Vertex Position and Ring Direction

There are three parts in the initial event vertex position and dominant ring fitting procedure: (1) point fit, (2) direction fit, and (3) TDC fit.

Point Fit

The event vertex position estimation is performed with the assumption that all of the light observed in the detector was emitted simultaneously from a single point source. To find the point that best matches this hypothesis, a three-dimensional grid of test points throughout the volume of the ID is considered, and the single best of these points is kept. Then the process is repeated with a finer granularity in a small region around the best point of the previous granularity level.

The best test point in the grid is determined by maximizing a goodness calculation, which is based on the hit times of the PMTs. The point fit goodness is defined as follows: Gpoint= 1 Nhit X i exp ₋ (ti− ¯t) 2 2 (a × σt)2 ! , (7.1)

where Nhit is the total number of hit PMTs, σtis the timing resolution of the PMTs

(2.5 ns), a = 1.5 is a factor to crudely account for light scattering, and ¯t represents the average value of the residual times of the PMTs, ti, which are defined:

ti = t0i −

di

v (di, qi)

, (7.2)

where t0

i is the recorded absolute time of the PMT hit, di is the distance from the test

vertex point to the PMT, and v is the effective velocity of light in water as a function of di and the recorded charge, qi, accounting for wavelength and acceptance effects.

This vertex fitting algorithm is guided by the assumption that all light in the event was originated simultaneously from the same point. In cases where this is not true, however, the reconstructed event vertex found by the point fit algorithm will tend to approximate an “average” event vertex (e.g., a point along the track over which a single muon radiates Cherenkov light, or a point between two spatially separated muons). The point fit vertex can still be safely used as a springboard for more accurate vertex fitting algorithms applied later in the reconstruction process, however.

Direction Fit

In the next step, the direction and Cherenkov angle of the dominant ring is esti- mated. The initial direction guess is found by calculating a charge-weighted vector sum of all light in the detector using the event vertex found by the point fit in the previous step. From there, a directional test grid in (θ, φ) space is generated, analo- gous to the test grid in the vertex point fit. A goodness calculation is again used, this time to find the best direction in the grid, along with the best Cherenkov opening angle.

The goodness used in the directional fit is defined as follows:

Gdir = RθC 0 Q(θ)dθ sin θC exp  −(θC − θmax) 2 σ2 a  , (7.3)

where θC is the test opening angle (allowed to vary), Q(θ) is the charge distribution

as a function of angle relative to the test direction, θmax = 42◦ is the maximum

Cherenkov angle assuming the particle’s velocity β = 1, and σa is the estimated rms

spread of PMT hits around θC. The test direction and opening angle which yield the

TDC Fit

The purpose of the TDC fitting step is to more precisely fit the vertex position by taking into account the finite track length of a massive charged particle and the effect of indirect light caused by scattered Cherenkov photons. The vertex, ring direction, and Cherenkov angle of the dominant ring found in the point fit and direction fit steps are used as inputs to this step.

In this step, a modified residual time calculation for the PMT hits inside the Cherenkov cone is used:

ti = t0i −

1

c| ~Xi− ~O| − n

c| ~Pi− ~Xi|, (7.4) where ~O is the test vertex position, ~Xi is the position along the track where photons

would be emitted toward the i-th PMT, n is the index of refraction for water, ~Pi is

the position of the i-th PMT, and t0

i is the recorded absolute time of the hit of the

i-th PMT.

The goodness for PMTs inside (GI) and outside (GO) the Cherenkov cone are

calculated slightly differently, as shown below:

GI = X i 1 σ2 i exp ₋ (ti− ¯t) 2 2 (a × ¯σ)2 ! , (7.5) GO = X i 1 σ2 i max " exp ₋ (ti− ¯t) 2 2 (a × ¯σ)2 ! , 0.8 exp  −ti− ¯t 20 ns # , (7.6)

where σi is the timing resolution of the i-th PMT, ¯σ is the average PMT timing

resolution, a = 1.5 is a factor to account for indirect scattered light, ti is the residual

time of the i-th PMT, ¯t is the average residual time of the PMTs, and 20 ns is the average time difference between direct and scattered light. The GI goodness value

and the GO goodness value are calculated separately for PMT hits inside (θi < θC or

ti < ¯t) and outside (θi > θC or ti > ¯t) of the Cherenkov opening angle, corresponding

to the direct and scattered light, respectively.

The final overall goodness for the test vertex position is then calculated using both GI and GO, as shown below:

Gtotal =

GI+ GO

P

iσi

. (7.7)

The track length used in the calculation is estimated by summing the charge within 70◦ _{of the direction of the ring and then calculating the corresponding muon}

momentum required to produce that amount of light. Using this track length, the total goodness, Gtotal, is maximized as a function of vertex position and ring direc-

tion. Then the procedure is repeated, calculating a new track length and varying the vertex position and ring direction to maximize the total goodness once again. This is repeated in an iterative process until a final, stable fit is reached.

The resolution of the TDC fit vertex, determined by the distance between the reconstructed vertex position and the true vertex position of Monte Carlo events, is shown in Fig. 7.1 for various subsamples of single-ring atmospheric neutrino events. The resolution falls between 50 cm and 90 cm, depending on the sample.