The work is reported in an open access paper in Science: Heterogeneous retreat and ice melt of Thwaites Glacier, West Antarctica which is very technical, thorough, and hard for me to understand.
The work is based on a substantial amount of radar data, both SAR (Synthetic Aperture Radar) from several satellites, and (from Pasadena Now) ...by ice-penetrating radar in NASA’s Operation IceBridge, an airborne campaign that began in 2010 and studies connections between the polar regions and the global climate.
Is it possible to explain the differences between the two kinds of radar data (space and airborne) and how they were combined and used together to establish the size/volume and growth of this unexpectedly large amount of liquid water below the glacier?
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.