Efficient Bloch-equation solvers for surface nuclear magnetic resonance.

Abstract:

Direct sensitivity to water content and links to pore structure make surface nuclear magnetic resonance (sNMR) a unique and promising hydrogeophysical tool. However, the nature of the measurement, chiefly the extreme magnetic field inhomogeneity during excitation, requires the simulation of a large range of physically significant scenarios. Computational intensity of modelling the full forward response is a current challenge for the method and limits the kinds of inversion schemes that are of practical use. Approximations, such as neglecting relaxation during pulse effects, have been used to circumvent this computational load – but are based on an incomplete description of the NMR physics. Alternatively, full-Bloch solutions have assumed the equality of longitudinal (T₁) and transverse (T₂) relaxation to reduce the model dimensionality. In this work, we consider numerical, analytical and hybrid approaches to solving the Bloch equation in inhomogeneous magnetic fields without the need for common approximations and simplifications. Our solver can predict the NMR signal at arbitrary times independent of the pulse sequence, making it a general modelling framework for all types of sNMR measurements. Using optimized solvers and exploiting GPUs for high parallelizability, we explore how these full-Bloch solutions depend on all the relaxation parameters, and how this dependence evolves with time, for single and multi-pulse sequences. We characterize and compress these model spaces by fitting polynomials to the solution ensembles, and thereby extend the original fast-mapping technique to include T₁ and arbitrary times. This permits the rapid and accurate reproduction of these solutions with a single matrix multiplication and opens up sNMR to time-series based inversion of single and multi-pulse data that adhere to the full-Bloch solution.