Goal: a more robust GridLeak calibration

Definitions:

Formulas:

Thus, when we look at data which has been corrected (and we have to, because the distortion is so large that looking at uncorrected data results in broken tracks due to the GridLeak, not allowing us to easily look at gap via the residuals), we have:

gap(z,L) = D_static + D_GL - C_GL
D_GL = g_GL z (L - GLO)
C_GL = g₂ z (L - SO) SC GL

Ideally, SO = GLO. But that is to be determined...

gap(z,L) = D_static + g_GL z (L - GLO) - g₂ z (L - SO_used) SC_used GL_used
gap(z,L) = D_static + z [g_GL (L - GLO) - g₂ (L - SO_used) SC_used GL_used]

We define the slope and intercept in z as gapf and gapi respectively, and we can write the formula for gapf in a few ways:

We disregard the intercept D_static, as it involves other possible distortions we don't care about here. We define the slope and intercept of gapf vs. L as Leak_S and Leak_I respectively:
gapf(L) = Leak_I + L * Leak_S
Leak_I = g₂ SO_used SC_usedGL_used - g_GL GLO
Leak_S = g_GL - g₂ SC_usedGL_used

And although I have written gapf as a function of L, when we are looking data for which a correction was applied, it is also a function of SC_usedGL_used and SO_used. We expect g_GL to be dependent on the colliding system and the selected luminosity scaler used for L, but g₂ should in principle be constant for all (the correction term C_GL takes the colliding system and the selected luminosity scaler into account via the SC term).

Methods:

We seek two conditions:

1. Leak_S → zero (no luminosity dependence for gapf)
2. Leak_I → zero (gapf becomes zero, thus no z dependence for gap)

The old approach: ignore the second condition and instead just use a linear fit of gapf vs. L to find Leak_S for each data pass (i.e. each different SC_usedGL_used), then linearly fit Leak_S vs. SC_usedGL_used to find the zero crossing, i.e. where SC GL = g_GL/g₂. This approach suffers from the lack of a strong constraint on the intercept of the fit of gapf vs. L: the fits could trade some of the slope for intercept, resulting in only a moderate global consistency across the different passes on the data.

The new approach: fit the data from all passes for three parameters in one shot: g₂, (g_GL/g₂), GLO. The fact that we are using multiple passes of data for a single fit provides additional constraint on the intercept in gapf vs. L, overcoming the weakness of the old approach.

The parameters are chosen such that the first should be a constant within error (something worth testing), the second automatically gives the desired SC GL and associated error without further calculations, and the third should be comparable to SO within errors (also something worth testing).

We define a 3 parameter function of 3 dimensions, analogous to the "Collected coefficients" form (ii) of gapf(L) above:

I have tried it and it works....more to come...

-Gene