Flux estimates for faint sources or transients are systematically biased high because there are far more truly faint sources than bright. Corrections which account for this effect are presented as a function of signal-to-noise ratio and the (true) slope of the faint-source number-flux relation. The corrections depend on the source being originally identified in the image in which it is being photometered. If a source has been identified in other data, the corrections are different; a prescription for calculating the corrections is presented. Implications of these corrections for analyses of surveys are discussed; the most important is that sources identified at signal-to-noise ratios of three or less are practically useless

Hogg, D W

Turner, E L

Flux estimates for faint sources or transients are systematically biased high because there are far more truly faint sources than bright. Corrections that account for this effect are presented as a function of signal‐to‐noise ratio and the (true) slope of the faint‐source number‐flux relation. The corrections depend on the source being originally identified in the image in which it is being photometered. If a source has been identified in other data, the corrections are different; a prescription for calculating the corrections is presented. Implications of these corrections for analyses of surveys are discussed; the most important is that sources identified at signal‐to‐noise ratios of 4 or less are practically useless

Hogg, David W.

Turner, Edwin L.

Caltech Authors - Main

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A maximum-likelihood method to improve faint source flux and color estimates
David W. Hogg1,2 & Edwin L. Turner2,3
ABSTRACT
Flux estimates for faint sources or transients are systematically biased high because
there are far more truly faint sources than bright. Corrections which account for
this effect are presented as a function of signal-to-noise ratio and the (true) slope of
the faint-source number-flux relation. The corrections depend on the source being
originally identified in the image in which it is being photometered. If a source has
been identified in other data, the corrections are different; a prescription for calculating
the corrections is presented. Implications of these corrections for analyses of surveys
are discussed; the most important is that sources identified at signal-to-noise ratios of
four or less are practically useless.
1. Introduction
Given a noisy photometric measurement of a very faint source, what is the best estimate of
its true flux?
The best answer to this question is “I don’t know—integrate longer to reduce the noise!”
However, in some cases, this is not possible. For example, the ultra-deep images of the Hubble
Deep Field (Williams et al 1996) represent so much HST observing time that in practice they
cannot be much improved. From the ground they cannot even in principle be improved because
any ground-based images significantly deeper than existing ones (e.g., Djorgovski et al 1995;
Metcalfe et al 1995; Smail et al 1995; Hogg et al 1997a, 1997b) would be totally confusion-limited
(e.g., Condon 1974). As another example, observations of transients, such as gamma-ray bursts or
supernovae, cannot be improved because they cannot be repeated, even in principle.
Given that in some cases deeper imaging is not an option, the reason the question does not
have a trivial answer is that the number counts of faint sources tend to rise with decreasing flux,
so more sources are available for “upscattering” to a given measurement than are available for
“downscattering.” A familiar analogy is with trigonometric parallaxes, where low signal-to-noise
ratio measurements are biased large, since given any observed parallax πo and associated error,
1 Theoretical Astrophysics, California Institute of Technology
2 School of Natural Sciences, Institute for Advanced Study, Olden Lane, Princeton NJ 08540; hogg@ias.edu
3 Princeton University Observatory, Princeton University, Peyton Hall, Princeton NJ 08544;
elt@astro.princeton.edu
– 2 –
there is a finite range of true parallaxes π consistent with it, but there are far more sources in
the sky with small parallaxes π < πo than large π > πo. The “Lutz–Kelker” corrections which
account for this are easy to compute and apply (Lutz & Kelker 1973; Hansen 1979); they have
been essential in providing unbiased distances in astronomy.
There is a conceptually similar set of corrections for low signal-to-noise ratio measurements
of faint source fluxes. In this article, these corrections are computed and discussed. As in the
