Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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Chapter 5 Adaptive Subtraction of Multiples 5.1 Introduction Noise is an inevitable problem in seismic data processing. All unwanted events that distort the signal are considered noise. I mentioned that random noise could be removed via Canales’ method in Chapter 4. Multiples in seismic data are examples of coherent noise. Multiples can be sorted according to their arrival times (Figure 5.1): short-path multiples that turn back soon after primaries and long- path multiples that turn back as distinct event (Sheriff, 2006). The reflected data contains both the primaries and the multiples. Energy of primaries have been reflected from source to receiver, while multiples have been reflected two or more times. Also multiples tend to obscure the primaries. The removal of multiples is a complicated problem and partially solved in seismic exploration. There are many methods for elimination of multiples and they are successful when their conditions are fulfilled (Weglein, 1999). Therefore, elimination of multi- 75
5.1. INTRODUCTION 76 ples remains as a problem in exploration seismology. In this Chapter, I present the adaptive subtraction methods based on Volterra series that can be used to attenuate the multiples. In the adaptive multiple subtraction problem, one has access to a distorted version of the multiples and the goal is to find an operator that eliminates the distortion before subtracting them from the data. There are many methods to annihilate multiple reflections such as inverse scattering and surface-related multiple attenuation (SRME) (autoconvolution in space) (Weglein et al., 1992; Verschuur et al., 1992; Berkhout and Verschuur, 1997; Spitz, 1999; Abma et al., 2005). I am not going to explain here how a multiple model can be produced; this subject is well understood and published in many studies (Verschuur et al., 1992; Weglein et al., 1992; Berkhout and Verschuur, 1997; Verschuur and Berkhout, 1997; Verschuur and Prein, 1999; Weglein, 1999). These methods are used to construct multiple models, which are subsequently utilized by adaptive subtraction algorithms. In this thesis I introduce a method for adaptive subtraction of multiples, using a f − x linear prediction filter. A nonlinear prediction filter based on a Volterra series for the removal of multiples from recorded data sets. The problem can also be tackled using linear prediction error filters in the f − x domain as suggested by Spitz (1999). The idea is to compute a f − x prediction error operator from the estimate of the multiples. Then, the estimated prediction error filter is applied to the data containing both primaries and multiples to annihilate the multiples. The procedure is equivalent to finding a notch filter with the notch positioned at the wave number representing the multiple. The procedure of Spitz (1999) can fail when the multiple events in the window of analysis cannot be modeled as linear
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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
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 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: 4.6. SUMMARY 74 4.6 Summary In this
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 and 106: 5.5. SUMMARY 90 5.5 Summary In this
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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