In inverse acoustics, the reconstruction of (distant) sound fields from microphone array measurements is fundamentally limited by the size of the array and by the microphone density. One approach to simulate large arrays and/or high microphone density is to scan the object of interest by sequentially moving a small prototype array. This typically requires a sufficient number of high-quality fixed references in order to preserve the phase relationships between consecutive snapshots or that measurements are made in the near-field in order to allow the successive reconstruction of local “patches” of the source by deliberately neglecting waves not radiating through the array (patch holography). In the present work, a solution is introduced to reconstruct the quadratic properties (e.g. source power, quadratic flux, sound intensity) of the radiating object from non-synchronous (sequential) measurements when references are insufficient in quantity or in quality. Remarkably, it also operates without any reference at all in certain conditions. It relies on the assumptions that the sound field is (i) stationary and (ii) spatially correlated and boils down to the factorization of a structured covariance matrix. This is undertaken within a Bayesian probabilistic approach where the source field is encoded by unobservable latent variables which are iteratively reconstructed by using the Expected-Maximization (EM) algorithm. The method is shown to return virtually similar results as if all data were captured simultaneously—or sequentially with sufficient high-quality references. As a consequence, it bears an advantage in terms of cost and simplicity. In addition, it is ultimately able to synthesize statistically equivalent realizations of the source field, which may turn out useful for numerical simulation of sound propagation. A solution is also provided within the Bayesian framework for automatically setting the regularization parameter to its optimal value.