Pile-up correction for the Swift-XRT observations in WT mode

The detector at the focal plane of the Swift X-ray Telescope (XRT) supports four readout modes, automatically changed on board, to cover the dynamical range of fluxes and rapid variability expected from GRB afterglows. The Windowed Timing (WT) mode is used for sources with flux higher than a few mCrab and is obtained by compressing 10 rows into a single row, and then reading out only the central 200 columns of the CCD. Point sources with a rate above ~300 c/s produce severe pile-up in the central region of the Point Spread Function. This paper presents three methods to correct the effects of the pile-up in WT mode. On ground calibration results and data from the very bright GRB 060124 are used to define and test these methods.