  We have developed a simulation-based method of spectral analysis for pile-up
affected data of X-ray CCDs without any loss of photon statistics. As effects
of the photon pile-up appear as complicated nonlinear detector responses, we
employ a detailed simulation to calculate the important processes in an X-ray
observation including physical interactions, detector signal generation,
detector readout, and a series of data reduction processes. This simulation
naturally reproduces X-ray-like and background-like events as results of X-ray
photon merging in a single pixel or in a chunk of adjacent pixels, allowing us
to construct a nonlinear spectral analysis framework that can treat pile-up
affected observation data. For validation, we have performed data analysis of
Suzaku XIS observations by using this framework with various parameters of the
detector simulation all of which are optimized for that instrument. We present
three cases of different pile-up degrees: PKS~2155-304 (negligible pile-up),
Aquila~X-1 (moderate pile-up), and the Crab Nebula (strong pile-up); we show
that the nonlinear analysis method produces results consistent with a
conventional linear analysis for the negligible pile-up condition, and
accurately corrects well-known pile-up effects such as spectral hardening and
flux decrease for the pile-up cases. These corrected results are consistent
with those obtained by a widely used core-exclusion method or by other
observatories with much higher timing resolutions (without pile-up). Our
framework is applicable to any types of CCDs used for X-ray astronomy including
a future mission such as XRISM by appropriate optimization of the simulation
parameters.
