Grid Fitter

The grid fitter is in essence a hybrid fitter which combines elements of the box fitter of T. Anderson and the maximum likelihood fitter of J. Klein. Like the time fitter, the fit proceeds in two sequential steps: first a vertex is fit and then the direction is fit, from the fitted vertex.

Only timing information is used in fitting the vertex, i.e. angular distributions are ignored. This choice was made in order to keep the grid fitter as general as possible, so that it can simultaneously fit different classes of events which can have very different angular distributions.

The first step in the fitting procedure is to search a coarse 3D grid of approximately one meter spacing across the entire detector volume, i.e. a sphere of approximately 10 meter radius. At each 3D grid point (fixed x,y and z coordinates) a log-likelihood function is maximized with respect to time, the only remaining free parameter of the vertex. Then, after the grid search has been completed, Ngrid best grid points (highest log-likelihoods) are used as starting points for maximizing the same log-likelihood function, but this time in 4D, i.e. with x,y,z, and t (of the vertex) all free parameters. Then, the highest log-likelihood from all the 4D searches is used to determine the best fit vertex.

The log-likelihood function is calculated in the dimension of residual time, $t_{res}$, which, for the ith PMT hit, is equal to the hit time, $t_{hit}(i)$, minus a fitted time, $t_{fit}$, and minus the time it takes light to travel from a fit vertex position, $r_{fit}$, to the ith PMT position, $r_{pmt}(i)$, at speed $c/n$:

$$t_{res}(i) = t_{hit}(i) - t_{fit} - \vert r_{fit} - r_{pmt}(i)\vert n/c$$ (17.1)

Thus, $t_{res}$ depends on the x,y and z values of $r_{fit}$ and on $t_{fit}$. If one fixes $r_{fit}$ at the true vertex position and fixes $t_{fit}$ at zero then the $t_{res}$ distribution, averaged over all PMT hits and over many events, is a gaussian centered on the true event time, with a slightly larger sigma than the PMT transit time spread and with long non-gaussian tails due to scattering, reflections and PMT noise. If one allows $t_{fit}$ to be fitted in 1D (see procedure below) for each event separately, whilst keeping $r_{fit}$ at the true vertex position, an almost identical distribution in $t_{res}$ is obtained apart from a shift in $t_{res}$ (it is now centered on $t_{res} =0$) and a very slight sharpening of the gaussian peak. This distribution ($P(t_{res})$) is used as the probablity density function (p.d.f) for calculating log-likelihoods.

At present an `average' p.d.f. is used which was generated using AV and PMT beta-gamma events. A future development, first suggested by C. Jillings, would be to make the p.d.f. itself a function of position (by generating different types of event in different locations). This will require the p.d.f. to be normalised since its shape would depend on the parameters which are being fitted.

Having obtained a p.d.f. ($P(t_{res})$) one can fit $t_{fit}$ in 1D (keeping $r_{fit}$ fixed) or $t_{fit}$ and $r_{fit}$ simultaneously in 4D by maximizing the log-likelihood function

$${\rm log-likelihood} = \Sigma_{i=1}^{\rm Nhits} \log(P(t_{res}(i)))$$ (17.2)

where $t_{res}$ is given by Eq. 1. Both types of maximization (1D and 4D) are performed using the CERNLIB MINUIT package routines, and in particular the routine MIGRAD with the strategy set to maximum reliability which also means slowest performance.

In order to speed up the initial grid search it is important (for each grid point) to start with a good initial guess for $t_{fit}$ before the 1D maximization. This is accomplished by an iterative process of averaging $t_{res}$ values and cutting out-of-time hits over several loops where the time window of accepted hits is successively narrowed until it is only $\pm 5$ ns. If, at any point during this process, the number of accepted hits falls below a prescribed fraction of the available hits, then the grid point is automatically discounted and the search moved on to the next grid point without performing the 1D maximization. Setting this fraction at 0.4 greatly increases the speed of the grid search and has practicaly no effect on the fitter's efficiency and reliability.

The fitter also has the option of using different p.d.f. in the fitting. There is a switch to turn on a process, which works similarily as the time fitter by throwing out the bad hits through iteration, after the 4D minimization converges.

The direction fit has two options: 1. uses the well known simple algorithmn of adding unit vectors (from the fitted vetex to each PMT hit) for all hits whose $t_{res}$ values fall within $\pm 5$ns of the final fitted time. 2.a maximum likelihood method based on a p.d.f. generated by the Monte Carlo.