825 .
In a controlled source one dimensional (1D) experiment, the explosion generates
strong compressional waves (P-waves) whose recorded travel times at
geophones are used estimate the velocity structure Vp. First arrival data time (T)
and the geophone location (X) for each P-wave is measured (see Figure 3).
Based on ray theory, our goal is to model the travel time data for a layered 1-D
Earth using straight lines,where the slopes of the lines represent velocities Vp of
each layer where the P-waves are travelling.
Ultimately our goal is to replace the optimization technique being
used by the Hole’s Algorithm [Ref. 3] to obtain the three
dimensional (3D) velocity model. We must first understand how to
develop the one dimensional model. Our contributions are to add
information on the bounds of the parameters and utilize different
data measurements to develop a single 3D earth model.
Experiment geometry
Mapping
model to
fuse data
assumed parameters
Fuse multiple
data types with
attributes
to obtain high
resolution
Earth model
measured parameters
mapping algorithm
Initial 3D velocity model
Vel 1 d
3-D Earth
model with
error and
sensitivity
estimates
output
parameters
a priori estimates of
parameters
Bound Constraints
Seismic Problem
Figure 2: One Dimensional Experiment
Seismic tomography allows geophysicists and geologists to observe the
velocities of seismic waves through structures of the earth’s crust. By creating
seismic waves through controlled source explosions and measuring the arrival
times of waves at geophone stations, a seismic velocity model of the earth’s crust
can be calculated by Hole’s nonlinear tomographic inversion procedure [H95,
H92, H00] and finite difference calculations [V88, V90]. The application for
such models is for earthquake analysis and oil exploration
Our research is to apply efficient optimization algorithms [Refs. 1-2] that allows
the addition of equality and / or inequality restrictions via interior-point
methodology. Currently, geophysical problems are usually posed as
unconstrained minimization problems. This was due to the lack of methodology
and computational techniques for solving more constrained problems. Now the
addition of bound conditions are to be considered for obtaining better
interpretation of real model problems.
Assignment of uncertainties
in a variety of ways
(covariance, interval, pdf,
fuzzy sets, etc.)
Optimization with ability to
handle multiple functions
of parameter uncertainty
and minimize error
Improved estimates
of “goodness of fit”
to measurements
Figure 4: Illustration of important components
of a physical mapping process
References
Unconstrained Problem
vs
m
i
n
i
m
i
z
efw
()
Constrained Problem
m
i
n
i
m
i
z
efw
()
s
u
b
j
e
c
tt
oa

w

b
Figure 3: One Dimensional Data
The data in Figure 3 [Ref. 4] can be divided into four intervals that may
represent four different layers of the earth. The data at each interval can be fit
into a linear model. The slope of the linear model is calculated using the two
point-slope formula that is the velocity of the layer. Then each layer has an
associated velocity that is calculated. We obtain the following slopes or Vp for
each interval using the specified points:
Interval 1:
Interval 2:
Interval 3:
Interval 4:
Figure 1: Our goal is to predict the three dimensional model
of the El Paso Region.
Proposed Modifications to Mapping Scheme
Points (Xi,Ti)
{(0,0), (25, 1.9)}
{(25, 1.9),(105,5)}
{(105, 5), (160,6)}
{(160,6),(240, 6.3)}
Slope= Vp
1.625 km/sec
3.125 km/sec
6.875 km/sec
8.33 km/sec
Layer
Basin
Shallow Crust
Crust
Mantle
The next step is to determine the depth of the layers using some geometrical
relationships. The final goal is that from each velocity spectrum, the material
properties of each layer are estimated: the bulk module β, shear module µ, and
density ρ by the following relationship:
Vp 
4
 
3

where a and b are upper and lower bound of the parameters w. Our goal is to add
restrictions to the parameters.
Data Fusion
Current approaches do not formally integrate a different types of data to develop
an Earth model. Analyses of individual data sets may lead to different models of
the earth due to mischaracterization of error. As part of this project, the
geoscientist wants to determine physical properties of the Earth utilizing
multiple data sets (data fusion). The mathematician (optimizer) will help
develop optimization techniques for integrating data with varying accuracy and
sensitivity. Later, the computer scientist will help model errors, sensitivities,
resolution, accuracies of data and output models of the Earth.
1. A Hybrid Optimization Approach for Automated Parameter Estimation
Problem, Miguel Argaez, Hector Klie, Carlos Quintero, Leticia
Velazquez and Mary Wheeler, Technical Report 2007.
2. Projected Conjugate Gradient for Constrained Optimization, Miguel
Argaez, Technical Report 2007.
3. Nonlinear High-Resolution Three-Dimensional Seismic Travel time
Tomography, John A. Hole’s, Journal of Geophysical Research, vol. 97,
No. B5, pp. 6553-6562, 1992.
4. Introduction to Seismology, Peter M. Shearer, Cambridge University
Press, 1999.
Acknowledgement
This work is being funded by NSF CyberShare Crest Center
Grant No. HRD-0734825.
Contact Information
Jose Islas, Graduate Student
The University of Texas at El Paso
Department of Mathematical Sciences
500 W. University Avenue
El Paso, Texas 79968-0514
Email: [email protected]