**2.2.3. Implementation of VELMOD**

The probability distribution
*P*(*m*|,
*cz*) (eq. [11]) is dependent on a number of free parameters, most
importantly
_{I}.
However, because
_{I}
enters at an earlier stage - in
the reconstruction of the underlying density and velocity fields from
*IRAS* (Appendix A) - it
is on a different footing from other parameters. Thus, rather than treating
_{I}
as a continuous free parameter, VELMOD is run sequentially for the 10
discrete values
_{I}
= 0.1, ..., 1.0 for the real data, and for the nine discrete values
_{I}
= 0.6, ..., 1.4 for the mock catalog data
(Section 3).
^{(8)} For each
_{I},
the probability is
maximized (_{forw}
is minimized) with respect to the remaining free parameters. These
parameters are as follows:

- The TF parameters
*A*,*b*, and_{TF}for each sample in the analysis. Here we limit the analysis to the Mathewson, Ford, & Buchhorn (1992, hereafter MAT) and Aaronson et al. (1982a, hereafter A82) samples, as we discuss in Section 4.1. Thus, there are a total of six TF parameters that are varied. Note that the TF scatters are not simply calculated a posteriori. The statistic_{forw}depends on their values, and they are varied to minimize it. - The small-scale velocity dispersion
_{v}. The quantities_{v}and_{TF}can trade off to a certain extent (cf. eq. [15]). However, their relative importance depends on distance. Sufficiently nearby (1000 km s^{-1}),_{v}is as large or a larger source of error than the TF scatter itself. Thus, it is determined in this local region. Beyond ~ 2000 km s^{-1}, the TF scatter dominates the error, and it is determined at these distances. Because the samples populate a range of distances, the two can be determined separately, with relatively little covariance. - We also allow for a
*Local**Group**random**velocity**vector***w**_{LG}. The*IRAS*peculiar velocity predictions are given in the Local Group frame (eq. [A1]). That is, the computed Local Group peculiar velocity vector has been subtracted from all other peculiar velocities. However, just as we expect all the external galaxies to have a noisy as well as a systematic component to their peculiar velocity, so we must expect the Local Group to have one as well, especially considering the uncertainties in the conversion from heliocentric to Local Group frame. We allow for this by writing*u*(*r*) =*u*_{IRAS}(*r*) -**w**_{LG}^{.}, where*u*_{IRAS}(*r*) is given as described in Appendix A, and the three Cartesian components of**w**_{LG}are varied in each VELMOD run at a given_{I}. We note briefly that this procedure is self-consistent only as long as |**w**_{LG}| is at most comparable to_{v}. In practice, we will find that for_{I}near its best value, the amplitude of**w**_{LG}is trivially small. - Finally, we allow for the existence of
a quadrupole velocity component that is not included in the
*IRAS*velocity field. The justification for such a velocity component will be discussed in Section 4.4 and Appendix B. The quadrupole is specified by five independent parameters, although we will not take them as free in the final analysis (we discuss this further in Section 4.4).

Thus, there are 3 × 2 + 1 + 3 = 10
free parameters that are varied for any given value of
_{I},
when the quadrupole is held fixed. Thus, for any value of
_{I}, we
give the data the fairest chance it possibly has to fit the *IRAS*
model. In particular, the TF relations for the two separate samples used
are not "precalibrated"
in any way. This ensures that TF calibration in no way prejudices the value
of _{I}
we derive.

^{8} The
choice of these values of
_{I}
was based on the need to bracket the "true"
value: 1.0 in the mock catalogs and, as it turns out, ~ 0.5 for
the real data. Back.