The air content of glacial firn determines the effect and attribution of observed changes in ice surface elevation, but is currently measurable only using labor-intensive ground-based techniques. Here a novel method is presented for using radar sounding measurements to decompose the total thickness of floating ice shelves into thicknesses of solid ice and firn air (or firn water). The method is applied to a 1997/98 airborne survey of Larsen Ice Shelf, revealing large spatial gradients in air content that are consistent with existing measurements and local meteorology. The gradients appear to be governed by meltwater-induced firn densification. We find sufficient air in Larsen C Ice Shelf for increased densification to account for its previously observed surface lowering, and the rate of lowering superficially agrees with published trends in melting. This does not preclude a contribution to the lowering from oceanic basal melting, but a modern repeat of the survey could conclusively distinguish atmosphere-led from ocean-led change. The technique also holds promise for the calibration of firn-density models, derivation of ice thickness from surface elevation measurements, and calculation of the sea-level contribution of changes in grounded-ice discharge.