Theoretical Basis for the Resolution and Noise of SWOT Estimates of Sea-Surface Height

This note clarifies two important issues related to ocean observations of sea-surface height (SSH) by SWOT. The analysis presented here is intended to supplement the discussion in the SWOT science requirements document, the onboard processing document, and the mission performance and error budget document. The first issue is the question of how the raw SWOT data will be filtered by the onboard processor and the related questions of what the resolution and decorrelation length scale are for the onboard pre-processed estimates of SSH. This information is provided in the onboard processing document, but without the mathematical basis for the conclusions. A rigorous development of the mathematical formalism for assessing the resolution and decorrelation length scale is presented in Section 2, along with a discussion of the error reduction that can be achieved by smoothing of the onboard pre-processed SWOT data in ground-based post-processing. Section 2 also includes a clarification of the relation between the feature resolution of smoothed SSH fields as defined in the onboard processing document and the wavelength resolution corresponding to the half-power filter cutoff that is often used by oceanographers to characterize resolution. The second issue is the question of the science requirement for the variance of the uncorrelated errors in the pre-processed estimates of SSH computed onboard the SWOT spacecraft. While this is important information for users since these data are fundamental to all oceanographic applications of SWOT data, the requirement for the noise variance of the onboard estimates of SSH is not explicitly specified in the present drafts of the SWOT documentation. The analysis in Section 3 clarifies the relationship between the uncorrelated errors in the onboard pre-processed estimates of SSH and the error specification as characterized in the SWOT documentation. The latter is expressed as a white-noise spectrum only for wavelengths longer than 15 km over which the estimated signal-to-noise ratio is required to exceed unity for 68% of the world ocean (see the cyan line in Figure 1). This uncorrelated error specification is furthermore based on hypothetical smoothing of the onboard SWOT data with an idealized 2-dimensional filter that eliminates all variability with scales shorter than 15 km. The 1-dimensional (e.g., along-track) white-noise spectral representation of the residual uncorrelated measurement errors after this hypothetical filtering is shown by the dashed red line in Figure 1.