A robust maximum likelihood method for gravity and magnetic interpretation

Historically, automatic curve-matching algorithms have employed the least squares method which minimizes the L₂ norm of the residuals. The least squares method, however, is very sensitive to the presence of outliers in data, and the alternative robust method employed so far has been the minimum absolute errors which minimizes the L₁ norm of the residuals.

Associating the type of residual norm being minimized with the sensitivity of the interpretation method to the presence of outliers may lead, however, to the false impression that there is no method more robust than the minimization of the L₁ norm.

Another viewpoint is to associate robustness with the probability density function describing the errors. Long-tailed distributions are related to more robust methods. Both least squares and minimum absolute errors may be derived from a maximum likelihood approach assuming that the errors present in the data follow a Gaussian or Laplace distribution, respectively. In this paper we present a maximum likelihood estimator based on the assumptions of errors following a Cauchy distribution which is more long-tailed than either the Gaussian or the Laplace distribution. As a result, the derived method is more robust than either the least squares or the minimum absolute errors.

The superior performance of the method as compared with the least squares method in the presence of Gaussian and geological noise is demonstrated using synthetic and field data. The results show that the method is less sensitive to the initial guess than the least squares. It produces better and more consistent solutions which are, therefore, less ambiguous. Also, as the method takes into account the presence of geological noise, it may be used to obtain a rough, but reliable outline of the sources, using simple interpretation models. This feature might be useful in designing automatic interpretation algorithms.