6.1 Magnitudes and Fluxes
The steep luminosity function and the rapid increase in the
source counts with magnitude means that estimates of surface densities,
and hence space densities, are sensitive to both errors in the
determination of the mean flux limit and to the dispersion in
individual flux measures. At bright apparent magnitudes, where
log (N) 
0.8
m, errors in the mean flux limit of 0.25,
0.5 and 1.0 mag produce corresponding errors in the integral number
density at m ~ 18 of factors 1.6, 2.5 and 6.5
respectively. Achieving the precision of photometry required, 
0.1 mag, over areas of the tens of square degrees required to define the
optical
source counts to high precision has become routine only in the last
five years. The steep slope of the source counts also results in a
significant Eddington (1913) bias when errors in flux determinations of
individual objects are substantial. In other words, while the mean flux
limit may be determined to very high precision, if a scatter in the
individual flux estimates is present, then a bias in the number of
objects detected will occur. The rising source counts result in more
objects intrinsically fainter than the flux limit being scattered into
the sample than intrinsically brighter objects being scattered out,
producing an overestimate of the true source counts. For number counts,
log (N) 0.8
m, and assuming that the photometric errors are
Gaussian with dispersion 
, overestimations of the surface
density of ~ 2%, ~ 11% and ~ 50% result for
observations characterized by photometric errors of = 0.1,
0.25 and 0.5 magnitudes, respectively.
The characteristics of the photometric errors must be known and incorporated into the calculation of the selection function in order to derive accurate surface densities. The effects of Eddington bias are equally important when samples are selected according to a property, such as emission line equivalent width, with an intrinsic distribution which increases steeply as a function of detection threshold.
Photometry of quasars has been
acquired using a wide variety of magnitude systems, and intercomparison
of results from surveys is desirable and necessary
(Section 2.1) and requires
the conversion of magnitudes from one system to another. Mean
transformations between the majority of color systems are
well-determined for stars, whose spectra are well-approximated by
blackbody-like energy distributions, but objects with spectra
exhibiting significant discontinuities and features that extend over a
substantial fraction of the passbands employed present greater
difficulties. Care is required, for example, when intercomparing
``R'' magnitudes of late-type M stars where prominent molecular
bands and steep gradient in flux as a function of wavelength mean that
particular passbands show large differences in effective wavelength
(Bessell
1986). Quasars, with their large equivalent width emission and
(BAL) absorption lines, and spectral discontinuities at ~ 1216 Å
and ~ 912 Å, present a more complex problem still. The strength
of the features varies significantly from object to object and the
spectrum moves through a specified passband as objects of higher
redshift are observed. A quasar at redshift z = 1.9 with a
rest-frame Lyman-
emission line equivalent width of 100 Å has a U
magnitude ~ 0.3 mag brighter than an otherwise identical quasar
with weak or undetectable emission. Many quasars have magnitudes
measured in the Johnson B band, or the BJ
system defined by the
Kodak IIIaJ emulsion combined with a GG395 filter as used by the UK
Schmidt Telescope in Australia. The color-equation relating the two
systems is well-determined for stars of intermediate spectral type and
is BJ = B - 0.28 (B - V)
(Blair and Gilmore
1982). Taking B - V = 0.3 for
a typical quasar results in a mean transformation of
BJ ~ B -0.1.
However, the red cutoffs of the two systems are quite different, with
Bmax ~ 4900 Å and
BJmax ~ 5350 Å. At redshifts
z ~ 3.1, the BJ magnitude can be a full
magnitude brighter,
with the Lyman- line and
adjacent continuum falling in the
BJ band, while the B band measures only the
continuum shortward of
1216 Å where suppression by the Lyman- forest occurs. Mean
color transformations provide a poor method for relating surveys
employing different passbands, with individual quasars showing
variations of ± 0.25 mag about the mean. These variations are
correlated with the redshift and SEDs of the individual quasars. A
precise specification of the passband used to define a
``flux-limited'' sample is required in order that the multiplication with
quasar SEDs at different redshifts can be performed to determine the
range of absolute magnitudes for which quasars of given SED and z
are included.
