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Astron. Astrophys. 336, 697-720 (1998)

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2. Power spectrum and phase distribution of molecular cloud images

Being generated predominantly by the turbulent internal cloud motion, the internal structure of molecular clouds is likely to be random to a large degree. All the information on the structural properties of a random function is contained in its power spectrum , the phase distribution being completely random. Thus, it is obviously important to study the power spectrum of observed cloud images. This has been done for HI clouds (Green 1993), deriving the azimuthally averaged power spectra directly from interferometer observations; he finds the power spectra to be smooth and well characterized by a power law with a spectral index around 2.6 to 2.8 for individual velocity channel maps.

2.1. The [FORMULA]-variance analysis: a generalized Allan-variance method

A very powerful tool to study the drift characteristics of a random time series has been introduced by Allan (1966), within the context of characterizing the stability of atomic clocks. It has been successfully applied more recently in the analysis of the drift contributions on the resulting signal-to-noise ratio from radio astronomy receivers and backends (Schieder et al. 1989). For a random time series, the Allan variance on a given time scale T is basically the variance of the differences of subsequent averages over time T. The definition can easily be carried over to Fourier space, where the full information on the statistical properties of the random function is contained in its power spectrum . The Allan variance then turns out to be the filtered average of the power spectrum of the noise function, weighted with the filter function [FORMULA]. This filtering function peaks at a frequency [FORMULA] (see Appendix A). Varying T thus allows to trace the average behavior of the power spectrum with frequency, which determines the drift characteristics of the signal. The Allan variance , representing an integral information over the power spectrum , however, gives much better statistical significance for the drift behavior than a single point in the power spectrum . In addition, it allows easy discrimination between different contributions to the random signal, such as purely white noise, flicker noise and the global drift characteristics. These properties make it useful in the analysis of drift behavior of instrumentation.

These same properties motivated us to develop a new procedure to characterize the observed 2-dimensional structure of molecular cloud images, based on the Allan variance concept. It is of particular importance that such a procedure should give a good discrimination between the white noise noise contribution resulting from the limited signal to noise ratio achievable for the observations and the information on the cloud structure contained in the data set.

As the literature on the Allan variance is rather sparse, Appendix A summarizes the definition of the Allan-variance for 1-dimensional random functions, discusses its relation to Fourier-analysis and in particular to the power spectrum in a consistent notation, and lists its characteristic properties with various types of drift behavior. With this background, extension to higher dimensions is rather straightforward. For the analysis of a 2-dim spatial structure, such as a molecular cloud image, we have to analyze the variance between differences of averages over adjacent areas with a typical spatial separation of size L. With no azimuthal direction preferred, this is naturally done in cylindrical symmetry, i.e. using a spatial filter function with a constant positive value of e.g. [FORMULA] on a circle of diameter L, enclosed by a ring of thickness L with a negative value of [FORMULA] (for equal weight of the two areas). In Fourier-space this corresponds to calculating the 2-dim variance as the average of the 2-dim power spectrum weighted with the corresponding power spectrum of the 2-dim filter function.

This is the concept behind the newly introduced [FORMULA]-variance , derived as a generalization of the 1-dim Allan variance in Appendix A, and defined for the analysis of random functions in higher dimensions in Appendix B. Corresponding to the [FORMULA]-behavior of the 1-dimensional Allan-variance , the 2-dimensional [FORMULA]-variance goes [FORMULA] for an image dominated by purely white noise. In general, if the image has a power spectrum [FORMULA] with [FORMULA] in the range [FORMULA], the 2-dimensional [FORMULA]-variance varies as [FORMULA] (see Appendix B).

For an observed image, the highest spatial frequency, i.e. shortest lag probed, will be close to the finite resolution, e.g. given by the Gaussian beam with a certain FWHM and a (close to) fully sampled spatial raster. The lowest spatial frequencies, i.e. largest lags, will be limited by the finite size of the cloud or of the area observed. In practice the useful limits fall at about a quarter of the image size, giving at least two independent difference measures in each spatial coordinate.

2.2. Application to observed cloud images

The result of such a [FORMULA]-variance analysis is shown in Fig. 1, which shows the [FORMULA]-variance as defined above for an integrated intensity image of the Polaris Flare to closely follow a power law corresponding to a power law spectral index of [FORMULA]. A small subsection of the cloud observed at higher angular resolution (observed as part of the IRAM key project "small scale structure of pre-star-forming molecular clouds", Falgarone et al. 1998) shows a very similar power law slope in the [FORMULA]-variance (Fig. 2). A second example is given in Fig. 3 for a subset of the FCRAO survey of the outer galaxy (Heyer et al. 1997), also giving [FORMULA]. The value of [FORMULA] is remarkably close to the one measured by Green 1993, for HI clouds directly from interferometer data. Similar results can be obtained for other observed images, although the analysis often does not give as straightforward an answer as in the three examples shown. The application of the [FORMULA]-variance analysis to several data sets and the limitations of deriving proper variance plots from observed data sets will be presented in a separate paper (Bensch et al., in prep. ).

[FIGURE] Fig. 1. [FORMULA]-variance analysis of an observed image of the Polaris Flare (CO [FORMULA], 8 arcmin per pixel; Heithausen & Thaddeus (1990), top left), showing a power law behavior with an index of 0.8, which, for the 2-dimensional image corresponds to a power spectrum with a spectral index [FORMULA] (bottom left). The turnover into the white noise behavior at small angular scales as well as the influence of the finite image size at large scales are weakly visible (compare Fig. 3). The right hand panels show a 92 by 128 pixel subsection of a synthesized fBm -image (see Sect. 4) with the same power spectrum spectral index (top) and its corresponding [FORMULA]-variance behavior (bottom).

