Satellite altimeters map wave heights with a "doughnut" footprint

The design and interpretation of satellite altimeters goes back to the 1970s: a radar shines a narrow beam of microwaves on the surface, and records the reflection of the radar pulses a few nanoseconds later. The recorded radar power rises around an "epoch" that can be translated into a  mean sea surface height (SSH), and the duration of the rising time gives a sense of the distribution of the surface elevation around the mean, which is generally translated into a "significant wave height" (SWH). The theory behind this interpretation of altimeter data relied on the assumption that the "sea state" (the statistical description of waves)  is spatially uniform at the scale of the radar beam.

All was nice and well when looking at SSH and SWH variations at scales of hundreds of kilometer along the satellite track.  Yet,  satellite data is typically averaged over 7 km. So, what is in the smaller scale fluctuations? Noise? What is the actual resolution limit of altimeters? One extreme case of small scale fluctuations is this schematic view of wave groups (series of high waves followed by lower waves) and its mapping by a satellite altimeter. In that schematic the groups are long compared to the effective footprint diameter ρc , and the altimeters reports a SWH that varies from 0.1 to 1.6 times the true significant wave height Hs. But what happens when groups get shorter?

The China-France Ocean Satellite (CFOSAT) has a very particular altimeter beam: because it flies much lower than other altimeters (530 km compared to 1350 km for Jason 3) its footprint is smaller and more sensitive to small scale fluctuations. Looking at wave heights in storms, we already found that the variability of SWH was mainly explained by 2 parameters: the mean value of SWH, which is the true significant wave height Hs, and a "peakedness" parameter Qkk that can be estimated from the wave spectrum (De Carlo et al. 2023), and this is all because the wave heights vary across the satellite footprint.