### ALMA Baseline Calibration Discussion:

*The following is a transcript of an email discussion that Al Wootten, Robert Lucas, and Jeff Mangum had in November 2003.*

*Regarding the method used to solve for a baseline measurements solution at PdBI, Robert Lucas dis:*
We do a baseline determination in good conditions (and short baselines to minimize atmospheric
phase terms). Typically we would measure ~20 sources in one to two hours. We include low and high elevation sources.

We do the following fit to the observed phases phi_i (i being either a baseline or an antenna index, j the index of measurement number):

phi_ij = C_i + X_i*cos(H_j) + Y_i*sin(H_j) + Z_i*sin(D_j) + E_i*cos(El_j)

...that is we compute X_i, Y_i, Z_I (equatorial coordinates of the baseline errors), and axes offset E_i to minimize the weighted sum of square phase residuals, where H_j, D_j, El_j are respectively the hour angle, declination and elevation of measurement number j.

The precision on E_i is typically only slightly worse that that of the Z_i (polar) terms since we can measure a range 20-90 degrees in elevation while we get declinations in the -25 to 90 range.

Now we could also have used horizontal coordinates for the baselines (we don't). Then it's clear that the azimuthal error of the baseline vector is decoupled from the axes offset, since it gives only azimuth dependent phase errors. The vertical baseline error z_i induces a z_i*sin(El_j) term while the axes offset a E_i*cos(El_j) as above. These are comfortably orthogonal functions of elevation. This is basically why it works.

Naturally we measure only axes offset differences from ona antenna to the other, while by going over the top you could measure the absolute values (by moving only a subset of the antennas over the top). But who cares for the absolute values?

*Al replied:*
I think what we need is an analysis of the technique and an estimate of the errors which we
might have on the baseline when we use the technique. How will we know if the antenna spec of <20 microns offset variation is met? I discussed this with Rick Perley concerning the VLA antennas. When the array is in the D configuration they make their most accurate measurements, which have an rms scatter of .3mm or so, some of which is thought to be due to variations in the axis offset (or errors in the model or....). Will ALMA be able to measure variations of 20 microns in the axis offset? If the spec isn't met by a large factor (the offset value is reportedly a factor of 5 off or so in an SMA antenna), will we be able to measure this?

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JeffMangum - 05 Aug 2004