Gradient field determination based on gravity measurements using spherical radial basis functions

Technical Summary: Gradient Field Determination Based on Gravity Measurements Using Spherical Radial Basis Functions

Problem Statement
Gravity gradients are critical for geodesy and geophysics, yet direct measurement via torsion balances is limited in spatial coverage. While global geopotential models (GPMs) provide a baseline, they are insufficient for detecting small-scale gradient anomalies due to their maximum degree limitations (typically up to degree 2190, e.g., EIGEN-6C4). Conversely, relying solely on high-resolution terrestrial gravity measurements to reconstruct gradients is challenging due to the ill-posed nature of the inverse problem and the risk of noise amplification. The study addresses the challenge of reconstructing regional gravity gradient fields solely from terrestrial gravity data in the absence of direct gradient measurements, specifically investigating how to balance signal recovery against noise amplification through spectral domain selection.

Methodology
The authors employed a regional modeling approach using Spherical Radial Basis Functions (SRBFs) combined with a "remove–restore" strategy.

1. Study Area and Data: The investigation focused on an approximately 800 km² area in central Hungary. The dataset included 1,195 raw terrestrial gravity measurements and 248 independent torsion balance measurement points (providing horizontal and curvature gradients) for validation.
2. Spectral Domain Definition:
   • Low-frequency removal: Components below degree $N_{min} = 1600$ were removed from the gravity anomalies using the EIGEN-6C4 global geopotential model. This lower bound was determined by the size of the model area ($w \approx 25.1$ km).
   • High-frequency modeling: The residual gravity anomalies were modeled using SRBFs. The theoretical upper bound ($N_{max}$) was calculated based on data point spacing, yielding an initial estimate of $N_{max} \approx 13,900$.
   • Filtering: A Shannon kernel was applied as a band-pass filter, where Legendre coefficients ($b_n$) were set to 1 between $N_{min}$ and $N_{max}$ and 0 elsewhere.
3. Inversion and Regularization: The SRBF coefficients were determined via least-squares inversion. To handle the ill-conditioned nature of the system and uneven data distribution, Tikhonov regularization was applied. Variance Component Estimation (VCE) was used iteratively to determine optimal weights for different data types (measured gravity vs. interpolated grid data).
4. Synthesis and Validation: After determining the SRBF coefficients, gravity gradients were synthesized. The low-degree components ($<1600$) were restored using the GPM. The resulting gradients were validated against the independent torsion balance measurements (which were excluded from the inversion process).

Key Results
The study systematically analyzed the impact of the maximum expansion degree ($N_{max}$) on solution accuracy:

• Optimal Spectral Range: While the theoretical $N_{max}$ based on data spacing was 13,900, empirical testing revealed that expanding beyond a specific range led to significant noise amplification. The optimal range was identified between $N_{max} = 6,500$ and $7,500$.
• Performance Metrics: At $N_{max} = 6,500$, the SRBF-derived gradients showed the best agreement with independent torsion balance data:
  • Correlation Coefficients: Reached approximately 0.796 for the $T_{xz}$ gradient and 0.747 for $T_{yz}$.
  • Coefficient of Determination ($R^2$): Improved to 0.628 for $T_{xz}$ and 0.551 for $T_{yz}$.
  • Comparison to Global Models: Solutions derived solely from the EIGEN-6C4 model (up to degree 2190) yielded significantly lower correlations (approx. 0.36–0.45).
  • Comparison to Over-expanded SRBF: Using the theoretical $N_{max} = 13,900$ resulted in poor correlations (approx. 0.32) and low $R^2$ values, indicating that the inversion was primarily modeling noise rather than signal.
• Spatial Resolution: The SRBF solution at $N_{max} = 6,500$ successfully reproduced regional trends and smaller-scale features observed in the torsion balance data, which were absent in the global model synthesis.

Significance and Claims
The paper claims that gravity gradients can be reliably reconstructed from gravity anomalies alone, provided that appropriate spectral constraints are applied. The primary contribution is the demonstration that frequency-domain regularization is critical in SRBF-based inversion.

• Balance of Signal and Noise: The study highlights that simply increasing the expansion degree to match data density does not improve accuracy; rather, an optimal "sweet spot" exists where signal content is maximized without amplifying noise.
• Independence from Gradient Data: The methodology confirms that high-resolution gradient fields can be generated in areas lacking direct gradient measurements, utilizing only terrestrial gravity data and a global geopotential model.
• Validation: The use of independent torsion balance measurements as a validation set (rather than input) provides robust evidence that the SRBF inversion captures physical signal content rather than fitting noise.

The authors conclude that while global models provide a necessary foundation for long-wavelength components, SRBF inversion with a carefully selected maximum degree offers a robust and effective approach for enhancing the spatial resolution of regional gravity gradient fields.