Photometric Flux Calibration
There is a separate page describing the conversion between SDSS magnitudes
and UBVRcIc, and ugriz colors of Vega and the
Sun.
Introduction
The objective of the photometric calibration process is to tie the
SDSS imaging data to an AB magnitude system, and specifically to the
"natural system" of the 2.5m telescope defined by the photon-weighted
effective wavelengths of each combination of SDSS filter, CCD
response, telescope transmission, and atmospheric transmission at a
reference airmass of 1.3 as measured at APO (see transmission curves for
SDSS 2.5m telescope).
The calibration process ultimately involves combining data from three
telescopes: the USNO 40-in on which our primary
standards were first measured, the
SDSS Photometric Telescope (or PT) , and the SDSS 2.5m telescope.
At the beginning of the survey it was expected that there would be a
single u'g'r'i'z' system. However, in the course of processing the
SDSS data, the unpleasant discovery was made that the filters in the
2.5m telescope have significantly different effective wavelengths from
the filters in the PT and at the USNO. These differences have been
traced to the fact that the short-pass interference films on the
2.5-meter camera live in the same vacuum as the detectors, and the
resulting dehydration of the films decreases their effective
refractive index. This results in blueward shifts of the red edges of
the filters by about 2.5 percent of the cutoff
wavelength, and consequent shifts of the effective
wavelengths of order half that. The USNO filters are in ambient air,
and the hydration of the films exhibits small temperature shifts; the
PT filters are kept in stable very dry air and are in a condition
about halfway between ambient and the very stable vacuum state. The
rather subtle differences between these systems are describable by
simple linear transformations with small color terms for stars of
not-too-extreme color, but of course cannot be so transformed for very
cool objects or objects with complex spectra. Since standardization is
done with stars, this is not a fundamental problem, once the
transformations are well understood.
It is these subtle issues that gave rise to our somewhat awkward
nomenclature for the different magnitude systems:
- magnitudes in the the USNO 40-in system are primed (u'g'r'i'z')
- magnitudes in the SDSS 2.5m system are unprimed (ugriz)
- magnitudes in the PT system only exist internally within the
Monitor Telescope Pipeline (mtpipe)
and have no official designation.
Previous reductions of the data, including that used in the EDR,
were based on inconsistent photometric equations; this is why we
referred to the 2.5m photometry with asterisks: u*g*r*i*z*. With the
DR1, the photometric equations are properly self-consistent, and we
can now remove the stars, and refer to u g r i z photometry with the
2.5m.
Overview of the Photometric Calibration in SDSS
The photometric calibration of the SDSS imaging data is a multi-step
process, due to the fact that the images from the 2.5m telescope
saturate at approximately r = 14, fainter than typical
spectrophotometric standards, combined with the fact that observing
efficiency would be greatly impacted if the 2.5m needed to interrupt
its routine scanning in order to observe separate calibration fields.
The first step involved setting up a primary standard
star network of 158 stars distributed around the Northern sky.
These stars were selected from a variety of sources and span a range
in color, airmass, and right ascension. They were observed repeatedly
over a period of two years using the US Naval Observatory 40-in
telescope located in Flagstaff, Arizona. These observations are tied
to an absolute flux system by the single F0 subdwarf star BD+17_4708,
whose absolute fluxes in SDSS filters are taken from
Fukugita et al. 1996 As noted above, the photometric system
defined by these stars is called the u'g'r'i'z' system. You
can look at the table containing
the calibrated magnitudes for these standard stars.
Most of these primary standards have brightnesses in the range r = 8 -
13, and would saturate the 2.5-meter telescope's imaging camera in
normal operations. Therefore, a set of 1520 41.5x41.5
arcmin2 transfer fields, called secondary patches,
have been positioned throughout the survey area. These secondary
patches are observed with the PT; their size is set by the field of
view of the PT camera. These secondary patches are grouped into sets
of four. Each set spans the full set of 12 scan lines of a survey
stripe along the width of the stripe, and the sets are spaced along
the length of a stripe at roughly 15 degree intervals. The patches
are observed by the PT in parallel with observations of the primary
standards and processed using the Monitor Telescope Pipeline (mtpipe).
The patches are first calibrated to the USNO 40-in
u'g'r'i'z' system and then transformed to the 2.5m
ugriz system; both initial calibration to the
u'g'r'i'z' system and the transformation to the ugriz
system occur within mtpipe. The ugriz-calibrated patches
are then used to calibrate the 2.5-meter's imaging data via the Final
Calibrations Pipeline (nfcalib).
