<p>Gap filling in geophysical time series is particularly challenging when observed data is non-stationarity and has sharp temporal transitions. Many existing approaches rely on global functional representations or stationarity assumptions, which can compromise the preservation and identification of physically meaningful local features. In this study, we introduce EigenSig, an eigenspace-based framework that reformulates gap filling as a coordinate-recovery problem in a reduced eigenspace. The method partitions a time series into epochs, embeds them in a vector space, and constructs an eigenspace using principal component analysis, where dominant eigenvectors capture within-epoch variability and eigenspace coordinates represent large-scale coherence across epochs. Missing epochs are addressed by estimating and analyzing these coordinates from neighboring epochs and projecting them back into the original data space. Unlike least-squares harmonic estimation (LSHE), which uses global Fourier series representation, EigenSig uses a fully data-driven orthogonal basis as the base of eigenspace and designed to preserve signal shape without stationary assumption. We also incorporate the widely used LSHE method to estimate coordinates of unknown epochs in eigenspace to enhance its robustness and uncertainty quantification. EigenSig is further extended for anomaly separation using a leave-one-out strategy, in which each epoch is reconstructed from all remaining data in eigenspace, and anomalies are quantified as the residual between the observed and reconstructed epochs. Applications to synthetic data gaps and geomagnetic time series demonstrate that EigenSig effectively reconstructs data gaps, separates localized anomalies while maintaining global structure, and has calculation efficiency, highlighting its methodological novelty and practical relevance for geophysical signal analysis.</p>