Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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4.2. ANALYSIS OF OPTIMUM FILTER LENGTH 51 I first define the observed data as (D), the noise free data or clean data (Dc) and the predicted data (Dp) . I now define the following two measures of goodness of fit: RMSE1 = RMSE2 = ||D − Dp|| 2 nx × nt ||Dc − Dp|| 2 nx × nt (4.9) (4.10) where nx × nt denotes the dimensions of the data: length of time series (nx) by number of traces (nt), respectively. RMSE2 is not computable for real cases but it can be used gain understanding about the problem of order selection. The best (optimum) operator length is given for the operator that minimizes RMSE2 (Fig- ure 4.1(a)). On the other hand, RMSE1 shows that increasing the filter length leads to a decrease of error that is only accounted by for fitting the noise (Figure 4.1(b)). In Figure 4.2(a), 2-D synthetic data consisting of three linear events yields pre- dictions for different filter lengths. The signal Dc is contaminated with additive noise (Figure 4.2(b)) and the signal-to-noise ratio (SNR) has been taken as 4. Figure 4.3(a) corresponds to the prediction of data presented in Figure 4.2(b) with param- eter p = 3. Noise is rejected but the data are also poorly modeled (Figure 4.3(a)). The prediction of linear events in the prediction panel is not satisfactory because a large amount of energy leaks to the noise panel (Figure 4.3(b)). The prediction with a filter length p = 6 provides good noise rejection and the data is properly modeled (Figure 4.4(a)). The noise panel contains a small amount of coherent en-
4.2. ANALYSIS OF OPTIMUM FILTER LENGTH 52 RMSE 2 0.14 0.13 0.12 0.11 0.1 0.09 0 5 10 15 Filter Length (a) RMSE 1 0.28 0.26 0.24 0.22 0.2 (b) 0 5 10 15 Filter Length Figure 4.1: Optimum filter length for the data in Figure 4.2. (a) RMSE2 . (b) RMSE1. ergy (Figure 4.4(b)). Note that this figure shows the prediction of the optimum filter length determined with RMSE2 in Figure 4.1(a). The minimum value for the RMSE in that example is 6. The prediction for filter length p = 15 is good but the result is not as good as in the optimum case with p = 6 (Figure 4.5(a)). Noise is also modeled and incorporated in the predicted data (Figure 4.5(b)). At this point some inferences are in order. First, it is clear that the filter length is very important both to model the data and to reject the noise. However, it is not possible to compute RMSE2 for real situations; it can be used to estimate optimum filter length when working with synthetic examples where one has accessed to data free of noise. Practical experience in seismic data processing shows that one can easily compute the optimum filter length by trying different lengths and observe the amount of coherent energy left in the error panel. At some point one can find a filter length
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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
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 and 64: 4.1. LINEAR PREDICTION IN THE F −
Page 65: 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 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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