The PT has two main functions: it measures the atmospheric extinction
on each clear night based on observations of primary standards at a
variety of airmasses, and it calibrates secondary patches in order to
determine the photometric zeropoint of the 2.5m imaging scans. The
extinction must be measured on each night the 2.5m is scanning, but
the corresponding secondary patches can be observed on any photometric
night, and need not be coincident with the image scans that they will
calibrate.
The Monitor Telescope Pipeline (mtpipe), so called for historical
reasons, processes the PT data. It performs three basic functions:
- it bias subtracts and flatfields the images, and performs
aperture photometry;
- it identifies primary standards in the primary standard
star fields and computes a transformation from the
aperture photometry to the primary standard star u'g'r'i'z' system;
- it applies the photometric solution to the stars in the
secondary patch fields, yielding u'g'r'i'z'-calibrated
patch star magnitudes, and then transforms these u'g'r'i'z'
magnitudes into the SDSS 2.5m ugriz system.
There is a document
describing photometric equations used in items 2 and 3 in detail .
The Final Calibration Pipeline
The final calibration pipeline (nfcalib) works much like mtpipe,
computing the transformation between psf photometry (or other
photometry) as observed by the 2.5m telescope and the final SDSS
photometric system. The pipeline matches stars between a camera
column of 2.5m data and an overlapping secondary patch. Each camera
column of 2.5m data is calibrated individually. There are of order
100 stars in each patch in the appropriate color and magnitude range
in the overlap.
The transformation equations are a simplified form of those used by mtpipe.
Since mtpipe delivers patch stars already calibrated to the
2.5m ugriz system, the nfcalib transformation equations have the following
form:
mfilter_inst(2.5m) = mfilter(patch) + afilter + kfilterX,
where, for a given filter, mfilter_inst(2.5m) is the
instrumental magnitude of the star in the 2.5m data [-2.5 log10(counts/exptime)],
mfilter(patch) is the magnitude of the same star in
the PT secondary patch, afilter is the photometric
zeropoint, kfilter is the first-order extinction
coefficient, and X is the airmass of the 2.5m observation. The
extinction coefficient is taken from PT observations on the same
night, linearly interpolated in time when multiple extinction
determinations are available. (Generally, however, mtpipe calculates
only a single kfilter per filter per night, so
linear interpolation is usually unnecessary.) A single zeropoint
afilter is computed for each filter from stars
on all patches that overlap a given CCD in a given run. Observations
are weighted by their estimated errors, and sigma-clipping is used to
reject outliers. At one time it was thought that a time dependent
zero point might be needed to account for the fact that the 2.5m
camera and corrector lenses rotate relative to the telescope mirrors
and optical structure; however, it now appears that any variations in
throughput are small compared to inherent fluctuations in the
calibration of the patches themselves. The statistical error in the
zeropoint is usually constrained to be less than 1.35 percent
in u and z and 0.9 percent in gri.
Assessment of Photometric Calibration
With Data Release 1 (DR1), we now routinely meet our requirements of
photometric uniformity of 2% in r, g-r, and r-i and of 3% in u-g and
i-z (rms).
This is a substantial improvement over the photometric uniformity
achieved in the Early Data Release (EDR), where the corresponding
values were approximately 5% in r, g-r, and r-i and 5% in u-g and i-z.
The improvements between the photometric calibration of the EDR and
the DR1 can be traced primarily to the use of more robust and
consistent photometric
equations by mtpipe and nfcalib and to improvements to the PSF-fitting algorithm and flatfield methodology in the Photometric Pipeline (photo).
Note that this photometric uniformity is measured based upon
relatively bright stars which are no redder than M0; hence, these
measures do not include effects of the
u band red leak (see caveats below) or the
model magnitude bug.
How to go from Counts in the fpC file to Calibrated
ugriz magnitudes?
First, note that all SDSS images ("corrected frames",
fpC*.fit files) have a "soft bias" of 1000 data numbers
(DN) added so they can be stored as unsigned integer. Secondly, since DR1,
the sky has not been subtracted from the corrected frames, but is
stored in the header keyword sky in units of DN. The
tsObj*.fit and related tables and the CAS database store
sky values as surface brightness (maggies/sq. arcsec., where one
maggie corresponds to 0 magnitudes/sq. arcsec).
Asinh and Pogson magnitudes
All calibrated magnitudes in the photometric catalogs are
given not as conventional Pogson astronomical
magnitudes, but as asinh
magnitudes. We show how to obtain both kinds of magnitudes from
observed count rates from the SDSS images and vice versa.
See further down for conversion of SDSS magnitudes to physical fluxes.
