Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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4.5. SYNTHETIC AND REAL DATA EXAMPLES 69 Time (s) Time (s) 0 0.5 1.0 1.5 2.0 0 0.5 1.0 1.5 2.0 Traces 20 40 (a) Traces 20 40 (d) Time (s) Time (s) 0 0.5 1.0 1.5 2.0 0 0.5 1.0 1.5 2.0 Traces 20 40 (b) Traces 20 40 (e) Time (s) Time (s) 0 0.5 1.0 1.5 2.0 0 0.5 1.0 1.5 2.0 Traces 20 40 (c) Traces 20 40 Figure 4.16: 2-D synthetic data. (a) Original data. (b) Prediction using a thirdorder Volterra series with parameters p = 6, q = 6, and r = 6. (c) Contribution from the linear part. (d) Contribution from the quadratic part. (e) Contribution from the cubic part. (f) Contribution from both quadratic and cubic parts (q = 6, and r = 6). Figure 4.17(a) shows a 2-D synthetic example with hyperbolic events and a linear event. These events have been synthesized using a forward apex-shifted hyperbolic Radon transform (Trad, 2003). I present this example to demonstrate that linear moveouts can be mostly predicted with linear terms in the Volterra series and nonlinear moveouts can be mostly (f)
4.5. SYNTHETIC AND REAL DATA EXAMPLES 70 predicted with quadratic and cubic nonlinear terms. Figure 4.17(b) is the prediction using a third-order Volterra series with p = 7, q = 7, and r = 7. In Figure 4.17(c) I portray the part of the prediction attributed to the linear kernel (p = 7). Figures 4.17(d) and 4.17(e) show the parts of the prediction associated with quadratic (q = 7) and cubic (r = 7) terms in the third-order Volterra series. In addition, Figure 4.17(f) illustrates the part of the prediction associated to both quadratic and cubic terms (q = 7 and r = 7). These terms give good predictions, especially for the apexes of events where linear terms cannot predict the data. Finally, I test the performance of the Volterra series with a marine data set. The data consist of 60 traces extracted from a marine common offset section acquired over a salt body in the Gulf of Mexico (Figures 4.18(a) and 4.19(a)). The real data set has a combination of diffractions, roughly linear events, and hyperbolic events. I used filters with order p = 9, q = 9, and r = 9 for a third-order Volterra prediction (Figures 4.18(b) and 4.19(b)). In this case the data is properly modeled. I also compute the linear prediction filter with parameter p = 9 and attempt to model the data (Figure 4.18(d)). The prediction error between the original data and the prediction with a third-order Volterra series is small (Figure 4.18(c)), whereas the difference between the original data and the predicted data via a linear prediction method is large. In particular, the diffractions are leaking to the error panel as a consequence of improper modeling (Figure 4.18(e)). It is clear that the linear prediction was not able to properly model the data. In Figures 4.19(c), 4.19(d) and 4.19(e) I examine the individual contributions of the linear and nonlinear parts of the Volterra series to the prediction. Linear terms mostly predict linear moveouts with parameter p = 9; nonlinear terms model
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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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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 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
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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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