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Hit errors in MarlinTPC [message #2013] 
Fri, 04 June 2010 10:12 
killenberg Messages: 125 Registered: July 2005 Location: CERN 



Hello,
I would like to discuss the implementation of hit errors in MarlinTPC.
Currently the RowBasedHitFinderProcessor is the only one which fills the covariance matrix at all. It uses the variance of the charge distribution, which for my understanding is not the value which should be in there (LCIO documentation: covariance of the position, not of the charge distribution).
If I understand it correctly the covariance matrix is supposed to hold the variances of the residual distributions, which are the errors of the hit position. These errors are definitely required in track fitting.
However, these values cannot so easily be computed from the charge distribution itself. The statistical answer for the error of the mean (sigma/sqrt(n)) does not work because the entries in the bins are not statistically independent measurements of the same value but ADC counts of the measurement of a suppositionally Gaussian shaped charge cloud.
If I assume no gas gain variations, electronics noise and infinite electronics resolution, the only error is the systematic error due to the binning onto the pads (pad response) when using centre of gravity to calculate the coordinates. After applying pad response correction the error is zero.
So the errors are all due to gas gain fluctuations, electronics noise and systematics which require additional parameters that are not in the hit itself. I see two solution to calculate the errors:
 Try to evaluate the error from the hit
x_bar = sum( x_i q_i) / sum (q_i) ;
sigma^2 = sum( (d x_bar / d q_i * delta q_i)^2 )
Here everything depends on the knowledge of the errors of the charge measurement on the pads.
 Advantages:
 Errors are hit dependent, actually measured charge is considered.

Disadvantages:
 Knowledge of the errors delta q_i strongly depends on gas mixture, gas gain, amplification structure, electronics etc.
 Requires extensive studies to determine the dependencies.
 Does not include errors due to pad response (systematics of the CoG method).
 Apply a mean error by parameterising the residual distributions as
sigma = sqrt( sigma_0^2 + d^2*z )
where d is the diffusion dependent contribution of the resolution. This is the way the resolution usually is parameterised in dependency on the drift distance.
 Advantages:
 Includes also systematic errors due to pad response
 Parameters easy to determine
 Disadvantages:
 Only mean error per hit is used, charge and shape of charge distribution are not considered
 A posteriori application of the error. First the track has to be fitted to calculate the residuals, only then the errors can be determined. (Two runs: First track fit with errors set to 1, second track fit with errors. Or errors determined externally by another run or MC simulation)
I am not sure which version I should implement, probably we want both to compare. Or make a more complicated parametrisation of the residual distribution in dependence on the charge, width of the hit etc.
On the technical implementation of the calculation (class design etc.) I will make another thread because I also have some questions on that.
Cheers
Martin
P.S. Does anyone know how to get formulas into the forum? Not even inserting individual images works since one can only upload one file per message. I attached one image with the two definitions. Not even forcing a line break is possible. Sorry

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[Updated on: Fri, 04 June 2010 10:21] Martin Killenberg
CERN
martin.killenberg@cern.ch


  
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