Snow depth on sea ice is a major constituent of the marine cryosphere. It is a key parameter for the derivation of sea-ice thickness from satellite altimetry. One way to retrieve the basin-scale snow depth on sea ice is by satellite microwave radiometry. There is evidence from measurements and inter-comparison studies that current retrievals likely under-estimate the snow depth over deformed, rough sea ice. We follow up on an earlier study, where satellite passive microwave data were combined with information on the sea-ice topography from the satellite laser altimeter on board the Ice, Cloud and land Elevation Satellite (ICESat) in a hybrid approach. Such topography information is spatiotemporally limited because of ICESat's operation mode. In this paper, we aim to derive a proxy for this topography information from satellite microwave radiometry. For this purpose, we co-locate parameters describing the sea-ice deformation taken from visual ship-based observations and the surface elevation standard deviation derived from ICESat laser altimetry with the microwave brightness temperatures (TB) measured via the Advanced Microwave Scanning Radiometer aboard Earth Observation Satellite (AMSR-E) and aboard Global Change Observation Mission-Water 1 (GCOM-W1) (AMSR2). We find that the TB polarization ratio at 6.9 GHz and the TB gradient ratio between 10.7 GHz (horizontal polarization) and 6.9 GHz (vertical polarization), might be suited as such a proxy. Using this proxy, we modify the above-mentioned hybrid approach and compute the snow depths on sea ice from the AMSR-E and AMSR2 data. We compare our snow depths with those of the commonly used approach, the hybrid approach, with the ship-based observations for the years 2002 through 2015 and with the measurements made by drifting buoys for the period of 2014 through 2018. We find a convincing overall agreement with the hybrid approach and some improvement over the common approach. However, our approach is sensitive to the presence of thin ice-here, the retrieved snow depths are too large; and our approach performs sub-optimally over old ice-here, the retrieved snow depths are too small. More investigations and, in particular, more evaluations are required to optimize our approach so that the snow depths retrieved for the combined AMSR-E/AMSR2 period could serve as a data set for sea-ice thickness retrieval based on satellite altimetry.