Structure analysis of interstellar clouds : I . Improving the ∆ - variance method

Context. The ∆-variance analysis, introduced as a wavelet-based measure for the statistical scaling of structures in astronomical maps, has proven to be an efficient and accurate method of characterising the power spectrum of interstellar turbulence. It has been applied to observed molecular cloud maps and corresponding simulated maps generated from turbulent cloud models. The implementation presently in use, however, has several shortcomings. It does not take into account the different degree of uncertainty of map values for different points in the map, its computation by convolution in spatial coordinates is very time-consuming, and the selection of the wavelet is somewhat arbitrary and does not provide an exact value for the scales traced. Aims. We propose and test an improved ∆-variance algorithm for two-dimensional data sets, which is applicable to maps with variable error bars and which can be quickly computed in Fourier space. We calibrate the spatial resolution of the ∆-variance spectra. Methods. The new ∆-variance algorithm is based on an appropriate filtering of the data in Fourier space. It uses a supplementary significance function by which each data point is weighted. This allows us to distinguish the influence of variable noise from the actual small-scale structure in the maps and it helps for dealing with the boundary problem in non-periodic and/or irregularly bounded maps. Applying the method to artificial maps with variable noise shows that we can extend the dynamic range for a reliable determination of the spectral index considerably. We try several wavelets and test their spatial sensitivity using artificial maps with well known structure sizes. Performing the convolution in Fourier space provides a major speed-up of the analysis. Results. It turns out that different wavelets show different strengths with respect to detecting characteristic structures and spectral indices, i.e. different aspects of map structures. As a reasonable universal compromise for the optimum ∆-variance filter, we propose the Mexican-hat filter with a ratio between the diameters of the core and the annulus of 1.5. When the main focus lies on measuring the spectral index, the French-hat filter with a diameter ratio of about 2.3 is also suitable. In paper II we exploit the strength of the new method by applying it to different astronomical data.