Empirical Optimization of the Source-Surface Height in the PFSS extrapolation

The potential field source surface (PFSS) method is a widely used magnetic field extrapolation technique in the space weather community. The only free parameter in the PFSS method is the source-surface height ($R_{\rm SS}$), beyond which all field lines are open. Although $R_{\rm SS}$ is known to vary with solar activity, there is no consensus on how to determine it for a given surface magnetic field distribution. In this study, we investigate the nature of $R_{\rm SS}$ using a long-period (2006-2023) data, covering two solar minima and one maximum. We adopt ADAPT-GONG magnetograms and determine $R_{\rm SS}$ by matching the open flux estimated from observations at 1 au with that calculated using the PFSS method. Our analysis reveals that $R_{\rm SS}$ increases slightly after the solar minima and around the solar maximum, and that it can be characterized by both the mean unsigned photospheric magnetic field strength and the dipolarity parameter $f_{\rm dip}$, defined as $f_{\rm dip} = B_{\rm dip}^2/(B_{\rm dip}^2 + B_{\rm quad}^2 + B_{\rm oct}^2)$, with $B_{\rm dip}$, $B_{\rm quad}$, and $B_{\rm oct}$ denoting the magnitudes of dipolar, quadrupolar, and octupolar components of photospheric radial magnetic field, respectively. Our results suggest that $R_{\rm SS}$ does not exhibit a simple monotonic dependence on the solar activity and must be determined by properly considering both surface magnetic field strength and global field structure.