  We propose and apply two methods to estimate pupil plane phase
discontinuities for two realistic scenarios on VLT and Keck. The methods use
both Phase Diversity and a form of image sharpening. For the case of VLT, we
simulate the `low wind effect' (LWE) which is responsible for focal plane
errors in the SPHERE system in low wind and good seeing conditions. We
successfully estimate the simulated LWE using both methods, and show that they
are complimentary to one another. We also demonstrate that single image Phase
Diversity (also known as Phase Retrieval with diversity) is also capable of
estimating the simulated LWE when using the natural de-focus on the SPHERE/DTTS
imager. We demonstrate that Phase Diversity can estimate the LWE to within 30
nm RMS WFE, which is within the allowable tolerances to achieve a target SPHERE
contrast of 10$^{-6}$. Finally, we simulate 153 nm RMS of piston errors on the
mirror segments of Keck and produce NIRC2 images subject to these effects. We
show that a single, diverse image with 1.5 waves (PV) of focus can be used to
estimate this error to within 29 nm RMS WFE, and a perfect correction of our
estimation would increase the Strehl ratio of a NIRC2 image by 12\%
