The atomic structure of the icosahedral Ti-Zr-Ni quasicrystal is studied using a canonical cell tiling approach. An investigation of the previously proposed structure for W-Ti-Zr-Ni, the 1/1 crystal approximant to the Ti-Zr-Ni icosahedral quasicrystal, has revealed improbable occupation of sites in the region between the Bergman clusters and at the cluster centre. The 1/1 structure is refined for a better-ordered sample of the approximant, using X-ray and neutron diffraction measurements and ab initio relaxation studies. Based on this refined structure, an atomic decoration model for large canonical cell tilings is developed. Calculated diffraction patterns from the generated structures are in reasonable agreement with those measured for the icosahedral quasicrystal.