The 1´´ pixel 2MASS Atlas Images (a.k.a. coadds) are derived from 2´´ pixel image frames. One must therefore correct for the resampling as well as the smoothing from the co-addition kernal. The following formulas include these corrections within the standard framework of the Poisson error modeling. Much of this work was carried out by Tom Chester during the 2MASS Prototype Observations phase of the survey (see Relevant Links for hyper-links to the gruesome details).
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Formal Signal-to-Noise
The calculation of the SNR is straightforward working from the individual frames that are averaged together to produce the coadd. If one considers only the individual frames, which are independent, and ignores the uncertainty due to the background fit (which will be considered below), the SNR is:In practice the gain and read-noise parameters are a function of the 2MASS hardware period and are not easily incorporated into the photometric uncertainties (see prescription in the Additional Sources of Uncertainty section below). The most robust method toward estimating photometric uncertainties is to use the measured RMS in the frame or coadd background (after large-scale background removal). The noise term is
Coadd Noise
As mentioned above, the pixels in the coadd are correlated since the sampling of the coadd is two times higher than the frame sampling and because of the smoothing function used to produce the coadd flux estimates. The coadd noise is a function of the background Poisson noise, the read-noise and high-frequency structure (e.g., "airglow"). The coadd noise, sigmac, must be corrected for the resampling and smoothing arising from the co-addition process. The typical number (Nc) of frames that are coadded is six. The resampling is carried out with a dither-optimized kernal, resulting in a slight smoothing of the data (see the 2MASS Explanatory Supplement). The coadd RMS noise can be written as
We can now re-write the coadd signal-to-noise using the measured coadd noise
Parametric Values
Example
Consider a 13th mag galaxy, whose size is 15´´ in circular diameter. Let the measured RMS of the background-removed coadd be 1.1 dn.
Uncertainty Due to the Background Fit
For extended sources, the coadd background is removed using a cubic-polynomial surface. Details of the background removal are given in the 2MASS Extended Source Catalog explanatory supplement.
The noise of the background determination depends on the structure of the background. For example, if there is no variation in background across the entire coadd, then all cubic polynomial coefficients will be zero except for the mean level, which will be determined from the mean of all the 8´ x 8´ median sections. The error in the determination of that mean is ~0.6% of the coadd noise (see Analysis of Photometric Noises for 2MASS Galaxies for additional details).
If the background variation requires a cubic, and is well-fit by a cubic, which is normally the case except for H-band observations with high airglow, then intuitively the coadd is effectively divided into 4 x 4 regions with the resulting errors then being 4 times higher. The resulting noise in the determination of the background is then
~2.4% of the coadd noise. If the background variation requires a higher-power polynomial than a cubic, then the error in the background will no longer be determined by the pixel noise. Instead, the background error will result from the residual error after a cubic polynomial is fit. This will be the case for airglow variation, which can vary on high spatial frequency scales. See Photometric Error Tree for more details.
We assume a net 2.4% of the coadd noise per pixel uncertainty in the background fit. The noise term is thus
Note that this term is negligibly small for most 2MASS galaxies, whose diameters are typically less than 20´´. For large galaxies, however, this term begins to dominate the S/N. Adding this term to the coadd signal-to-noise equation, we have
For large galaxies this noise term dominates to the total flux uncertainty. This is demonstrated in the plots below, the first showing a very bright galaxy (similar to NGC891) with an exponential surface brightness profile and the second a bright galaxy more typical to the 2MASS survey. The radial profile follows f0exp(-r/alpha). The total noise delta_mag is shown with and without this noise correction. Note that the radial size limit for catalogued 2MASS galaxies is ~120´´, while for the Large Galaxy Atlas there is no size limit.
Additional Sources of Uncertainty
Airglow & Background Structure
The coadd noise is also a function of high-frequency structure such as "airglow" that is not removed by the background-fitting process. Airglow presents structure on all size scales, increasing both the background level and the RMS across the coadd. We can quantify the airglow structure by comparing the measured coadd noise with the expected coadd noise due to Poisson effects. The expected S/N is given by the first equation in this document. Adapting the S/N to a relative (noise, background coefficients) scale, the expected noise is
The gain and read noise are constants that change with 2MASS hardware period (a hardware period is defined to be the time during which the survey was carried out with a particular hardware/instrumentation configuration). The coefficients and gain/readnoise as a function of hardware period are given in Coadd Noise Parameters.
If the measured noise is significantly higher than the expected coadd noise, then it is likely that airglow is present. In that case, the flux measurements will have an additional error that cannot be characterized by solely adding in a noise due to airglow. This is because most 2MASS galaxies are small enough so that the airglow "noise" is constant over their aperture, leading to a direct flux bias. For a rough measurement of this effect, see H Photometric Error Due To Airglow.
Confusion Noise
Confusion arises from the appearance of an interloping star within close proximity to the "beam" or point spread function pattern that induces a significant flux bias (or "deflection") from the non-contaminated flux due to the intended source (in our case, galaxy) near the beam center. A discussion of the confusion noise relevant to the 2MASS survey is given in Stellar Number Density and Confusion Noise. A plot of the equivalent confusion noise in surface brightness units (delta_mag per arcsec^2) as a function of the stellar number density is given here.
Handy Equations
