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Traditional methods of deriving temporal variability of Antarctic ice-shelf elevation from satellite altimetry use a fixed (“Eulerian”) reference frame, where the measured changes include advection of ice thickness gradients between measurement epochs. We present a new method which removes advection effects by using an independent velocity field to compare elevations in a moving (“Lagrangian”) reference frame. Applying the technique to ICESat laser altimetry for the period 2003–2009 over the two largest Antarctic ice shelves, Ross and Filchner-Ronne, we show that the Lagrangian approach reduces the variability of derived elevation changes by about 50% compared to the Eulerian approach and reveals clearer spatial patterns of elevation change. The method simplifies the process of estimating basal mass budget from the residual of all other processes that contribute to ice-shelf elevation changes. We use field data and ICESat measurements over ice rises and the grounded ice sheet to account for surface accumulation and changes in firn air content, and remove the effect of ice-flow divergence using surface velocity and ice thickness data. The results show highest basal melt rates (>5 m a−1) near the deep grounding lines of major ice streams, but smaller melt rates (<5 m a−1) near the ice-shelf fronts are equally important to total meltwater production since they occur over larger areas. Integrating over the entire ice-shelf areas, we obtain basal mass budgets of −50 ± 64 Gt a−1 for Ross and −124 ± 66 Gt a−1 for Filchner-Ronne, with changes in firn air content as the largest error source.