In interstellar clouds the deposition of water ice onto grains only occurs at visual extinctions above some threshold value A_th. At extinctions greater than A_th there is a (near-linear) correlation between the inferred column density of the water ice and A_V. For individual cloud complexes such as Taurus, Serpens and Rho-Ophiuchi, A_th and the gradients of the correlation are very similar along all lines of sight. We have investigated the origin of this phenomenon, with careful consideration of the various possible mechanisms that may be involved and have applied a full chemical model to analyse the behaviours and sensitivities in quiescent molecular clouds. Our key results are: (i) the ubiquity of the phenomenon points to a common cause, so that the lines of sight probe regions with similar, advanced, chemical and dynamical evolution, (ii) for Taurus and Serpens; A_th and the slope of the correlation can be explained as resulting from the balance of freeze-out of oxygen atoms and photodesorption of H2O molecules. No other mechanism can satisfactorily explain the phenomenon, (iii) A_th depends on the local density, suggesting that there is a correlation between local volume density and column density, (iv) the different values of A_th for Taurus and Serpens are probably due to variations in the local mean radiation field strength, (v) most ice is accreted onto grains that are initially very small (<0.01 microns), and (vi) the very high value of A_th observed in Rho-Ophiuchi cannot be explained in the same way, unless there is complex microstructure and/or a modification to the extinction characteristics.
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