We apply epsilon-machine spectral reconstruction theory to analyze structure and disorder in four previously published zinc sulfide diffraction spectra and contrast the results with the most common alternative theory, the fault model. In each case we find that the reconstructed epsilon-machine provides a more comprehensive and detailed understanding of the stacking structure, often detecting stacking structures not previously found. Using the epsilon-machines reconstructed for each spectrum, we calculate a number of physical parameters - such as configurational energies, configurational entropies and hexagonality - and several quantities - including statistical complexity and excess entropy - that describe the intrinsic computational properties of the stacking structures.