Comparison and understanding of sparse magnetization vector inversion

Magnetization vector inversion (MVI) is an effective method for interpreting magnetic data without knowing the magnetization directions. Nevertheless, the serious nonuniqueness problem makes it difficult to obtain satisfactory results without proper constraints. Several constrained methods have been applied to the magnetization vector inversion to produce reliable results. To better understand these issues and provide some improvements, we compared and evaluated different forms of magnetization vector inversion: (1) magnetization vector inversion in Cartesian coordinates (MVI-C); (2) magnetization vector inversion in spherical coordinates (MVI-S); and (3) compact magnetization vector inversion with magnitude constraints. Magnetization vector inversion incorporates prior information or assumptions about subsurface geological structures into the model objective function and solves the optimal problem with respect to the data and the model objective function to recover the desired features. We first analyze different model objective functions and then test these methods against synthetic and real datasets. Theoretical analysis and tests reveal that the linear relationship in the rectangular coordinate system simplifies the calculation process, but it is difficult to apply reasonable constraints, which results in a lack of correlation in the direction of magnetization; moreover, the distribution is not concentrated. It is easy to constrain the magnetization magnitude and direction in the spherical coordinate system, and better results can be obtained. However, due to the nonlinear relationship, the calculation complexity increases, and the inversion results are heavily dependent on the initial model. The method based on the modulus constraint establishes the relation between components in the Cartesian coordinate system, but the direction cannot be constrained. Therefore, we believe that the magnitude and direction of magnetization should be constrained simultaneously in the rectangular coordinate system to obtain a fast, stable, and accurate inversion method.