The combined effects of non-isoplanatic conditions and telescopic aberrations on speckle imaging are analyzed. The general form of the two-point source wave-structure function for upward viewing through the turbulent atmosphere is used as a basis for developing descrip-tions of the second-order statistics used in speckle imaging. An analysis of the information recoverable from astronomical speckle imagery shows that diffraction limited resolution can be obtained. Two important assumptions which are usually satisfied have been made: 1) the isoplanatic patch encompasses the average spread function; 2) the greatest contribution to the atmospheric distortion comes from turbulence at low altitudes. Under these conditions, the aberrations of the telescope do not affect the second-order statistics and the integral equation relating the object and measured second-order statistics is well conditioned.