  The search for Earth-like exoplanets requires high-contrast and high-angular
resolution instruments, which designs can be very complex: they need an
adaptive optics system to compensate for the effect of the atmospheric
turbulence on image quality and a coronagraph to reduce the starlight and
enable the companion imaging. During the instrument design phase and the error
budget process, studies of performance as a function of optical errors are
needed and require multiple end-to-end numerical simulations of wavefront
errors through the optical system.
  In particular, the detailed analysis of long-exposure images enables to
evaluate the image quality (photon noise level, impact of optical aberrations
and of adaptive optics residuals, etc.). Nowadays simulating one long but
finite exposure image means drawing several thousands of random frozen phase
screens, simulating the image associated with each of them after propagation
through the imaging instrument, and averaging all the images. Such a process is
time consuming, demands a great deal of computer resources, and limits the
number of parametric optimization.
  We propose an alternative and innovative method to directly express the
statistics of ground-based images for long but finite exposure times. It is
based on an analytical model, which only requires the statistical properties of
the atmospheric turbulence. Such a method can be applied to optimize the design
of future instruments such as SPHERE+ (VLT) or the planetary camera and
spectrograph (PCS - ELT) or any ground-based instrument.
