This paper is concerned with the spectral content of short-pulse field scattered from an inhomogeneous medium. Direct diffraction tomography type relations between the time-dependent data and the medium are determined via two spectral formulations: a time-dependent plane wave spectrum and a local phase-space spectrum. The former is formulated in the Radon space, wherein the time-dependent plane wave spectrum of the data is directly related to the Radon transform of the medium. The latter is formulated as a local Radon transform in the space-time domain, wherein scattered pulsed beams (space-time wave packets) are directly related to localized scattering subdomains. This result leads to the formulation of a local inverse scattering scheme, wherein pulsed beams are back propagated and focused to form local images of any desirable region.