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. Polar ice sheets are important components of the Earth system. As the geometries of land, ocean and ice sheets evolve, they must be consistentlycaptured within the lexicon of geodesy. Understanding the interplay between the processes such as ice-sheet dynamics, solid-Earth deformation, andsea-level adjustment requires both geodetically consistent and mass-conserving descriptions of evolving land and ocean domains, grounded ice sheetsand floating ice shelves, and their respective interfaces. Here we present mathematical descriptions of a generic level set that can be used totrack both the grounding lines and coastlines, in light of ice–ocean mass exchange and complex feedbacks from the solid Earth and sea level. We nextpresent a unified method to accurately compute the sea-level contribution of evolving ice sheets based on the change in ice thickness, bedrockelevation and mean sea level caused by any geophysical processes. Our formalism can be applied to arbitrary geometries and at all timescales. Whileit can be used for applications with modeling, observations and the combination of two, it is best suited for Earth system models, comprising icesheets, solid Earth and sea level, that seek to conserve mass.

A kinematic formalism for tracking ice–ocean mass exchange on the Earth's surface and estimating sea-level change