By performing molecular dynamics simulations we have generated several forms of high density amorphous (HDA) water (rhogreater than or equal to1.1 g/cm3), either by pressurizing cubic ice or by quenching liquid water maintained under pressure. It is shown that the temperature of amorphization is a key variable in understanding the thermodynamic and structural properties of the resulting amorphous phase. Thus, the higher the temperature of amorphization the denser the amorphous ice recovered at low temperature. Two limiting forms of high density are identified, a less dense one obtained by pressure-induced amorphization of cubic ice at low temperature which can be assigned to the HDA phase commonly investigated experimentally, and a denser form obtained by vitrifying the liquid under pressure whose structure is analogous to the VHDA phase recently discovered. Upon heating, both phases transform into low density phases (LDA(I) and LDA(II)), respectively, with slightly different properties. In particular, it is found that the LDA(I) phase exhibits thermodynamic, structural, and dynamical properties very close to that of hyperquenched glassy water. These findings can be understood by analyzing the relative displacements of the molecules with respect to their original positions in the sample at the beginning of the amorphization process. This study supports the idea of a continuity of metastable amorphous phases in low temperature water. (C) 2003 American Institute of Physics.