Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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4.3. NONLINEAR PREDICTION OF COMPLEX WAVEFORMS 55 Time (s) 0 0.2 0.4 0.6 Traces 20 40 (a) Time (s) 0 0.2 0.4 0.6 Traces 20 40 Figure 4.4: (a) Prediction of Figure 4.2(b) (p = 6). (b) The error between original data and predicted data. with additive noise. The prediction for different filter lengths cannot model the data (p = 3, 5, and 15) in Figures 4.8, 4.9, and 4.10; p = 5 rejects noise but cannot model the data; p = 15 models the data better than the optimum filter length but it is also not a perfect solution because it overfits noise in the prediction panel (Figure 4.10(b)). Events with nonlinear moveout can be modeled with a Volterra series. I begin by considering equation (4.8) as a Volterra series expansion by appending nonlinear coefficients. Remember that although the data vector m in equation (3.25) contains linear and nonlinear prediction coefficients, the problem is linear in the coefficients. (b)
4.3. NONLINEAR PREDICTION OF COMPLEX WAVEFORMS 56 Time (s) 0 0.2 0.4 0.6 Traces 20 40 (a) Time (s) 0 0.2 0.4 0.6 Traces 20 40 Figure 4.5: (a) Prediction of Figure 4.2(b) (p = 15). (b) The error between original data and predicted data. Equation (4.8) will be changed for the f − x nonlinear predictions as an expression with terms of the following form as an illustration: Sj = a Sj−1 + b Sj−1Sj−2 + c Sj−1Sj−2Sj−3 . (4.11) I explored the feasibility of using linear plus nonlinear quadratic and nonlinear cubic prediction filters to model waveforms that exhibit moveout curves that are not linear with synthetic and real data examples in the next section. The idea is, again, to operate in the f − x domain with one temporal frequency at a time and (b)
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Name of Author: Soner Bekleric Univ
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University of Alberta Faculty of Gr
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Abstract Linear filter theory has p
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Contents 1 Introduction 1 1.1 Appli
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List of Tables 3.1 Filter length an
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4.4 (a) Prediction of Figure 4.2(b)
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List of symbols Symbol Name or desc
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List of abbreviations Abbreviation
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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 and 40: 2.6. SUMMARY 24 Relative PSD Relati
Page 41 and 42: Chapter 3 Nonlinear Prediction 3.1
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: 4.3. NONLINEAR PREDICTION OF COMPLE
Page 73 and 74: RMSE 2 4.3. NONLINEAR PREDICTION OF
Page 75 and 76: 4.3. NONLINEAR PREDICTION OF COMPLE
Page 77 and 78: 4.4. NOISE REMOVAL AND VOLTERRA SER
Page 79 and 80: 4.5. SYNTHETIC AND REAL DATA EXAMPL
Page 91 and 92: 5.1. INTRODUCTION 76 ples remains a
Page 93 and 94: 5.3. SYNTHETIC DATA EXAMPLES 78 Not
Page 95 and 96: 5.3. SYNTHETIC DATA EXAMPLES 80 Tim
Page 97 and 98: 5.3. SYNTHETIC DATA EXAMPLES 82 Tim
Page 99 and 100: 5.4. REAL DATA EXAMPLES 84 Time [s]
Page 107 and 108: series in this thesis has been trun
Page 109 and 110: 6.1. FUTURE DIRECTIONS 94 Noise red
Page 111 and 112: BIBLIOGRAPHY 96 Canales, L. L. (198
Page 113 and 114: BIBLIOGRAPHY 98 Marple, S. L. (1980
Page 115 and 116: BIBLIOGRAPHY 100 Trad, D. (2003). I
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Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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