Closed-Form Formulae for Fractional Gradient Tensor of Magnetic Dipole

Fractional calculus can be regarded as an important supplement to
integer calculus, and has been gradually applied in physics, engineering
and so on. In this paper, we define the fractional magnetic gradient
tensor of a magnetic dipole, and derive its analytic expressions by
using the rule of the composition of fractional-order and integer-order
derivatives. Then we verify the analytic expressions by comparing with
the results of the numerical method. When the order α of fractional
derivatives approaches zero, the fractional magnetic gradient tensor of
a magnetic dipole becomes the matrix composed of three magnetic field
components. When α approaches one, the fractional magnetic gradient
tensor becomes the standard magnetic dipole tensor. This trend shows
that fractional derivatives and integer derivatives are consistent.
Magnetic gradient tensors have larger attenuation with higher derivative
orders when increasing the distance between the observation point and
the magnetic source. Therefore, the limited resolution of the magnetic
sensors causes a large blind area in a survey, which can be compensated
by measuring the fractional magnetic gradient tensor. In addition, each
component of the fractional tensor is independent and has great
potential of solving the multiple solution problems of the localization
of a magnetic dipole.