  The kinetic freeze-out for the hydrodynamical description of relativistic
heavy ion collisions is discussed using a background-fluctuation splitting of
the hydrodynamical fields. For a single event, the particle spectrum, or its
logarithm, can be written as the sum of background part that is symmetric with
respect to azimuthal rotations and longitudinal boosts and a part containing
the contribution of fluctuations or deviations from the background. Using a
complete orthonormal basis to characterize the initial state allows one to
write the double differential harmonic flow coefficients determined by the
two-particle correlation method as matrix expressions involving the initial
fluid correlations. We discuss the use of these expressions for a mode-by-mode
analysis of fluctuating initial conditions in heavy ion collisions.
