  We employ classical molecular dynamics simulations to investigate the
molecular-level structure of water during the isothermal compression of
hexagonal ice (I$h$) and low-density amorphous (LDA) ice at low temperatures.
In both cases, the system transforms to high-density amorphous ice (HDA) via a
first-order-like phase transition. We employ a sensitive local order metric
(LOM) [Martelli et. al., Phys. Rev. B, 97, 064105 (2018)], that can
discriminate among different crystalline and non crystalline ice structures and
is based on the positions of the oxygen atoms in the first and/or second
hydration shell. Our results confirm that LDA and HDA are indeed amorphous,
i.e., they lack of polydispersed ice domains. Interestingly, HDA contains a
small number of domains that are reminiscent of the unit cell of ice IV,
although the hydrogen-bond network (HBN) of these domains differ from the HBN
of ice IV. The presence of ice IV-like domains provides some support to the
hypothesis that HDA could be the result of a detour on the HBN rearrangement
along the I$h$-to-ice IV pressure induced transformation. Both nonequilibrium
LDA-to-HDA and I$h$-to-HDA transformations are two-steps processes where a
small distortion of the HBN first occurs at low pressures and then, a sudden,
extensive re-arrangement of hydrogen bonds at the corresponding transformation
pressure follows. Interestingly, the I$h$-to-HDA and LDA-to-HDA transformations
occur when LDA and I$h$ have similar local order, as quantified by the
site-averaged LOMs. Since I$h$ has a perfect tetrahedral HBN, while LDA does
not, it follows that higher pressures are needed to transform I$h$ into HDA
than that for the conversion of LDA to HDA. In correspondence with both
first-order-like phase transitions, the samples are composed of a large HDA
cluster that percolates within the I$h$/LDA samples.
