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

AbstractAn 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 .

References

¹ Numerical Optimization Group

Department of Mathematical Sciences

Contact Information

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.

Data Measurements

Seismic ProblemSeismic 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

² Department of Geological Sciences

³ Department of Computer Science 

Figure 2: One Dimensional Experiment

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

This work is being funded by NSF CyberShare Crest Center Grant No. HRD-0734825.

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.

Future Goal

Bound Constraints

minimize ( ) minimize ( )

subject to

f w f w

a w b

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 Initial 3D velocity modelVel 1 d

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:

Figure 3: One Dimensional Data

Figure 1: Our goal is to predict the three dimensional model

of the El Paso Region.

Data Fusion

Fuse multipledata types withattributes to obtain high resolutionEarth model

3-D Earth model with error and sensitivity estimates

Mapping model to fuse data

43

pV

Acknowledgement

Jose Islas, Graduate StudentThe University of Texas at El PasoDepartment of Mathematical Sciences500 W. University AvenueEl Paso, Texas 79968-0514

Email: islas.jose@gmail.com

measured parameters

a priori estimates ofparameters

assumed parameters

output parameters

mapping algorithm

Proposed Modifications to Mapping Scheme

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

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

where a and b are upper and lower bound of the parameters w. Our goal is to add restrictions to the parameters.

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.

Figure 4: Illustration of important components

of a physical mapping process

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: