Characterizing Hit Residuals

The position uncertainties ("errors") returned by the TPC hitfinder are notoriously unreliable. I believe this to be a generic problem, and one way to supply better errors is to use the data itself, as well as some physical model of what is causing the errors (diffusion, crossing angle effects, hitfinding algorithms), to characterize the errors via the observed residuals (and their dependences on local parameters), and feed this information back into the tracking.

So a while ago Raimond and I did this-- using his QACosmicMaker, we characterized the residuals and did a fit with a physical model, and wrote up a little function that could be used in the tracking, that gave the expected uncertainty, given the drift distance and crossing angles. Somewhat strangely, this has become called "using the tpt errors".

The residuals and physical fit from that first study can be found here. I think this is what is currently being used in reco.

There is some feeling that perhaps this needs to be redone, and, indeed, it is in principle an iterative proceedure. Also, one wonders about possible differences between primary and global tracks, or different tracking algorithms in general.

For this sort of study, it is natural to interface with StEvent, so I have written a little maker that characterizes the residuals (in z and local-x, i.e. along the padrow) in terms of driftlength and crossing angles (α for local-x, λ for z).

Low vs high pT - significantly different?

For a run-through of about 500 events the results are in the plots below. The residuals and their dependence on drift and crossing angle look good at least for outer sector hits.
Primary Tracks
 Inner SectorOuter Sector
X-residual pT=0.1-0.6
pT=1.0-5.0		 pT=0.1-0.6
pT=1.0-5.0
Z-residual pT=0.1-0.6
pT=1.0-5.0		 pT=0.1-0.6
pT=1.0-5.0

Global Tracks
 Inner SectorOuter Sector
X-residual pT=0.1-0.6
pT=1.0-5.0		 pT=0.1-0.6
pT=1.0-5.0
Z-residual pT=0.1-0.6
pT=1.0-5.0		 pT=0.1-0.6
pT=1.0-5.0
Although high pT tracks provide less of dynamic range in x-crossing angle, I think one can see a difference in residuals for low and high pT tracks.

I do not see a big difference in the track-wise chi-square value for tracks with pT=0.1-0.6 and tracks with pT=1.0-5.0, although I expected one from tracking discussions. There is a difference in confidence levels, however.

A next step is to fit these with the physical model, essentially regenerating new constants for our little function used in the tracking.

This is currently in progress. As a preview, you can see a few of my fits so far in this directory

(It is a 3-parameter fit, where diffusion, "intrinsic resolution" and a crossing-strength term are the parameters. When the fits are all good, we can check the parameters against expectations (e.g. diffusion const) to see how much sense they make physiscally. At some level, however, that is not the point here.)

02Mar2001 - Fits to Residuals from P00hm production

We fit the residual widths as a function of drift distance of the hit, and of crossing angle. If it is σx-local, the crossing angle is α (angle of local transverse momentum vector relative to normal to padrow), and if it is σz, the crossing angle is λ, the dip angle.

The functional form used is from Blum & Rolandi

Where
  • σint quantifies an "intrinsic" resolution
  • σdiff quantifies the effect of diffusion
  • σcros quantifies the effect finite crossing-angle and wire coupling
    Note that for simplicity, other terms have been absorbed into the σ's, so that σdiff is not directly the diffusion constant in transverse (longitudinal) direction for σlocal-x (σz), but is related to it.

    Anyhow, the fits mostly but not always do a good job. Adequate, I would say, for inputting back into the reco. An exception is really the x-residuals for inner sector; these are not looking too reasonable I would say.

    RMS Distributions
      σ2int σ2diff σ2cros Plot of fit
    X, Inner Sector, Global 2.16x10-2 1.73x10-4 3.62x10-7 BAD!
    X, Inner Sector, Primary 1.13x10-2 1.93x10-4 2.12x10-2 OK
    X, Outer Sector, Global 8.24x10-3 1.45x10-4 2.81x10-1 OK
    X, Outer Sector, Primary 5.60x10-3 1.04x10-4 8.89x10-2 OK
    Z, Inner Sector, Global 6.32x10-2 3.15x10-4 1.58x10-6 OK
    Z, Inner Sector, Primary 3.36x10-2 4.29x10-4 1.24x10-2 OK
    Z, Outer Sector, Global 7.36x10-2 1.09x10-4 1.49x10-1 OK
    Z, Outer Sector, Primary 2.68x10-2 2.18x10-4 8.52x10-2 OK

    Gaussian Sigma

    Instead of RMS of the hit residuals, Al suggested extracting the Gaussian sigma, which is in general smaller. These fits are shown here.

    Gaussian Sigma Distributions
      σ2int σ2diff σ2cros Plot of fit
    X, Inner Sector, Global 2.70x10-4 6.23x10-5 1.70x10-2 OK
    X, Inner Sector, Primary 2.33x10-3 8.57x10-5 6.00x10-2 OK
    X, Outer Sector, Global 3.88x10-7 2.84x10-5 2.19x10-1 OK
    X, Outer Sector, Primary 5.75x10-4 4.20x10-5 9.29x10-2 OK
    Z, Inner Sector, Global 9.19x10-3 2.64x10-5 4.02x10-2 OK
    Z, Inner Sector, Primary 7.75x10-3 1.13x10-4 8.65x10-2 OK
    Z, Outer Sector, Global 1.62x10-2 2.36x10-5 1.40x10-1 OK
    Z, Outer Sector, Primary 7.75x10-3 1.13x10-4 8.64x10-2 OK

    13 Mar 01

    Helen suggested to refine the above study by
  • Only considering those global tracks which have as well primary nodes -- this way, any differences between global and primary is due to the fitting itself, not also due to the fact that the track sets under consideration are not identical.
  • Only look at those hits which are used in the fit. This should not be a huge deal, since outlier rejection by kalman doesn't throw away that much.

    Not too much changed...

    Gaussian Sigma Distributions
      σ2int σ2diff σ2cros Plot of fit
    X, Inner Sector, Global 1.36x10-6 4.09x10-5 8.39x10-2 OK
    X, Inner Sector, Primary 2.80x10-3 8.28x10-5 4.80x10-2 OK
    X, Outer Sector, Global 1.69x10-8 1.90x10-5 2.64x10-1 OK
    X, Outer Sector, Primary 5.44x10-4 4.02x10-5 9.04x10-2 OK
    Z, Inner Sector, Global 1.12x10-2 2.60x10-5 3.73x10-2 OK
    Z, Inner Sector, Primary 7.69x10-3 1.82x10-4 4.39x10-2 OK
    Z, Outer Sector, Global 1.92x10-2 3.33x10-9 1.30x10-1 OK
    Z, Outer Sector, Primary 7.58x10-3 1.08x10-4 8.30x10-2 OK

    Similar analysis done on /star/data05/reco/central/DEV01b/ data

    This data has the "hit indexing" problem fixed. This problem affects in principle both the globals and the primaries, as the hits "used in fit" were mis-labelled prior to the fix. As outlier rejection does not throw away huge numbers of hits, the effect of this fit is not expected to be large.

    Other fixes have also been applied to this data, as compared to the hm data analyzed above, which was run in December 2000.

    Gaussian Sigma Distributions
      σ2int σ2diff σ2cros Plot of fit
    X, Inner Sector, Global 1.54801e-07 3.12966e-05 8.57816e-02 Not so hot
    X, Inner Sector, Primary 1.68243e-03 5.23272e-05 5.75341e-02 OK
    X, Outer Sector, Global 6.83660e-08 1.65249e-05 2.28662e-01 so-so
    X, Outer Sector, Primary 2.02775e-04 3.55219e-05 6.45610e-02 OK
    Z, Inner Sector, Global 6.65408e-03 5.69640e-05 2.51232e-02 OK
    Z, Inner Sector, Primary 3.12735e-03 1.51055e-04 2.43806e-02 OK
    Z, Outer Sector, Global 1.68737e-02 4.33648e-09 1.00589e-01 OK
    Z, Outer Sector, Primary 8.15800e-03 5.69582e-05 4.48440e-02 OK

    Issues

    I would say that the Sigma fits should be input to reco, to see if Kalman and its confidence levels, etc. are any happier. Fits are not perfect, but should be good enough.

    Not understood:

  • In the residuals themselves (not the fit to them), why is there essentially no diffusion effect in the z-residuals for global tracks?? And only on the outer sector????
    Michael A. Lisa
    Last modified: Wed Mar 14 09:27:24 EST 2001