A low noise, high resolution Shack-Hartmann wavefront sensor was included in the ABLE-ACE instrument suite to obtain direct high resolution phase measurements of the 0.53 micrometers pulsed laser beam propagated through high altitude atmospheric turbulence. The wavefront sensor employed a Fired geometry using a lenslet array which provided approximately 17 sub-apertures across the pupil. The lenslets focused the light in each sub-aperture onto a 21 by 21 array of pixels in the camera focal plane with 8 pixels in the camera focal plane with 8 pixels across the central lobe of the diffraction limited spot. The goal of the experiment was to measure the effects of the turbulence in the free atmosphere on propagation, but the wavefront sensor also detected the aberrations induced by the aircraft boundary layer and the receiver aircraft internal beam path. Data analysis methods used to extract the desired atmospheric contribution to the phase measurements from the data corrupted by non-atmospheric aberrations are described. Approaches which were used included a reconstruction of the phase as a linear combination of Zernike polynomials coupled with optical estimator sand computation of structure functions of the sub-aperture slopes. The theoretical basis for the data analysis techniques is presented. Results are described, and comparisons with theory and simulations are shown. Estimates of average turbulence strength along the propagation path from the wavefront sensor showed good agreement with other sensor. The Zernike spectra calculated from the wavefront sensor data were consistent with the standard Kolmogorov model of turbulence.