II. User's Guide to the WISE Preliminary Data Release

3. Atlas Images

f. Aperture Photometry Using the Atlas Images

First, we strongly advise you search the WISE Source Catalog for objects of interest, and use photometric measurements therein if available. Otherwise, for objects that didn't satisfy the catalog selection criteria, or for extended sources/emission where PSF-fit photometry is unreliable (see section IV.5.c.iv), we outline below a recommended method for performing aperture photometry with uncertainty estimation using the Atlas Image products. These are primarily the intensity and uncertainty maps. Particular attention is given to uncertainty estimation in the presence of correlated noise - a feature of the Atlas Images. The uncertainty estimation methodology can be applied to (by rescaling) the outputs from most photometry packages, including those that perform PSF-fit photometry. Here we summarize the main equations.

In general, the flux of a source (in image pixel units) within an aperture containing N_A pixels can be computed from:

(Eq. 1)

where F_tot is the sum of all pixel fluxes in the source (measurement) aperture and B is a robust measure of the local background level per pixel (robust against outliers and other contaminating sources, e.g., a mode, trimmed mean or trimmed median), usually estimated from an annulus around the source aperture. f_apcor is the aperture correction factor, whose equivalent in magnitudes is: m_apcor = -2.5log₁₀(f_apcor).

The corresponding 1-σ uncertainty in F_src (Eq. 1) is given by the square root of the following noise-variance expression:

(Eq. 2)

where

N_A = number of pixels in source aperture.

N_B = number of pixels in source annulus.

σ_i = flux uncertainty for pixel i from uncertainty map, rescaled if necessary to ensure consistency with local RMS fluctuations at fixed depth-of-coverage as described in section II.3.e.i.

F_corr = correlated noise correction factor for flux variance (see section II.3.f.i below).

B = robust background estimate per pixel in annulus (e.g., trimmed mean or trimmed median).

k = 1 if B = mean background/pixel, or mean-related measure; k = π/2 if B = median background/pixel, or median-related measure.

σ²_B/pix = Variance in sky-background annulus. Can estimate from the squared median or trimmed mean of σ_i values over pixels i in annulus off the uncertainty map. In regions with low confusion and a spatially uniform background, can also compute from the square of some robust estimator of the RMS fluctuation, e.g., using inter-percentile ranges: ≈ [0.5(P_84% - P_16%)]² ≈ (P_50% - P_16%)², or the Median Absolute Deviation from the median (MAD): [1.4826median{|p_i - median{p_i}|}]².

σ²_conf = Confusion noise-variance on scale represented by measurement aperture (see section II.3.f.ii below).

Note that we have assumed the uncertainty in the aperture correction (f_apcor) is negligible since usually it can be determined to good accuracy. The aperture corrections for a limited set of aperture sizes used on Atlas Images in constructing the WISE source catalog can be found here. For WISE Atlas Image (co-add) products, the pixel units are in DN. Equations (1) and (2) will therefore respectively give source fluxes and uncertainties in units of DN. To convert to calibrated magnitudes (relative to WISE standards) with corresponding uncertainties, the following equations should then be used:

(Eq. 3)

where F_src and σ_src are computed using Equations (1) and (2) and have units of DN. The "1.179" factor comes from (2.5/log_e[10])². MAGZP is the magnitude zero-point with corresponding uncertainty σ_MAGZP. These are represented by FITS keywords MAGZP and MAGZPUNC respectively in the headers of Atlas Images. The values for MAGZP and MAGZPUNC are the same for all Atlas Images of a given band (see Table 1). This is because all input single exposure frames were throughput (gain)-matched to a common relative photometric zero-point. The input frame photometric zero points (section IV.3.g.iii) were observed to be stable and the MAGZPUNC values reflect their scatter from internal repeatability in relative photometry. Absolute flux densities (e.g., in Jy) can be computed from: f = f₀10^-0.4MAG, where f₀ is the absolute flux for MAG = 0 for your band of interest (see also Table 1). Alternatively, you could simply apply a DN-to-Jy conversion factor to the outputs from Eqs (1) and (2) directly (more precisely, to the square root of Eq. 2). These factors are given by f₀10^-0.4MAGZP and summarized in Table 1.

