Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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Chapter 6 Conclusions and Future Directions This thesis presented a method of signal modeling based on Volterra series. Specif- ically, I have developed an autoregressive method based on a nonlinear prediction algorithm. The prediction coefficients of the first-, second-, and third-order Volterra model are obtained with a least squares solution. I also bring a new approach to solving two fundamental problems in seismic exploration: modeling of complex waveforms in the f − x domain and adaptive subtraction of multiples. Primary issues and available methods in modeling time series are examined in Chapters 1 and 2. First, the nonlinear system is introduced as a higher-order extension of the usual linear convolution method. Former linear prediction methods are also investigated to obtain prediction coefficients. Comparisons between a first- order Volterra series and Yule-Walker equations and Burg’s algorithm have shown that linear prediction methods model the data quite similarly. On the contrary, all linear methods can not properly model complex data sets (waveforms) unless a large number of coefficients is used. In Chapter 3 a Volterra series and its properties are analyzed. The Volterra 91
series in this thesis has been truncated to a third-order Volterra series. Prediction coefficients are found via least squares solutions. The real and synthetic data examples showed that Volterra series modeling modeling is more versatile than linear prediction at the time of representing signals such as those arising in the context of exploration seismology. Events that exhibit strong curvature cannot be modeled via linear f − x prediction filters. I can model these events by implementing the Volterra algorithm. The optimum filter length of a linear model of order (p) or a nonlinear Volterra model of order (p, q, r) can be computed via inspection of the residuals. A more practical method is desirable but at this stage looking for evidence of under-over fitting using sophisticated statistical analysis method was beyond the scope of this study. Different synthetic and real data examples showed that the modeling of events with linear moveout can be predicted mainly via linear prediction coefficients and cubic prediction coefficients. The contribution of quadratic parts to events with linear moveouts is negligible. In addition to this, events that exhibit curvature can be modeled properly with nonlinear terms of a Volterra series. Finally, it is important to stress that waveforms that exhibit linear moveout can be predicted with a linear system. Conversely, when waveforms exhibit nonlinear moveout, which translates in the f − x domain as non stationary signals, the nonlinear part of the Volterra system helps in predicting the signal. Another application of the nonlinear Volterra model in this thesis is adaptive subtraction of multiples. Elimination of multiples is a common problem in exploration seismology. In this case the prediction coefficients have been designed as prediction error filters (PEF). I used this algorithm to annihilate multiples from 92
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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
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3.1. NONLINEAR PROCESSES VIA THE VO
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3.1. NONLINEAR PROCESSES VIA THE VO
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3.2. NONLINEAR MODELING OF TIME SER
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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 89 and 90: 4.6. SUMMARY 74 4.6 Summary In this
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 105: 5.5. SUMMARY 90 5.5 Summary In this
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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