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.

The magnetic method, perhaps the oldest of geophysical exploration techniques, blossomed after the advent of airborne surveys in World War II. With improvements in instrumentation, navigation, and platform compensation, it is now possible to map the entire crustal section at a variety of scales, from strongly magnetic basement at regional scale to weakly magnetic sedimentary contacts at local scale. Methods of data filtering, display, and interpretation have also advanced, especially with the availability of low-cost, high-performance personal computers and color raster graphics. The magnetic method is the primary exploration tool in the search for minerals. In other arenas, the magnetic method has evolved from its sole use for mapping basement structure to include a wide range of new applications, such as locating intrasedimentary faults, defining subtle lithologic contacts, mapping salt domes in weakly magnetic sediments, and better defining targets through 3D inversion. These new applications have increased the method's utility in all realms of exploration — in the search for minerals, oil and gas, geothermal resources, and groundwater, and for a variety of other purposes such as natural hazards assessment, mapping impact structures, and engineering and environmental studies.

The paper extends to three dimensions (3-D) the two‐dimensional (2-D) Hilbert transform relations between potential field components. For the 3-D case, it is shown that the Hilbert transform is composed of two parts, with one part acting on the X component and one part on the Y component. As for the previously developed 2-D case, it is shown that in 3-D the vertical and horizontal derivatives are the Hilbert transforms of each other. The 2-D Cauchy‐Riemann relations between a potential function and its Hilbert transform are generalized for the 3-D case. Finally, the previously developed concept of analytic signal in 2-D can be extended to 3-D as a first step toward the development of an automatic interpretation technique for potential field data.