  The ice shelves buttressing the Antarctic ice sheet determine the rate of
ice-discharge into the surrounding oceans. The geometry of ice shelves, and
hence their buttressing strength, is determined by ice flow as well as by the
local surface accumulation and basal melt rates, governed by atmospheric and
oceanic conditions. Contemporary methods resolve one of these rates, but
typically not both. Moreover, there is little information of how they changed
in time. We present a new method to simultaneously infer the surface
accumulation and basal melt rates averaged over decadal and centennial
timescales. We infer the spatial dependence of these rates along flow line
transects using internal stratigraphy observed by radars, using a kinematic
forward model of internal stratigraphy. We solve the inverse problem using
simulation-based inference (SBI). SBI performs Bayesian inference by training
neural networks on simulations of the forward model to approximate the
posterior distribution, allowing us to also quantify uncertainties over the
inferred parameters. We demonstrate the validity of our method on a synthetic
example, and apply it to Ekstr\"om Ice Shelf, Antarctica, for which newly
acquired radar measurements are available. We obtain posterior distributions of
surface accumulation and basal melt averaging over 42, 84, 146, and 188 years
before 2022. Our results suggest stable atmospheric and oceanographic
conditions over this period in this catchment of Antarctica. Use of observed
internal stratigraphy can separate the effects of surface accumulation and
basal melt, allowing them to be interpreted in a historical context of the last
centuries and beyond.
