Source Surface Height Optimisation for Improved Solar Wind Velocity Forecasting Across Solar Cycles 23, 24 and 25

Technical Summary: Source Surface Height Optimisation for Improved Solar Wind Velocity Forecasting Across Solar Cycles 23, 24, and 25

Problem Statement
The Potential Field Source Surface (PFSS) model is a fundamental component of state-of-the-art space weather forecasting frameworks, serving as the input for empirical solar wind models like the Wang–Sheeley–Arge (WSA) model. The PFSS model relies on a single physical free parameter: the source surface (SS) height, traditionally fixed at 2.5 $R_\odot$. While previous studies have investigated SS height optimization using various datasets and temporal intervals, most have focused on short-term analyses or specific phases of the solar cycle (SC). Furthermore, earlier efforts primarily aimed to improve estimates of heliospheric open magnetic flux rather than direct solar wind velocity predictions at L1. There remains a lack of comprehensive, long-term investigation into how the optimal SS height varies across multiple solar cycles (specifically SCs 23, 24, and 25) when integrated into the WSA framework for solar wind velocity forecasting. Additionally, the performance of different magnetogram products (Ground-based GONG vs. Space-based SDO/HMI) and the specific impact of zero-point correction (ZPC) in GONG maps on SS optimization require further clarification.

Methodology
The authors conducted a comprehensive analysis spanning Solar Cycles 23, 24, and 25 (approximately 2006–2026). The study utilized synoptic magnetic field maps from two primary sources:

1. Ground-based: Global Oscillation Network Group (GONG) maps, specifically comparing Standard (STD) and Zero-Point-Corrected (ZPC) versions.
2. Space-based: Solar Dynamics Observatory/Helioseismic and Magnetic Imager (SDO/HMI) radial synoptic maps.

To ensure comparability, the higher-resolution HMI maps (3600 $\times$ 1440) were downsampled to match the GONG resolution (360 $\times$ 180). The study covered 254 Carrington Rotations (CRs) for GONG data and 204 CRs for HMI data.

The modeling framework followed a standard pipeline:

1. PFSS Extrapolation: The coronal magnetic field was extrapolated from photospheric maps to the source surface using the PFSS model. The study tested eight discrete SS heights ranging from 1.5 $R_\odot$ to 3.25 $R_\odot$ in 0.25 $R_\odot$ increments.
2. WSA Model: Empirical solar wind velocities were estimated at the SS based on the expansion factor ($f_s$) and the angular distance from the nearest coronal hole boundary ($\theta_b$).
3. HUX Propagation: The Heliospheric Upwind eXtrapolation (HUX) model propagated these velocities to the L1 Lagrange point.
4. Validation: Modeled solar wind velocities at L1 were compared against in-situ observations from the OMNI database. The primary metric for optimization was the Pearson correlation coefficient ($cc$) between modeled and observed velocities. The "optimized SS height" was defined as the value yielding the maximum $cc$ for each Carrington Rotation.

Key Contributions

• Long-Term Scope: This study represents one of the longest-term investigations of SS height variability, covering nearly three solar cycles (SC23–25) rather than isolated intervals or single cycles.
• Multi-Source Comparison: It provides the first long-term comparison of GONG ZPC maps (widely used in WSA models) against HMI maps specifically for SS height optimization in the context of solar wind velocity forecasting at L1.
• Granular Sampling: Unlike previous studies that tested only a few discrete SS heights, this work employed a finer sampling (eight values) to better characterize the distribution of optimized heights.
• Phase-Specific Analysis: The analysis explicitly correlates optimized SS heights with specific phases of the solar cycle (minimum, ascending, maximum, descending) and distinguishes between periods of high and low activity within those phases.

Results

• Solar Cycle Phase Dependence: A clear relationship exists between the optimal SS height and the phase of the solar cycle.
  • Solar Minimum/Ascending/Descending Phases: Higher SS heights ($\ge$ 2.5 $R_\odot$) consistently yield better solar wind speed predictions at L1. For example, during the minimum phases of SC23 and SC24, the distribution of optimized heights skewed significantly toward values $\ge$ 2.5 $R_\odot$, with 3.25 $R_\odot$ often being the best fit.
  • Solar Maximum: During active phases, the trend reverses. Lower SS heights (< 2.5 $R_\odot$) generally provide better predictions. During the SC25 maximum, the optimized distribution skewed strongly toward lower values (e.g., 1.5–1.75 $R_\odot$ for HMI/ZPC maps).
• Magnetogram Performance:
  • GONG ZPC vs. STD: GONG ZPC maps consistently outperformed GONG STD maps, particularly during SC24 and the ascending phase of SC25. The STD maps showed degraded performance, likely due to observational limitations and quality issues reported in earlier literature.
  • GONG ZPC vs. HMI: ZPC and HMI maps performed similarly overall. However, during solar maximum, HMI maps showed a stronger skew toward lower SS heights compared to ZPC maps.
• Anomalies and Nuances:
  • During the SC24 maximum, a specific interval of significantly reduced solar activity (a "local minimum") showed a preference for higher SS heights, contrasting with the general trend for solar maximum. Excluding this interval shifted the overall SC24 maximum distribution toward lower SS heights.
  • The optimized SS height distribution exhibits significant scatter, particularly during solar maximum. The authors attribute this to the limited applicability of the PFSS model (which assumes a potential field) during highly active, multipolar periods, as well as the non-spherical nature of the true source surface.

Significance and Claims
The paper claims that optimizing the SS height in the PFSS model significantly improves the performance of the WSA solar wind speed prediction model at L1 compared to the conventionally fixed 2.5 $R_\odot$ height. The study establishes that the optimal SS height is not a static value but varies systematically with the solar cycle phase: higher values are required during quiet solar conditions, while lower values are preferable during active conditions.

The authors emphasize that while the absolute optimized height varies between cycles and datasets, the relative trend (higher SS at minimum, lower SS at maximum) is robust. They conclude that these findings support the physical interpretation that the effective source surface height is lower during solar maximum (due to the rapid decrease of transverse magnetic energy density in a multipolar field) and higher during solar minimum (dipolar field).

The study suggests that space weather forecasting frameworks can be refined by adopting a variable SS height strategy—specifically using heights slightly higher than 2.5 $R_\odot$ near solar minimum. However, the authors modestly note that the PFSS model's assumptions limit its accuracy during high-activity periods and propose that future improvements may require non-symmetric, non-spherical source surface models to account for the intrinsically non-spherical nature of the solar atmosphere.