Radar-inference of the bulk properties of glacier beds, most notably identifying basal melting, is, in general, derived from the basal reflection coefficient. On the scale of an ice-sheet, unambiguous determination of basal reflection is primarily limited by uncertainty in the englacial attenuation of the radio wave, which is an exponential function of temperature. Most existing radar algorithms assume stationarity in the attenuation rate, which is not feasible at an ice-sheet wide scale. Here we introduce a new framework for deriving englaical attenuation and basal reflection, and, to demonstrate its efficacy, we apply it to the Greenland Ice-Sheet. A central feature is the use of a prior Arrhenius temperature model to estimate the spatial variation in englaical attenuation as a first guess input for the radar algorithm. We demonstrate regions of solution convergence for two input temperature fields, and for independently analysed field campaigns. The coverage achieved is a trade-off with uncertainty and we propose that the algorithm can be "tuned" for discrimination of basal melt (attenuation loss uncertainty ∼ 5 dB). This is supported by our physically realistic (∼ 20 dB) range for the basal reflection coefficient. Finally, we show that the attenuation solution can be used to predict the temperature bias of thermomechanical ice-sheet models.