Intrinsic resolution in a tracking element

Foreword: This has probably been worked out in a textbook somewhere, but I wanted to write it down for my own sake. This is a re-write (hopefully more clear, with slightly better notation) of Appendix A of my PhD thesis (I don't think it was well-written there)...



Let's establish a few quantities:
  • Eintr : error on the measurement by the element in question
    • σintr2 = <Eintr2> : intrinsic resolution of the element, and its relation to an ensemble of errors in measurement
  • Eproj : error on the track projection to that element (excluding the element from the track fit)
    • σproj2 = <Eproj2> : resolution of the track projection to an element, and its relation to an ensemble of errors in track projections
  • Etrack : error on track fit at an element including the element in the fit
  • Rincl = Eintr - Etrack : residual difference between the measurement and the inclusive track fit
    • σincl2 = <(Eintr - Etrack)2> : resolution from the inclusive residuals
  • Rexcl = Eintr - Eproj : residual difference between the measurement and the exclusive track fit
    • σexcl2 = <(Eintr - Eproj)2> : resolution from the exclusive residuals
Let us further assume that the projection from the track fit excluding the element Eproj is uncorrelated with the intrinsic error of the measurement from the element: <Eproj Eintr> = 0. This implies that we can write:

σexcl2 = <Eintr2> + <Eproj2> = σintr2 + σproj2

Our goal is to determine σintr given that we can only observe σincl and σexcl.

To that end, we utilize a guess, σ'intr, and write down a reasonable estimation of Etrack using a weighted average of Eintr and Eproj, where the weights are wproj = 1/σproj2, and wintr = 1/σ'intr2:

Etrack = [(wintr Eintr) + (wproj Eproj)] / (wintr + wproj)
 = [(Eintr / σ'intr2) + (Eproj / σproj2)] / [(1/σ'intr2) + (1/σproj2)]
 = [(σproj2 Eintr) + (σ'intr2 Eproj)] / (σ'intr2 + σproj2)

Substituing this, we find...

σincl2 = <(Eintr - Etrack)2>
 = <Eintr2> - 2 <Eintr Etrack> + <Etrack2>
 = σintr2 - 2 <Eintr {[(σproj2 Eintr) + (σ'intr2 Eproj)] / (σ'intr2 + σproj2)}> + <{[(σproj2 Eintr) + (σ'intr2 Eproj)] / (σ'intr2 + σproj2)}2}>

Dropping terms of <Eintr Eproj>, replacing terms of <Eproj2> and <Eintr2> with σproj2 and σintr2 respectively, and multiplying through such that all terms on the right-hand-side of the equation have the denominator (σ'intr2 + σproj2)2, we find

σincl2 = [(σintr2 σ'intr4) + (2 σintr2 σ'intr2 σproj2) + (σintr2 σproj4) - (2 σintr2 σ'intr2 σproj2) - (2 σ'intr2 σproj4) + (σ'intr4 σproj2) + (σintr2 σproj4)] / (σ'intr2 + σproj2)2
 = (σintr2 σ'intr4 + σ'intr4 σproj2) / (σ'intr2 + σproj2)2
 = σ'intr4intr2 + σproj2) / (σ'intr2 + σproj2)2

We can substitute for σproj2 using σexcl2 = σintr2 + σproj2:

σincl2 = σ'intr4 σexcl2 / (σ'intr2 + σexcl2 - σintr2)2
σincl = σ'intr2 σexcl / (σ'intr2 + σexcl2 - σintr2)

And solving for σintr2 we find:

σintr2 = σ'intr2 + σexcl2 - (σ'intr2 σexcl / σincl)
σintr = √{ σexcl2 - σ'intr2 [(σexcl / σincl) - 1] }

This is an estimator of σintr. Ideally, σintr and σ'intr should be the same. One can iterate a few times starting with a good guess for σ'intr and then replacing it in later iterations with the σintr found from the previous iteration until the two are approximately equal.