Generalized Hilbert transforms of the effect of single magnetic sources

Abstract

The generalized Hilbert transforms of potential fields, particularly magnetic fields, provide a useful resource for improving interpretation. Even though 𝑥 - and 𝑦 -Hilbert transforms of a potential field on a plane can be approximately computed from the whole observed field, they are locally independent of the observations when working on reduced areas of the plane, and therefore, on those areas, Hilbert transforms can be combined with them to better constrain and stabilize local inversions. Extended Euler deconvolution, based on Euler’s homogeneity equation, makes extensive use of this principle. We investigated closed-form expressions for evaluating in the space domain the generalized 𝑥 - and 𝑦 -Hilbert transforms of magnetic fields generated by single point sources, given in the form of dipoles, monopoles, and Newtonian potential type sources. Apart from providing a way to calculate Hilbert transforms via equivalent sources, this approach, adding little computer effort, can be applied to solve local inverse problems as a way to improve the definition of a model previous to an inversion over a large area; the solution is modeled by point sources, and it enables matching not only the observations, but also the Hilbert transforms, thus providing stronger constraints. A synthetic example with noisy data demonstrated the resolving power of this approach in locally inverting for magnetization intensities of prismatic bodies using data windows either above or displaced from the sources. In an example from a magnetic anomaly over Sierra de San Luis, Argentina, the magnetization intensities of the basement are locally inverted for at each solution point of an extended Euler deconvolution. Results indicated a more coherent pattern when the Hilbert transforms were incorporated to the inversion, illustrating how local inversions with Hilbert transforms, which do not involve time-consuming operations, can accompany extended Euler deconvolutions to better outline characteristics of a model.