Optimization Algorithm for Determining the Source Surface Radius Based on Parker Solar Probe in situ Measurements from Encounters 1 to 19
The Potential Field Source Surface (PFSS) extrapolation is a method for estimating the large scale coronal magnetic field from photospheric magnetograms. The source surface serves as the outer boundary of its solution domain, and is typically a spherical surface. An appropriate source surface radius ($R_{ss}$) enables more accurate identification of the coronal magnetic field topology and estimation of the open flux, thereby potentially enhancing the accuracy of space weather modeling. We prove the well-posedness of the PFSS forward problem and establish the existence and uniqueness of the optimal source surface by combining compactness of the admissible set with continuity of the objective functional. The objective functional is the mean squared error (MSE) between PFSS extrapolation and Parker Solar Probe (PSP) radial magnetic field measurements after Parker spiral backmapping and radial scaling for Encounters 1-19. The optimization algorithm is validated with an analytical solution, and Advanced Composition Explorer (ACE) in situ measurements are used as an independent cross-validation dataset. Additional evaluation metrics and Pareto analysis are used to identify the dominant metrics between open flux and polarity prediction accuracy. Our results show that the optimal $R_{ss}$ derived from the algorithm generally increase from solar minimum into the ascending phase of solar cycle 25. The optimized solution improves open flux agreement while preserving or improving polarity prediction accuracy relative to $2.5R_{s}$. The Pareto frontiers show a transition for dominant metrics from open flux during solar minimum to polarity prediction accuracy during the ascending phase.