  AWAKE, the Advanced Proton-Driven Plasma Wakefield Acceleration Experiment,
is a proof-of-principle R&D experiment at CERN using a 400 GeV/c proton beam
from the CERN SPS (longitudinal beam size sigma_z = 12 cm) which will be sent
into a 10 m long plasma section with a nominal density of approx. 7x10^14
atoms/cm3 (plasma wavelength lambda_p = 1.2mm). In this paper we show that by
measuring the time integrated transverse profile of the proton bunch at two
locations downstream of the AWAKE plasma, information about the occurrence of
the self-modulation instability (SMI) can be inferred. In particular we show
that measuring defocused protons with an angle of 1 mrad corresponds to having
electric fields in the order of GV/m and fully developed self-modulation of the
proton bunch. Additionally, by measuring the defocused beam edge of the
self-modulated bunch, information about the growth rate of the instability can
be extracted. If hosing instability occurs, it could be detected by measuring a
non-uniform defocused beam shape with changing radius. Using a 1 mm thick
Chromox scintillation screen for imaging of the self-modulated proton bunch, an
edge resolution of 0.6 mm and hence a SMI saturation point resolution of 1.2 m
can be achieved.
