2. Structure measure by the -variance
The -variance was comprehensively introduced by Stutzki et al. (1998). We will repeat here only the formalism essential for the further analysis in this paper.
The -variance is a type of averaged wavelet transform that measures the variance in an E-dimensional structure filtered by a spherically symmetric down-up-down function of varying size (Zielinsky & Stutzki 1999). It is defined by
where, the * stands for a convolution and describes the down-up-down function with the length l of each step
with being the volume of the E-dimensional unit sphere.
Thus, the -variance measures the amount of structural variation on a certain scale, e.g. in a map or three-dimensional distribution. A familiar, slightly different kind of variance defined for one-dimensional problems is the Allan-variance commonly used for stability investigation (Schieder et al. 1989). In contrast to the -variance, it works with a non-symmetric up-down filter.
Instead of convolving the structure in ordinary space with a filter function one can carry out the -variance analysis in Fourier space by simple multiplication. This directly relates the -variance to the power spectrum of a structure. If is the radially averaged power spectrum of the structure , the -variance is given by
where is the Fourier transform of the E-dimensional down-up-down function with the scale length l, and we are using k to denote the spatial frequency or wavenumber.
If the power spectrum is given by a simple power law, , the -variance also follows a power law with within the exponential range . (To avoid confusion with the ratio between thermal and magnetic pressure we use for the power spectral index rather than the variable used by Stutzki et al. 1998 and Bensch et al. 1999.) The main advantages of the -variance compared to the direct computation of the Fourier power spectrum are the clear spatial separation of different effects influencing observed structures like noise or finite observational resolution, and the robustness against singular variations due to the regular filter function.
Furthermore, for most astrophysical structures, where a periodic continuation is not possible, Bensch et al. (1999) has shown that the periodicity artificially introduced by the Fourier transform can lead to considerable errors. Here, even the -variance has to be determined in ordinary space. For the simulations examined in this paper, periodic wrap around is not a problem because it is already explicitly assumed, so we apply the faster Fourier method to determine their -variance spectra.
In Fig. 1 we compare the power spectrum and -variance for three simulations described below. They have different driving scales, and therefore each shows a different characteristic scale visible as a turn-over at large lags in the -variances and at small wavenumbers in the power spectrum, respectively. At smaller lags and higher wavenumbers power laws can be seen in both cases (with their slopes related by the analytic relation given above). A steep drop-off follows at the smallest scales indicating the resolution limit of the simulation. The power laws are equivalent in both cases, but the characteristic scale at one end and the resolution limit at the other end of the spectrum can be more clearly seen in the -variance. The smooth spatial filter function in the -variance analysis still provides a good measure for the behavior at large scales whereas the power spectrum suffers from the low significance of the few remaining points there.
The -variance analysis of astronomical maps was extensively discussed and demonstrated by Bensch et al. (1999). In all the observations analyzed by them, the total cloud size was the only characteristic scale detected by means of the -variance. Below that size they found a self-similar scaling behaviour reflected by a power law with index corresponding to a Fourier power spectral index . The analysis of further maps will be discussed in future work.
2.2. Two-dimensional maps and three-dimensional structures
In the application to molecular cloud structure simulations we have to restrict the analysis either to the three-dimensional structure or to the two-dimensional projection of the structure which would be astronomically observed, e.g. in optically thin lines or the FIR dust emission.
Stutzki et al. (1998) have shown that the spectral index of the power spectrum for an spatially isotropic structure remains constant on projection. This means that the projected map of three-dimensional density structures shows the same as the original structure as long as we assume that the astronomical structure is on the average isotropic. Consequently the slope of the -variance grows by 1 in projection.
In Fig. 2 we demonstrate this for a simulation where we determined the -variance of the three-dimensional structure and the -variances of the three perpendicular projections. The dashed lines show the three projected two-dimensional -variances. We multiply the three-dimensional -variance by the abscissa values to obtain the dotted line, which has the same local slope as the two-dimensional -variances. However, there is still an offset, as we have not corrected for the scale length of the measure. The length of an arbitrary three-dimensional vector is reduced on projection to two dimensions by a factor on the average. Therefore, we adjust the local scale by this factor for the -variance determined in three dimensional space. The resulting plot is shown as the thick solid line in Fig. 2. We obtain exactly the same general behaviour as for the projected maps. The equivalent plot for numerous other models verified this as a general behavior. Hence, we can either consider the three-dimensional variance or the projected variances and can simply translate them into each other.
The treatment of the three-dimensional variances has the advantage that it measures the scales exactly as they occur in the density structure. The projected maps, on the other hand, allow direct comparison to astronomical observations. As the comparison to the observations is our first priority, we will show in the following the variances transformed to the two-dimensional behaviour and we will only mention the physical three-dimensional scales if they appear to be especially prominent.
In this paper, we will restrict ourselves to simple projections taking them as representations of the integrated map of optically thin lines or optically thin continuum emission. We will not treat the full radiative transfer problem which has to be solved for a general treatment. Optical depths effects in fractal and random structures were discussed by Ossenkopf et al. (1998) and they will be taken into account in a subsequent paper dealing with the simulation of certain molecular clouds.
As a side-result of this comparison we find however, that the treatment of maps instead of three-dimensional cubes by observers can easily lead to a misinterpretation of the structure scaling. Fig. 3 compares the three-dimensional -variances computed in 3-D and the same variance corrected for 2-D projection as it could be measured by an observer for a hydrodynamic decaying turbulence model. Whereas the plot for 3-D only shows a broad distribution of structures, the human eye tries to see in the 2-D curve at least a reasonable range with a power law between 0.03 and 0.2. Except for the smallest lags dominated by numeric viscosity as discussed below the plot is quite similar to variances obtained e.g. by Stutzki et al. (1998) for molecular clouds. Hence, noncritical observers might be forced to see a self-similar behaviour even if there is no strong indication for a power law.
© European Southern Observatory (ESO) 2000
Online publication: December 8, 1999