6.2 Spectroscopic Identification
Following definition of a quasar candidate list the next phase is spectroscopy to produce unambiguous classifications. The limitations imposed by large numbers of candidates and small amounts of telescope time has meant that most spectroscopy has been of low signal-to-noise ratio and low spectral resolution. The low resolution enables a broader wavelength range to be accessed, increasing the chances of determining an unambiguous redshift through identification of several emission lines. However, this strategy is not always effective for identifying quasars with weak or absent emission lines. Many published quasar surveys contain fractions of unidentified candidates representing ~ 10% of the number of quasars identified: e.g., 54 unidentified objects and 420 confirmed quasars (13%) (Boyle et al. 1990), 10 unidentified objects and 127 confirmed quasars (8%) (Crampton, Cowley and Hartwick 1987). Some of the unidentified objects will be metal-poor hot stars that show no strong absorption features or characteristic breaks in their continua; the almost absorption line-free DC white dwarfs are particularly hard to eliminate. Reduction of the fraction of unidentified objects would be most effectively achieved by re-observation at higher resolution and at wavelengths where the strongest stellar absorption features occur, thus optimizing the chances of eliminating stars.
The probability of a quasar's remaining unidentified is a strong function
of both the SED and the redshift, the latter dependence arising in part
because of the absence of strong emission lines within the
observed-frame wavelength interval at certain redshifts. To achieve a
quantitative description of the spectral properties, or evolution, of
the quasar population, the variable probability of spectroscopic
identification should be incorporated in the selection function.
To assume that the catalog of unidentified objects contains quasars with
the same distribution of redshift and SED as the confirmed object is
unrealistic. For optical surveys, only at bright magnitudes such as in
Palomar-Green, is the spectroscopic identification of candidates
complete. At fainter flux levels the Large Bright Quasar Survey (LBQS;
Hewett, Foltz and Chaffee
1993) is the first major optical survey where the fraction of unidentified
objects has been reduced to nearly zero: 8 objects out of more than 2000
candidates observed remain unclassified, representing 0.8% of the
confirmed quasars. A number of the weak-lined objects in the LBQS of
magnitude mB ~ 18 required integrations of ~ 2000 s with a
4.5-metre telescope (albeit with a detector that is relatively
inefficient in comparison to the best available today). Achieving such
a high rate of identification at faint magnitudes, m 
20, is
not practical, and incorporating the probability of successful
identification as a function of magnitude, redshift and SED remains the
only viable way to proceed.
6.3 Quasar Photometric Variability
The variability of the energy output of quasars has provided powerful constraints on the inferred size of the continuum-producing and broad emission line regions. Very short timescale variability in the X-ray regime places limits on the size of the continuum-emitting region (e.g., Kunieda et al. 1990) and the detailed monitoring of the response of emission line fluxes to changes in continuum flux has resulted in a substantial revision in the estimate of the extent of the broad line region (Peterson 1993). Until the late 1970s, attention was concentrated on objects which exhibited the most extreme variability, in terms of both amplitude and timescale. The properties of the variability behavior of these extreme objects, BL Lac objects and the Optically Violently Variable quasars, are now well characterized (e.g., Carini et al. 1992) if not well understood. However, they form a very small fraction of the quasar population. Of more direct relevance to surveys of the population as a whole is the less extreme variability. The first systematic studies (Bónoli et al. 1979, Netzer and Sheffer 1983) indicated that over rest-frame time intervals of years, photometric variability of tens of percent was exhibited by essentially all quasars. Usher and collaborators (Usher, Warnock and Green 1983) employed a combination of variability and an ultraviolet excess criterion to compile a sample of mB < 18 quasars that, until recently, provided one of the few samples with magnitudes mB ~ 17. More recently, Hawkins (1986) has acquired a unique database of more than 100 deep UK Schmidt Telescope plates taken over a period of 20 years, and a long-term program to identify quasars through their variability is well advanced. The results (Hawkins and Véron 1993) confirm that surface densities comparable to those achieved using other selection techniques are obtained, although Hawkins and Véron claim that the turnover in the quasar source counts for objects fainter than mB ~ 20 is not evident in their sample. The results are based on small numbers of confirmed quasars, and the problems of matching samples defined using single-epoch magnitudes with that of Hawkins and Véron, who use a magnitude based on the minimum observed magnitude over a period of years, should not be underestimated. Further work is required to confirm the differences.