For both kinds of magnitudes, there are two ways to obtain the
zeropoint information for the conversion.
A little slower, but gives the final calibration and works
for all data releases
Here you first need the following information from the tsField
files:
aa = zeropoint
kk = extinction coefficient
and airmass
To get a calibrated magnitude, you first need to determine the
extinction-corrected ratio of the observed count rate to the
zero-point count rate:
- Convert the observed
number of counts to a count rate using the exposure time
exptime
= 53.907456 sec ,
- correct counts for atmospheric extinction using the
extinction coefficient
kk and the
airmass , and
- divide by the zero-point count rate, which is given by
f0 = 10-0.4*aa counts/second
both for asinh and conventional
magnitudes.
- In a single step,
f/f0 = counts/exptime * 100.4*(aa + kk *
airmass)
Then, calculate either the conventional ("Pogson") or the SDSS
asinh magnitude from f/f0 :
- Pogson
mag = -2.5 * log10(f/f0)
error(mag) = 2.5 / ln(10) * error(counts) /
counts
To get the error on the counts, see the note on computing count
errors below.
- asinh
mag =
-(2.5/ln(10))*[asinh((f/f0)/2b)+ln(b)]
error(mag) = 2.5 / ln(10) * error(counts)/exptime *
1/2b *
100.4*(aa + kk *
airmass) / sqrt(1 + [(f/f0)/2b]2) ,
where b is the softening parameter
for the photometric band in question and is given in the
table of b
coefficients below (for details on the asinh
magnitudes, see the paper by Lupton,
Gunn, and Szalay 1999 [AJ 118, 1406]).
asinh Softening Parameters (b coefficients)
Band | b | Zero-Flux Magnitude [m(f/f0 = 0)] | m(f/f0 = 10b) |
u | 1.4 × 10-10 | 24.63 | 22.12 |
g | 0.9 × 10-10 | 25.11 | 22.60 |
r | 1.2 × 10-10 | 24.80 | 22.29 |
i | 1.8 × 10-10 | 24.36 | 21.85 |
z | 7.4 × 10-10 | 22.83 | 20.32 |
Note: These values of the softening
parameter b are set to be approximately 1-sigma of the sky
noise; thus, only low signal-to-noise ratio measurements are affected
by the difference between asinh and Pogson magnitudes. The final
column gives the asinh magnitude associated with an object for which
f/f0 = 10b; the difference between
Pogson and asinh magnitudes is less than 1% for objects brighter than
this.
The calibrated asinh magnitudes are given in the tsObj
files. To obtain counts from an asinh magnitude, you first need to
work out f/f0 by inverting the asinh
relation above. You can then determine the number of counts from
f/f0 using the zero-point, extinction
coefficient, airmass, and exposure time.
The equations above are exact for DR1 and later
releases. Strictly speaking, for EDR photometry, the corrected counts
should include a color term cc*(color-color0)*(X-X0) (cf.
equation 15 in section 4.5 in the EDR paper), but it turns out
that generally, cc*(color-color0)*(X-X0) < 0.01 mag and the
color term can be neglected. Hence the calibration looks
identical for EDR and DR1.
Faster magnitudes via "flux20"
The "flux20" keyword in the header of the corrected frames
(fpC files) approximately gives the net number of
counts for a 20th mag object. So instead of using the zeropoint
and airmass correction term from the tsField file,
you can determine the corrected zero-point flux as
f/f0 = counts/(108 * flux20)
Then proceed with the calculation of a magnitude from
f/f0 as above.
The relation is only approximate because the final calibration
information (provided by nfcalib) is not available at the
time the corrected frames are generated. We expect the error
here (compared to the final calibrated magnitude) to be of order
0.1 mag or so, as estimated from a couple of test cases we have
tried out.
Note the counts measured by photo for each object are given
in the fpObjc files, as e.g., "psfcounts", "petrocounts", etc.
On a related note, in DR1 one can also use relations
similar to the above to estimate the sky level in magnitudes per
sq. arcsec (1 pixel = 0.396 arcsec). Either use the header keyword
"sky" in the fpC files, or remember to first subtract "softbias" (=
1000) from the raw background counts in the fpC files. Note the sky
level is also given in the tsField files. This note only
applies to the DR1 and later data releases. Note also that the calibrated
sky brightnesses reported in the tsField values have been
corrected for atmospheric extinction.
