Robust Inversion using Biweight norm Jun Ji Hansung
Robust Inversion using Biweight norm Jun Ji, Hansung University ( visiting the University of Texas at Austin ) SEG 2011 San Antonio Introduction Least-squares (l 2) inversion: Sensitive to outliers Least-absolute (l 1) inversion Resistant to outliers (i. e. Robust) Variants of l 1 : - Huber norm - Hybrid norm etc. IRLS Review A linear system LS solution WLS solution (Weighted LS) IRLS algorithm implementation using nonlinear Conjugate Gradient (NCG) method (Claerbout, 1991) robust weights require a nonlinear inversion such as IRLS (Iteratively Reweighted LS) Compute residual Compute weighting Solve WLS to find model Iterate until satisfy
Robust norm : l norm 1 l 1 norm function : Weighting : Robust norm : Huber norm (Huber, 1981) Huber norm function : Weighting : ε = 1. 345 x MAD/0. 6746 ( ~95% of efficiency for Gaussian Noise) Robust norm : Hybrid norm 1977) Hybrid l 1 / l 2 norm function : (Holland & Welsch, 1977) (Bube &Langan, Weighting : ε ~ 0. 6 x σ (Bube &Langan, 1977) Robust norm : Biweight norm (Beaton & Tukey, 1974) Tukey’s Biweight (Bisquare Weight) norm function : Weighting : ε = 4. 685 x MAD/0. 6745 ( ~95% of efficiency for Gaussian Noise) (Holland & Welsch, 1977) Problems for Biweight norm IRLS • Local minimum (due to noncovex measure) good initial guess (e. g. Huber norm sol. ) would be helpful • Carefully choose threshold (ε) and do not change during iteration
Properties of different norms Single parameter estimation problem with N observations di 2 Minimize squares of error (l norm) : 1 Minimize absolute of error (l norm): Example data : ( 2, 3, 4, 5, 66 ) => Mean : 16, Median : 4, More robust estimation : ~ 3. 5 Examples - Line fitting BG noise : N(µ, σ)=(0, 0. 02) Outliers (20% of data) : 2 spikes(4. 5, 5) + 8 points with N(3, 0. 1) Example : Hyperbola fitting BG noise : N(0, 0. 4) Outliers 1) Three spikes of 10 times of signal amplitude 2) A bad trace with N(0, 1) BG noise : N(0, 0. 4) Outliers 1) Three spikes : 10 times of signal amplitude 2) A bad trace with N(0, 1) 3) 12 bad traces with U(10, 2) ~ 10 % of data
Real data Example Conclusions 1 • IRLS using Biweight norm provides a robust inversion method like the variants of l norm 1 approaches such as l , Huber, and Hybrid norms. 1 • Biweight norm inversion sometimes demonstrates better estimation than the one of l norm variants when outliers are not simple. • For optimum performance • need a good initial guess (e. g. Huber norm solution) to converge to the global minimum • carefully choose threshold (ε) based on noise distribution and do not change during iteration
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