APEX computes two kinds of point-source flux uncertainties, as well as aperture uncertainties. The extract table output contains the following columns: delta_flux, SNR (signal-to-noise ratio), and ap_unc (aperture uncertainties).
8.9.1 delta_flux
delta_flux is the uncertainty in the PRF_fitting parameters, determined by the Hessian matrix:
Equation 8.27
where ’s are the optimal fitting parameters:
Provided the fit is successful, the covariance matrix Cab of the fitting parameters is defined by the inverse Hessian matrix:
Equation 8.28
In particular the positional (delta_x, delta_y) and flux (delta_flux) uncertainties computed by MOPEX are equal to delta_x = Cxx, delta_y = Cyy, delta_flux = Cff.
Often, delta_flux will underestimate the true flux uncertainty because of correlated errors. The flux uncertainty is correlated with the positional uncertainty, and the formal uncertainties do not reflect this.
8.9.2 SNR Computation
In addition to the formal uncertainty in the PRF fit, APEX provides a rough estimate of the signal-to-noise ratio (SNR) of the source flux. This estimate is given in the output column "SNR". Three options are available, controlled by the Use Photerr for SNR parameter in the Source Estimate module. In addition, the user can swap in either APEX's background noise estimates (Gaussnoise), or the data uncertainties, with the APEX Settings parameter, Use Data Uncertainties for PRF-fitted SNR (this is use_data_unc_for_fitted_snr in namelists), where on (= 1) means use the data uncertainties. The SNR choices are given below. In namelists, use the number, e.g. Use_Photerr_for_SNR = 2 is equivalent to “yes, with gain” in the GUI.
0. no: This is APEX's old SNR. Scale the peak pixel uncertainty by that pixel's contribution to the total central PRF.
1. yes, no gain: Use the sum (in quadrature) of the data uncertainties within a box roughly the size of the core of the PRF.
2. yes, with gain: Use the background noise estimate from Gaussnoise within a box roughly the size of the core of the PRF, and add the source photon noise calculated from the flux and the gain factor.
Each option is described below. The basic problem is that in the general case of a multi-source, multi-parameter fit, the true flux uncertainty limits lie in a multi-dimensional plane that is difficult to describe with a simple analytic expression. In addition, APEX was designed to handle undersampled data, and some simple expressions for flux uncertainty break down when the true center of the point source cannot be pinned down accurately. All of the following should be taken as rough estimates, and have no formal statistical significance. Uncertainties in background level or due to confusion are not included. Best estimates of true uncertainties come from simulations and/or multiple measurements.
0. No Photerr: This option is held over from previous versions of APEX. It has been shown to perform poorly for undersampled (IRAC channels 1 and 2) data. The noise is estimated by scaling the noise in the pixel nearest to the source by that pixel's contribution to the total "central" PRF. The central PRF is that corresponding to a center-of-pixel hit. The pixel noise should be from Gaussnoise, which produces images called, e.g. coadd_Tile_*_Image_noise.fits. Defining the factor, NP, as the ratio of the total central PRF volume to that under the peak pixel, the SNR is the extracted flux Fps for a point source divided by the scaled noise σ(xps,yps), at the peak pixel (xps,yps):
Equation 8.29
This calculation assumes only one PRF, the central one. It becomes poor when the pixels are large relative to the PRF width (e.g. IRAC channels 1 and 2 data), and generally gives an uncertainty estimate that is too low. It does not take into account the positional uncertainty. However, it is reasonable in background-limited, well-sampled cases.
1. Photerr, no gain: Simulations of Spitzer data have shown that a better estimate of the SNR is obtained by summing the data uncertainties within a box roughly the size of the core of the PRF. The box extends out to about 10% of the peak and was estimated from the simulations; no sub-pixel summing is done. For this option, the data uncertainties are used. The data uncertainties include the photon noise from the source itself, so this method is more general than the previous estimate, based solely upon the background. This is usually the best option.
2. Photerr, with gain: If the data uncertaintes are unavailable, or unreliable, then the source photon noise can be put in explicitly. This option brings APEX in line with noise estimates from other standard software (e.g. SExtractor; Bertin & Arnouts 1996).
In the case with gain, the noise for the SNR is:
Equation 8.30
where σi are the pixel uncertainties in the same box described above, F is the source flux in data units, and g is the gain (electrons per data unit). The final noise is converted to flux units. The input σi in this case should be the estimates obtained from the Gaussnoise module. The gain to use is the gain for a single image. It will take into account the coverage to estimate an effective gain. In the case of integration-time weighting, it should be for unit time.
8.9.3 Aperture Uncertainties
Aperture uncertainties are calculated in the Aperture module. It uses the data uncertainties (or "gain-estimated" ones, if enabled). The aperture flux uncertainty is a sum over the aperture area:
Equation 8.31
The sum takes into account fractional pixels. If any have bad uncertainties, ap_unc is set to -9.99. Uncertainty in the background subtraction is not included.
The results are written out in the aperture.tbl and extract_raw.tbl as "ap_unc1", "ap_unc2", etc. The APEX module, Select, can select on them when creating the final extract.tbl.
Uncertainties will only be calculated if the uncertainty file is found, and is the right size. If not found, or not the right size, Aperture Photometry will proceed but set the aperture uncertainties to -9.99.
It is possible to run Aperture Photometry by itself on an extract list, but if the extract list already contains aperture columns, then Aperture Photometry will append new columns.
’s are the optimal fitting parameters: 
