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.