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$\begin{figure} \par\includegraphics[angle=90,width=5.8cm,clip]{9106_fig1a.eps}\i... ...ig1b.eps}\includegraphics[angle=90,width=5.8cm,clip]{9106_fig1c.eps}\end{figure}$ Figure 1: Examples of periodic data sets used to test the $\Delta$ -variance algorithm. The fBm structure on the left is characterised by a spectral index $\zeta =3.0$ . The chess board has a characteristic structure size of about 0.13. The sine wave field on the right consists of a single Fourier component at k=8, the corresponding characteristic distance is 0.09.
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In the text

$\begin{figure} \par\includegraphics[angle=90,width=7.3cm,clip]{9106_fig2.eps}\end{figure}$ Figure 2: Example of a filled circle structure used to test the $\Delta$ -variance algorithm. The diameter of the circle is 0.25.
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In the text

$\begin{figure} \par\includegraphics[angle=90,width=8.6cm,clip]{9106_fig3a.eps}\\ [2mm] \includegraphics[angle=90,width=8.6cm,clip]{9106_fig3b.eps}\end{figure}$ Figure 3: $\Delta$ -variance spectra computed with three different types of the edge treatment for one periodic fBm structure ( upper plot) and one non-periodic subset from a fBm structure ( lower plot). The fBms are characterised by a spectral index $\zeta =3.0$ so that the $\Delta$ -variance should show a slope $\alpha =1.0$ . The measured slopes are indicated for each graph in the plot.
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In the text

$\begin{figure} \par\includegraphics[angle=90,width=8.7cm,clip]{9106_fig4a.eps}\\ [2mm] \includegraphics[angle=90,width=8.7cm,clip]{9106_fig4b.eps}\end{figure}$ Figure 4: Average and standard deviation of the spectral indices of the $\Delta$ -variance spectra determined for a set of fBm structures (upper plot) and for a set of sub-maps from fBm structures (lower plot) depending on the spectral index of the fBms. All spectra are computed at the same power spectral indices $\zeta =2.0, 2.4, 2.8, 3.2, 3.6,$ and 4.0, but for a better visibility, the standard deviations for the truncated filter and for the mirror continuation are displaced in the plot relative to these values. In the case of the periodic analysis of the fBm maps the error bars giving the standard deviation of the statistical samples are practically zero. The dotted line indicates the theoretical index for infinitely large fBm structures $\alpha =\zeta -2$ .
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In the text

$\begin{figure} \par\includegraphics[angle=90,width=8.6cm,clip]{9106_fig5.eps}\end{figure}$ Figure 5: $\Delta$ -variance spectra for the filled-circle map where the circle has a diameter of 1/8 of the map size and is shifted by three diameters from the centre of the map. The values $\hat{l}$ indicate the lag of the measured peak in the spectrum.
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In the text

$\begin{figure} \par\includegraphics[angle=90,width=8.5cm,clip]{9106_fig6.eps}\end{figure}$ Figure 6: Factor translating the filter core diameter l into the average distance measured by the filter as a function of the diameter ratio between annulus and core of the filter. The average distance is computed by a double integral over the core and the annulus of the filter.
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In the text

$\begin{figure} \par\includegraphics[angle=90,width=8.5cm,clip]{9106_fig7.eps}\end{figure}$ Figure 7: $\Delta$ -variance spectra computed for the sine wave field with a wavenumber k=8. The filter truncation is applied in both cases for the edge treatment. The peak scale is given as $\hat{l}$ , the logarithm of the ratio between upper and lower lag where the $\Delta$ -variance has dropped to 1/2 of the peak value is given as w.
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In the text

$\begin{figure} \par\includegraphics[angle=90,width=8.5cm,clip]{9106_fig8a.eps}\\ [2mm] \includegraphics[angle=90,width=8.5cm,clip]{9106_fig8b.eps}\end{figure}$ Figure 8: Logarithmic FWHM of the $\Delta$ -variance peak and contrast between the peak value and the values at l=0.014and l=1/3 for the sine wave map analysed in Fig. 7.
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In the text

$\begin{figure} \par\includegraphics[angle=90,width=8.6cm,clip]{9106_fig9a.eps}\\ [2mm] \includegraphics[angle=90,width=8.6cm,clip]{9106_fig9b.eps}\end{figure}$ Figure 9: Logarithmic FWHM of the $\Delta$ -variance peak and contrast between the peak value and the values at l=0.014and l=1/3 for the filled circle map discussed in Fig. 5. The widths measured for the French hat at v<1.25 are lower limits because the spectrum did not fall below half of the peak value at the upper end of the spectrum.
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In the text

$\begin{figure} \par\includegraphics[angle=90,width=8.7cm,clip]{9106_fig10.eps}\end{figure}$ Figure 10: $\Delta$ -variance spectra computed for a sub-map from an fBm structure with $\zeta =3$ . Filter truncation is used for the edge treatment. $\alpha$ denotes the power-law slope fitted over the full range.
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In the text

