Robust inversion of seismic data using the Huber norm

The “Huber function” (or “Huber norm” ) is one of several robust error measures which interpolates between smooth (l2) treatment of small residuals and robust (l1) treatment of large residuals. Since the Huber function is differentiable, it may be minimized reliably with a standard gradient‐based optimizer. We propose to minimize the Huber function with a quasi‐Newton method that has the potential of being faster and more robust than conjugate‐gradient methods when solving nonlinear problems. Tests with a linear inverse problem for velocity analysis with both synthetic and field data suggest that the Huber function gives far more robust model estimates than does a least‐squares fit with or without damping.

[1]  M. Turhan Taner,et al.  Velocity spectra-digital computer derivation and applications of velocity functions , 1969 .

[2]  R. Fletcher,et al.  A New Approach to Variable Metric Algorithms , 1970, Comput. J..

[3]  D. Shanno.  Conditioning of Quasi-Newton Methods for Function Minimization , 1970 .

[4]  D. Goldfarb.  A family of variable-metric methods derived by variational means , 1970 .

[5]  P. J. Huber.  Robust Regression: Asymptotics, Conjectures and Monte Carlo , 1973 .

[6]  J. Claerbout,et al.  Robust Modeling With Erratic Data , 1973 .

[7]  H. L. Taylor,et al.  Deconvolution with the l 1 norm , 1979 .

[8]  Ian Barrodale,et al.  Algorithm 552: Solution of the Constrained I1 Linear Approximation Problem [F4] , 1980, TOMS.

[9]  J. Nocedal.  Updating Quasi-Newton Matrices With Limited Storage , 1980 .

[10]  Philip E. Gill,et al.  Practical optimization , 1981 .

[11]  N. R. Chapman,et al.  Deconvolution of marine seismic data using the l1 norm , 1983 .

[12]  Jon F. Claerbout,et al.  Velocity-stack and slant-stack stochastic inversion , 1985 .

[13]  J. Scales,et al.  Robust methods in inverse theory , 1988 .

[14]  Sven Treitel,et al.  Robust Optimization Methods In Geophysical Inverse Theory , 1988 .

[15]  Jorge Nocedal,et al.  On the limited memory BFGS method for large scale optimization , 1989, Math. Program..

[16]  Kaj Madsen,et al.  Algorithms for non-linear huber estimation , 1989 .

[17]  David J. Thuente,et al.  Line search algorithms with guaranteed sufficient decrease , 1994, TOMS.

[18]  Mauricio D. Sacchi,et al.  High‐resolution velocity gathers and offset space reconstruction , 1995 .

[19]  David Lumley,et al.  Amplitude-preserved multiple suppression , 1995 .

[20]  Clement Kostov,et al.  Moveout-discriminating Adaptive Subtraction of Multiples , 1995 .

[21]  K. Bube,et al.  Hybrid l 1 /l 2 minimization with applications to tomography , 1997 .

[22]  Wu Li,et al.  The Linear l1 Estimator and the Huber M-Estimator , 1998, SIAM J. Optim..

[23]  K. Marfurt,et al.  Toward true amplitude multiple removal , 1999 .

[24]  Carl Tim Kelley,et al.  Iterative methods for optimization , 1999, Frontiers in applied mathematics.

[25]  Philippe Herrmann,et al.  De-aliased, High-Resolution Radon Transforms , 2000 .

[26]  R. Chunduru,et al.  Determining bed boundaries from inversion of EM logging data using general measures of model structure and data misfit , 2000 .