Retrieving the impulse response of the Earth due to random electromagnetic forcing

A derivation is presented of Green's function in an arbitrary, heterogeneous conductive medium subject to random, ambient, uncorrelated noise sources. The approach for extracting Green's function is based on the correlation of time series of magnetic field components at two independent locations. As in the related case for the electric field, where the volume distribution of noise sources must be spatially correlated with the heterogeneous conductivity distribution, Green's function for magnetic field requires noise sources to be spatially correlated with the volume distribution of magnetic permeability. For applications of electromagnetic imaging of Earth's deep subsurface, the effect of magnetic permeability variations in the subsurface is often assumed to be negligible when compared to the effect of conductivity variations. Hence, the expressions derived here may be useful for passive electromagnetic subsurface imaging, in apparent contrast to their electric field counterparts. Numerical validation exercises are described which validate this theory for Green's function estimation in the low-frequency limit.