Subglacial roughness can be determined at a variety of length scales from radio-echo sounding (RES) data either via statistical analysis of topography or inferred from basal radar scattering. Past studies have demonstrated that subglacial terrain exhibits self-affine (power law) roughness scaling behaviour, but existing radar scattering models do not take this into account. Here, using RES data from northern Greenland, we introduce a self-affine statistical framework that enables a consistent integration of topographic-scale roughness with the electromagnetic theory of radar scattering. We demonstrate that the degree of radar scattering, quantified using the waveform abruptness (pulse peakiness), is topographically controlled by the Hurst (roughness power law) exponent. Notably, specular bed reflections are associated with a lower Hurst exponent, with diffuse scattering associated with a higher Hurst exponent. Abrupt waveforms (specular reflections) have previously been used as a RES diagnostic for basal water, and to test this assumption we compare our radar scattering map with a recent prediction for the basal thermal state. We demonstrate that the majority of thawed regions (above pressure melting point) exhibit a diffuse scattering signature, which is in contradiction to the prior approach. Self-affine statistics provide a generalised model for subglacial terrain and can improve our understanding of the relationship between basal properties and ice-sheet dynamics.