$\begin{figure} \par\includegraphics[angle=90,width=8.5cm,clip]{9106_fig11a.eps}\\ [2mm] \includegraphics[angle=90,width=8.5cm,clip]{9106_fig11b.eps}\end{figure}$ Figure 11: Distribution of fitted $\Delta$ -variance exponents for fBm maps ( upper plot) and submaps from fBm structures ( lower plot) with $\zeta =2.5$ , 3.0, and 3.5 as a function of the filter diameter ratio. The French-hat filter combined with the filter truncation method is used. Dotted lines indicate the expected $\Delta$ -variance exponents corresponding to the fBm spectral indices. The error bars indicate the scatter obtained for structures with the same parameters in terms of the standard deviation.
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In the text

$\begin{figure} \par\includegraphics[angle=90,width=8.5cm,clip]{9106_fig12.eps}\end{figure}$ Figure A.1: $\Delta$ -variance spectra computed from an fBm structure with a spectral index $\zeta =3.1$ and a superimposed circular pattern of white noise. The solid line shows the $\Delta$ -variance spectrum of the original fBm, the dotted line the spectrum obtained from the direct analysis of the noisy map, and the dashed line the spectrum computed with a reliability weighting given by the inverse noise rms.
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In the text

$\begin{figure} \par\includegraphics[angle=90,width=8.5cm,clip]{9106_fig13.eps}\end{figure}$ Figure A.2: Scale range (ratio between maximum and minimum lag) within which the derived noisy spectrum agrees within 10% with the original spectrum when the $\Delta$ -variance spectrum is computed either with or without a weighting function given by the inverse noise rms. The original structure is given by an fBm structure with $\zeta =3.0$ and the noise pattern by a chess board structure with four fields and a signal-to-noise ratio of 5.0. On the abscissa we have changed the amplitude of the noise level variation between the different fields of the chess board. The error bars show the absolute minimum and maximum of the matching range size found in a sample of 30 different fBms.
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In the text

$\begin{figure} \par\includegraphics[angle=90,width=5.8cm,clip]{9106_fig1a.eps}\i... ...ig1b.eps}\includegraphics[angle=90,width=5.8cm,clip]{9106_fig1c.eps}\end{figure}$	Figure 1: Examples of periodic data sets used to test the $\Delta$ -variance algorithm. The fBm structure on the left is characterised by a spectral index $\zeta =3.0$ . The chess board has a characteristic structure size of about 0.13. The sine wave field on the right consists of a single Fourier component at k=8, the corresponding characteristic distance is 0.09.
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$\begin{figure} \par\includegraphics[angle=90,width=7.3cm,clip]{9106_fig2.eps}\end{figure}$	Figure 2: Example of a filled circle structure used to test the $\Delta$ -variance algorithm. The diameter of the circle is 0.25.
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$\begin{figure} \par\includegraphics[angle=90,width=8.6cm,clip]{9106_fig5.eps}\end{figure}$	Figure 5: $\Delta$ -variance spectra for the filled-circle map where the circle has a diameter of 1/8 of the map size and is shifted by three diameters from the centre of the map. The values $\hat{l}$ indicate the lag of the measured peak in the spectrum.
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$\begin{figure} \par\includegraphics[angle=90,width=8.5cm,clip]{9106_fig6.eps}\end{figure}$	Figure 6: Factor translating the filter core diameter l into the average distance measured by the filter as a function of the diameter ratio between annulus and core of the filter. The average distance is computed by a double integral over the core and the annulus of the filter.
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$\begin{figure} \par\includegraphics[angle=90,width=8.5cm,clip]{9106_fig7.eps}\end{figure}$	Figure 7: $\Delta$ -variance spectra computed for the sine wave field with a wavenumber k=8. The filter truncation is applied in both cases for the edge treatment. The peak scale is given as $\hat{l}$ , the logarithm of the ratio between upper and lower lag where the $\Delta$ -variance has dropped to 1/2 of the peak value is given as w.
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$\begin{figure} \par\includegraphics[angle=90,width=8.5cm,clip]{9106_fig8a.eps}\\ [2mm] \includegraphics[angle=90,width=8.5cm,clip]{9106_fig8b.eps}\end{figure}$	Figure 8: Logarithmic FWHM of the $\Delta$ -variance peak and contrast between the peak value and the values at l=0.014and l=1/3 for the sine wave map analysed in Fig. 7.
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$\begin{figure} \par\includegraphics[angle=90,width=8.6cm,clip]{9106_fig9a.eps}\\ [2mm] \includegraphics[angle=90,width=8.6cm,clip]{9106_fig9b.eps}\end{figure}$	Figure 9: Logarithmic FWHM of the $\Delta$ -variance peak and contrast between the peak value and the values at l=0.014and l=1/3 for the filled circle map discussed in Fig. 5. The widths measured for the French hat at v<1.25 are lower limits because the spectrum did not fall below half of the peak value at the upper end of the spectrum.
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$\begin{figure} \par\includegraphics[angle=90,width=8.7cm,clip]{9106_fig10.eps}\end{figure}$	Figure 10: $\Delta$ -variance spectra computed for a sub-map from an fBm structure with $\zeta =3$ . Filter truncation is used for the edge treatment. $\alpha$ denotes the power-law slope fitted over the full range.
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