**2.2. VELMOD**

The approach we take in this paper, "VELMOD,"
is a maximum likelihood method designed to surmount several of the
difficulties that face POTIRAS and ITF. VELMOD generalizes and improves
upon the Method II approach to velocity analysis. Method II takes as its
basic input the *TF* *observables* (apparent magnitude and
velocity width) and redshift of a galaxy, and asks the following: what is
the probability of observing the former, given the value of the latter? It
then maximizes this probability over the entire data set with respect to
parameters describing the TF relation and the velocity field. The
underlying assumption of Method II is that a galaxy's redshift, in
combination with the correct model of the velocity field, yields its true
distance, which then allows the probability of the TF observables to be
computed. This analytic approach was developed originally by
Schechter (1980)
and was used later by
Aaronson et al. (1982b),
Faber & Burstein
(1988),
Strauss (1989),
Han & Mould (1990),
Hudson (1994),
Roth (1994), and
Schlegel (1995),
among others.

The main problem with Method II is its assumption that a unique redshift-distance mapping is possible. This assumption breaks down for two reasons. First, it is only approximately true that the redshift of a galaxy is equal to the sum of its distance and its predicted peculiar velocity (cf. eq. [A1]); there is true velocity noise generated on very small (1 Mpc) scales, as well as inaccuracy of the velocity model (even for the correct ) due to nonlinear effects and shot noise in the density field. Second, even in the absence of noise, the redshift-distance relation, in principle, can be multivalued: more than one distance along the line of sight can correspond to a given redshift. VELMOD accounts for all of these effects statistically by replacing the unique distance of Method II with the joint probability distribution of redshift and distance. This distribution is constructed to allow for both noise and multivaluedness. The distance dependence is then integrated out (Section 2.2.1), yielding the correct probability distribution of the TF observables given redshift.

There are two additional advantages to the
VELMOD approach. First, it requires neither a priori calibration of the TF
relations (as does POTENT) nor matching of the input data from disparate
samples (as does ITF). An individual TF calibration for each independent
sample occurs naturally as part of the analysis. Second, it does not
require smoothing of the input TF data, and thus allows as high-resolution
an analysis as the data intrinsically permit. This second feature, along
with its allowance for velocity noise and triple-valued zones, makes VELMOD
well suited for probing the local (*cz*
3000
km s^{-1}) velocity field. An analysis of local data is desirable
because random and systematic errors in both the *IRAS* and TF data
are less important nearby than far away.