XMM-Newton Science Analysis System


ewavelet (ewavelet-3.12) [xmmsas_20230412_1735-21.0.0]

The detection threshold

Figure 2: Histograms of wavelet convolved images. The wavelet scale was $2\sqrt 2$ and the background value were respectively $b = 0.01$/$q \simeq 0.5$, $b = 0.05$/$q \simeq 2.5$, $b = 0.1$/$q \simeq 5.0$ and $b = 1$/$q \simeq 50$ (cf. Damiani et al. [2]).

In order to determine the significance of a detection, we have to calculate what the chance is that we find a peak in the wavelet convolved image in the absence of real sources. The fluctuations in the wavelet map is given by eq. (8). For large number of (background) photons we may expect that the fluctuations in $W$ approach a Gaussian distribution, with standard deviation $\sqrt{\frac{b}{2\pi\sigma^2}}$:

  $$P(C_{max} > C_0) = \frac{\sigma}{\sqrt{b}} \int_{C_0}^{\infty} \exp\Bigl( -\frac{1}{2}\frac{2\pi\sigma^2C^2}{b}\Bigr) dC.$$ (9)

Because we used a different normalization constant in eq. (5), this equation is different from Damiani et al. ([2]) and Dobrzycki et al. ([1]). However, using the transformation $q = 2\pi \sigma^2 b$ and $C' = 2\pi\sigma^2$ we obtain:

  $$P(C_{max}' > C_0') = \frac{1}{\sqrt{2\pi q}} \int_{C_0'}^{\infty}... ...) dC' = \frac{1}{2}\Bigl[ 1 - {\rm erf}\Bigl(\frac{C'}{\sqrt{2q}}\Bigr)\Bigr].$$ (10)

The transformation is useful, as both Dobrzycki et al. ([1]) and Damiani et al. ([2]) have made Monte Carlo simulations to approximate the statistical behaviour in case the Gaussian approximation is not valid. They observed that this is the case for $\log q < 3.25$. To determine significant thresholds for the case the Gaussian approximation is not valid Damiani et al. ([2]) fitted the results of Monte Carlo simulations to the following formula:

  $$C_0'(k,q) = C_0 2\pi\sigma^2 = k\sqrt{q} + (c_1 + c_2k + c_3 k^2),$$ (11)

where $C_0'$ and $C_0$ are the detection threshold for MH wavelets with and without normalization constants respectively, and $k$ is the number of Gaussian sigmas. The numerical constants were found to be: $c_1 = -0.2336$, $c_2 = 0.0354$ and $c_3 = 0.1830$. Clearly, this fit has the correct asymptotic behaviour.

Note that all the above apply to correlation values of individual pixels. In reality the probabilities to detect spurious sources above the background is a little bit smaller, as detections are defined as local maxima above the threshold. Furthermore, since we are only interested in maxima not in minima eq 11 refers to the “single tail” probability of a Gaussian distribution. As an example, in a $1024 \times 1024$ image, $1363 = 0.0013 \times 1024^2$ pixels are expected above the $3\sigma$ threshold.