Probabilistic estimation of model parameters through grid search approaches: applications to geomagnetic anomaly source estimations

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Abstract Model parameters, extracted from observed data that inherently contain uncertainties, necessitate estimation as probability distributions. In geophysical problem-solving, especially when dealing with a few model parameters, the conventional approach employing a grid search is widely used to determine model parameters that explain observed data. However, the metrics of the results derived from the grid search approach are predominantly based on residuals between observed data and the model’s anticipated response, such as the root mean square misfit, which lacks representation as a probability distribution. This study introduces a straightforward technique to transform the distributions of root mean square misfits acquired via grid search into probability distributions, facilitating a statistical evaluation grounded in a Bayesian framework. The outcomes of this methodology are effectively visualized through marginal probability distributions. Employing this method, we investigated synthetic geomagnetic anomaly datasets to evaluate the location and magnitude of magnetic moments of the source. The synthetic tests showed that the method is applicable not only for well-posed problems, but also for ill-posed problems, which are challenging to evaluate solely using root mean square misfits. Subsequently, we applied this methodology to real geomagnetic anomaly data reflecting temporal magnetic fluctuations induced by volcanic activity within the Nishinoshima volcano. The method’s versatility allows its broad application across various geophysical problems, including identification of earthquake epicenters, analysis of gravity anomalies and surface geodetic deformation, and their concurrent analyses. Furthermore, this approach easily utilizes prior grid search outcomes to evaluate the probability of model parameters. Graphical Abstract