Reconstruction of Frontal Gradients Using Radial Basis Function Interpolation

The accurate representation of frontal zones—characterized by sharp scalar gradients—remains a critical challenge in regional objective analysis and data assimilation, particularly when utilizing sparse or stochastically distributed observations. This study evaluates the efficacy of Multiquadric Radial Basis Functions (RBFs) as a high-order alternative to standard spatial mapping operators frequently used in machine learning atmospheric emulators. We contrast the performance of the regularized, C∞-continuous RBF approach against nearest neighbor and linear mesh interpolation schemes using both synthetic baroclinic wave profiles and an operational case study of the intense extratropical cyclone that impacted the East Coast of North America in mid-March 1993. To mitigate characteristic boundary artifacts and geometric clipping in bounded regional domains, we implement a targeted numerical stabilization framework combining localized boundary mirroring with four-corner domain anchoring. Our quantitative results demonstrate that the optimized RBF framework substantially improves gradient fidelity and reduces Root Mean Square Error across a wide range of observation densities. Furthermore, we evaluate the computational scalability of RBFs on high-performance computing architectures, demonstrating how Algebraic Multigrid solvers and Graphics Processing Unit acceleration mitigate the foundational O(N3) computational bottleneck. We conclude that RBF interpolation provides a physically consistent, analytically differentiable manifold that addresses the derivative discontinuities of traditional linear methods, offering a stable pre-processing framework for high-resolution meteorological analysis and machine learning optimization.