A really simple noise simulation to nail down the constants in verification that determine how fast sigma should grow away from the quad center, without radial distortion.

I sampled a 100-pixel diameter quad (AB distance = 100 pixels) whose center was the center of the image and then sampled 1000 stars along a line through the center of the image. These are the field stars.

Next, I added noise to only the four stars in the quad, and copied the rest of the stars without noise. These are the index stars before transformation.

I then fit a rigid Procrustes transformation from the index quad to the field quad, and projected the index stars through this transformation.

In the plots I consider the following quantities:

L - the distance from the field star to the field quad center of mass.
H - the "radius" of the quad: the RMS distance of the four field quad stars to the field quad center of mass.
R - L / H
E - the error: distance from the field star to the projected position of the index star.

Here are the locations of the field stars and index stars after transformation (transformed according to the matched quad). Notice the slight tilt of the line due to noise in the quad.

 http://trac.astrometry.net/attachment/wiki/NoiseSim/FI.png?format=raw

Here's how the noise changes as a function of distance from the quad center.

 http://trac.astrometry.net/attachment/wiki/NoiseSim/EvsR.png?format=raw

Here's the fit I get if I do a naive fit for E^2 = C1 + C2 * R^2.

 http://trac.astrometry.net/attachment/wiki/NoiseSim/E2vsR2.png?format=raw

Next, I added one-pixel Gaussian noise to all the index stars (not just the quad), and sampled 10k stars. (This is a different quad sample also).

 http://trac.astrometry.net/attachment/wiki/NoiseSim/FIb.png?format=raw

 http://trac.astrometry.net/attachment/wiki/NoiseSim/EvsRb.png?format=raw

Notice how the distance distribution changes with radius:

 http://trac.astrometry.net/attachment/wiki/NoiseSim/Ehistsb.png?format=raw

With the same naive fit, I get:

 http://trac.astrometry.net/attachment/wiki/NoiseSim/E2vsR2b.png?format=raw

But the residuals are fairly structured:

 http://trac.astrometry.net/attachment/wiki/NoiseSim/Residhistsb.png?format=raw

Next, I sampled the stars in 2D instead:

 http://trac.astrometry.net/attachment/wiki/NoiseSim/FIc.png?format=raw

 http://trac.astrometry.net/attachment/wiki/NoiseSim/EvsRc.png?format=raw

 http://trac.astrometry.net/attachment/wiki/NoiseSim/Ehistsc.png?format=raw

With the same naive fit, I get:

 http://trac.astrometry.net/attachment/wiki/NoiseSim/E2vsR2c.png?format=raw

 http://trac.astrometry.net/attachment/wiki/NoiseSim/Residhistsc.png?format=raw

However, the fit parameters I get vary a lot from run to run, and don't seem to correlate with the code distance (ie, amount of noise in the matched quad).