Generalized Hilbert transform of deffect of single magnetic sources

GUSPI, F.; NOVARA, I.; CARUGATI, G.

FOZ DE IGUAZU

Congreso; MEETING OF THE AMERICAS 2010; 2010

The generalized Hilbert transforms of potential fields, particularly magnetic fields, provide a useful resource for improving the interpretation. Even though both x and y Hilbert transforms of a potential field on a plane are determined and can be approximately computed from the whole observed field, they are locally independent of the observations on restricted areas of the plane, and therefore can be combined with them to better constrain local inversions. The generalized Euler deconvolution, based on Euler's homogeneity equation, makes extensive use of this principle. This work, starting from the equations that define the magnetic field itself, develops space-domain expressions that, apart from providing an alternative method for calculating Hilbert transforms via equivalent sources, allow for local inversions, based on a single or a few magnetic sources, to jointly match the observed magnetic field and its Hilbert transforms, incorporating this way a stronger constraint. On a cartesian coordinate system, closed form expressions are obtained for the x and y generalized Hilbert transforms of the magnetization tensor defined on the x-y plane by a point magnetic source located at depth z, and three types of sources are considered: dipole, monopole, and Newtonian or potential-type source, which roughly correspond to structural indices 3, 2 and 1 respectively. With those tensor expressions, the transforms of a magnetic field observed on a given direction, and generated by the source from other magnetization direction, can be easily calculated. In a numerical example with noisy data, it is shown that the intensity of magnetization within a known prismatic body can be better estimated if the Hilbert transforms of the field are taken into account. In an example with real data, the generalized Hilbert transforms of a magnetic anomaly over Sierra de San Luis, Argentina, are calculated from Newtonian equivalent sources, and the results compared to those provided by frequency-domain expressions.