The Effects of Noise and Sampling on the Spectral Correlation Function

Abstract

The effects of noise and sampling on the spectral correlation function (SCF) introduced by Rosolowsky and coworkers are studied using observational data, numerical simulations of magneto-hydrodynamic turbulence, and simple models of Gaussian spectral line profiles. The most significant innovations of this paper are (1) the normalization of the SCF based on an analytic model for the effect of noise and (2) the computation of the SCF as a function of the spatial lag between spectra within a map.A new definition of the "quality" of a spectrum, Q, is introduced, which is correlated with the usual definition of signal-to-noise ratio. The prenormalization value of the SCF is a function of Q. We derive analytically the effect of noise on the SCF, and then normalize the SCF to its analytic approximation.By computing the dependence of the SCF on the spatial lag, S(0)(Deltar), we have been able to conclude the following : (1) S(0)(Deltar) is a power law, with slope a, in the range of scales l(i) < l < l(o). (2) The correlation outer scale, l(o), is determined by the size of the map, and no evidence for a true departure from selfl similarity on large scales has been found. (3) The correlation inner scale, l(i), is a true estimate of the smallest self-similar scale in a map. (4) The spectral slope, alpha, in a given region, is independent of velocity resolution (above a minimum resolution threshold), spatial resolution, and average spectrum quality. (5) Molecular transitions that trace higher gas density yield larger values of alpha (steeper slopes) than transitions tracing lower gas density. (6) Nyquist sampling, bad pixels in detector arrays, and referencesharing data acquisition need to be taken into account for a correct determination of the SCF at Deltar = 1. The value of alpha, however, can be computed correctly without a detailed knowledge of observational procedures.