
In our test case, the process is leptokurtic, that is the nongaussianity is characterised by a positive excess of kurtosis. We quote, in the left panel of Table 3 the median excesses of kurtosis computed with the multiscale gradient coefficients of the 100 maps. This is a more suitable quantity to characterise a nonGaussian process, than the mean , as there is an important dispersion of the k values with a clear excess towards large values. The , which represents the rms excess of kurtosis with respect to the median for one realisation, takes naturally into account the non symmetric distribution of the multiscale gradients. This results in a lower boundary () for the confidence interval smaller that the upper boundary (). The latter is biased towards large values as we are studying a leptokurtic process. Therefore, the comparison between the values of k and indicates the detectability of nongaussianity. When for one realisation differs from zero by a value of the order of, or larger than, this suggests that the signal is nonGaussian. For the tophat profiles, there is an obvious excess of kurtosis at all scales.
In order to test the nonGaussian signature arising form different processes with same power spectra, we analyse a set of 100 nonGaussian maps made of the superposition of Gaussian profiles of different sizes and amplitudes. We compute the median value of the excess of kurtosis and the corresponding confidence intervals (Table 3, right panel). We note that k is different from zero at all scales, exhibiting the nonGaussian nature of the studied process. However, it is smaller than in the case of the tophat profiles. This decrease is due to the superposition of smoother profiles.
We now add one representative Gaussian map to 100 nonGaussian maps (tophat profiles). As the nonGaussian signal is very strongly dependent on the studied map, it is necessary to span a large set of nonGaussian statistical realisations in order to have a reliable statistical specification of nongaussianity. The Gaussian and nonGaussian signals were summed with different mixing ratios represented by the ratio of their rms amplitudes ().
After wavelet decomposition, we compute the multiscale gradient coefficients
of the summed maps and derive the normalised median excess of kurtosis with
respect to a Laplace distribution together with the confidence intervals. The
results are quoted in Table 4 as a
function of the mixing ratio and the wavelet decomposition scale.
For , the excess of kurtosis is larger than that of the Gaussian
test map and it is smaller than that of the purely nonGaussian signal. The
summation of
the two processes has therefore, as expected, smoothed the gradients
and diluted
the nonGaussian signal. For a nonGaussian signal half that of the
Gaussian signal, only the first three scales indicate an excess of kurtosis
different from the Gaussian one. For the ratio , only the first
scale has an excess marginally different from the Gaussian signal. For larger
ratios, the nonGaussian signal is quite blurred.


In Fig. 4, the solid line represents the values of the excess of kurtosis of each nonGaussian realisation. The dashed line represents the same quantity for the Gaussian test maps. We first note the overall shift of the values towards nonzero positive values (leptokurtic signal), with some very large values compared to the median. A second characteristic worth noting is the difference in amplitudes between, on the one hand, the excess of kurtosis of the coefficients associated with and, on the other hand, those associated with and . The former are indeed smaller. As the excess of kurtosis of the first derivative coefficients is of the same order, we compute one median k over and coefficients, and compare it to the excess of kurtosis of the cross derivative. At the first two decomposition scales there is an important and noticeable difference between the two sets of values and ,and . At the third and fourth decomposition scales, the difference decreases but is still present.

For the nonGaussian process made of the superposition of Gaussian profiles,
we compute the median excess of kurtosis associated with the wavelet
coefficients of the first and cross derivatives (Table 6). As for
the multiscale gradient coefficients, we find that the excess of kurtosis
is smaller for this type of nonGaussian maps but it is still significantly
different from zero at all scales except the fourth.
.D..1

We analyse the sum of a representative Gaussian map and the set of 100 nonGaussian maps (tophat profile). The sum of the two processes has been performed again with different mixing ratios. The results we obtain are given in Table 7. An accompanying figure (Fig. 5) illustrates the corresponding results for a mixing ratio (solid line). In this figure, the dashed line represents the Gaussian process alone. For , we find that nongaussianity is detected at all decomposition scales for both first and cross derivative coefficients. For a mixing ratio , we observe a significant excess of kurtosis only at the first decomposition scale. For , the excess becomes marginal for both and coefficients at the first decomposition scale, all other scales showing no departure from gaussianity. The same tendency is noted for the coefficients associated with .
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