  We use particle-in-cell (PIC) simulations to study the effects of variations
of the incoming 400 GeV proton bunch parameters on the amplitude and phase of
the wakefields resulting from a seeded self-modulation (SSM) process. We find
that these effects are largest during the growth of the SSM, i.e. over the
first five to six meters of plasma with an electron density of $7
\times10^{14}$ cm$^{-3}$. However, for variations of any single parameter by
$\pm$5%, effects after the SSM saturation point are small. In particular, the
phase variations correspond to much less than a quarter wakefield period,
making deterministic injection of electrons (or positrons) into the
accelerating and focusing phase of the wakefields in principle possible. We use
the wakefields from the simulations and a simple test electron model to
estimate the same effects on the maximum final energies of electrons injected
along the plasma, which are found to be below the initial variations of
$\pm$5%. This analysis includes the dephasing of the electrons with respect to
the wakefields that is expected during the growth of the SSM. Based on a PIC
simulation, we also determine the injection position along the bunch and along
the plasma leading to the largest energy gain. For the parameters taken here
(ratio of peak beam density to plasma density $n_{b0}/n_0 \approx 0.003$), we
find that the optimum position along the proton bunch is at $\xi \approx -1.5
\; \sigma_{zb}$, and that the optimal range for injection along the plasma (for
a highest final energy of $\sim$1.6 GeV after 10 m) is 5-6 m.
