Towards a new potential field theory of fractal objects

Abstract

The Potential Field Anomaly (PFA) data of the self similar Fractal Objects (FOs) include gravity and magnetic fields and potentials along with the related derivatives. These elements are calculated on grids due to buried FOs at different fractal orders. The objects have variable physical property distributions; while in magnetic, the orientation and magnitude of polarization or earth magnetic field is arbitrary. Using the structural index as Universal Fractal Order Invariant Measure, one of the contributions of this work is expressing the elements of the PFA data at any measuring point on a grid as geometric sequences in terms of the fractal order. We found that the common ratio of the sequences is equal to the Fractal Mass Ratio (FMR), a physical quantity characterizing the object. Therefore, we can interpolate the PFA data backward or forward from one fractal order to the other. This in turn allows us to directly calculate PFA data of FOs from the zero order objects equivalent to the solid sources or initiators. We conclude that the patterns of PFA data due to a self similar FO are scale-invariant and reflect the nature of this object. We express the FMR of a FO in a new equation describing the difference between the topological and fractal dimensions in terms of a linear scale.