Obtaining Spectra of Turbulent Velocity from Observations
  We discuss a long-standing problem of how turbulence can be studied using observations of Doppler broadened emission and absorption lines. The focus of the present review is on two new techniques, the Velocity-Channel Analysis (VCA), which makes use of the channel maps, and the Velocity Coordinate Spectrum (VCS), which utilizes the fluctuations measured along the velocity axis of the Position-Position Velocity (PPV) data cubes. Both techniques have solid theoretical foundations based on analytical calculations as well as on numerical testings. Among the two the VCS, which has been developed quite recently, has two unique features. First of all, it is applicable to turbulent volumes that are not spatially resolved. Second, it can be used with absorption lines that do not provide good spatial sampling of different lags over the image of the turbulent object. In fact, recent studies show that measurements of absorption line along less than 10 absorption directions are sufficient for a reliable recovering of the underlying spectrum of the turbulence. Moreover, both weak absorption lines and absorption lines in saturated regime can be used, which extends the applicability of the technique. Our comparison of the VCA and the VCS with a more traditional technique of Velocity Centroids shows that the former two techniques recover reliably the spectra of supersonic turbulence, while the Velocity Centroids may be used only for studying subsonic turbulence. We discuss spectra of astrophysical turbulence obtained with the VCA and the VCS techniques. 
A. Lazarian