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Algorithm Descriptions

# The model magnitude

Important Note for EDR and DR1 data ONLY:Comparing the model (i.e., exponential and de Vaucouleurs fits) and Petrosian magnitudes of bright galaxies in EDR and DR1 data shows a systematic offset of about 0.2 magnitudes (in the sense that the model magnitudes are brighter). This turns out to be due to a bug in the way the PSF was convolved with the models (this bug affected the model magnitudes even when they were fit only to the central 4.4" radius of each object). This caused problems for very small objects (i.e., close to being unresolved). The code forces model and PSF magnitudes of unresolved objects to be the same in the mean by application of an aperture correction, which then gets applied to all objects. The net result is that the model magnitudes are fine for unresolved objects, but systematically offset for galaxies brighter than at least 20th mag. Therefore, model magnitudes should NOT be used in EDR and DR1 data. This problem has been corrected as of DR2.

Just as the PSF magnitudes are optimal measures of the fluxes of stars, the optimal measure of the flux of a galaxy would use a matched galaxy model. With this in mind, the code fits two models to the two-dimensional image of each object in each band:

1. a pure deVaucouleurs profile:
I(r) = I0exp{-7.67[(r/re)1/4]}
(truncated beyond 7re to smoothly go to zero at 8re, and with some softening within r=re/50.

2. a pure exponential profile
I(r) = I0exp(-1.68r/re)
(truncated beyond 3re to smoothly go to zero at 4re.

Each model has an arbitrary axis ratio and position angle. Although for large objects it is possible and even desirable to fit more complicated models (e.g., bulge plus disk), the computational expense to compute them is not justified for the majority of the detected objects. The models are convolved with a double-Gaussian fit to the PSF, which is provided by psp. Residuals between the double-Gaussian and the full KL PSF model are added on for just the central PSF component of the image.

These fitting procedures yield the quantities

• r_deV and r_exp, the effective radii of the models;
• ab_deV and ab_exp, the axis ratio of the best fit models;
• phi_deV and phi_exp, the position angles of the ellipticity (in degrees East of North).
• deV_L and exp_L, the likelihoods associated with each model from the chi-squared fit;
• deVMag and expMag, the total magnitudes associated with each fit.

Note that these quantities correctly model the effects of the PSF. Errors for each of the last two quantities (which are based only on photon statistics) are also reported. We apply aperture corrections to make these model magnitudes equal the PSF magnitudes in the case of an unresolved object.

In order to measure unbiased colors of galaxies, we measure their flux through equivalent apertures in all bands. We choose the model (exponential or deVaucouleurs) of higher likelihood in the r filter, and apply that model (i.e., allowing only the amplitude to vary) in the other bands after convolving with the appropriate PSF in each band. The resulting magnitudes are termed modelMag. The resulting estimate of galaxy color will be unbiased in the absence of color gradients. Systematic differences from Petrosian colors are in fact often seen due to color gradients, in which case the concept of a global galaxy color is somewhat ambiguous. For faint galaxies, the model colors have appreciably higher signal-to-noise ratio than do the Petrosian colors.

Due to the way in which model fits are carried out, there is some weak discretization of model parameters, especially r_exp and r_deV. This is yet to be fixed. Two other issues (negative axis ratios, and bad model mags for bright objects) have been fixed since the EDR.

Caveat: At bright magnitudes (r <~ 18), model magnitudes may not be a robust means to select objects by flux. For example, model magnitudes in target and best imaging may often differ significantly because a different type of profile (deVaucouleurs or exponential) was deemed the better fit in target vs. best. Instead, to select samples by flux, one should typically use Petrosian magnitudes for galaxies and psf magnitudes for stars and distant quasars. However, model colors are in general robust and may be used to select galaxy samples by color. Please also refer to the SDSS target selection algorithms for examples.