Mathematical and Computational Studies of Geomagnetic Anomaly Detections

Description

The project is mainly concerned with the mathematical and computational studies on the identification of anomalies beneath the Earth using the geomagnetic monitoring. Consider a collection of certain unknown anomalies presented in the shell of the Earth. The anomalies interrupt the geomagnetic field. Hence, one can monitor the variation of the geomagnetic field and use it to identify the underlying anomalies. We propose to investigate two scenarios that are motivated by the practical applications. In the first one, the variation of the magnetic field is caused by the presence of certain unknown anomalies, and as a practical example, this may be the presence of submarines. In the second one, the variation of the magnetic field is caused by the change of the unknown anomalies, and as practical examples, this may be the growth of a certain mineral or the expansion of an underground lake. In our study, we propose to consider two geomagnetic models, with the first one described by a Maxwell system and the other one described by a hydromagnetic dynamo system. We formulate the geomagnetic anomaly detection as nonlinear inverse problems associated with the aforementioned geomagnetic models. The unknowns in these inverse problems are the locations, shapes and material parameters of the underlying anomalies, and the measurement data for these inverse problems are the variations of the geomagnetic field due to the presence or the change of the anomalies. Furthermore, from a practical point of view, we shall not assume any a-priori knowledge on the geomagnetic configuration of the Earth's core. We aim to establish the unique recovery results for those proposed inverse problems in these challenging but practically significant scenarios. That is, we shall establish sufficient conditions under which the anomalies can be uniquely identified by the corresponding measurement data. Based on the theoretical results, we aim to further develop efficient numerical recovery schemes for the anomaly identification. The proposed theoretical study shall lay out a rigorous mathematical theory to the geomagnetic detection technology that has been used in practice. The corresponding computational study shall provide novel and efficient reconstruction methods for the detections that are of significant real values.