The analytic signal of two-dimensional magnetic bodies with polygonal cross-section; its properties and use for automated anomaly interpretation

Abstract

This paper presents a procedure to resolve magnetic anomalies due to two-dimensional structures. The method assumes that all causative bodies have uniform magnetization and a cross-section which can be represented by a polygon of either finite or infinite depth extent. The horizontal derivative of the field profile transforms the magnetization effect of these bodies of polygonal cross-section into the equivalent of thin magnetized sheets situated along the perimeter of the causative bodies. A simple transformation in the frequency domain yields an analytic function whose real part is the horizontal derivative of the field profile and whose imaginary part is the vertical derivative of the field profile. The latter can also be recognized as the Hilbert transform of the former. The procedure yields a fast and accurate way of computing the vertical derivative from a given profile.For the case of a single sheet, the amplitude of the analytic function can be represented by a symmetrical function maximizing exactly over the top of the sheet. For the case of bodies with polygonal cross-section, such symmetrical amplitude functions can be recognized over each corner of each polygon. Reduction to the pole, if desired, can be accomplished by a simple integration of the analytic function, without any cumbersome transformations. Narrow dikes and thin flat sheets, of thickness less than depth, where the equivalent magnetic sheets are close together, are treated in the same fashion using the field intensity as input data, rather than the horizontal derivative. The method can be adapted straightforwardly for computer treatment. It is also shown that the analytic signal can be interpreted to represent a complex "field intensity", derivable by differentiation from a complex "potential". This function has simple poles at each polygon corner.Finally, the Fourier spectrum due to finite or infinite thin sheets and steps is given in the Appendix.