Point Spread Function (PSF) Project
started January, 2002
last updated September 5, 2002
The Goal
Improve clustering mechanism by exploiting the shape of clusters, depending on the energy and radial position, to distinguish between one and two cluster hypothesis.The Plan
- A:
- Use single photon simulation to guess a form of PSF that will describe the shower shape depending on energy and direction.
- Make 1-cluster fitter based on PSF that depends only on (x0,y0,E), or (theta,phi,E).
- B:
- Obtain a good sample of isolated clusters from real data.
- Test the shape of PSF on real data, modify if necessary.
- Compare resolution and efficiency with current clusterizer, and decide on further action.
- C:
- Design a multi-cluster fitter using MC data.
- Refine fitter/finder using real data.
- Compare resolution/efficiency with present clusterizer.
- Make recommendation for further analysis based on cluster-finding efficiency table.
The first guess
The very first guess was single cylindrical symmetric Gaussian. Of course, it was clear that this function cannot describe the shower shape. The next first guess was Gauss-like function with elliptic width and center of gravity at focal point. It was working for low energy photons, but it was not able to reproduce higher (>2.0 GEV) showers.The next try
I again used Gaussian, elliptically symmetric with constant widths and with center of gravity at the pole of ellipse. I also tried the function proposed by Scott: a linear combination of exponential and Gaussian, parametrized according to sheared energy between them. There was some improvement, but still not satisfactory. There were always blocks where difference between MC data and PSF was substantially larger then errors.The failure to fit data with analytic function is probably connected with the fact that MC data are peaked at central block with usually one or two blocks sharing most of the shower energy. I calculated shower size from hits list, and got similar values as Scott got for cluster widths for different energies and angles. However, sigmas from fits are still 1.5-2.5 smaller then cluster widths, and accidentally or not, coincide with uncertainty in shower position. This basically confirms that MC showers are not Gauss-like.
Fitter
It can fit event by event both MC and real data. The package is build to use PAW interface. However, it should be flexible enough to be used as stand alone program. The concept it follows is:- put data from an event into memory
- initialize starting values for E,X0,Y0 (the shower energy and center in the LGD plane)
- take any additional parameters from data file
- fit, currently using just MIGRAD
- report
, with
,
where r is the radial distance from the shower center,
L size of the block and E total shower energy.
The block energy error used to find
has been taken as
, with A=0.08,
while B has still to be determined. 
is the energy of the block and 
Gaussian at the center of the block.
The Gaussian width (sigma) in G(x,y) is fixed by choosing large error offset B,
in order to make all blocks equal during fitting (small errors make fitter
to adjust width for just one "relevant" block, the one with maximum energy).
Here is an example of fit for two different
initial widths:
Average fit value of the sigma from the above sample was 1.7. When sigma is fixed, B is reduced to the size that gives reasonable Chi2 for 1.0 GeV showers (~ 1.) Here are the results for Sigma=1.7 and B=0.008:
- energy and chi2
- ndof and chi2
- box view of Data and PSF
- color view with errors and Psf-Data difference
,
where r is again distance from the center of the block
to the center of the shower.
May 25, 2002
The importance of 2-nd order correction can be seen from following table. When Gaussian width is lees then or comparable to the block half-size 2-nd order corrections are significant. When this correction is taken into account the Gaussian width droped to ~ 1.4 cm (fitted with large B). Then errors are set back to A=0.08, B=0.01. Resulting fit widh fixed width independently on shower energy looks little worse comparing to the previous case. One might expect that if the shape of the showers is not the single Gaussian. The more accurate we are in representing Gaussian the more inaccurate we reproduce data. The last step in this (first step) study of MC showers is to try two Gaussian function fixing one by one parameter (Gauss+Exp function with 2-nd order corrections might be to complicating to implement).June 10, 2002
With integrated function used as PSF, radial width droped to ~1.1 cm. The central blocks are fitted slightly better, but overall fit did not change substantially (energy is stil biased to lower values). For such a small width both zero and 2-nd order expansioan of the integral differ from I.A summary of this study of isolated clusters from MC resulted in technical note
August 8, 2002
To check fitter on real data Richard generated large sample of isolated clusters, with theta=(5, 10 , 15, 20, 25) and phi=12. The first look at the shape of clusters with single Gaussian shows that real photons are similar in shape to MC shower but with larger radial width: the first attempt ,
the second attempt .
August 19, 2002
Energy comparison between the fitter (single Gaussian) and present clusterizer for 500 events.
Profile of Radial width as a function
of fitted energy from 500 events.
