An Optimization Method on Joint Inversion of Different Types of Seismic Data M. Argaez¹, R. Romero 3, A. Sosa¹, L. Thompson² L. Velazquez¹, A. Velasco² ¹ Mathematical Science, Computational Science Program, ²Geological Sciences, ²Computer Science Cyber-ShARE Steering Committee Meeting‏ December 8, 2009

may have infinitely many solutions is numerically unstable (ill conditioned) Implement an efficient solver for the linear system Forward and Inverse Problems Forward Problem Inverse Problem Given an operator and find “Easy” to solve Given an operator and find the true such that Hard to solve Inverse Problem Difficulties

Goal To characterize the Earth structure underneath a station by using velocity distributions

Experiment Design Body waves First arrivals Longer periods Smaller amplitudes Surface waves Slower velocities Shorter periods Higher amplitudes We use data from different EQs at a given station Each EQ can be recorded in a 3 component seismogram

From the Body waves we have What does a seismogram tell us? (Developed by L. Braile)

From the Surface waves What does a seismogram tell us? (Developed by L. Braile)

How do we find the velocities x? From the body waves Gives more resolution for deeper layers Helps to define layer thicknesses From the surface waves Gives high resolution of velocities near the surface VALUABLE Each data set provides different VALUABLE information Question : How to join each data set to obtain a better characterization of the given site ? Joint Inversion!!! Answer : Joint Inversion!!!

Joint Inversion methods Uses direct methods: –QR Factorization –Cholesky Factorization –SVD Factorization* Uses indirect method : –Conjugate Gradient* * We are using the fortran 77 code provided by C. Ammon et al. with synthetic data sets

Numerical Results Inversion SVD

Numerical Results Inversion our CG

Future Work To study the choice of the regularization parameter to improve the rate of convergence of the algorithm To implement the code for solving large scale real geophysical problems To implement state-of-the-art optimization techniques using the norm (using current research Dr. Argaez). That is:

References 1.Joint inversion of receiver function and surface wave dispersion observations. J. Julia, C. J. Ammon, R. Hermann, M. Correig. Geophysics Int. J (2000) References On the Nonuniqueness of Receiver Function Inversion. C. J. Ammon, G. Randall, G. Zandt. Journal of Geophysical Res. Vol. 95. (1990) 3.An Introduction to Seismology Earthquakes and Earth Structure, S. Stein and M. Wysession, Blackwell Publishing, Numerical Optimization. J. Nocedal, and S. J. Wright. Springer. Second Edition Acknowledgements Computational Science and Department of Mathematical Sciences at University of Texas at El Paso (UTEP) This work is being funded by NSF Crest Cyber-ShARE HRD