3.8 Fitting and optimization

Our basic approach is to minimize $\chi ^2$ between the model predictions and the data, summing values for the three independent Stokes parameters, I, Q and U. The value of the ``noise'' on the observed images is important in the optimization process, as sums of $\chi ^2$ over different areas need to be added with the appropriate weights to ensure that the data are fitted sensibly over the full range of resolutions available. The ``noise'' is dominated by small-scale intensity fluctuations - knots and filaments - whose amplitude is unknown a priori. Our best guess at their level comes from a measure of the deviation from axisymmetry. The ``noise'', $\Sigma$, is taken to be $1/\sqrt 2$ times the rms difference between the image and a copy of itself reflected across the jet axis. This is always much larger than the off-source rms. Any contribution from deconvolution artefacts will also be included in this estimate. Some components of the small-scale structure will result in mirror-symmetric features in the brightness distribution (e.g. the bright arc in the main jet; Fig. 1), and we will therefore underestimate $\Sigma$.

We fit to the 0.25-arcsec images over the area covered by the model from 0.5 to 4.1 arcsec from the core. This excludes the core itself, and covers all of the area where significant polarized emission is detected at this resolution. Further out, the signal-to-noise ratio for these images (especially in linear polarization) is too low to provide an effective constraint, so we fit to the 0.75-arcsec images. Fits made using the high-resolution images alone are consistent with those that we describe here, but are less well constrained. $\chi ^2$ is computed only over the model area and we evaluate it at at grid-points chosen to ensure that all values are independent. There are 1346 independent points, each with measurements in 3 Stokes parameters. Of these, 44, 162 and 1140 are in the inner, flaring and outer regions, respectively.

We have optimized the models over the whole area and with one or more of the brightest small-scale features excluded from the $\chi ^2$ calculation. The derived parameters did not vary by appreciable amounts, but exclusion of the obvious ``arc'' in the main jet (Fig. 1) somewhat reduced the final $\chi ^2$. Given that we are effectively averaging over many small-scale filaments in the brightness distribution, we have no physical reason to remove the brightest few, but it is reassuring that the results are insensitive to their exclusion.

In order to optimize the model parameters, we use the downhill simplex method of Nelder & Mead (Press et al.1992). This usually converges in 150 - 200 iterations given reasonable starting parameters.

2002-06-13