Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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2.6. SUMMARY 25 ward and forward predictions by minimizing the error between the predicted data and original data. Finally, I presented a least squares method that uses only the available data and avoid truncation effects by properly using all the available in- formation at the time of setting the system of linear prediction equations. Because of practical considerations, I will use the least squares approach presented in section 2.3.4 to solve for the coefficients of the nonlinear model that I will present in Chapter 3.
Chapter 3 Nonlinear Prediction 3.1 Nonlinear Processes via the Volterra Series: Background The failure of linear systems (prediction techniques) to accurately model all phys- ical systems leads to the creation of nonlinear prediction methods (Wiener, 1942; Bracalari and Salusti, 1994). Examples of nonlinear systems in which nonlinear modeling techniques have been applied range from communication to nonlinear in- teractions of waves (Coker and Simkins, 1980; Benedetto and Biglieri, 1983; Koh and Powers, 1985; Kim et al., 1994; Flioriani et al., 2000). The potential of nonlinear systems in seismic data processing, however, is relatively underutilized. In this section the Volterra series will be introduced by extending a classical linear prediction technique to nonlinear prediction technique. The goal is to ad- dress the modeling waveforms with variable curvature in the t − x domain with nonlinear prediction theory implemented via a Volterra series and provide a set of 26
Page 1 and 2: Name of Author: Soner Bekleric Univ
Page 3 and 4: University of Alberta Faculty of Gr
Page 5 and 6: Abstract Linear filter theory has p
Page 7 and 8: Contents 1 Introduction 1 1.1 Appli
Page 9 and 10: List of Tables 3.1 Filter length an
Page 11 and 12: 4.4 (a) Prediction of Figure 4.2(b)
Page 13 and 14: List of symbols Symbol Name or desc
Page 15 and 16: List of abbreviations Abbreviation
Page 17 and 18: In applied seismology predictive de
Page 19 and 20: 1.1. APPLICATIONS 4 applied to vide
Page 21 and 22: 1.3. THESIS OUTLINE 6 • Chapter 4
Page 23 and 24: 2.2. LINEAR PROCESS 8 Finally, I pr
Page 25 and 26: 2.3. LINEAR PREDICTION 10 E(z) 1 X(
Page 27 and 28: 2.3. LINEAR PREDICTION 12 In my the
Page 29 and 30: 2.3. LINEAR PREDICTION 14 which can
Page 31 and 32: 2.3. LINEAR PREDICTION 16 ∂ρ fb
Page 33 and 34: 2.3. LINEAR PREDICTION 18 I have an
Page 35 and 36: 2.4. 1-D SYNTHETIC AND REAL DATA EX
Page 39: 2.6. SUMMARY 24 Relative PSD Relati
Page 43 and 44: 3.1. NONLINEAR PROCESSES VIA THE VO
Page 45 and 46: 3.1. NONLINEAR PROCESSES VIA THE VO
Page 47 and 48: 3.2. NONLINEAR MODELING OF TIME SER
Page 61 and 62: 3.4. SUMMARY 46 3.4 Summary In this
Page 63 and 64: 4.1. LINEAR PREDICTION IN THE F −
Page 65 and 66: 4.2. ANALYSIS OF OPTIMUM FILTER LEN
Page 67 and 68: 4.2. ANALYSIS OF OPTIMUM FILTER LEN
Page 69 and 70: 4.3. NONLINEAR PREDICTION OF COMPLE
Page 73 and 74: RMSE 2 4.3. NONLINEAR PREDICTION OF
Page 77 and 78: 4.4. NOISE REMOVAL AND VOLTERRA SER
Page 79 and 80: 4.5. SYNTHETIC AND REAL DATA EXAMPL
Page 89 and 90: 4.6. SUMMARY 74 4.6 Summary In this
Page 91 and 92:
5.1. INTRODUCTION 76 ples remains a
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5.3. SYNTHETIC DATA EXAMPLES 78 Not
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5.3. SYNTHETIC DATA EXAMPLES 80 Tim
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5.3. SYNTHETIC DATA EXAMPLES 82 Tim
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5.4. REAL DATA EXAMPLES 84 Time [s]
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5.5. SUMMARY 90 5.5 Summary In this
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series in this thesis has been trun
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6.1. FUTURE DIRECTIONS 94 Noise red
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BIBLIOGRAPHY 96 Canales, L. L. (198
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BIBLIOGRAPHY 98 Marple, S. L. (1980
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BIBLIOGRAPHY 100 Trad, D. (2003). I
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Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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