Closed-form expressions of the magnetic and magnetic gradient tensor due to an elliptical cylinder

Magnetic gradient tensors obtained from multiple magnetic sensors have been increasingly applied in various near-surface explorations. Accurate interpretation of high-resolution magnetic gradient tensor data requires analytical expressions due to simple geometric bodies. In this study, analytical expressions for the magnetic field and magnetic gradient tensor responses due to an elliptical cylinder are derived. An elliptical cylinder is geologically relevant, as igneous intrusions such as kimberlite pipes commonly exhibit elliptical cross-sections with axial symmetry and anisotropic radial extents in the strike and transverse directions. The magnetic responses are obtained by transforming the previously derived gravity gradient tensor of an elliptical cylinder using Poisson’s relation. The gravitational potential, defined as a triple integral, is differentiated twice with respect to each coordinate axis to obtain the gravity gradient tensor. And the gravity gradient tensors are then converted into magnetic responses in the real domain. The magnetic gradient tensor expressions in the real domain are integrated along the symmetry axis (z-direction) to reduce them to double integrals. By introducing complex variables, the real double integrals are transformed into complex integrals. Finally, using the complex form of Green’s theorem, the magnetic gradient tensors due to the elliptical cylinder are expressed as a one-dimensional line integral evaluated along the elliptical boundary.

 Acknowledgements: This work was supported in part by research project from KIGAM and G-LAMP project based on a National Research Foundation of Korea grant from the Ministry of Education (No. RS-2023-00301938), S. Korea.