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Source Extent and Errors

The apparent sizes and associated errors of sources reported in version 2 of the Chandra Source Catalog are determined using a Mexican-Hat optimization method described in the memo "Measuring Detected Source Extent Using Mexican-Hat Optimization", with some minor changes documented below. The method uses a wavelet transform to define elliptical source regions. This is a refinement of the source extent results produced by wavdetect, the source detection algorithm which identifies source candidates in each observation in catalog processing. The basic idea is as follows: given wavdetect sizes for the source an the PSF at the location, one can derive the intrinsic size of a source by deconvolving its observed size. In order to decide if a source is extended, srcextent evaluates if the intrinsic size of a source is different than zero at a \(5\sigma\) confidence level. All catalog sources are run through the srcextent algorithm, except if they have less than 15 counts, in which case no extent information is provided.

Source Region | Modified Source Region | Source Extent | Point Spread Function Extent | Deconvolved Source Extent

Changes with Respect to Earlier Versions

Source Region

Stacked Observation Detection Table:
ra_aper, dec_aper, mjr_axis_aper, mnr_axis_aper, pos_angle_aper, mjr_axis1_aperbkg, mnr_axis1_aperbkg, mjr_axis2_aperbkg, mnr_axis2_aperbkg, pos_angle_aperbkg

The spatial regions defining a source and its corresponding background are determined by scaling and merging the individual source detection regions that result from all of the spatial scales and source detection energy bands in which the source is detected during the source detection process (wavdetect). The result is a single elliptical source region which excludes any overlapping source regions, and a single, co-located, scaled, elliptical annular background region. The parameter values that define the source region and background region for each source are the ICRS right ascension and signed ICRS declination of the center of the source region and background region; the semi-major and semi-minor axes of the source region ellipse and of the inner and outer annuli of the background region ellipse; and the position angles of the semi-major axes defining the source and background region ellipses.

Note

In the first catalog release, the source region is defined on a tangent plane projection. The 0 deg position angle reference is defined on that tangent plane to be parallel to the true North direction at the location of the tangent plane reference (refer to the tangent plane reference right ascension (ra_nom), declination (dec_nom), and roll angle (roll_nom)).

Modified Source Region

Per-Observation Detections Table:
area_aper, area_aperbkg

The modified source region and modified background region for each source are defined as the areas of intersection of the source region and background region for that source with the field-of-view, excluding any overlapping source regions.

Convolved Source Extent

Per-Observation Detections Table:
mjr_axis_raw, mjr_axis_raw_lolim, mjr_axis_raw_hilim, mnr_axis_raw, mnr_axis_raw_lolim, mnr_axis_raw_hilim, pos_angle_raw, pos_angle_raw_lolim, pos_angle_raw_hilim

In order to estimate the intrinsic extent of a source in the sky, one first needs to realize that the measured extent of the source on the detector is the result of a convolution between the source itself and the PSF corresponding to that particular observation. It is therefore necessary to estimate the convolved extent of the source and of the PSF, and then perform a deconvolution.

The extent of the convolved source is estimated in a given science energy band with a rotated elliptical Gaussian parametrization of the raw extent of a source, i.e., the extent of a source before deconvolution has been performed. The corresponding ellipse has the following form:

\[ s(x,y;c_{1},c_{2},\phi) = \frac{s_{0}}{c_{1}c_{2}} \exp\left[-\pi\left(\mathcal{C}\mathbf{x}\right)^{2}\right] \ , \]

Where

\[ \mathcal{C} = \left[\begin{array}{cc} c_{1}^{-1} \quad 0 \\ 0 \quad c_{2}^{-1} \end{array} \right] \left[\begin{array}{cc} \cos{\phi} \quad \sin{\phi} \\ -\sin{\phi} \quad \cos{\phi} \end{array} \right] ,\ \mathbf{x} = \left[\begin{array}{cc} x \\ y \end{array} \right] \ . \]

Here, \(\phi\) (pos_angle_raw) is the clockwise angle between the positive x-axis and the ellipse major axis; \(c_{1}\) and \(c_{2}\) are the \(1\sigma\) radii along the major and minor axes of the source ellipse (mjr_axis_raw, mnr_axis_raw); \(s_{0}\) is the amplitude of the source elliptical Gaussian distribution.

For source extent purposes, the parameters of the ellipse are estimated by performing a spatial transform with a Mexican-Hat wavelet (also known as Ricker wavelet) directly on the counts in the raw source region, provided that more than 15 counts have been detected (for less than 15 counts, the error in the determination of the source size is comparable to the size itself). Note that this region describes the raw size of the source, and it is therefore different from the source region derived by wavdetect. Below we describe how that region is fitted to the observed distribution of counts.

The idea is simple: the two-dimensional correlation integral (i.e., the transform) between the wavelet function \(W\) and the ellipse function \(S\) is defined as:

\[ C(x,y;\mathbf{\alpha}) = \int_{-X}^{X} \int_{-Y}^{Y} W( x-x^{\prime}, y-y^{\prime}; \mathbf{\alpha} ) S( x^{\prime}, y^{\prime}; \mathbf{\alpha} ) dy^{\prime} dx^{\prime} \]

where \(\mathbf{\alpha} = (c_{1},c_{2},\phi)\) are the semi-major axis, semi-minor axis, and rotational angle of the Mexican-Hat wavelet. This correlation should be maximized when the scale and position of the wavelet coincide with that of the source. Spcifically, the quantity \(\psi(x,y;\mathbf{\alpha}) = C(x,y;\mathbf{\alpha})/\sqrt{c_{1} c_{2}}\) is maximized if the dimensions of the ellipse and the Mexican-Hat wavelength are related as: \(c_{i} = \sqrt{3} \sigma_{i} \) and \(\phi = \phi_{0}\). We can therefore estimate the parameters of the source extent ellipse by maximizing \(\phi(x,y;\mathbf{\alpha})\). Note that this assumes that sources can always be described as elliptical Gaussians. In practice, the maximization is evaluated as a discrete version of the equations above on the pixels of the image. In CSC2, the optimization of the correlation integral is performed using the Sherpa fitting tool.

Point Spread Function Extent

Per-Observation Detections Table:
psf_mjr_axis_raw, psf_mjr_axis_raw_lolim, psf_mjr_axis_raw_hilim, psf_mnr_axis_raw, psf_mnr_axis_raw_lolim, psf_mnr_axis_raw_hilim, psf_pos_angle_raw, psf_pos_angle_raw_lolim, psf_pos_angle_raw_hilim

The same approach as for the convolved source extent is used to estimate the elliptical parameters that best represent the instrumental point spread function (PSF) in each science band at the location of the source. The inputs are the PSF counts in the source region. The parameterization of the PSF can be compared with the parameterization of the detected source to determine whether the latter is consistent with a point source (see below).

The point spread function extent is a rotated elliptical Gaussian parameterization of the raw extent of the point spread function (PSF) at the location of the source. The parameterization of the PSF is computed from a wavelet transform analysis of the PSF counts in the source region in a given science energy band, and can be compared with the parameterization of the detected source to determine whether the latter is consistent with a point source. The point spread function extent is defined by the values and associated errors of the \(1\sigma\) radii along the major and minor axes, and position angle of the major axis of the point spread function ellipse that the detection process would assign to a monochromatic PSF at the location of the source, and whose energy is the effective energy of the given energy band. The point spread function has the following form:

\[ p(x,y;b_{1},b_{2},\psi) = \frac{p_{0}}{b_{1}b_{2}} \exp{\left[-\pi(\mathcal{B} \mathbf{x})^{2}\right]} . \]

Here, \(\psi\) (psf_pos_angle_raw) is the clockwise angle between the positive x-axis and the ellipse major axis; \(b_{1}\) and \(b_{2}\) are the \(1\sigma\) radii along the major and minor axes of the PSF ellipse (psf_mjr_axis_raw, psf_mnr_axis_raw); \(p_{0}\) is the amplitude of the PSF elliptical Gaussian distribution, and

\[ \mathcal{B} = \left[\begin{array}{cc} b_{1}^{-1} \quad 0 \\ 0 \quad b_{2}^{-1} \end{array} \right] \left[\begin{array}{cc} \cos{\psi} \quad \sin{\psi} \\ -\sin{\psi} \quad \cos{\psi} \end{array} \right] ,\ \mathbf{x} = \left[\begin{array}{cc} x \\ y \end{array} \right] . \]

Deconvolved Source Extent

Stacked Observation Detections Table:
major_axis, major_axis_lolim, major_axis_hilim, minor_axis, minor_axis_lolim, minor_axis_hilim, pos_angle, pos_angle_lolim, pos_angle_hilim

For stacked observation detections, the deconvolved source extent is a parameterization of the best estimate of the flux distribution defining the PSF-deconvolved source, which is determined in each science energy band from a variance-weighted mean of the deconvolved extent of each source measured in all contributing observations. The parameterization represents the best estimate values and associated errors for the \(1\sigma\) radius along the major axis, the \(1\sigma\) radius along the minor axis, and the position angle of the major axis of a rotated elliptical Gaussian source that has been fitted to the observed source spatial event distribution deconvolved with the ray-trace local PSF at the location of that source event distribution.
Per-Observation Detections Table:
major_axis, major_axis_lolim, major_axis_hilim, minor_axis, minor_axis_lolim, minor_axis_hilim, pos_angle, pos_angle_lolim, pos_angle_hilim

Deconvolving the raw source image with the PSF produces the intrinsic shape of the source. In order to determine whether a source is extended, we first need to realize that the intrinsic size of a point source is, by definition, zero. By performing the fitting described above, srcextent estimates the raw sizes and errors of both the source and the PSF. In what follows, we derive the size and error for the intrinsic size of the source, and then we establish a criterion to flag a source as extended.

In principle, one can determine the parameters of the intrinsic source ellipse, \(\{a_{1},a_{2},\phi\}\), by solving a non-linear system of equations involving the PSF parameters, \(\{a_{1},a_{2},\psi\}\), and the measured source parameters, \(\{\sigma_{1},\sigma_{2},\delta\}\). However, because these equations are based on a crude approximation and because the input parameters are often uncertain, such an elaborate calculation seems unjustified.

A much simpler and more robust approach makes use of the identity:

\[ \sigma_{1}^{2} + \sigma_{2}^{2} = a_{1}^{2} + a_{2}^{2} + b_{1}^{2} + b_{2}^{2} \ , \]

which applies to the convolution of two elliptical Gaussians having arbitrary relative sizes and position angles. Using this identity, one can define a root-sum-square intrinsic source size:

\[ a_{\mathrm{rss}} \equiv \sqrt{a_{1}^2 + a_{2}^{2}} = \sqrt{ \max\{0,(\sigma_{1}^{2} + \sigma_{2}^{2}) - (b_{1}^{2} + b_{2}^{2}) \}} \ , \]

that depends only on the sizes of the relevant ellipses and is independent of their orientations. This expression is analogous to the well-known result for convolution of 1D Gaussians and for convolution of circular Gaussians in 2D.

Using the equation above, one can derive an analytic expression for the uncertainty in \(a_{\mathrm{rss}}\) in terms of the measurement errors associated with \(\sigma_{i}\) and \(b_{i}\). Because \(\sigma_{i}\) and \(b_{i}\) are non-negative, evaluating the right-hand side of the equation using the corresponding mean values should give a reasonable estimate of the mean value of \(a_{\mathrm{rss}}\). A Taylor series expansion of the right-hand side evaluated at the mean parameter values, therefore, yields the uncertainty:

\[ \Delta a_{\mathrm{rss}} = \frac{1}{a} \sqrt{ \sigma_{1}^{2} \left(\Delta \sigma_{1} \right)^{2} + \sigma_{2}^{2} \left(\Delta \sigma_{2} \right)^{2} + b_{1}^{2} \left(\Delta b_{1} \right)^{2} + b_{1}^{2} \left(\Delta b_{1} \right)^{2} } \]

where \(\left(\Delta X \right)^{2}\) represents the variance in \(X\) and where

\[ a \equiv \left\{ \begin{array}{ll} a_{\mathrm{rss}} \ , & a_{\mathrm{rss}} > 0 \ , \\ \sqrt{b_{1}^{2} + b_{2}^{2}} \ , & a_{\mathrm{rss}} = 0 \ . \end{array} \right. \]

A source is extended if its root-sum-square intrinsic size is larger than the root-sum-square error of this size. Namely, if its derived intrinsic size, that depends on the raw source and PSF sizes and errors, is larger than the fluctuations due to the uncertainty in the determination (different from zero). If \(a_{\mathrm{rss}} > f \Delta a_{\mathrm{rss}}\), then the source is extended at the \(f\sigma\) level. In CSC2, we use \(f = 5\), which implies that sources are flagged as extended if the intrinsic size is determined with a significance of \(5\sigma\).

Changes with Respect to Earlier Versions

With respect to earlier versions of the catalog, a number of improvements have been included in Release 2 of the catalog to improve the source extent estimate. The main changes were:

We use the results from wavdetect to set the initial parameter guess for the source size. The correlation integral is maximized using the Nelder-Mead Simplex optimization method, but only the scale and orientation of the Mexican-Hat wavelet are free parameters. The centroid position is estimated prior to the fit by maximizing a simplified version of the wavelet that uses the initial guesses for \(a_{i}\) (from wavdetect), and \(\phi = 0\). Therefore, the position of the pixel where the maximum occurs is found first, and then the orientation and size of the ellipse are optimized for.
Adding the effect of aspect blur to the PSF. Both the aspect solution and detector effects add an aspect blur to the instrumental PSF that effectively increase its extent. We have added an estimated blur in quadrature to the PSF extent in order to improve our estimate of the deconvolved source extent.
Improving the PSF image fitting by adjusting image centering and size, and using sub-pixelated PSFs where appropriate.

Caveats

When using the srcextent results, users should keep in mind the following two caveats:

The algorithm is not designed to separate blended sources and is unlikely to generate optimal source regions in such cases. Users should use caution in interpreting srcextent results in very crowded regions.
The algorithm makes no attempt to detect cases in which no significant source is present above background within the ellipse that was initially provided by wavdetect. In such cases, subsequent optimization of \(\psi\) may yield a meaningless result, such as an ellipse of maximum size of an ellipse of random size centered on a noise peak.

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