Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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4.3. NONLINEAR PREDICTION OF COMPLEX WAVEFORMS 53 that models the data and produce an error panel containing an important amount of incoherent energy. 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 4.2: Synthetic data example for different filter lengths. (a) Original noise free data. (b) Data contaminated with additive noise (SNR = 4). 4.3 Nonlinear Prediction of Complex Waveforms in the f − x Domain Analysis of the linear events presented above is not valid for events that exhibit curvature in the t − x domain. These events map to the f − x domain as chirp-like (b)
4.3. NONLINEAR PREDICTION OF COMPLEX WAVEFORMS 54 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 4.3: (a) Prediction of Figure 4.2(b) (p = 3). (b) The error between original data and predicted data. signals. It is clear that linear prediction will fail in modeling such events. A solution to the problem is to use linear prediction techniques in small windows or to resort to nonstationary linear prediction operators (Sacchi and Kuehl, 2001). I chose a 2-D synthetic data example consisting of 5 different hyperbolic events. The example does not satisfy the f − x assumption of constant ray parameter waveforms in the t − x domain. In this case I do not have a superposition of complex exponentials in the f − x domain. Therefore the minimum value for the filter length (p = 5) in Figure 4.7(a) will be an approximated value for the hyperbolic event presented in Figure 4.6(b). Again, the data in Figure 4.6(a) is contaminated (b)
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Name of Author: Soner Bekleric Univ
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University of Alberta Faculty of Gr
Page 5 and 6:
Abstract Linear filter theory has p
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Contents 1 Introduction 1 1.1 Appli
Page 9 and 10:
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
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 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: 4.2. ANALYSIS OF OPTIMUM FILTER LEN
Page 71 and 72: 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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