Computing errors on counts (converting counts to photo-electrons)
The fpC (corrected frames) and fpObjc
(object tables with counts for each object instead of magnitudes)
files report counts (or "data numbers", DN). However, it is
the number of photo-electrons which is really counted by the CCD
detectors and which therefore obeys Poisson statistics. The number of
photo-electrons is related to the number of counts through the gain
(which is really an inverse gain):
photo-electrons = counts * gain
The gain is reported in the headers of the tsField and
fpAtlas files (and hence also in the field
table in the CAS). The total noise contributed by dark current and
read noise (in units of DN2) is also reported in the
tsField files in header keyword
dark_variance (and correspondingly as
darkVariance in the field table in the CAS),
and also as dark_var in the fpAtlas
header.
Thus, the error in DN is given by the following expression:
error(counts) = sqrt([counts+sky]/gain + Npix*(dark_variance+skyErr)),
where counts is the number of object counts,
sky is the number of sky counts summed over the same
area as the object counts, Npix is the area covered
by the object in pixels, gain and
dark_variance and skyErr are the gain, dark
variance, and the error on the estimate of the average sky level in
the frame, respectively, from the corresponding tsField
file.
Conversion from SDSS ugriz magnitudes to AB
ugriz magnitudes
The SDSS photometry is intended to be on the AB system (Oke
& Gunn 1983), by which a magnitude 0 object should have the
same counts as a source of Fnu =
3631 Jy. However, this is known not to be exactly true, such that the
photometric zeropoints are slightly off the AB standard. We continue
to work to pin down these shifts. Our present estimate, based on
comparison to the STIS standards of Bohlin,
Dickinson, & Calzetti~(2001) and confirmed by SDSS photometry and
spectroscopy of fainter hot white dwarfs, is that the u
band zeropoint is in error by 0.04 mag, uAB =
uSDSS - 0.04 mag, and that g, r, and
i are close to AB. These statements are certainly not
precise to better than 0.01 mag; in addition, they depend critically
on the system response of the SDSS 2.5-meter, which was measured by
Doi et al. (2004, in preparation). The z band zeropoint is
not as certain at this time, but there is mild evidence that it may be
shifted by about 0.02 mag in the sense zAB =
zSDSS + 0.02 mag. The large shift in the
u band was expected because the adopted magnitude of the
SDSS standard BD+17 in Fukugita
et al.(1996) was computed at zero airmass, thereby making the
assumed u response bluer than that of the USNO system
response.
We intend to give a fuller report on the SDSS zeropoints, with
uncertainties, in the near future. Note that our relative
photometry is quite a bit better than these numbers would imply;
repeat observations show that our calibrations are better than 2%.
Conversion from SDSS ugriz
magnitudes to physical fluxes
As explained in the preceding section, the SDSS system is nearly an
AB system. Assuming you know the correction from
SDSS zeropoints to AB zeropoints (see above), you can turn the AB
magnitudes into a flux density using the AB zeropoint flux
density. The AB system is defined such that every filter has a
zero-point flux density of 3631 Jy (1 Jy = 1 Jansky = 10-26
W Hz-1 m-2 = 10-23 erg s-1
Hz-1 cm-2).
- To obtain a flux density from SDSS data, you need to work out
f/f0 (e.g. from the asinh magnitudes in
the tsObj files by using the inverse of the
relations given above). This number is
then the also the object's flux density, expressed as fraction of the
AB zeropoint flux density. Therefore, the conversion to flux
density is
S = 3631 Jy * f/f0 .
Then you need to apply the correction for the zeropoint offset
between the SDSS system and the AB system. See the description of SDSS to AB conversion
above.
Transformation Equations Between SDSS magnitudes and UBVRcIc
There is a separate page describing the conversion between SDSS magnitudes
and UBVRcIc, and ugriz colors of Vega and the
Sun.
Improved photometric calibration ("Übercal")
Ubercal is an algorithm to photometrically calibrate wide field
optical imaging surveys, that simultaneously solves for the
calibration parameters and relative stellar fluxes using overlapping
observations. The algorithm decouples the problem of relative
calibrations from that of absolute calibrations; the
absolute calibration is reduced to determining a few numbers for the
entire survey. We pay special attention to the spatial structure of
the calibration errors, allowing one to isolate particular error modes
in downstream analyses. Applying this to the Sloan Digital Sky Survey
imaging data, we achieve ~1% relative calibration errors across 8500
sq.deg. in griz; the errors are ~2% for the u band. These errors are
dominated by unmodelled atmospheric variations at Apache Point
Observatory. For a detailed description of ubercal, please see the Ubercal
paper (Padmanabhan et al. 2007, ApJ submitted [astro-ph/0703454]).
This improved calibration is available only through the ubercal
table in the CAS
Last modified: Mon Jun 25 22:01:30 CEST 2007
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