**2.2.2. Further Discussion of VELMOD Likelihood**

The physical meaning of the VELMOD
likelihood expressions is clarified by considering them in a suitable
limit. If we take
_{v}
to be "small,"
in a sense to be more made precise below, the integrals in equations
(11) and (12) may be approximated using standard techniques. In addition, if
we neglect sample selection (*S* = 1) and density variations
[*n*(*r*) = constant], and assume that the redshift-distance
relation is single-valued, we find, for the forward relation,

(15) |

where *w* is the solution to the equation *cz* = *w* +
*u*(*w*), i.e., it is the distance inferred from the redshift and
peculiar velocity
model; _{v}
_{v} /
[*w*(1 + *u*')], where *u*'
= (*u* /
*r*)_{r=w},
is the effective logarithmic velocity dispersion; and

(16) |

is the effective TF scatter, including the contribution due
to _{v}.
An analogous result holds for the inverse relation. The
criterion _{v}^{2} << 1,
which quantifies the statement that
_{v}
is "small," must be satisfied to derive equation (15).

Equation (15) shows that the probability distribution
*P*(*m*|,
*cz*) preserves the Gaussian character of the real-space TF
probability distribution
*P*(*m*|,
*r*) in this limit. However, the expected value of *m* is shifted
from the "naïve" value
*M*()
+ 5 log *w* by an amount ~
4.3_{v}^{2}.
This shift is in fact nothing more than the homogeneous Malmquist bias due
to small-scale velocity noise; it differs in detail from the
usual Malmquist expression (i.e., that which affects a Method I analysis)
because it arises from the Gaussian (rather than lognormal)
probability distribution (eq. [9]). Furthermore, the effective scatter
_{e}
is larger
than _{TF},
because the velocity dispersion introduces additional distance error and
thus magnitude scatter. However, the effects associated with velocity noise
diminish with distance
(_{v}
*r*^{-1});
the velocity Malmquist effect vanishes in the limit of large distances, in
contrast with the distance-independent Malmquist effect for Method I, and
the effective scatter approaches the TF scatter. At large enough distance,
the VELMOD likelihood approaches a simple Gaussian TF distribution
with expected apparent magnitude
*M*()
+ 5 log *w*, and VELMOD reduces to the standard Method II.

Indeed, equation (15) enables us to define
the regime in which VELMOD represents a significant modification of Method
II. The distance *r*_{II} at which the velocity noise
effects become unimportant is determined by *r*_{II} >>
_{v} /
_{TF}(1
+ *u*'), where
_{TF}
= ln10_{TF}/5
is the fractional distance error due to the TF scatter
(_{TF}
0.2
for the samples used here).
For _{v}
= 125 km s^{-1}, the value we find for the real data
(Section 4.5), this shows that in the
*unperturbed* Hubble flow, where
*u*' = 0, velocity noise effects become unimportant beyond
~ 1500 km
s^{-1}. However, at about this distance, in many directions, the
Local Supercluster significantly retards the Hubble flow,
*u*' -0.5,
so that the effective
_{v}
is about twice its nominal value. Thus, VELMOD in fact differs
substantially from Method II to roughly twice the Virgo distance. This fact
guided our decision to apply VELMOD only out to 3000 km s^{-1}
(cf. Section 4).

Equation (15) also demonstrates that maximizing the likelihood
(minimizing
_{forw})
is not equivalent to
^{2}
minimization, even under the adopted assumptions of constant density and
negligible selection effects, because of the factor
_{e}^{-1}
in front of the exponential factor. This factor couples the velocity
model [i.e., the values of *w* and *u*'(*w*)]
to the velocity noise. In particular, *maximizing* *the*
*VELMOD* *likelihood* *is* *not* *equivalent*
*to* *minimizing* *TF* *scatter* (cf.
Section 4.5), except in the limit that
_{v} is
set to zero.

