Theory of surface nuclear magnetic resonance with applications to geophysical imaging problems

The general theory of nuclear magnetic resonance (NMR) imaging of large electromagnetically active systems is considered. We emphasize particularly noninvasive geophysical applications such as the imaging of subsurface water content. We derive a general formula for the NMR response voltage, valid for arbitrary transmitter and receiver loop geometry and arbitrary conductivity structure of the medium in which the nuclear spins reside. It is shown that in cases where the conductivity is large enough such that the electromagnetic skin depth at the Larmor frequency is of the same order or smaller than the measurement depth, there are diffusive retardation time effects that significantly alter the standard NMR response formula used in the literature. The formula now includes the full complex response, the imaginary part of which has previously been observed but not modeled. These differences are quantified via numerical investigation of various effectively one-dimensional model inverse problems with a horizontally stratified nuclear spin and conductivity distribution. It is found that inclusion of the imaginary part of the response significantly stabilizes the inversion. Large quantitative differences are found between conducting and insulating cases in physically relevant situations. It is shown also that the diffusive long time tail of the signal may be used to infer the distribution of time constants ${T}_{1},$ normally not measurable in geophysical applications. Although in present applications the signal due to this tail is immeasurably small, this relationship may become useful in the future.