On Optimization Techniques for the One-Dimensional Seismic Problem

1
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
² Department of Geological Sciences ³
Department of Computer Science 
¹ Numerical Optimization Group Department of
Mathematical Sciences

Future Goal
Data Measurements
3-D Earth model with error and sensitivity
estimates
Ultimately our goal is to replace the
optimization technique being used by the Holes
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.
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.
Abstract
Mapping model to fuse data
assumed parameters
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
one-dimensional 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 .
Fuse multiple data types with attributes to
obtain high resolution Earth model
measured parameters
mapping algorithm
Experiment geometry
Initial 3D velocity model
output
Vel 1 d
parameters
a priori estimates of
parameters
Bound Constraints
Proposed Modifications to Mapping Scheme
Assignment of uncertainties
Optimization with ability to
Improved estimates
in a variety of ways
handle multiple functions
of goodness of fit
Seismic Problem
(covariance, interval,
pdf
,
of parameter uncertainty
to measurements
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. Unconstrained Problem
vs Constrained Problem
fuzzy sets, etc.)?
and minimize error
Figure 2 One Dimensional Experiment
Figure 4 Illustration of important components
of a physical mapping process
Seismic tomography allows geophysicists and
geologists to observe the velocities of seismic
waves through structures of the earths 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 earths crust can be calculated by
Holes 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
References

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. Holes,
   Journal of Geophysical Research, vol. 97, No. B5,
   pp. 6553-6562, 1992.
4. Introduction to Seismology, Peter M. Shearer,
   Cambridge University Press, 1999.

Figure 3 One Dimensional Data
where a and b are upper and lower bound of the
parameters w. Our goal is to add restrictions to
the parameters.
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
Acknowledgement
Data Fusion
This work is being funded by NSF CyberShare Crest
Center Grant No. HRD-0734825.
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.
Points (Xi,Ti)
Slope Vp Layer Interval 1
(0,0), (25, 1.9) 1.625 km/sec
Basin Interval 2 (25, 1.9),(105,5)
3.125 km/sec Shallow Crust Interval 3
(105, 5), (160,6) 6.875 km/sec
Crust Interval 4 (160,6),(240, 6.3)
8.33 km/sec Mantle
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 islas.jose_at_gmail.com

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 ß, shear module µ,
and density ? by the following relationship
Figure 1 Our goal is to predict the three
dimensional model of the El Paso Region.