Updated Thursday August 9th: Well, that did and did not work. Most of the noise went away but enough stayed to make it tough to analyze. Not only that, the noise moved around!
Now we try -noise 3

As of this morning (Friday August 10th, 2012) the convert -noise 3 function did its job and produced a much cleaner image.

2012 August 9th
convert -noise 2

2012 August 10th
convert -noise 3

In addition we moved from using the lossly .JPG format to the lossless .PNG format.