Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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4.1. LINEAR PREDICTION IN THE F − X DOMAIN 49 frequency: Sj = A e −ik(j−1)δx j = 1, . . . , N . (4.5) One can write a linear event in the f − x domain as a one-step-ahead prediction given by: Sj−1 = A e −ik(j−2)δx = A e −ik(j−1)δx e ikδx = Sj e ikδx (4.6) I can write a recursive form for the prediction of the signal recorded at receiver j as a function of the signal at receiver j − 1 (along the spatial variable x) as follows: Sj = Sj−1 e −ikδx = a Sj (4.7) The equation above is the basis for the f − x prediction/deconvolution and SNR enhancement (Canales, 1984; Gulunay, 1986; Sacchi and Kuehl, 2001). Similarly, it can be proved that the superposition of p linear events (p complex harmonics) is in a recursive form:
4.2. ANALYSIS OF OPTIMUM FILTER LENGTH 50 Sj = p A e n=1 −ikn(j−1)δx = a1Sj−1 + a2Sj−2 + · · · + apSj−p Recursion of order p (4.8) The coefficients of the recursion are also called prediction error coefficients when related to the wave number of each linear event. These coefficients can be found using a least squares solution as presented in Chapters 2 and 3. The f − x domain noise prediction algorithm can simply be summarized to predict both data and noise as follows: • Original data in t − x, • Transform data to the f − x domain, • For each frequency f, • Find m that solves d=Am+ e, • Use m to predict data and noise, d = Am e = d- d • Transform back to the t − x domain. 4.2 Analysis of Optimum Filter Length In a AR model of order (p), the best filter length p is not usually known. To continue our analysis I will define two measures of goodness of fit. For that purpose
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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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Page 15 and 16: 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: 4.1. LINEAR PREDICTION IN THE F −
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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 and 94: 5.3. SYNTHETIC DATA EXAMPLES 78 Not
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Page 97 and 98: 5.3. SYNTHETIC DATA EXAMPLES 82 Tim
Page 99 and 100: 5.4. REAL DATA EXAMPLES 84 Time [s]
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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
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BIBLIOGRAPHY 100 Trad, D. (2003). I
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Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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