The assumptions required for
deriving equation (15) remind us that there are two other factors
that distinguish VELMOD from standard Method II. First, for realistic
samples one cannot assume that *S* = 1. The presence of the selection
function in equations (6) and (7) is essential for evaluating true
likelihoods, and we have fully
incorporated these effects into
our analysis. ^{(7)} Second, the
galaxy density *n*(*r*) is not effectively constant along most
lines of sight. Thus, VELMOD, like Method I but unlike Method II, requires
that *n*(*r*) be modeled. We do so here by using the *IRAS*
density field itself, which is a good approximation to the number density
of the spiral galaxies in the TF samples. The density field has a
nonnegligible effect on the VELMOD likelihood whenever it changes rapidly
on the scale of the effective velocity dispersion
_{v} /
(1 + *u*').

The most significant differences between
VELMOD and Method II thus occur in regions where *u*'
- 1
(flat or triple-valued zones), or when the density varies
particularly sharply. In practice, both these effects occur in the vicinity
of large density enhancements such as the Virgo Cluster. We illustrate this
in Figure 1, which shows
the redshift-distance relation and the corresponding value
of *P*(*r*|*cz*)
*P*(*cz*|*r*)*P*(*r*) in the
vicinity of triple-valued zones. When looking at these panels, keep in mind
that the VELMOD likelihood is given by
multiplying *P*(*r*|*cz*) and the TF probability factor
*P*(*m*|,
*r*) and integrating over the entire line of sight.
Figures 1*a* and
1*b*
depict the situation near the core of a strong cluster, and
Figures 1*c* and
1*d*
depict it farther from the center. In each case, the cloud of points
represents the velocity noise, here taken to be
_{v}
= 150 km s^{-1}. In Figure 1*a*, the
redshift of 1200 km s^{-1} crosses the redshift-distance diagram at
three distinct distances. The quantity
*P*(*r*|*cz*)
shows three distinct peaks. The highest redshift one is the strongest
because of the *r*^{2} weighting in equation (8). In
Figure 1*b*, the redshift of
1700 km s^{-1} is such that the object just misses being
triple-valued; however, the finite scatter in the redshift-distance diagram
means that there is still appreciable probability that the galaxy is
associated with the near crossing at *cz* ~ 900 km
s^{-1}. In Figure 1*c*,
the redshift-distance diagram goes nearly flat for almost 600
km s^{-1}; a redshift that comes close to that flat zone has
a probability distribution that is quite extended.
Finally, Figure 1*d* shows a galaxy whose
redshift
crosses the redshift-distance diagram in a region in which it is quite
linear, and the probability distribution has a single, narrow peak without
extensive tails.

Two final details deserve brief mention.
First, the integrals over *m* and
that
appear in the denominators of equations (11) and (12) may be done
analytically for the case of "one-catalog selection"
studied by Willick (1994,
Section 4.1), which
indeed applies for the samples used in this paper
(Willick et al. 1996).
The numerical integrations required to evaluate equations
(11) and (12) are thus one dimensional only. Second, as noted
above, the velocity width distribution function
()
drops out of equation (11), but the luminosity function
(*M*)
does not drop out of equation (12) Thus, inverse VELMOD requires that we
model the
luminosity function of TF galaxies. This is an annoyance at best and could
introduce biases, if we model it incorrectly, at worst. Thus, we have
chosen to implement only forward VELMOD in this paper. On the other hand,
inverse VELMOD enjoys the virtue that inverse Method II approaches do
generally: to the degree that the selection function *S* is
independent of
and
*r*, it drops out of equation (12). In a future paper, we will
apply the
small-_{v}
approximation to VELMOD for more extensive samples to larger distances. For
that analysis, the inverse approach will be used as well.

^{7}
Selection
effects are not specific to VELMOD per se, however. They can and should be
modeled in any Method II-like
analysis. In particular, they do not vanish in the
_{v}
0 limit.
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