Definitions:

Formulas for GridLeak:

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 = g₁ z (L - GLO)
C_GL = g₂ z (L - SO) SC GL = g₂ z sc GL

...and using sc as defined in the next section...

Ideally, SO = GLO, and we can force this. But that ought to be determined. Anyhow...

gap(z,L) = D_static + g₁ z (L - GLO) - g₂ z sc_used GL_used
gap(z,L) = D_static + z [g₁ (L - GLO) - g₂ 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 gapi = D_static, as it involves other possible distortions we don't wish to deal with here. And we will use the "Collected coefficients" form for the calibration.
 

Formulas for SpaceCharge:

Our formalism for SpaceCharge is as follows:

sc(L) = SC (L - SO)
sc(L) (g₃ + g₄ GL) = D_static + D_SCGL   # Note that there is an assumption here of the luminosity offset (L - SO) applying for both SpaceCharge & GridLeak!
sc(L) = (D_static + D_SCGL) / (g₃ + g₄ GL)
...but what we observe has distortion corrections applied, and we interpret the distortions (using StMagUtilities::PredictSpaceChargeDistortion(), assuming the GL_used that we applied, and with some g₃' and g₄' that are built into the code, which we hope to be very close to their true counterparts g₃ and g₄)...
sc_observed(L) = (D_static + D_SCGL - C_SCGL) / (g₃' + g₄' GL_used)
D_SCGL = g₃ sc(L) + g4 GL sc(L) = sc(L) (g₃ + g₄ GL)
C_SCGL = sc_used (g₃ + g₄ GL_used)

We record (write out) the observed + used sc as our measurement:

sc_measured(L) = sc_observed + sc_used

The sc_used term cancels with that from C_SCGL only if g₃' = g₃ and g₄' = g₄ (i.e. g₃r = g₄r = 1). Otherwise we are left with:

sc_measured(L) = sc_used + [D_static + sc(L) (g₃ + g₄ GL) - sc_used (g₃ + g₄ GL_used)] / (g₃' + g₄' GL_used)
sc_measured(L) = sc_used + [D_static + SC (L - SO) (g₃ + g₄ GL) - sc_used (g₃ + g₄ GL_used)] / (g₃' + g₄' GL_used)

Let us modify SO to include the static distortions, as we do not have enough observables to separate these:
SO - { D_static / [ SC (g₃ + g₄ GL) ] } → SO   # This uses only fixed numbers (i.e. it doesn't depend on what we used)

sc_measured(L) = sc_used + [SC (L - SO) (g₃ + g₄ GL) - sc_used (g₃ + g₄ GL_used)] / (g₃' + g₄' GL_used)
sc_measured(L) = sc_used + [ SC (L - SO) (g₅ + GL) - sc_used (g₅ + GL_used)] / [g₄r (g₅r g₅ + GL_used)]

This is the primary form we will use for the calibration. Only if g₃r = g₄r = g₅r = 1 does this condense to:

sc_measured(L) = [ SC (L - SO) (g₅ + GL) ] / (g₅ + GL_used)

Let us define an effective coefficient:
SC_effective = SC (g₅ + GL)   # This uses only fixed numbers

sc_measured(L) = SC_effective (L - SO_effective) / (g₅ + GL_used)

This is a secondary form we will use in the calibration to help get approximate values in an initial fit, as it does not allow deviation from g₃r = g₄r = g₅r = 1. To come full circule, for the sake of clarity, we could still define another effective coefficient:
SC_effective,2 = SC_effective / (g₅ + GL_used)   # Actually a function of GL_used

sc_measured(L) = SC_effective,2 (L - SO)

This compares in form to our original formulation of sc = SC (L - SO). But we will not use this form.

A caveat:

There is a simple scale dependence of all four coefficients g_1-4 on the TPC wt through a variable which we have labeled in StMagUtilities as Const1. The dependence cancels in ratios, g₁/g₂, g₃/g₄ = g₅, so the only place this constant appears is in the bare g₂ in the GridLeak formula. Thus, it is slightly modified to make it correct:

gapf(L) = (Const1_used / Const1_reference) g₂ [ (g₁/g₂) (L - GLO) - sc_used GL_used ]

Fitting The Data:

We do three fits. The first two fits are for SpaceCharge and GridLeak distortions separately to help obtain initial values in a third, unifying fit. These fits can be described as follows: