next previous
Up: The ATESP radio survey


  
Appendix A: Master equations for error derivation

As discussed in Sect. 4, internal errors for the ATESP source parameters are well described by Condon (1997) equations of error propagation derived for two-dimensional elliptical Gaussian fits in presence of Gaussian noise. In order to get the total rms error on each parameter, a calibration term should be quadratically added. Using Eqs. (21) and (41) in Condon (1997), total percentage errors for flux densities ( $\frac{\sigma(S_{\rm peak})}{S_{\rm peak}}$) and fitted axes ( $\frac{\sigma(\theta_{\rm maj})}{\theta_{\rm maj}}$, $\frac{\sigma(\theta_{\rm min})}{\theta_{\rm min}}$) can be calculated from:

 \begin{displaymath}
\sqrt{\frac{2}{\rho^2} + \epsilon^2}
\end{displaymath} (A1)

where $\epsilon$ is the calibration term and the effective signal-to-noise ratio, $\rho$, is given by:

 \begin{displaymath}
\rho^2\!\! =\!\! \frac{\theta_{\rm maj} \theta_{\rm min}}{4 ...
...!
\right]^{\alpha_{\rm m}} \frac{S_{\rm peak}^2}{\sigma^2}\!
\end{displaymath} (A2)

where $\sigma$ is the image noise ($\sim 79$ $\mu$Jy on average for ATESP images), $\theta_{\rm N}$ is the FWHM of the Gaussian correlation length of the image noise (assumed $\simeq$ FWHM of the synthesized beam) and the exponents are:
$\displaystyle \alpha_{\rm M} = 3/2 \; \; {\rm and}$ $\textstyle \alpha_{\rm m} = 3/2$ $\displaystyle {\rm for} \;
\; \sigma(S_{\rm peak})$ (A3)
$\displaystyle \alpha_{\rm M} = 5/2 \; \; {\rm and}$ $\textstyle \alpha_{\rm m} = 1/2$ $\displaystyle {\rm for} \;
\; \sigma(\theta_{\rm maj})$ (A4)
$\displaystyle \alpha_{\rm M} = 1/2 \; \; {\rm and}$ $\textstyle \alpha_{\rm m} = 5/2$ $\displaystyle {\rm for} \;
\; \sigma(\theta_{\rm min}) \; .$ (A5)

Similar equations hold for position rms errors (Condon 1997; Condon et al. 1998):
  
$\displaystyle \sigma^2(\alpha) =$ $\textstyle \epsilon^2_{\alpha} + \sigma^2(x_0)
\sin^2({\rm PA}) + \sigma^2(y_0) \cos^2({\rm PA})$   (A6)
$\displaystyle \sigma^2(\delta) =$ $\textstyle \epsilon^2_{\delta} + \sigma^2(x_0) \cos^2({\rm PA})
+ \sigma^2(y_0) \sin^2({\rm PA})$   (A7)

where $\sigma^2(x_0) =\sigma^2(\theta_{\rm maj})/8\ln 2$ and $\sigma^2(y_0) =\sigma^2(\theta_{\rm min})/8\ln 2$ represent the rms lengths of the major and minor axes respectively.

Calibration terms are in general estimated from comparison with consistent external data of better accuracy than the one tested. Unfortunately there are no such data available in the region of sky covered by the ATESP survey. Nevertheless, from our typical flux and phase calibration errors, we estimate calibration terms of about $5-10\%$ for both flux densities and source sizes.

As a caveat we remind (see discussion in Paper I) that the 500 m baseline cutoff applied to our data makes the ATESP survey progressively insensitive to sources larger than $30\hbox{$^{\prime\prime}$ }$: assuming a Gaussian shape, only $50\%$of the flux for a $30\hbox{$^{\prime\prime}$ }$ large source would appear in the ATESP images. It is important to have this in mind when dealing with flux densities and source sizes of the largest ATESP sources.

Right ascension and declination calibration terms have been estimated from the astrometry results reported in Sect. 4.3.2. As already discussed, the ATESP astrometry can be considered accurate within a small fraction of an arcsec, even though the scarcity of (accurate) external data available in the ATESP region makes it difficult to quantify this statement. Nevertheless from the rms dispersion of the median offsets found between ATESP and the external comparison samples (see Sect. 4.3.2) we can tentatively estimate $\epsilon_{\alpha}\simeq 0.1\hbox{$^{\prime\prime}$ }$ and $\epsilon_{\delta}\simeq 0.4\hbox{$^{\prime\prime}$ }$.

It can be easily demonstrated that the master Eqs. (A1), (A6) and (A7) reduce to Eqs. (3) - (7) in Sect. 4 (where the calibration term is neglected) in the case of ATESP point sources ( $\theta_{\rm maj}\times \theta_{\rm min} \simeq 14\hbox{$^{\prime\prime}$ }\times 8\hbox{$^{\prime\prime}$ }$, PA $\simeq +2\hbox{$^\circ$ }$), assuming $\theta_{\rm N}\simeq 11\hbox{$^{\prime\prime}$ }$ (or $\theta_{\rm N}^2 \sim \theta_{\rm maj}
\theta_{\rm min}$).

For a complete and detailed discussion of the error master equations of source parameters obtained through elliptical Gaussian fits we refer to Condon (1997) and Condon et al. (1998).


next previous
Up: The ATESP radio survey

Copyright The European Southern Observatory (ESO)