case of parallaxes, the corrections depend on how the sources are selected, and on the intrinsic
distribution of the measured quantity, in this case the true number-flux relation. Unfortunately
the number-flux relation is not exactly known in most cases of interest, since the faint source
photometry is usually being performed in order to determine this very relation! Furthermore,
in many cases of interest, the correction presented here does not represent the largest source
of systematic error. However, unlike the other sources of error, this correction applies to all
flux-selected sources, independent of instrumentation or analysis technique.
One note of terminology: The systematic bias discussed here is often improperly referred to
as the “Malmquist bias.” The Malmquist bias is the effect that in a flux-limited sample, there is a
larger-than-representative fraction of high-luminosity sources because they can be seen to greater
distances and hence over a larger volume (Malmquist 1924; Mihalas & Binney 1981). It is due to
the intrinsic scatter in source luminosities. Malmquist bias is removed, e.g., when one computes a
luminosity function from star counts. It does not involve any kind of measurement error; it does
not go away if one obtains more precise photometry! The bias corrected-for here results from the
observational scatter in fluxes; the measurement errors. It does indeed go away when the fluxes
are re-measured at much higher precision; it only needs to be considered when low signal-to-noise
data are being used. What is discussed in this paper is closely related to Eddington bias, the effect
of low signal-to-noise flux measurements on faint source number-magnitude relations. Statistical
corrections to observed number-magnitude relations are computed by Eddington (1913); flux
corrections for individual survey sources are computed here.
2. Computation of corrections
Consider the simplest case, in which a source is being photometered in the image in which
it was first detected. That is, it is being measured in the data in which it was selected. The
likelihood p(S|So) (probability per unit flux) that a source has true flux S given that it is observed
to have flux So is related to the likelihood p(So|S) that it is observed to have So when it has S by
Bayes’s theorem
p(S|So) ∝ p(So|S) p(S) (1)
where a proportionality is used because the normalization is being ignored (for now) and p(S)
(probability per unit flux) is the true underlying distribution of fluxes, given by the (true, not
observed) number-flux relation. If the number of sources N(< m) brighter than magnitude m as a
– 3 –
function of m is a power law
2.5
d logN
dm
= −
d logN
d log S
= q (2)
then the conditional probability becomes
p(S|So) ∝
(
S
So
)
−(q+1)
exp
[
−
(S − So)
2
2σ2
]
(3)
where it is assumed that the observational error is gaussian-distributed and σ is the uncertainty
in the observed flux So, or So/σ is the signal-to-noise ratio r. Figure 1 shows these likelihood
curves for number-flux exponent q = 2.0, 1.5, 1.0 and 0.5 and signal-to-noise ratios r = 3, 5
and 10. This Figure demonstrates that measurements at a signal-to-noise ratio of three do not
strongly constrain the true flux, whatever the slope of the number counts, but particularly if the
counts have the Euclidean4 slope of q = 1.5 (or greater). It is worth emphasizing that the above
equation and the curves plotted in Figure 1 are essentially identical to those computed for parallax
corrections (Lutz & Kelker 1973; Hansen 1979) except that the parallax corrections are computed
for only one particular exponent value.
If the flux measurement was unbiased, the peak in the likelihood function p(S|So) would be
at S/So = 1. However, taking the derivative dp/dS, it is found that the maximum-likelihood true
flux SML is in fact
SML
So
=
1
2
+
1
2
√
1−
4 q + 4
r2
(4)
where q is the number-magnitude exponent defined above and r is the signal-to-noise ratio. There
is no finite maximum-likelihood value at all if r2 < 4 q+4; an example is the q = 1.5, r = 3 curve in
Figure 1. The above equation specifies a correction which should in principle be applied to all flux
measurements in a flux-limited sample. When the signal-to-noise is good enough (r2 ≫ 4 q + 4)
the correction can be approximated as
SML
So
≈ 1−
q + 1
r2
when r2 ≫ 4 q + 4 (5)
or in terms of the magnitude correction ∆m ≡ mML −mo
∆m ≈
1.086 q + 1.086
r2
when r2 ≫ 4 q + 4 (6)
Things change slightly if the likelihood is computed in the magnitude (i.e., log flux rather
than flux) domain; after all, maximum-likelihood techniques are sensitive to the “metric” of the
