  We develop a general decomposition of an ensemble of initial density profiles
in terms of an average state and a basis of modes that represent the
event-by-event fluctuations of the initial state. The basis is determined such
that the probability distributions of the amplitudes of different modes are
uncorrelated. Based on this decomposition, we quantify the different types and
probabilities of event-by-event fluctuations in Glauber and Saturation models
and investigate how the various modes affect different characteristics of the
initial state. We perform simulations of the dynamical evolution with KoMPoST
and MUSIC to investigate the impact of the modes on final-state observables and
their correlations.
