  Context. The diffusion of adaptive optics systems in astronomical
instrumentation for large ground-based telescopes is rapidly increasing and the
pyramid wavefront sensor is replacing the Shack-Hartmann as the standard
solution for single conjugate adaptive optics systems. The pyramid wavefront
sensor is typically used with a tip-tilt modulation to increase the linearity
range of the sensor, but the non-modulated case is interesting because it
maximizes the sensor sensitivity. The latter case is generally avoided for the
reduced linearity range that prevents robust operation in the presence of
atmospheric turbulence.
  Aims. We aim to solve part of the issues of the non-modulated pyramid
wavefront sensor by reducing the model error in the interaction matrix. We
linearize the sensor response in the working conditions without extending the
sensor linearity range.
  Methods. We developed a new calibration approach to model the response of
pyramid wave front sensor in partial correction, whereby the working conditions
in the presence of residual turbulence are considered.
  Results. We use in simulations to show how the new calibration approach
allows for the pyramid wave front sensor without modulation to be used to sense
and correct atmospheric turbulence and we discuss when this case is preferable
over the modulated case.
