Gravi-magnetic Anomalies of Uniform Thin Polygonal Sheets

Thin planar sheets are useful gravitational and magnetic models of dykes and veins treated as two-dimensional geophysical structures on the scale of the survey. Thus, the anomaly of a polygonal thin sheet of uniform surface density or magnetization in arbitrary orientation has practical interest. The limiting thin-sheet anomaly can be approached from the corresponding polyhedral parallelepiped under decreasing thickness, though the numerical limit cannot be reached this way on account of the floating point finite precision. We derive the analytical zero thickness limit for the gravity potential while maintaining finite total mass. We use the concept of gravi-magnetic similarity to extend the thin-sheet potential formula to include the potential, field and field gradient in both gravity and magnetic cases, thereby generalising other studies that have obtained isolated polygonal thin-sheet anomaly solutions. We compare the anomalies computed by the new formulae to those of corresponding finite thickness targets, and to the finite difference estimates of the field and field gradient obtained from numerically differentiated thin-sheet potentials. In both cases a second order rate of approach to the limit is observed, verifying the correctness of the new formulae. Thin-sheet solutions are attractive for their reduced computational burden compared to full parallelepiped solutions, while the stacking of thin sheets may be used to simulate variable density or magnetization targets. It is anticipated that thin-sheet solutions presented here will find wide application in gravi-magnetic modelling.