Formulas for the statistical moments of a spherical sound wave propagating in a medium with arbitrary (anisotropic) spectra of temperature and medium velocity fluctuations are obtained. These statistical moments are: the variances of log-amplitude and phase fluctuations, the correlation functions of log-amplitude and phase fluctuations, the mean sound field, and the coherence function of the sound field. Then, the statistical moments of a spherical sound wave are calculated analytically and numerically for Gaussian spectra of temperature and medium velocity fluctuations. It is shown that the temperature and medium velocity contributions to these statistical moments can differ not only quantitatively but also qualitatively. The useful relationships between the statistical moments of spherical and plane waves propagating in a moving random medium are derived. Some of the theoretical results obtained are compared with experimental data from the literature on sound propagation through the turbulent atmosphere.