4Another note on terminology: What is called the “Euclidean” slope really ought to be called the “no-evolution,
non-expanding” slope, because even in a Euclidean space, the number counts have q 6= 1.5 at large distance if either
the Universe is expanding or the sources are evolving.
– 4 –
space in which the likelihood is computed. When computed purely in the magnitude or log-flux
domain, the correction is
∆m ≈
1.086 q + 2.171
r2
when r2 ≫ 4 q + 8 (7)
The fact that the solution depends on the type of data space (log or linear) demonstrates that
the specific value of the correction is not completely specified, because it only provides a “best
guess” (a subjective estimate) for the true flux. It is worthy of note that the Lutz-Kelker parallax
corrections are similarly subjective, as are essentially all statistical estimators.
More robust than maximum-likelihood estimates are confidence intervals, because these do
not depend on the choice of “metric.” Confidence intervals are found by integrating the likelihood
curves. Unfortunately, the areas under the curves shown in Figure 1 do not converge; the
likelihood distributions are not normalizable! This non-normalizability comes from the divergence
of p(S|So) as S → 0 (not visibile in some of the curves in Figure 1 simply because at high r
the divergence happens at very small S/So). There are two respects in which this divergence or
non-normalizability is unphysical: First, there cannot be an infinite number of sources in the
visible Universe; there aren’t even an infinite number of atoms in the Universe! Secondly, most
ultra-deep images of the sky, including the HDF, are close to their confusion limits, beyond which
the observed number counts have to “cut off” no matter how much integration time is employed.
Neither of these effects can be simply taken into account in general; they depend on the data
quality and the sources under study.
The equations in this Section have assumed that observational errors are gaussian-distributed,
which is not true for all photometric measurements. The equations are easily generalized (although
they do not necessarily remain analytic) with the gaussian in equation (3) replaced by whatever
error distribution is appropriate for the measurement in question.
3. An empirical test
The correction can be tested with any imaging data in which the number-flux relation is
known. Here, the HST HDF data in the F606W (0.6 µm) bandpass are used. Noise was added
to the 1024 × 1024 “Version 2” mosiacs of the HST images of the HDF (Williams et al 1996) to
make the pixel-to-pixel sky noise ten times as bad as in the original mosaics. The higher-noise
mosaics will be referred to as the “bad” images and the originals as the “good” images. A catalog
of sources was chosen in the bad images down to very faint levels using the “SExtractor” source
detection package (Bertin & Arnouts 1996) in essentially its default mode: Smooth with a 2-pixel
FWHM triangular filter and select take sources whose central pixel in the smoothed image is above
a given threshold. These sources were then photometered with the NOAO “IRAF” software in
matched 0.16 arcsec (2 pixel) diameter apertures in both the bad and good images. The bad/good
flux ratios are plotted against signal-to-noise ratio in Figure 2 along with the expected correction
– 5 –
computed with equation (4) and the (known) count slope q = 0.5 (Williams et al 1996). The
correction does very well down to signal-to-noise ratios r ≈ 3. At r < 4, a significant number
of spurious (zero-flux in the good image) sources start to appear. Figure 2 shows, as with the
Lutz-Kelker corrections, that the corrections are on the same order as the intrinsic scatter due to
measurement error, so some sources have under- rather than overestimated fluxes. However the
correction is still necessary if an unbiased estimator of the true flux is desired.
4. Changing the selection technique
The next case to consider is photometry of a source in one image (say I-band) after it is
detected (and its position is known) in another image (say V -band). In this case, Bayes’s theorem
is still used but for p(S), the true distribution of V − I colors is used rather than the I-band
number counts. Actually, rather than the color distribution, it is better to think of the conditional
I-band flux distribution p(S(I)|S(V )) (probability per unit flux) given that the V -band flux S(V )
is known (in what follows it is assumed that the V -band detection is at very high signal-to-noise
ratio so the V -band flux is well known). Because, unlike the number counts, these conditional
distributions are not generally power laws, the flux correction depends not only on the shape of
