Time reversal is a widely used technique in wave physics, for both imaging purposes and experimental focusing. In this paper, a complete double-layer time reversal imaging process is proposed for in situ acoustic characterization of non-stationary sources, with perturbative noise sources and reverberation. The proposed method involves the use of a hemispherical array composed of pressure-pressure probes. The complete set of underlying optimizations to sonic time reversal imaging is detailed, with regard to space and time reconstruction accuracy, imaging resolution and sensitivity to reverberation, and perturbative noise. The proposed technique is tested and compared to more conventional time reversal techniques through numerical simulations and experiments. Results demonstrate the ability of the proposed method to back-propagate acoustic waves radiated from non-stationary sources in the volume delimited by the measurement array with a high precision both in time and space domains. Analysis of the results also shows that the process can successfully be applied in strongly reverberant environments, even with poor signal-to-noise ratio.