Maximum-Likelihood Deconvolution

1 - Introduction.- 1.1 Introduction.- 1.2 Our Approach.- 1.3 Likelihood Versus Probability.- 1.4 Maximum-Likelihood Method.- 1.5 Comments.- 2 - Convolutional Model.- 2.1 Introduction.- 2.2 The Seismic Convolutional Model.- 2.3 Input.- 2.3.1 Gaussian White Sequences.- 2.3.2 Bernoulli White Sequences.- 2.3.3 Bernoulli-Gaussian White Sequences.- 2.3.4 Bernoulli-Gaussian Plus Backscatter Sequences.- 2.4 Channel Model IR (Seismic Wavelet).- 2.5 Measurement Noise.- 2.6 Other Effects.- 2.7 Mathematical Model.- 2.8 Summary.- 3 - Likelihood.- 3.1 Introduction.- 3.2 Loglikelihood.- 3.3 Likelihood Function.- 3.4 Using Given Information.- 3.5 Message for the Reader.- 3.6 Mathematical Likelihood Functions.- 3.7 Mathematical Loglikelihood Functions.- 3.8 Summary.- 4 - Maximizing Likelihood.- 4.1 Introduction.- 4.2 A Rationale.- 4.3 Block Component Search Algorithms.- 4.4 Mathematical Fact.- 4.5 Separation Principle.- 4.6 Update Random Parameters.- 4.7 Binary Detection.- 4.7.1 Threshold Detector.- 4.7.2 Single Most-Likely Replacement Detector.- 4.7.3 Multiple Most-Likely Replacement Detector.- 4.7.4 Single Spike Shift Detector.- 4.7.5 Other Detectors.- 4.8 Update Wavelet Parameters.- 4.9 Update Statistical Parameters.- 4.10 Message for the Reader.- 4.11 Summary.- 5 - Properties and Performance.- 5.1 Introduction.- 5.2 Minimum-Variance Deconvolution.- 5.3 Detectors.- 5.3.1 Threshold Detector.- 5.3.2 SMLR Detector.- 5.4 A Modified Likelihood Function.- 5.5 An Objective Function.- 5.6 Marquardt-Levenberg Algorithm.- 5.7 Convergence.- 5.8 Entropy Interpretation.- 5.9 Summary.- 6 - Examples.- 6.1 Introduction.- 6.2 Some Real Data Examples.- 6.3 Minimum-Variance Deconvolution.- 6.4 Detection.- 6.5 Block Component Method.- 6.6 Backscatter.- 6.7 Noncausal Channel Models.- 6.8 Summary.- 7 - Mathematical Details for Chapter 4.- 7.1 Introduction.- 7.2 Mathematical Fact.- 7.3 Separation Principle.- 7.4 Minimum-Variance Deconvolution.- 7.5 Threshold Detector.- 7.6 Single Most-Likely Replacement Detector.- 7.7 Single Spike Shift Detector.- 7.8 SSS-SMLR Detector.- 7.9 Marquardt-Levenberg Algorithm.- 7.10 Calculating Gradients.- 7.10.1 Gradients of M with Respect to a and b.- 7.10.2 Gradients of L with Respect to a and b.- 7.10.3 Derivatives of M with Respect to Variances.- 7.10.4 Derivatives of L with Respect to Variances.- 7.11 Calculating Second Derivatives.- 7.11.1 Pseudo-Hessian of M with Respect to a and b.- 7.11.2 Pseudo-Hessian of L with Respect to a and b.- 7.11.3 Second Derivatives of M with Respect to Variances.- 7.11.4 Second Derivatives of L with Respect to Variances.- 7.12 Why vr Cannot be Estimated: Maximization of L or M is an Ill-Posed Problem.- 7.13 An Algorithm for ?.- 8 - Mathematical Details for Chapter 5.- 8.1 Introduction.- 8.2 MVD Filter Properties.- 8.2.1 Derivation of F(?).- 8.2.2 Undershoot Property.- 8.3 Threshold Detector.- 8.4 Modified Likelihood Function.- 8.5 Separation Principle for P and Derivation of N from P.- 8.6 Why vr Cannot be Estimated: Maximization of P or N is not an Ill-Posed Problem.- 8.7 SMLR1 Detector Based on N.- 8.8 Quadratic Convergence of the Newton-Raphson Algorithm.- 8.9 Wavelet Identifiability.- 8.10 Convergence of Adaptive SMLR Detector.- 9 - Computational Considerations.- 9.1 Introduction.- 9.2 Recursive Processing.- 9.2.1 A Recursive Wavelet Model.- 9.2.2 Recursive MVD Algorithm.- 9.2.2.1 Input Estimator.- 9.2.2.2 Backward-Running Filter.- 9.2.2.3 Innovations Process.- 9.2.2.4 Kalman Predictor.- 9.2.3 Detection.- 9.2.4 Likelihood and Objective Functions.- 9.2.5 Gradients of L and M.- 9.2.5.1 Gradients of L.- 9.2.5.2 Gradients of M.- 9.2.6 Pseudo-Hessians of L and M.- 9.2.6.1 Pseudo-Hessian of L.- 9.2.6.2 Pseudo-Hessian of M.- 9.2.7 Computational Requirements for Recursive Processing.- 9.3 Summary.- References.