Soner Bekleric Title of Thesis: Nonlinear Prediction via Volterra Ser

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2.3. LINEAR PREDICTION 17 x1 = a1x2 + a2x3 + a3x4 + ε1 x2 = a1x3 + a2x4 + a3x5 + ε2 x3 = a1x4 + a2x5 + a3x6 + ε3 x4 = a1x5 + a2x6 + a3x7 + ε4 x4 = a1x3 + a2x2 + a3x1 + ε5 x5 = a1x4 + a2x3 + a3x2 + ε5 x6 = a1x5 + a2x4 + a3x3 + ε6 x7 = a1x6 + a2x5 + a3x4 + ε7. (2.27) Notice that only data that has been recorded is used to compute the prediction coefficients ap. In other words, no assumption about samples of non-recorded data is made. In addition, I are conveniently using forward and backward prediction to avoid any type of truncation or aperture artifact. Data that cannot be predicted with equation (2.12) is predicted via equation (2.13) and and vice versa (Marple, 1987). The equations in (2.27) can be written in matrix form as follows: ⎡ ⎤ ⎡ ⎢ xl ⎥ ⎢ xl+1 ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ . ⎥ ⎢ ⎥ ⎢ . ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ xm ⎥ = ⎢ xm ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ . ⎥ ⎢ ⎥ ⎢ . ⎢ ⎥ ⎢ ⎣ ⎦ ⎣ xn xn−1 ⎛ ⎞ xl+2 . .. xm . xn−2 ⎛ . . . . . .. . . . . . . . . . . .. . ⎞ xl+p . xm . xn−p ⎥ ⎢ a1 ⎥ ⎢ εl ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ a2 ⎥ ⎢ εl+1 ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ × ⎢ . ⎥ + ⎢ . ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ . ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎦ ⎣ ⎦ ⎣ ⎦ ap εn ⎛ ⎞ ⎛ ⎞ ⎜ ⎝ d ⎟ ⎠ N × 1 ⎜ ⎝ A N × P ⎟ ⎠ ⎜ m ⎟ ⎜ ⎟ ⎝ ⎠ P × 1 ⎜ ⎝ ε ⎟ ⎠ N × 1 ⎤ ⎡ ⎤ ⎡ ⎤ (2.28)
2.3. LINEAR PREDICTION 18 I have an overdetermined system (where the number of observations is larger than the number of unknowns (Menke, 1989)) of equations which I will solve using the method of least squares with zero-order quadratic regularization (damped least squares method) (Lines and Treitel, 1984). In this case, I form the following cost function by using the l2 norm: J = ||Am − d|| 2 2 + µ||m|| 2 2 . (2.29) The first term of J indicates the modeling error or misfit. This term defines how well the prediction filter can reproduce the data. The second term is a stability or regularization term. The parameter µ is the trade-off parameter that accounts for the amount of weight given to each one of the terms in the cost function (Figure 2.2). The minimum of the cost function is found by taking derivatives with respect to the unknown parameters and setting them to zero. The solution, the damped least squares solution (or the minimum quadratic norm solution), is given by: where I denotes the identity matrix. m = (A T A + µI) −1 A T d (2.30) The main goal in linear prediction is to model the data with a small set of coefficients. These coefficients can be used to reconstruct (model) a clean version of the data (Canales, 1984) (see Chapter 4) , to compute the AR spectral estimator (Marple, 1980) and to design data compression algorithms (Makhoul, 1975). I are interested in the predictability of seismic events in the spatial domain, not only to
Page 1 and 2: Name of Author: Soner Bekleric Univ
Page 3 and 4: University of Alberta Faculty of Gr
Page 5 and 6: Abstract Linear filter theory has p
Page 7 and 8: Contents 1 Introduction 1 1.1 Appli
Page 9 and 10: List of Tables 3.1 Filter length an
Page 11 and 12: 4.4 (a) Prediction of Figure 4.2(b)
Page 13 and 14: 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: 2.3. LINEAR PREDICTION 16 ∂ρ fb
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
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4.5. SYNTHETIC AND REAL DATA EXAMPL
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4.6. SUMMARY 74 4.6 Summary In this
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5.1. INTRODUCTION 76 ples remains a
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5.3. SYNTHETIC DATA EXAMPLES 78 Not
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5.3. SYNTHETIC DATA EXAMPLES 80 Tim
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5.3. SYNTHETIC DATA EXAMPLES 82 Tim
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5.4. REAL DATA EXAMPLES 84 Time [s]
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5.5. SUMMARY 90 5.5 Summary In this
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series in this thesis has been trun
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6.1. FUTURE DIRECTIONS 94 Noise red
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BIBLIOGRAPHY 96 Canales, L. L. (198
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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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