[FIGURE] Fig. 2. [FORMULA]-variance analysis of a subsection of the Polaris Flare (top) observed at higher angular resolution with the IRAM 30m-telescope (CO [FORMULA]; Falgarone et al. 1998). This map covers 48 by 64 pixels, each of 7.5" extent. The image is thus zoomed in by a factor of 64 from the image in Fig. 1. The 2-dimensional [FORMULA]-variance analysis shows the same slope as the large scale image (Fig. 1), so that the power spectrum with [FORMULA] continues to these higher spatial frequencies (middle). The [FORMULA]-variance of the 1-dimensional projection of the same image (bottom) shows a slope by about 1 higher than that of the 2-dimensional image, as expected theoretically.

[FIGURE] Fig. 3. [FORMULA]-variance analysis of a subset of the FCRAO outer galaxy survey (13CO [FORMULA]; Heyer et al. 1997). This very large scale data set (384 by 128 pixels; 50" per pixel) with its relatively higher noise than the examples in Figs. 1 and 2 shows a clear turnover into the white noise behavior at the smallest angular scales, and a turnover at about a 30 pixel length scale due to the influence of the typical size of main structures in the image.

2.3. Phase distribution of observed cloud images

Fig. 4 shows that the phases of the observed image are indeed randomly distributed. For one, they are uniformly distributed over the interval from 0 to [FORMULA] (due to the image being real valued, its Fourier transform is hermitian and the phases are odd, so that the phase distribution at negative values is symmetric to that at positive values). Secondly, the [FORMULA]-variance of the phases has a slope close to -2 as expected for a completely uncorrelated, white noise distribution (see Appendix C). Deviations from the white noise behavior may actually be significant with regard to a multifractal characteristic of the cloud image; however, they may also result from the fact that the observed image possibly has some correlation due to a (though small) extended error beam pickup. Our numerical simulations (see Sect. 4) also indicate, that the finite size of the observed image may result in deviations from the pure white noise behavior.

[FIGURE] Fig. 4. The phases (top) of the Polaris Flare image shown in Fig. 1: the distribution of the phases is uniform (center); the [FORMULA]-variance of the phases shows a close to white noise behavior, i.e. a slope of -2, corresponding to [FORMULA] (bottom).

2.4. Discussion

The way the [FORMULA]-variance is defined, it implicitly averages over the angular variation in the power spectrum . This is due to the circular symmetry of the 2-dimensional filter function. The [FORMULA]-variance is thus sensitive only to the circular symmetric part of the power spectrum of the cloud image. The [FORMULA]-variance analysis shows over which range in spatial frequencies the azimuthally averaged power spectrum is consistent with a power law (by the straightness of the [FORMULA]-variance plot); and it allows, over this range, to directly derive the numerical value of the power law spectral index with high signal to noise, as it implicitly averages over the directional variation. It does not provide a measure of the degree of deviation of the power spectrum from spherical symmetry. Direct analysis of the azimuthally averaged power spectrum in principle gives the same result but makes the discrimination against the white noise contribution and finite size effects of observed images much less obvious.

One should emphasize, however, that this does not imply that the observed image has to have circular symmetry: the phases are arbitrary (and to a large degree random) and will thus cause a non circular symmetric image (the simulations presented in Figs. 1 and 3 show nice examples for this). One should keep in mind, however, that the [FORMULA]-variance method should be applied with care to images which obviously are dominated by systematic large scale structure, e.g. showing a preferred direction, like a large scale filament broken up into a series of clumps on preferred length scales, or a sharp cloud boundary resulting from e.g. shock interaction.

Instead of analyzing the 2-dimensional image, one can investigate its 1-dimensional projection. The projection results from the integration along a given direction, e.g. along the x-axis. The Fourier transform of the projected 1-dimensional strip is the value of the 2-dimensional Fourier transform at the [FORMULA] axis. For a power law power spectrum [FORMULA] in 2 dimensions, the 1-dimensional projection thus has a power spectrum [FORMULA]. The power law index of the power spectrum of the 1-dimensional projection is thus the same as that of the 2-dimensional image. This result holds equally well for higher dimensional images: the [FORMULA]-dimensional projection of an E-dimensional structure with a power law power spectrum has again a power law power spectrum with the same spectral index. With the general relation between the power spectrum index [FORMULA] and the power law index of the E-dimensional [FORMULA]-variance [FORMULA] as given in Appendix C, we thus expect the projection of a 2-dimensional image with power spectrum index [FORMULA], i.e. 2-dimensional [FORMULA]-variance index [FORMULA], to have a 1-dimensional [FORMULA]-variance index [FORMULA]. Fig. 2 (bottom) shows as an example the result for the high angular resolution data set of the Polaris Flare observed with the IRAM 30m telescope. We note, however, that other clouds analyzed in the same way show [FORMULA]-variance results for the 1-dimensional projections that are not consistent with the values expected from the analysis of the 2-dimensional images; this is likely due to real data sets being limited by finite size and other systematic effects, as we will discuss in detail in Bensch et al. (in prep. ).

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© European Southern Observatory (ESO) 1998

Online publication: July 20, 1998
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