The paper basically combines multiple datasets to figure out the changes in ice surface elevation and ice thickness at tidal and multi-year timescales. That way they were able to study grounding line migration, and the melt rate of the glacier and its corresponding ice shelf from above and below. Then they were able to make inferences of the processes producing glacier melting, specially basal melting that is difficult to measure directly.
They actually used three datasets:
DInSAR: Standing for Differential Interferometric Synthetic Aperture Radar. It is basically a technique that compares the phase of two SAR images acquired at different time to compute uplift/subsidence of the surface with great accuracy (usually millimeters). So they were able to see the surface of the ice shelf rise and lower with the tides, and also compute surface flow velocities.
IceBridge MCoRDS: Airborne radar depth sounder MCoRDS (Multichannel Coherent Radar Depth Sounder) operated by NASA mission IceBridge. This is a much more powerful radar operated close to the surface and designed to penetrate the ice, allowing to see the bedrock topography beneath the ice and some of the structure above the bedrock, such as water/ice transitions or layers of ice with distinct dielectric properties. This allowed them to know the ice thickness and bedrock topography.
TDX DEM: TanDEM-X Digital Elevation Model. This are Surface topography models generated using single pass InSAR data provided by two SAR satellites that orbit in tandem formation (one immediately behind the other). This allows precise surface topography measurements, that are also needed for the DInSAR processing.
I hope that helps. It is a very interesting work. Although the "giant cavern" part is not the most relevant part of it at all, but it was great for catchy headlines.