best fit mode coefficients $a^\alpha$, the slope, $s$, and the zero point $\eta_0$, are found by minimizing the $\chi^2$ statistic $$\label{eq:chi}
\chi^2=\frac{1}{\ssigint}\sum_{i=1}^{N_{\rm g}}
 {\left(s M_{0i}+ s P^M_i+\eta_0-\eta_i\right)^2}+
 \sum_{\alpha=1}^{N_{\rm a}}  (a^\alpha)^2\; ,$$ where $\ssigint$ is the $rms$ of the intrinsic scatter in $\eta$ about the ITF relation, and $N_{\rm g}$ is the number of galaxies in the sample. The second term of the sum over the squares of $a^\alpha$ is introduced in order to regularize the solution especially in regions of poor data coverage. In the appendix we derive this term from a Bayesian formulation. The solution to the equations $\partial \chi^2/\pa a^\alpha=0$, $\partial \chi^2/\pa s=0$ and $\partial \chi^2/\pa \eta_0=0$ is straightforward. The coefficients $a^\alpha$ will be used in equation (\[eq:linc\]) to get $\vv(\vx)$ everywhere in a region of space large enough to contain the data. For each field $\vv^\alpha(\vx)$ we compute its corresponding bulk flow, $\vB^\alpha(r)$, according to equation (\[eq:bdef\]) and write our   bulk flow as $$\label{eq:best}
\vB_{_{\rm ASCE}}(r)=\sum_\alpha a^\alpha \vB^\alpha(r)\; .$$

The Maximum Likelihood Estimate (MLE) {#sec:mle}
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For comparison we will present estimates of the bulk flow obtained with the standard MLE [@K88]. This method approximates the bulk flow of a sphere of radius $r$ as the vector 