Seismological data provide evidence of a depth-dependent rupture behaviour of earthquakes occurring at the megathrust fault of subduction zones, also known as megathrust earthquakes(1). Relative to deeper events of similar magnitude, shallow earthquake ruptures have larger slip and longer duration, radiate energy that is depleted in high frequencies and have a larger discrepancy between their surface-wave and moment magnitudes(1-3). These source properties make them prone to generating devastating tsunamis without clear warning signs. The depth-dependent rupture behaviour is usually attributed to variations in fault mechanics(4-7). Conceptual models, however, have so far failed to identify the fundamental physical causes of the contrasting observations and do not provide a quantitative framework with which to predict and link them. Here we demonstrate that the observed differences do not require changes in fault mechanics. We use compressional-wave velocity models from worldwide subduction zones to show that their common underlying cause is a systematic depth variation of the rigidity at the lower part of the upper plate-the rock body overriding the megathrust fault, which deforms by dynamic stress transfer during co-seismic slip. Combining realistic elastic properties with accurate estimates of earthquake focal depth enables us to predict the amount of co-seismic slip (the fault motion at the instant of the earthquake), provides unambiguous estimations of magnitude and offers the potential for early tsunami warnings.