The spatial pattern of accumulation rate can be inferred from internal layers in glaciers and ice sheets. Non-dimensional analysis determines where finite strain can be neglected (‘shallow-layer approximation’) or approximated with a local one-dimensional flow model (‘local-layer approximation’), and where gradients in strain rate along particle paths must be included (‘deep layers’). We develop a general geophysical inverse procedure to infer the spatial pattern of accumulation rate along a steady-state flowband, using measured topography of the ice-sheet surface, bed and a ‘deep layer’. A variety of thermomechanical ice-flow models can be used in the forward problem to calculate surface topography and ice velocity, which are used to calculate particle paths and internal-layer shapes. An objective tolerance criterion prevents over-fitting the data. After making site-specific simplifications in the thermomechanical flow algorithm, we find the accumulation rate along a flowband through Taylor Mouth, a flank site on Taylor Dome, Antarctica, using a layer at approximately 100 m depth, or 20% of the ice thickness. Accumulation rate correlates with ice-surface curvature. At this site, gradients along flow paths critically impact inference of both the accumulation pattern, and the depth-age relation in a 100 m core.