Forward modeling of single-sided magnetic resonance and evaluation of T2 fitting error based on geometric analytical method

Single-sided magnetic resonance (SSMR) offers advantages of portability and noninvasive measurement for water detection, with significant potential applications in groundwater exploration, petroleum well logging, and soil moisture monitoring. However, the inherent highly inhomogeneous static magnetic field and radiofrequency (RF) field in SSMR necessitate the utilization of the Carr–Purcell–Meiboom–Gill (CPMG) sequence measurement scheme. To accelerate forward modeling during pulse excitation, we introduce a Geometric Analysis Method (GAM) and assess $T_2$ error using its primary parameters. The GAM involves applying spatial geometric rotations on the magnetization vector, leading to an analytical solution to the Bloch equation that disregards relaxation effects. Compared with the rotation matrix (RM) method, the GAM demonstrates high accuracy and reduces computational time by approximately 20.9%. By analyzing the primary parameters governing the magnetization vector in the analytical formula, we evaluated their impact on the transverse relaxation time ($T_2$) obtained through fitting the SE signal. Ultimately, the forward modeling results of the CPMG sequence within the region of interest (ROI) of a single-sided Halbach magnet array are validated. The $T_2$ fitting error increases as the primary parameters deviate from the ideal values, highlighting their significant role in the $T_2$ fitting results. This study provides a theoretical foundation for optimizing the design of SSMR magnets and RF coils.