On Identifying Magnetized Anomalies Using Geomagnetic Monitoring

Abstract

We are concerned with the inverse problem of identifying magnetic anomalies with varing parameters beneath the Earth using geomagnetic monitoring. Observations of the change in Earth's magnetic field-the secular variation-provide information about the anomalies as well as their variations. In this paper, we rigorously establish the unique recovery results for this magnetic anomaly detection problem. We show that one can uniquely recover the locations, the variation parameters including the growth or decaying rates as well as their material parameters of the anomalies. This paper extends the existing results in [8] by two of the authors to varying anomalies.

On Identifying Magnetized Anomalies Using Geomagnetic Monitoring Within a Magnetohydrodynamic model

“…We have (4.3) by using Lemma 4.1. Then by following the proof of Theorem 4.2 in [13] (see also Theorem 7.8 in [2]), one immediately has z (1) l = z (2) l , l = 1, 2, . .…”

“…We believe the strategy developed here can be extended to dealing with other coupled-physics inverse problems associated with coupled PDEs. This work is a continuation as well as an extension of our recent work [13] on the identification of magnetized anomalies using geomagnetic monitoring, which aims to establish a rigorous mathematical theory for the geomagnetic detection technology. The geomagnetic model considered in [13] is a linear Maxwell system, which is of different mathematical features from the magnetic system in this work.…”

“…To that end, one needs to derive the low-frequency asymptotics of the magnetic fields H 0 and H. However, there are several new challenges as well as the corresponding novel technical developments in the current setup of study. The geomagnetic model is a linear Maxwell system in [13], and hence the low-frequency asymptotics therein were conducted globally for the whole system. However, the MHD system is a coupled nonlinear system and similar global low-frequency asymptotics of the Maxwell system (1.17) in coupling with the whole MHD system shall be fraught with various difficulties.…”

Determining a fractional Helmholtz equation with unknown source and scattering potential

“…Though it is believed that one can establish the simultaneous recovery result in a certain general scenario, the uniqueness results in [7,12,14] were only established for certain restrictive configurations. We would also like to mention in passing that the simultaneous recovery of unknown medium parameters and internal sources were also considered for the electromagnetic wave phenomena arising from brain imaging [6] and geomagnetic detection technique [5]. It is readily observed that in the exterior measurement, the information of the unknown sound speed is coupled with that of the unknown source, and this makes the simultaneous recovery radically challenging.…”

Unique determinations in inverse scattering problems with phaseless near-field measurements

“…Over the past several decades, the inverse scattering problems with full measured data (both phase and intensity) have been mathematically and numerically studied intensively in the literature (see, e.g. the research monographs [1,13,23] and [14,15,34,35,53,54] for recently proposed interesting applications). Recently, a great deal of effort has been devoted to phaseless inverse scattering problems [2,5,6,24,25,29,30].…”