A large effort has been carried out to characterize the background of sodium iodide crystals within the ANAIS (Annual modulation with NaI Scintillators) project. In this paper, the background models developed for three 12.5-kg NaI(Tl) scintillators produced by Alpha Spectra Inc. and operated at the Canfranc Underground Laboratory are presented together with an evaluation of the background prospects for the full experiment. Measured spectra from threshold to high energy in different conditions are well described by the models based on quantified activities. At the region of interest, crystal bulk contamination is the dominant background source. Contributions from 210Pb, 40K, 22Na and 3H are the most relevant. Those from 40K and 22Na could be efficiently suppressed thanks to anticoincidence operation in a crystals matrix or inside a Liquid Scintillator Veto (LSV), while that from 210Pb has been reduced by improving crystal production methods and 3H production could be reduced by shielding against cosmic rays during production. Assuming the activities of the last characterized detector, for nine crystals with a total mass of 112.5 kg the expected background rate is 2.5 counts/(keV kg d) in the region from 1 to 4 keV, which could be reduced at 1.4 counts/(keV kg d) by using a LSV.
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