the distribution but where in the distribution the observed flux So lies.
Fortunately, when the signal-to-noise ratio r is large enough, it is possible to linearize Bayes’s
formula around the observed flux S
(I)
o so only the local power-law slope
Q ≡ −
d log p(S(I)|S(V ))
d log S(I)
∣
∣
∣
∣
∣
S
(I)
o
(8)
of the distribution of source fluxes is important. The likelihood function (probability per unit flux)
for the true I-band flux S(I) given the observed flux S
(I)
o and the known V -band flux S(V ) is then
p(S(I)|S(I)o , S
(V )) ∝
(
S(I)
S
(I)
o
)
−(Q+1)
exp
[
−
(S(I) − S
(I)
o )2
2σ2
]
(9)
which leads to the maximum-likelihood correction
S
(I)
ML
S
(I)
o
≈ 1−
Q+ 1
r2
when r2 ≫ 4Q+ 4 (10)
or in terms of the magnitude correction ∆m ≡ mML −mo
∆m ≈
1.086Q + 1.086
r2
when r2 ≫ 4Q+ 4 (11)
Note that the correction can be positive or negative, depending on the sign of the local slope Q.
Corrections applicable when more complicated selection procedures have been used can be
computed in analogous ways.
– 6 –
5. Summary and discussion
Maximum-likelihood corrections for faint source flux measurements have been computed
for the case in which the sources are measured at low signal-to-noise in the data in which they
were originally selected. It is found that since the number-flux relation tends to be rising at the
faint end, the low signal-to-noise flux measurements are usually overestimates of the true flux.
At signal-to-noise ratios r < 4, flux measurements (of this type—i.e. in the data in which the
sources were selected) are almost meaningless because they are consistent with almost any true
flux between zero and the measured value.
The bias considered here tends to “steepen” measured number-flux relations at the faint
end; i.e. the measured d logN/d log S is more negative than the true value because the very
numerous faint sources are scattered up to brighter levels. This effect is only significant at very
faint levels, where it is usually mitigated or in fact canceled out by incompleteness. The best way
to correct measured number-flux relations for both the flux bias and incompleteness is to perform
full completeness simulations, which, if done correctly, will account for both effects simultaneously
(e.g., Smail et al 1995; Hogg et al 1997b), and of coures a full accounting for all systematic errors
requires detailed modeling of every stage in the observing and analysis procedures. Although the
corrections presented here do not comprehensively account for most of these systematic biases,
they are very general, improving flux estimates for individual sources independent of observational
technique.
These corrections ought to be applied to the source fluxes at the faint end of the catalogs
from all huge (and therefore difficult-to-improve-upon) surveys, such as the Palomar Observatory
Sky Surveys, the Infrared Astronomical Sattelite (IRAS) survey and from future huge surveys
such as the 2-Micron All Sky Survey and the Sloan Digital Sky Survey. In fact the IRAS catalogs
were corrected at the faint end for some related biases but not this bias per se (Beichman et al
1988). Also, all transients discovered at low signal-to-noise ratios in transient searches, such as
faint gamma-ray bursts, of which no additional measurements can be made after the fact, should
have these corrections applied.
We thank Roger Blandford, John Gizis, Gerry Neugebauer, Neill Reid and Yun Wang for
useful discussions, and the HDF team for planning, taking, reducing and calibrating the HDF
data. Some financial support was provided by NSF grant AST–9529170.
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Washington
– 7 –
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This preprint was prepared with the AAS LATEX macros v4.0.
– 8 –
Fig. 1.— Likelihood curves for several number-magnitude exponents q and signal-to-noise ratios
r = 3 (worst), 5 (middle) and 10 (best).
– 9 –
Fig. 2.— The ratios of flux measurements in the “bad” image, in which sources in the HDF were
detected, to the flux measurements in the “good” image. The bad image is simply the good image
plus additional noise (see text). The solid line is the expected ratio SML/So of the maximum-
likelihood flux to the observed flux given by equation (4) for number-magnitude exponent q = 0.5
(see text).


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A Maximum Likelihood Method to Improve Faint‐Source Flux and Color Estimates

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A maximum-likelihood method to improve faint source flux and color estimates

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