  Several telescopes include large Deformable Mirrors (DM) located directly
inside the telescope. These adaptive telescopes trigger new constraints for the
calibration of the Adaptive Optics (AO) systems as they usually offer no access
to an artificial calibration source for the interaction matrix measurement.
Moreover, the optical propagation between the DM and the Wave-Front Sensor
(WFS) may evolve during the operation, resulting in mis-registrations that
highly affect the AO performance and thus the scientific observation. They have
to be measured and compensated, for instance by updating the calibration. A new
strategy consists of estimating the mis-registrations and injecting them into
synthetic models to generate noise-free interaction matrices. This
pseudo-synthetic approach is the baseline for the Adaptive Optics Facility
working with a Shack-Hartmann WFS and seems particularly suited for the future
Extremely Large Telescope as the calibration will have to be regularly updated,
for a large numbers of actuators. In this paper, the feasibility of a pseudo
synthetic calibration with Pyramid WFS at the Large Binocular Telescope (LBT)
is investigated. A synthetic model of the LBT AO systems is developed, and the
procedure to adjust the mis-registrations parameters is introduced, extracting
them from an experimental interaction matrix. We successfully tested an
interaction matrix generated from the model on the real system in high-order AO
mode. We recorded a slightly better performance with respect to the
experimental one. This work demonstrates that a high accuracy calibration can
be obtained using the pseudo synthetic approach with pyramid WFS.
