  We investigate how the initial geometry of a heavy-ion collision is
transformed into final flow observables by solving event-by-event ideal
hydrodynamics with realistic fluctuating initial conditions. We study
quantitatively to what extent anisotropic flow (v_n) is determined by the
initial eccentricity epsilon_n for a set of realistic simulations, and we
discuss which definition of epsilon_n gives the best estimator of v_n. We find
that the common practice of using an r^2 weight in the definition of
varepsilon_n in general results in a poorer predictor of v_n than when using
r^n weight, for n > 2. We similarly study the importance of additional
properties of the initial state. For example, we show that in order to
correctly predict v_4 and v_5 for non-central collisions, one must take into
account nonlinear terms proportional to (epsilon_2)^2 and
(epsilon_2)*(epsilon_3), respectively. We find that it makes no difference
whether one calculates the eccentricities over a range of rapidity, or in a
single slice at z=0, nor is it important whether one uses an energy or entropy
density weight. This knowledge will be important for making a more direct link
between experimental observables and hydrodynamic initial conditions, the
latter being poorly constrained at present.
