1 OPTIMUM DECONVOLUTION OF SEISMIC TRANSIENTS 7 L" A MODEL-BASED SIGNAL PROCESSING APPROACH A Thesis Presented to The Faculty of the Russ College of Engineering and Technology Ohio University In Partial Fulfillment of the Requirement for the Degree Master of Science by Kerry D. -.Schutz/ June, 1994 Acknowledgments I would first like to thank my parents and my grandfather "Dave" who just turned 91. Even though they had no direct influence on this research, they all played a big role in my personal development. I mentioned my grandfather particularly, because he taught me what work ethic really means. Whether he was coaching me on the finer points of baseball, showing me how to start a stubborn tractor, chop firewood, or cook a good breakfast, he always gave me a ton of time and love to go with it. For that and more, I will never be able to repay him. I also wish to thank my advisor Dr. John A. Tague who provided constant support and guidance when I needed it most. It was also comforting to know that someone else understood my dreaded fear of the "high pollen" spring here in Athens as I consistently fell victim to it. I would also like to thank the members of my graduate committee -- Dr. Jeff Dill, Dr. Jeff Giesey, Dr. Larry Snyder, and Dr. James V. Candy -- for their gracious participation. I owe a very special thanks to Beth and C. Paul Stocker for the fellowship that I received while in graduate school. Without their support, tackling the demands of graduate school would have been much more difficult. I would also like to thank Dr. James V. candy ( Bad Hops!) and Fred E. Followill of Lawrence Livermore National Labs who were responsible for initiating this research during my internship and for their time out of their busy schedules when I called upon them fiom Ohio. I also wish to thank Ying-Wei Jan who provided constant expertise concerning the finer points of word processing and the latest in software tools. Lastly, I would like to put a big thanks out to all the people with whom I shared a graduate office. Our insightfbl and sometime nonsense discussions added the much needed color to what otherwise could have been just a routine. TABLE OF CONTENTS Acknowledgments .....................................................................i Table of Contents .....................................................................ii List of Figures ........................................................................ v Chapter 1 Introduction ..........................................................1 1.1 Introduction .................................................................1 1.2 Problem Overview ...........................................................2 1.3 Past Work ...................................................................5 1.4 Innovative Approach ........................................................9 1.5 Issues ..................................................................... 9 1.6 Thesis Overview ........................................................... 12 Chapter 2 Signal and Seismometer Modeling ................................ 13 2.1 Introduction ................................................................ 13 2.2 Seismometer Background .................................................. 13 2.3 Feedback Seismometers ..................................................... 16 2.3.1 Displacement Feedback Seismometers ...............................16 2.3.2 Velocity Feedback Seismometers .................................... 17 2.4 Kinemetrics Force Balance Accelerometer .................................. 18 2.5 State-Space Seismometer Model ............................................ 22 2.6 Gauss-Markov Representation .............................................. 22 2.7 Input Signal Models ........................................................26 2.8 Background Earth Noise and Seismometer Process Noise ................... 28 ... 111 2.9 Colored Noise Models with the Kalman Filter ...............................29 2.10 Summary ...................................................................33 Chapter 3 The Optimum Deconvolution Algorithm .........................34 3.1 Introduction ................................................................ 34 3.2 Kalman Filtering ............................................................34 3.2.1 Block Diagram Analysis .............................................35 3.2.2 Predictor-Corrector Equations ......................................36 3.2.3 Critiquing the Innovations Sequence ................................39 3.3 Modified Model-Based Deconvolver: Derivation ...........................40 3.3.1 Overview ...........................................................40 3.3.2 State Augmentation .................................................42 3.3.3 Selecting the Input Model Matrix ...................................45 3.4 Summary ................................................................... 50 Chapter 4 Critical Evaluation of Model-Based Deconvolution ............. 51 4.1 Introduction ................................................................ 51 4.2 Generating Simulated Data ................................................. 52 4.3 Defining the Signal-to-Noise Ratios ......................................... 52 4.4 PCA vs TSA Deconvolution ................................................55 4.4.1 Slowly Time-Varying Input Signal ..................................55 4.4.2 Exponentially Damped Input Signal ................................. 62 4.4.3 Exact Deconvolution ................................................67 4.5 l/f2 Deconvolution ........................................................68 iv 4.6 Conclusions ................................................................72 Chapter 5 Conclusion ...........................................................73 5.1 Summary of Research ......................................................73 5.2 Summary of Results ........................................................74 5.3 Future Investigations .......................................................74 Bibliography ......................................................................76 LIST OF FIGURES Figure 1.1 The Deconvolution Problem ............................................. 3 Figure 1.2 The Stochastic Deconvolution Problem .................................. 3 Figure 1.3 Signal Generation and Deconvolution .................................... 3 Figure 1.4 Physical Diagram of Deconvolution Problem ............................. 6 Figure 1.5 Magnitude 5.3 Earthquake from Kanto-Tokai Region in Japan ........... 11 Figure 2.1 Vertical Seismometer ...................................................15 Figure 2.2 Displacement Feedback Accelerometer .................................. 15 Figure 2.3 DFBA Frequency Response ............................................. 15 Figure 2.4 Velocity Feedback Accelerometer .......................................19 Figure 2.5 VFBA Frequency Response ............................................. 19 Figure 2.6 Force-Balance Accelerometer ........................................... 20 Figure 2.7 FBA-11 Seismometer Model ............................................20 Figure 2.8 Exponentially Damped Seismometer Input Signal ........................27 Figure 2.9 Piecewise Smooth Seismometer Input Signal ............................ 27 Figure 2.10 Earth Noise Modeling .................................................. 30 Figure 2.1 1 Continuous-Time State Space Seismometer and Noise Models .......... 31 Figure 3.1 Discrete-Time Kalman Filter Block Diagram ............................ 36 Figure 3.2 Conventional Approach to Seismic Deconvolution ...................... 42 Figure 3.3 Reduced Model-Dependence Seismic Deconvolution ....................42 Figure 3.4 Second-order Taylor Series Deconvolution Block Diagram ..............47 Chapter 1 Introduction 1.1 Introduction This thesis presents new optimal stochastic deconvolution algorithms with applications to seismometry. Specifically, we study the problem of deconvolving transient signals appearing at a seismometer's input. This approach differs from past work that attempted to deconvolve signals back to the source of earth motion. We show how to deconvolve a broad variety of input signals when both background earth noise and measurement noise are present. A modified Kalman linear discrete-time estimator forms the basis of each structure. Our results demonstrate that this is a promising technique. Deconvolving a transient signal fiom noisy measurement data has been a problem of major interest for many years [12]. Originally motivated by geophysicists using signal processing for oil exploration, deconvolution techniques have matured from educated guessing by the field engineer to a well-defined science incorporating detailed knowledge of the physics into the processing algorithms. The modem geophysicist wants to know the reflectivity fbnction with high precision. For example, large reflections may indicate potential hydrocarbon reservoirs, and the estimated thickness of these layers strongly influences the monetary investment made in the drilling rights [26]. More recently, the applications have expanded to include earthquake understanding, target tracking, anti-distortion in communications, military intelligence gathering, and nuclear test ban treaty verification. In this thesis, a number of commercially available seismometers were analyzed from a control-theoretic perspective in order to find a signal processing algorithm up to the challenging task of