The impact of two-dimensional filtering on SWOT observations with white noise
The Surface Water and Ocean Topography (SWOT) mission provides two-dimensional observations of sea surface height (SSH) at unprecedented spatial resolution, enabling exploration of ocean variability down to scales of $O(10~\mathrm{ km})$. At these scales, however, interpreting SSH variability is challenging because ocean dynamical signals overlap with measurement noise, and their respective spectral signatures are not yet fully understood. Recent analyses of SWOT 2-km posting observations have shown that along-track spectra transition to flatter but still red spectra at scales around 30 km. These flatter portions of the spectra have a power-law-like behavior and spectral slopes of approximately $-1$ or steeper, and their magnitudes and slopes are correlated with SWOT measurement noise magnitude. Here, we investigate the hypothesis that these flatter but still red along-track small-scale spectra can arise from two-dimensional filtering and aliasing of spatially uncorrelated (white) noise. Using synthetic experiments, we show that the resulting one-dimensional along-track spectra exhibit a similar transition to red, power-law-like behavior at scales of 15--50 km, qualitatively similar to spectral behavior reported in SWOT observations. The transition scale and apparent spectral slopes depend on the noise level, its cross-track variability, and the background ocean signal. This finding highlights the importance of carefully accounting for measurement noise and processing effects when interpreting SWOT spectra, and suggests that a white noise model should serve as a baseline null hypothesis for small-scale spectral analyses.