Interpolation-free method to recover spectra from sparse observations of random fields

Observational data are fundamental for understanding geophysical dynamics, yet constraints such as cost or environmental conditions often result in sparse and noisy data. Recovering physical quantities such as energy spectra from such data constitutes a classic ill-posed inverse problem. Traditional approaches typically rely on interpolation to regular grids, which can introduce errors, especially for shallow spectra.

This study proposes a random recovery framework that infers spectra from the second-order statistics of observations under suitable assumptions. Observation noise is reduced using the Best Linear Unbiased Estimator (BLUE), while shrinkage techniques are employed to obtain stable and invertible covariance estimates under limited sampling. To achieve robust solutions without interpolation, we introduce a high-order L² regularization, using the discrepancy principle to determine the optimal regularization parameter. For high-dimensional settings, we apply hard clustering to group similar spectra, thereby reducing the number of unknowns and enhancing recovery efficiency.

Numerical experiments demonstrate that this method offers a robust and practical approach for spectral recovery without interpolation, making it particularly suitable for sparse and noisy observations.