Goal: a more robust SpaceCharge calibration

Definitions:

Formulas:

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::PredictSpaceCharge()) assuming the GL that we applied...
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 the observed + used sc as our measurement:

sc_measured(L) = sc_observed + sc_used

The sc_used term cancels with that from C_SCGL, and we are left with:

sc_measured(L) = (D_static + D_SCGL) / (g₃ + g₄ GL_used)
sc_measured(L) = (D_static + sc(L) (g₃ + g₄ GL)) / (g₃ + g₄ GL_used)
sc_measured(L) = (D_static + SC (L - SO) (g₃ + g₄ GL)) / (g₃ + g₄ GL_used)

Let us define an effective offset:
SO_effective = SO - { D_static / [ SC (g₃ + g₄ GL) ] }   # This uses only fixed numbers (i.e. it doesn't depend on what we used)

sc_measured(L) = [ SC (L - SO_effective) (g₃ + g₄ GL) ] / (g₃ + g₄ GL_used)
sc_measured(L) = [ SC (L - SO_effective) (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)

Let us 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_effective)

This compares in form to our original formulation of sc = SC (L - SO).

Methods

The old approach: Fit sc_measured(L) for SC_effective,2 and fit abitrarily as a second order polynomial function of GL_used (i.e. SC_effective,2 = A + B GL_used + C GL_used²). Determine (SC GL) as a constant (i.e. SC GL = D) from fits to gapf(L) and subsequently Leak_S as a function of SC_usedGL_used. Then solve for either SC or GL from the two formulas (two equations and two unknowns). This involves taking roots of nonlinear functions, so the appropriate root must be guessed, and the condition of non-intersection of the two equations must be considered.

The new approach: The formulas above, really for sc_measured(L,GL_used), allow us to define a 3 parameter function of 2 dimensions which can be used to fit all of the data at once without special considerations:

Notes:

Note 1: When GL_used = GL (the true GridLeak parameter), we find as we expect that we are measuring the true SC parameter:
sc_measured(L) = SC (L - SO_effective)

Note 2: Observe that (g₃) and (g₄ GL) determine the relative contributions of SpaceCharge and GridLeak to the signed DCA distortions respectively, or one can compare g₅ to GL. If g₅ is large compared to GL, then GridLeak would have no bearing on this distortion. Likewise, if g₅ is small, then GridLeak dominates this distortion. When we learned about the existence of GridLeak, we quickly learned that these contribiutions are comparable, so we expect g₅ to be similar to GL.

Note 3: There should be no reason for g₃ and g₄ to change for different data. In other words, g₅ should be the same constant for every calibration, and this will be worth checking.

Different Offsets:

Let's take one step back and ask, "What if SpaceCharge and GridLeak do really have different luminosity scaler offsets?" This can be written using GLO for the GridLeak offset to differentiate it from the one for SpaceCharge. We then have:

D_SCGL = [ g₃ SC (L - SO) ] + [ g₄ GL SC (L - GLO) ]
sc_measured = { [ SC (g₃ + g₄ GL) ] L + [D_static - SC (g₃ SO + g₄ GLO GL)]} / (g₃ + g₄ GL_used)

This time, we define:
SO_effective,2 = [ (g₃ SO + g₄ GLO GL) - (D_static / SC) ] / (g₃ + g₄ GL)   # This uses only fixed numbers

sc_measured(L) = SC (L - SO_effective,2) (g₃ + g₄ GL) / (g₃ + g₄ GL_used)
sc_measured(L) = SC (L - SO_effective,2) (g₅ + GL) / (g₅ + GL_used)
sc_measured(L) = SC_effective (L - SO_effective,2) / (g₅ + GL_used)
sc_measured(L) = SC_effective,2 (L - SO_effective,2)

But this is exactly the same functional form as before. When GLO = SO, we have SO_effective,2 = SO_effective. When D_static = 0 also, then SO_effective,2 = SO_effective = GLO. This would be a desirable outcome, but is perhaps a condition which should not be imposed.

Unifying Fit:

In principle, using the similar formulas for an alternative GridLeak calibration, one can fit the gapf and sc data simulatneously, such that 6 parameters can be fit: g₂, g₅, SC, SO_effective,2, GL, GLO. This can be done by substituting (SC GL) for (g_GL/g₂) [parameter p₁ in f_GL] and reverting back to SC (g₅ + GL) instead of SC_effective [parameter p₁ in f_SC]. Because we have no way of differentiating the effects of D_static from that of SO, from here on we just use SO instead of SO_effective,2. One could make this a 5 parameter fit by further imposing that SO = GLO.

Fitting both the gapf and sc data simultaneously with f_GL and f_SC this way lets the fitter deliver errors on SC and GL individually, rathern than having to do the math of determining where the separate constraints on SC and GL from f_GL and f_SC intersect, and propogating errors manually.

More on this in my next post.

-Gene