oa Conditional and marginal probabilities in AEM inversions using multivariate Gaussian statistics

We summarise and extend the concept of Gaussian, or normal, distributions into multivariate statistics over many dimensions. We demonstrate how multivariate statistics can be applied to probability distributions. Through assumptions in the linearisation of the inverse problem, we show that the best-fit inverse model parameters are normally distributed with mean values and associated variance and covariance values that obey Gaussian statistics. Variance and covariance values describe how the model parameters interact with each other. By changing one value in the model parameter vector, other parameters are changed through the covariance that links them. We apply Gaussian statistics over many dimensions to query our models for statistically meaningful questions that can only be answered by taking the integral of the multivariate distribution over the multidimensional space that contains the model parameter values. We illustrate this with an example of aquifer detection, using resistivity limits, for an electromagnetic transect adjacent to the Gascoyne River near Carnarvon, Western Australia.