Linear-Algebraic Audit of Geophysical Fields and Inverse Deconvolution: Three Case Studies in Regularized Inference

Observational geophysics, climate dynamics, and experimental spectroscopy routinely rely onEmpirical Orthogonal Functions (EOF / SVD) and Tikhonov-regularized deconvolution to extractcoherent physical modes and sharp spatial profiles from noisy sensor grids. Lacking analyticalground truth for complex natural fields, practitioners frequently interpret orthogonalspatial multipoles or smooth inversion curves as genuine physical oscillations. Grounded in inverseproblem theory, this technical report subjects three geophysical and physical datasets to asequential four-step linear-algebraic audit evaluating operator conditioning (κ), Effective Degreesof Freedom (DoF), singular value decay (Σ), and eigenvector Participation Ratios (PR). Acrossthree distinct regimes—NOAA Pacific sea-surface temperature grids (61 × 12), two-dimensionaltopographic surfaces (87×61), and ill-posed instrumental deconvolution (N = 200)—we exposehow orthogonality constraints manufacture seasonal dipoles and spatial quadrupoles (DoF =1.66/61), while smoothness priors either smear sharp physical discontinuities or induce Gibbsringing near boundaries (κ(ATA) = 6.00×1033). These empirical diagnostics establish a transparentstandard for separating physical structure from operator geometry in continuous fieldanalysis.