and easier to implement. Assume that the TF catalogue contains $i=1\dots N_{\rm g}$ galaxies with measured redshifts (in $\kms$), $cz_i$, apparent magnitudes, $m_i$, and line width parameters, $\eta_i$. We write the absolute magnitude of a galaxy, $$M_i = M_{0i} + P_i\; ,$$ where $$M_{0i} = m_i + 5{\rm log}(cz_i)-15 \;$$ and $$\label{eq:utp}
P_i = -5 {\rm log}(1- u_i/cz_i)$$ with $u_i$ the radial peculiar velocity of the galaxy. Both $cz_i$ and $u_i$ are defined in the frame of the cosmic microwave background radiation (CMB). Assume that an estimate of the underlying cosmological velocity field, $\vv(\vx)$, can be written as a linear combination of $$\label{eq:linc}
\vv(\vx)=\sum_\alpha^{N_{\rm a}} a^\alpha \vv^\alpha(\vx)\; ,$$ where the $N$ fields, $\vv^\alpha(\vx)$ ($\alpha=1\cdots N_{\rm a}$), are gaussian random velocity fields generated using a cosmologically viable power spectrum. In practice these basis velocity fields will be extracted from a linear cosmological velocity field generated in a very large box using the power spectrum of the $\Lambda$CDM model. We then compute $u^\alpha_i$, the radial component of the $\vv^\alpha$ at the redshift space positions of the observed galaxies, and define the $P-$basis functions, $P^{\alpha}_i=-5{\rm log}(1-u^\alpha_i/cz_i)$ for the observed galaxies only. The model $P$ is then written as $$P^M_i=\sum_\alpha a^\alpha P^\alpha_i\; .$$ The 