Robust and least-squares estimation of acoustic attenuation from well-log data

Several least-squares attenuation (Q) estimation algorithms are tested on various types of models. These algorithms include the spectral ratio method and methods based upon eigenvector decomposition and Wiener filtering. The eigenvector decomposition and Wiener filter methods prove to be unsatisfactory even on model data, while the spectral ratio method yields fairly poor results. Tests indicate that the underlying Q error distribution is non-Gaussian; hence more robust methods are needed.The errors in Q estimation have an asymptotically Cauchy distribution, with a reasonable noise model and Gaussian input noise. On noise models with Gaussian errors slightly contaminated by Cauchy or Laplacian noise, a maximum-likelihood (ML) estimator based on Gaussian noise performed best. On heavily contaminated models, the ML estimator based on Laplacian noise performed best; but simple, robust estimators such as the median also did well. On more realistic models with noise, the median and alpha-trimmed mean (ATM) appear to be the best.