Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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5.3. SYNTHETIC DATA EXAMPLES 79 multiples from linear and nonlinear prediction error filters in Figures 5.4(e) and 5.4(f), respectively. Linear prediction error filter eliminated less multiples than the nonlinear prediction error filter. However, the nonlinear prediction filter removes more multiples, but it also affects the amplitude response of the primaries. In Figure 5.5(a) I present three primaries (linear events) and two multiples (two multiples with sharp apexes) that cannot be properly modeled by linear events. Figure 5.5(b) shows the distorted estimation of the multiples and Figure 5.5(c) is the solution via classical linear prediction. Again, the method can not separate multiples from primary and also distorts primaries. Figure 5.5(d) is the solution with the third order Volterra series. However, the prediction error filter was able to model the curved multiples and produce an acceptable result when applied to the data panel. Time (s) 0 0.2 0.4 0.6 0.8 Traces 20 40 (a) 0 0.2 0.4 0.6 0.8 Traces 20 40 (b) 0 0.2 0.4 0.6 0.8 Traces 20 40 Figure 5.2: Synthetic data example. (a) Original data. (b) Multiples. (c) Adaptive multiple attenuation using f − x linear prediction error operators computed from (b). (c)
5.3. SYNTHETIC DATA EXAMPLES 80 Time (s) 0 0.2 0.4 0.6 Traces 0 50 (a) Time (s) 0 0.2 0.4 0.6 Traces 0 50 (b) Time (s) 0 0.2 0.4 0.6 Traces 0 50 (c) Time (s) 0 0.2 0.4 0.6 Traces 0 50 Figure 5.3: Synthetic data example with two multiples and one primary. (a) Original data. (b) Prediction of multiples. (c) Adaptive multiple attenuation via f − x linear prediction error filtering. (d) Adaptive multiple attenuation obtained with a f − x nonlinear prediction error operator (third order Volterra series). (d)
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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
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1.1. APPLICATIONS 4 applied to vide
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1.3. THESIS OUTLINE 6 • Chapter 4
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2.2. LINEAR PROCESS 8 Finally, I pr
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2.3. LINEAR PREDICTION 10 E(z) 1 X(
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2.3. LINEAR PREDICTION 12 In my the
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2.3. LINEAR PREDICTION 14 which can
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2.3. LINEAR PREDICTION 16 ∂ρ fb
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2.3. LINEAR PREDICTION 18 I have an
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2.4. 1-D SYNTHETIC AND REAL DATA EX
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2.4. 1-D SYNTHETIC AND REAL DATA EX
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2.6. SUMMARY 24 Relative PSD Relati
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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 55 and 56: 3.3. 1-D SYNTHETIC AND REAL DATA EX
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 91 and 92: 5.1. INTRODUCTION 76 ples remains a
Page 93: 5.3. SYNTHETIC DATA EXAMPLES 78 Not
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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