Inverse Problems in Geomagnetism
Geologic materials containing ferromagnetic minerals can record the magnetic field of the Earth at the time rocks were formed. This is done through their remanent magnetization. Such magnetizations preserve information from ancient magnetic fields for long periods of time (upwards of a billion years). These rocks then have their own magnetic fields, mimicking those of an ancient Earth, providing us with records of the intensity and orientation of the field at the time. The study of paleomagnetism relies heavily on the measurements of the field external to a magnetized sample. Recent developments of scanning magnetic microscopy (SMM) has expanded the pool of samples we are able to analyze; giving us the ability to examine weakly magnetized geologic materials.
However, with new technology, novel analytical techniques are required to infer the magnetization of such samples. This yields an inverse problem on how to determine the magnetization from the set of measurements taken from the magnetic field of these weakly magnetized samples. The collaborative work from Vanderbilt University and MIT have obtained several fundamental theoretical results, including the reconstruction of unidirectional samples.
In this project, we’ll use MATLAB and data provided by the SQUID magnetometer at MIT to investigate the preference of algorithms based on Tikhonov and total variation regularizations, as well as the inversion based on measurements of the magnetic fields taken from both sides of a sample. In particular, we focus on non-unidirectional reconstruction from both synthesized and and real measurements.