Probability distributions of model parameters obtained by grid search – application to the geomagnetic anomaly source estimation problem

Abstract

Grid search is widely used in geophysical studies to determine model parameters that explain observed data. The residual between observed data and the model-predicted response, such as root mean square misfit, is usually used as a criterion to select the optimal model parameters. However, root mean square misfit is not a direct criterion for evaluating the reliability of a solution. In this study, we present a method to convert distributions of root mean square misfits obtained from a grid search into probability distributions to evaluate the results statistically based on a Bayesian framework. We applied the proposed method to synthetic geomagnetic anomaly datasets to evaluate the source position and magnitude of magnetic moments. The results are effectively visualized using marginal probability distributions for both well- and ill-posed problems, which are difficult for only root mean square misfits to evaluate. Then we applied the method to real geomagnetic anomaly data reflecting temporal magnetic variations due to volcanic activity in the Nishinoshima volcano. The resultant probability distributions indicate that the source must be in a narrow area northwest of the summit of the volcano with a large magnitude of demagnetization. The method is convenient and thus can be widely applied to multiple geophysical problems, including searches for the source locations of earthquakes, surface geodetic deformation, and magnetic change, and to their joint analyses. In addition, the method easily utilizes previous grid search results to evaluate the probability of model parameters.