An instrument allowing the quantitative analysis of X-ray pulsed wavefronts is presented and its processing method explained. The system relies on the X-ray speckle tracking principle to accurately measure the phase gradient of the X-ray beam from which beam optical aberrations can be deduced. The key component of this instrument, a semi-transparent scintillator emitting visible light while transmitting X-rays, allows simultaneous recording of two speckle images at two different propagation distances from the X-ray source. The speckle tracking procedure for a reference-less metrology mode is described with a detailed account on the advanced processing schemes used. A method to characterize and compensate for the imaging detector distortion, whose principle is also based on speckle, is included. The presented instrument is expected to find interest at synchrotrons and at the new X-ray free-electron laser sources under development worldwide where successful exploitation of beams relies on the availability of an accurate wavefront metrology.