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Sound or acoustic tomography is a type of inverse problem. The idea of estimating physical quantities that influence sound propagation by measuring the parameters of sound propagation has proven to be successful in several practical domains, including medicine, seismology, oceanography. The use of acoustic tomography for estimating temperature and wind fields in the atmosphere has been shown to be possible as well and, moreover, its potentials have been demonstrated in field experiments. However, in most of the previous work, the algorithms used have not been proven to be the mathematically correct solution to the inverse problem. This paper considers the problem of reconstructing 2D temperature and wind fields using acoustic tomography setups. Primarily, it is shown that the classical time-of-flight measurements are not sufficient for the reconstruction of wind fields. As a solution, an additional set of measurements is suggested. The proposed set is related solely to the parameters of sound propagation, namely to the angle-of-departure/arrival of sound waves, and together with the time-of-flights enables complete temperature and wind recovery in a general case. Special cases are also discussed, emphasizing the situations for which it is possible to reduce the required measurements to only one of the two proposed sets. Specifically, it is proven that when a temperature and a source-free 2D wind are observed on a bounded domain, the time-of-flight measurements are sufficient for the complete reconstruction. Conversely, the angle-of-departure/arrival measurements are sufficient to reconstruct a temperature and a curl-free 2D wind fields observed on bounded domains. Further, an iterative reconstruction algorithm that covers both the general and the special setups is proposed and possible variations to the main scheme are discussed. In order to evaluate the reconstruction a qualitative error analysis is given. Finally, the simulation results confirm the theoretical results and the iterative algorithm demonstrates fast convergence. Also, the simulation shows that the adopted bent ray model for sound propagation always outperform the straight ray model. The unknown temperature and wind fields are reconstructed with a high accuracy.