Forward Modeling Steady-State Free Precession in Surface NMR

In efforts to map water at depth, steady-state free precession (SSFP) sequences promise to rapidly increase data acquisition rates in the practice of surface nuclear magnetic resonance (NMR). Unlike conventional surface NMR excitation schemes, pulses in SSFP are transmitted so frequently that the nuclear magnetization of hydrogen in water cannot return to its natural alignment with the Earth’s ambient magnetic field, and instead achieve a steady-state; a dynamic equilibrium between pulses. Unfortunately, the traditional formulations of SSFP sequences and the full surface NMR forward models are not immediately compatible with each other. First, the traditional analysis of SSFP sequences assumes that relaxation during pulse (RDP) effects is negligible, which is not always valid in surface NMR. Second, even for single-pulse measurement, the surface NMR forward model can be computationally demanding; this challenge scales with the number of pulses. Here, we investigate the incorporation of RDP effects on the dynamic equilibrium of SSFP measurement. This is then incorporated into the full surface NMR forward model by deriving the analytical expressions to directly predict processed surface NMR data. The model is validated by jointly inverting an extensive and diverse suite of SSFP measurement, 12 distinct sequences each with 16 pulse moments. The inverted model has a data misfit of 0.99 and is consistent with models derived from standard NMR data. The ability of our forward model to reproduce diverse signals and jointly invert them is a strong indication of its validity.