Wave types employed for mathematical modeling of propagation and scattering in fairly general environments usually exhibit frequency dispersion, which may arise either from the material properties of the media or from boundary shapes and inhomogeneities even when bulk media are nondispersive. This feature creates difficulties for recovering transient fields from the harmonic constituents. At high frequencies, effects of dispersion are minimal and may be neglected in an asymptotic sense. For local plane wave spectral integrals representing ray-type fields, it is then possible to effect the inversion into the time domain explicitly, in closed form. Two principal methods, by Cagniard-DeHoop and by Chapman, have been employed in this context. The former has limited scope. The latter, while more broadly applicable, requires real values for the wavenumbers in the local plane wave spectra; this restriction forces results for certain ray fields into an inconvenient form, dissimilar from that for Cagniard-DeHoop inversion. The two methods are examined here from the perspective of a unified Spectral Theory of Transients that allows spectral wavenumbers to be real or complex and is shown to accommodate both formulations within the same spectral framework. The theory is described and illustrated on examples for which the Cagniard-DeHoop method is inapplicable and the Chapman method less convinient.