As with proper-motion-based selection it has been suggested that a survey based on variability will be more ``complete'' than other techniques, and that the selection function is inherently simple. In fact, the selection function requires an extra independent variable, i.e., P (M, z, SED, Var), where Var represents a specification of the variability behavior of a quasar. This in turn will include at least two factors: timescale and amplitude, and these may not be separable. The amplitude and timescale of the variability exhibited by a quasar will also show a dependence on rest-frame wavelength, because at a given wavelength, contributions are present from different regions of the quasar (continuum source, accretion disc, broad emission line region) that have different variability properties. For example, should one component of the quasar SED (say the underlying power-law) vary on a different timescale than the Big Blue Bump, then the variability properties of quasars will show a dependence on redshift that reflects the different proportion of flux from the power-law and Big Blue Bump contributing to the observed-frame passband in which the variability is measured (Giallongo and Trevese 1990).
The probability of detection as a function of redshift also includes a term proportional to (1 + z)-1} to take account of the decreasing rest-frame time interval covered by fixed observed-frame epoch differences. Thus, it is still necessary to calculate the probability that a quasar at a given redshift with a specified SED and variability properties as a function of time will be detected. In another investigation, Hook et al. (1991, 1994) have examined the variability properties of ~ 400 quasars, again employing UK Schmidt Telescope plates. Their results suggest a significant difference in the variability behavior as a function of luminosity. If confirmed, this trend should be included in the P (M, z, SED, Var) calculation, as it has important implications for the interpretation of luminosity functions derived from variability selection.
The completion of surveys like that of Hawkins, including the
quantification of the selection function will represent an important
step, offering the prospect of deriving information of direct relevance
to models of quasars themselves. Our assessment of the requirements
for a quantitative survey employing variability should be contrasted
with the view of Hawkins and Véron (1993), where in their
introduction they claim, in our terminology, that P (M, z, SED,
Var) = 0.95
for redshifts 0 
z
3.5, and an extended, but
unspecified, range
of SED and Var. There is disagreement among workers active in the
field on this point, as Borgheest and Schramm (1993), using the results
of the Hamburg Quasar Monitoring Program, come to a conclusion contrary
to that of Hawkins and Véron.
The variability behavior of quasars has implications for surveys that use flux estimates that are not nearly simultaneous. In the multicolor work of Warren et al. (1994), where some photographic plate material was obtained with observed-frame epoch differences of ~ 5 years, the derived colors of quasars are affected significantly by variability. Warren et al. had to include a determination of the effects of variability on the observed quasar fluxes to quantify the selection function. There are a number of surveys employing fluxes obtained years apart, where the photometric variability affects the selection function. In the LBQS (Hewett, Foltz and Chaffee 1993) any quasars whose magnitudes vary by more than 1 magnitude between the epochs of the broadband direct plates and the objective-prism plates have been excluded. The ultraviolet excess survey of Boyle et al. (1990) employs plate pairs with epoch separations of up to 9 years, and intrinsic photometric variability effectively blurs the well-defined ultraviolet excess color criterion by ~ 0.25 mag. Boyle et al. (1987) inferred a very low ``incompleteness'', ~ 2%, due to variability. The figure will be appropriate for objects with U - B colors well to the blue of the discrimination boundary. However, quasars with redshifts and/or SEDs that cause them to lie close to the U - B discrimination line will be affected to a much greater extent, in much the same way as found by Warren et al. (1994), and no quantitative discussion of this effect has been presented.