Time-Domain Electromagnetic Fractional Anomalous Diffusion Over a Rough Medium

The theory of magnetic-source electromagnetic anomalous diffusion is developed in this article. As the geological medium is spatially rough, the late-time tail phenomenon can be explained as a multiscale electromagnetic anomalous diffusion, which is a union of fractional subdiffusion and regular diffusion. The roughness parameter β and weighting parameter W are introduced into the generalized electrical conductivity to indicate the roughness level and ratio. The characteristics and superiority of the improved generalized electrical conductivity are analyzed in the time domain. A 3-D modeling method is proposed based on the fractional finite-difference time-domain method. The introduced Caputo fractional derivative in the time domain is calculated by the Alikhanov superconvergent approximation method, so that the stability of the solution is improved. Several half-space models are designed to verify the 3-D modeling efficiency. It turns out that the improved generalized electrical conductivity can model the electromagnetic late-time tail efficiently, which is characterized by responses that coincide with the classical attenuation law at earlier times and the departure at later times. Moreover, this approach is also applied to anomalous models with 3-D bodies; the results indicate that the proposed method can recognize 3-D anomalous bodies efficiently. The improved method can model the complex electromagnetic propagation in the rough geologic medium more accurately, providing a theoretical basis for multiscale and refined detection.