  During the summer melt period, ponds form on the surface of Arctic sea ice as
it melts, with important consequences for ice evolution and marine ecology. Due
to the ice-albedo feedback, these melt ponds experience uneven heat absorption,
and exhibit complex patterns, which has motivated the development of modelling
and data analysis to understand their particular dynamics. We provide a
multiscale shape analysis using tools from computational algebraic topology,
simultaneously capturing convexity, proximity, integrity, and feature size
complementing existing single-scale quantification. Of particular interest in
modelling the ponds is a percolation threshold at which local pond structure
begins merging into macroscopic features. This percolation threshold has
previously been observed using fractal dimension techniques. The signed
Euclidean distance transform (SEDT) is a topological encoding of heterogeneous
shape in binary images, and has been previously applied to porous media for
percolation as well as other material behaviours. Here we adapt the SEDT for
Arctic melt pond data to give a rich characterization and computation of shape,
quantifying overall melt pond development in several complementary ways, and
from which classical percolation and dimension results can be extracted. This
orientation-invariant topological approach distinguishes different dynamical
network models of melt pond evolution of varying complexity.
