  Modeling the amplitudes and shapes of the X-ray pulsations observed from hot,
rotating neutron stars provides a direct method for measuring neutron-star
properties. This technique constitutes an important part of the science case
for the forthcoming NICER and proposed LOFT X-ray missions. In this paper, we
determine the number of distinct observables that can be derived from pulse
profile modeling and show that using only bolometric pulse profiles is
insufficient for breaking the degeneracy between inferred neutron-star radius
and mass. However, we also show that for moderately spinning (300-800 Hz)
neutron stars, analysis of pulse profiles in two different energy bands
provides additional constraints that allow a unique determination of the
neutron-star properties. Using the fractional amplitudes of the fundamental and
the second harmonic of the pulse profile in addition to the amplitude and phase
difference of the spectral color oscillations, we quantify the signal-to-noise
ratio necessary to achieve a specified measurement precision for neutron star
radius. We find that accumulating 10^6 counts in a pulse profile is sufficient
to achieve a <5% uncertainty in the neutron star radius, which is the level of
accuracy required to determine the equation of state of neutron-star matter.
Finally, we formally derive the background limits that can be tolerated in the
measurements of the various pulsation amplitudes as a function of the system
parameters.
