Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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4.5. SYNTHETIC AND REAL DATA EXAMPLES 67 Time (s) Time (s) 0 0.2 0.4 0.6 0 0.2 0.4 0.6 Traces 10 20 (a) Traces 10 20 (d) Time (s) Time (s) 0 0.2 0.4 0.6 0 0.2 0.4 0.6 Traces 10 20 (b) Traces 10 20 (e) Time (s) Time (s) 0 0.2 0.4 0.6 0 0.2 0.4 0.6 Traces 10 20 (c) Traces 10 20 Figure 4.14: 2-D synthetic data. (a) Original data. (b) Prediction using a thirdorder Volterra series with parameters p = 3, q = 3, and r = 3. (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 = 3, and r = 3). In Figure 4.16(c) I portray the contribution due to the linear terms of a third- order Volterra series to predicted data in Figure 4.16(b) due to the linear terms of third order Volterra series. Note that this contribution is negligible. Figures 4.16(d) and 4.16(e) illustrate the parts of the prediction associated with quadratic (q = 6) and cubic (r = 6) terms in the third-order Volterra series. Figure 4.16(f) shows the (f)
4.5. SYNTHETIC AND REAL DATA EXAMPLES 68 contribution of nonlinear terms associated with quadratic and cubic terms (q = 6 and r = 6). It is clear that nonlinear quadratic and cubic terms are required to properly model the full aperture. . In summary, I are able to model the data by adding quadratic and cubic terms to improve the predictability of the model. Time (s) 0 0.5 1.0 1.5 2.0 Traces 20 40 Time (s) 0 0.5 1.0 1.5 Traces 20 40 2.0 2.0 (a) (b) (c) Time (s) 0 0.5 1.0 1.5 2.0 Traces 20 40 Time (s) Time (s) 0 0.5 1.0 1.5 Traces 20 40 2.0 (e) (d) (f) (e) 0 0.5 1.0 1.5 Traces 20 40 Figure 4.15: 2-D synthetic data for comparison of prediction between linear prediction theory and a third-order Volterra series. (a) Original data. (b) Prediction using a third order Volterra series with parameters p = 6, q = 6, and r = 6. (c) Error between original data and predicted data via a third-order Volterra series. (d) Prediction using linear prediction theory with parameter p = 6. (e) Error between original data and predicted data via linear prediction theory.
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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
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 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 81: 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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