Optimization Algorithm for Determining the Source Surface Radius Based on Parker Solar Probe in situ Measurements from Encounters 1 to 19

Technical Summary: Optimization Algorithm for Determining the Source Surface Radius Based on Parker Solar Probe in situ Measurements

Problem Statement
The Potential Field Source Surface (PFSS) extrapolation is a standard method for estimating the large-scale coronal magnetic field from photospheric magnetograms. A critical parameter in this model is the source surface radius ($R_{ss}$), the spherical boundary where the magnetic field is assumed to become radial. While $R_{ss} = 2.5 R_s$ is a conventional reference, previous studies indicate that a single value cannot universally satisfy all observational constraints (e.g., coronal hole extent, open magnetic flux, and magnetic polarity). The optimal $R_{ss}$ is known to vary with solar activity, yet determining a rigorous, observationally constrained value remains challenging. This paper addresses the inverse problem of determining the optimal $R_{ss}$ by minimizing the discrepancy between PFSS extrapolations and in situ magnetic field measurements, specifically utilizing data from the Parker Solar Probe (PSP) during Encounters 1–19.

Methodology
The authors approach the determination of $R_{ss}$ as a shape optimization problem within a free boundary framework.

1. Mathematical Formulation:

   • The forward PFSS problem is formulated as a second-order elliptic boundary value problem with mixed Dirichlet and Neumann conditions. The authors prove the well-posedness (existence and uniqueness) of the forward solution in the Sobolev space $H^1(\Omega)$.
   • The inverse problem is cast as minimizing an objective functional over an admissible set of domains. The authors prove the compactness of the admissible set and the continuity of the objective functional, thereby establishing the existence of a global minimum for the inverse problem. These proofs apply to both spherical and nonspherical source surface geometries.

2. Data and Preprocessing:

   • Inputs: Synoptic photospheric magnetograms from the Global Oscillation Network Group (GONG) serve as the inner boundary condition.
   • Observations: Radial magnetic field ($B_r$) and radial solar wind velocity ($v_r$) from PSP (FIELDS and SWEAP instruments) for Encounters 1–19.
   • Processing: PSP data undergoes outlier removal (Interquartile Range criterion) and temporal averaging (20-minute cadence). Solar wind velocity gaps are filled via linear interpolation to enable continuous Parker spiral backmapping.
   • Validation: Independent in situ measurements from the Advanced Composition Explorer (ACE) near 1 AU are used for cross-validation.

3. Optimization Algorithm:

   • Objective Functional: The algorithm minimizes the Mean Squared Error (MSE) between the observed PSP $B_r$ and the PFSS-predicted $B_r$. The PFSS prediction is derived by tracing the spacecraft position back to the source surface along Parker spirals, sampling the PFSS field, and radially scaling it to the spacecraft distance ($r^2 B_r = \text{const}$).
   • Normalization: To remove encounter-dependent amplitude differences, both observed and modeled fields undergo Z-score normalization before MSE calculation.
   • Solver: A gradient-based iterative scheme (Algorithm 1) is employed. The gradient is approximated using centered finite differences. The algorithm iterates until convergence or a maximum iteration count is reached.
   • Validation: The algorithm is validated against an analytical potential field solution and synthetic data with added Gaussian noise, demonstrating the ability to recover a prescribed $R_{ss}$.

4. Multiobjective Analysis:

   • To understand trade-offs, the authors introduce two additional metrics: the magnetic field scaling factor ($\alpha_k$, related to open flux) and polarity prediction accuracy ($P^k_a$).
   • A Pareto analysis (Algorithm 2) is performed to identify the Pareto frontier between the MSE objective and the polarity prediction error. This quantifies whether the optimization is driven primarily by amplitude agreement or polarity accuracy.

Key Results

• Optimal $R_{ss}$ Trend: The optimized $R_{ss}$ values for PSP Encounters 1–19 generally increase from solar minimum into the ascending phase of Solar Cycle 25. This trend is corroborated by independent ACE data.
• Comparison with Reference: Compared to the fixed reference $R_{ss} = 2.5 R_s$, the optimized solutions generally show better agreement with observed magnetic field amplitudes (reducing the open flux discrepancy) while preserving or improving magnetic polarity prediction accuracy.
• Solar Minimum vs. Ascending Phase:
  • During solar minimum (early encounters), the optimization is primarily driven by amplitude agreement (open flux). Lower $R_{ss}$ values are selected to increase the modeled open flux to match observations.
  • During the ascending phase, the Pareto frontiers become more clustered, indicating that polarity prediction accuracy becomes the dominant metric. The optimal $R_{ss}$ increases to better capture the complex sector structures and polarity reversals observed.
• Local Deviations: Specific encounters (e.g., 9 and 13) show local departures from the general trend. For Encounter 13, which crossed a complex Interplanetary Coronal Mass Ejection (ICME), the optimization selected a very low $R_{ss}$ ($1.27 R_s$) to capture complex polarity inversions, highlighting the limitations of the Parker spiral assumption in disturbed environments.

Significance and Claims
The paper claims to provide a mathematically rigorous foundation for the PFSS inverse problem, proving the existence of a unique minimizer for the source surface radius estimation. By constructing a specific optimization algorithm validated against analytical solutions and in situ data, the study demonstrates that $R_{ss}$ is not a static constant but a dynamic parameter that evolves with solar activity.

The authors emphasize that there is no single "globally optimal" $R_{ss}$; rather, the optimal value depends on the specific observational constraints and evaluation metrics prioritized (e.g., open flux vs. polarity). The study highlights a shift in the dominant metric from amplitude agreement during solar minimum to polarity agreement during the ascending phase. Furthermore, the paper acknowledges the "open flux problem," noting that while the optimization improves agreement with in situ flux, it does not resolve the fundamental discrepancy between coronal hole-based flux estimates and in situ measurements, suggesting that future improvements may require better polar magnetic field observations or more advanced coronal models. The work sets a framework for future shape optimization of nonspherical source surfaces as data from missions like Solar Orbiter becomes available.