1. Estimated from the internal repeatability in relative photometry of WISE calibration standards. Does not include absolute calibration errors used to derive the baseline single-exposure MAGZP values. For the latter, see section IV.3.g.vii.
2. From Table 4 in section IV.3.g.v. These are specific to the effective wavelengths and spectral energy distribution (SED) described therein [see also note 4 below].
3. Computed from f₀10^-0.4MAGZP.
4. Beware that these calibration factors will depend on the SED of your sources or emission of interest. Color corrections are given in section IV.3.g.vi (Table 6 therein).
5. Even though the factors in Table 1 were mostly derived using point/compact-source photometry, they are also appropriate for calibrating extended/diffuse emission that can be measured in-excess of the local background level in an image.

i. Correlated Pixel Noise

Correlated noise usually occurs in re-sampled and interpolated images with the degree of correlation depending on the extent of the interpolation kernel and ratio of input-to-output pixel size (for details, see section IV.5.a.vii). The smoothing (interpolation) kernel used to create a WISE Atlas Image has moved noise-power from high to low spatial frequencies and therefore needs to be recaptured to properly quantify the uncertainty in the flux summed over a region. Ignorance of correlated-noise will lead to an underestimate of the final photometric uncertainty, sometimes by an appreciable amount. This is the purpose of the correlated-noise correction factor F_corr in Eq. (2).

Since the Atlas Images were created using the detector PRF as the interpolation kernel (section IV.5.a.vii), it can be shown that F_corr ≈ N_p(s_in/s_out)² where N_p is the effective number of noise-pixels characterizing the PRF, in terms of the number of native detector pixels, and s_in/s_out is the ratio of input (detector) pixel scale to output (Atlas Image) pixel scale. This ratio is ≈2 for W1,W2,W3, and ≈4 for W4. This is a good approximation for apertures containing N_A >~ 50N_p pixels. For significantly smaller apertures, F_corr is smaller and can be approximated to ~10% accuracy using Figures 1-4 for a given aperture radius measured in co-add (Atlas Image) pixels. These figures apply only to the WISE-band PRFs and output co-add pixel size (1.375 arcsec), and were derived using Monte-Carlo simulations of co-added frames with input Gaussian noise.

Figure 1 - Fcorr vs aperture radius for W1 [click to enlarge].	Figure 2 - Fcorr vs aperture radius for W2 [click to enlarge].
Figure 3 - Fcorr vs aperture radius for W3 [click to enlarge].	Figure 4 - Fcorr vs aperture radius for W4 [click to enlarge].

ii. Confusion Noise term

The confusion-noise term in Eq. (2), σ_conf, represents the contribution to the image-noise from the superposition of faint and unresolved sources on scales equivalent to the measurement aperture. Confusion noise becomes important in regions of high source density, typically when the surface density down to the instrumental and background-Poisson noise limit for the image at hand exceeds ~1/30 sources beam^-1 (Condon, 1974), although this drops to ~1/50 sources beam^-1 when the number-flux relation is Euclidean or steeper (e.g., Hogg, 2001). Taking a "beam" to be the solid angle spanned the FWHM of the effective PSF for WISE Atlas Images (~8.5 arcsec in bands W1,W2,W3, and ~17 arcsec in W4), the 1/30 beam^-1 criterion translates to ~2 sources arcmin^-2 for W1,W2,W3, and ~0.5 sources arcmin^-2 for W4.

The confusion noise component is measured as an excess above any correlated instrumental and background-Poisson pixel-noise on these scales. It can be derived empirically from the background image RMS fluctuation on measurement-aperture scales if one has prior knowledge the instrumental and background-Poisson components in their region at the given depth-of-coverage (i.e., as provided by the uncertainty maps). For example, if σ_apbck is the background RMS fluctuation on measurement-aperture scales, we have

(Eq. 4)

where <σ²_i>_bck is a robust estimate (e.g., median or mode) of the noise-variance per pixel over the background region of interest from the uncertainty map. Note: the noise-variance is the square of pixel values in this map. N_A is the number of pixels in your measurement aperture. The estimation of σ_aptot involves throwing many apertures of size N_A at random, summing the pixel flux in each (to yield a flux of F_Aj for aperture j), and then retaining those apertures that (i) don't significantly overlap with another (so spatial correlations are minimized), and (ii) fall within regions with a relatively uniform, source-free background. σ_aptot is then the RMS over all remaining F_Aj values. The difficulty here is having enough random samples (apertures) that are not significantly contaminated and cover a spatially-uniform background.

Alternatively, the confusion noise can be predicted using star or galaxy count models for your field and knowledge of the WISE beam or measurement aperture (e.g., Väisänen et al. 2001). Note that the Atlas Image derived confusion noise and its comparison to predictions from star/galaxy count models for WISE has not yet been studied in detail.

Last update: 2012 January 24