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Public Reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comment regarding this burden estimates or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302, and Finite-difference, time-domain ͑FDTD͒ calculations are typically performed with partial differential equations that are first order in time. Equation sets appropriate for FDTD calculations in a moving inhomogeneous medium ͑with an emphasis on the atmosphere͒ are derived and discussed in this paper. Two candidate equation sets, both derived from linearized equations of fluid dynamics, are proposed. The first, which contains three coupled equations for the sound pressure, vector acoustic velocity, and acoustic density, is obtained without any approximations. The second, which contains two coupled equations for the sound pressure and vector acoustic velocity, is derived by ignoring terms proportional to the divergence of the medium velocity and the gradient of the ambient pressure. It is shown that the second set has the same or a wider range of applicability than equations for the sound pressure that have been previously used for analytical and numerical studies of sound propagation in a moving atmosphere. Practical FDTD implementation of the second set of equations is discussed. Results show good agreement with theoretical predictions of the sound pressure due to a point monochromatic source in a uniform, high Mach number flow and with Fast Field Program calculations of sound propagation in a stratified moving atmosphere.

Stochastic inversion is a well known technique for the solution of inverse problems in tomography. It employs the idea that the propagation medium may be represented as random with a known spatial covariance function. In this paper, a generalization of the stochastic inverse for acoustic travel-time tomography of the atmosphere is developed. The atmospheric inhomogeneities are considered to be random, not only in space but also in time. This allows one to incorporate tomographic data ͑travel times͒ obtained at different times to estimate the state of the propagation medium at any given time, by using spatial-temporal covariance functions of atmospheric turbulence. This increases the amount of data without increasing the number of sources and/or receivers. A numerical simulation for two-dimensional travel-time acoustic tomography of the atmosphere is performed in which travel times between sources to receivers are calculated, given the temperature and wind velocity fields. These travel times are used as data for reconstructing the original fields using both the ordinary stochastic inversion and the proposed time-dependent stochastic inversion algorithms. The time-dependent stochastic inversion produces a good match to the specified temperature and wind velocity fields, with average errors about half those of the ordinary stochastic inverse.