In this paper, we explore the use of synthetic aperture processing for optimizing the spatial covariance estimation capabilities of a moving linear co-prime sensor array. The linear co-prime sensor array geometry is a thinned linear array that is constructed by nesting uniform linear arrays with inter-related element spacing factors. The application of synthetic aperture processing in this setting is designed to create virtual sensors at missing half-wavelength intervals up to the degree required to produce a hole-free difference co-array across the full aperture of the synthetic array. Once a full set of spatial covariances are estimated covariance matrix based array processing methods can be applied as if the synthetic array were a uniform linear array. In this sense, the synthetic array is designed to approximate a uniform linear array while it retains a thinned linear array structure. We show simulation results examining the quality of the uniform linear array approximation afforded by this application synthetic aperture processing.