Discrimination and Classification of UXO Using Magnetometry: Inversion and Error Analysis Using Robust Statistics

Geophysical inversion using least-squares has been a very successful method for UXO discrimination of magnetics. However, the residuals do not follow a normal distribution, which means that the assumptions underlying the inverse problem are violated. Two consequences of this are (i) model bias and (ii) incorrect estimates of model parameter uncertainties. We have found that the residuals are better modeled by an Ekblom distribution which can be designed to be more tolerant of statistical outliers than the Gaussian. To reformulate the inverse problem we are left with the issue of determining the two parameters that define the Ekblom distribution. We appeal to the concept of self-consistency to resolve these parameters; i.e. the statistical distribution used to determine the model parameters should be the same as the distribution of the residuals. To achieve this aim, we set the problem up as a two parameter inverse problem and use the maximum difference between the cumulative density functions of the Ekblom distribution and the residuals as a misfit measure. For magnetic data collected in Montana, the recovered dipole parameters using the Ekblom distribution could be significantly different than those obtained by least-squares. In one case considered in detail, the 95.4% confidence regions for the different solutions didn’t even overlap.