11 B. Event Position Reconstruction FIG. 16: (Color online) A typical event showing S 1 and S 2 and their hit patterns on the two arrays of PMTs (from [41]) than 0 . 4 µs it will be considered as an S2 pulse and its lo- cation and boundaries will be saved. The recursive search for S2 pulses then continues within the interval (exclud- ing the regions where any pulse might already have been found). The parameters of the four largest S2 pulses are kept. A key difference in the MATLAB-based analysis was the development of a custom double box filter [46], with the width of the two boxes set to 10 and 100 digi- tizer samples, respectively. These choices correspond to the approximate width of S1 and S2 signals. This fil- ter proved extremely adept at correctly identifying the smallest S2 pulses. The S1 pulse finding algorithm searches the total wave- form for signal excursions of at least 12 mV above the baseline. The boundaries of the potential pulse are de- fined as the points where the signal drops below 0.1% of its maximum. If the 0 . 5 µs preceding the pulse and the 50 ns following the pulse respectively have an average signal less than 0.5% and 4% of the maximum, and if the decay time of the pulse is faster than 200 ns, then its location and boundaries will be saved. The parameters of the two largest S1 pulses identified are kept. Once all the pulses have been identified, the reduction enters its last stage, computing the quantities of inter- est. Each PMT waveform is integrated over the bound- aries of the S1 and S2 pulses identified and the values are converted into photoelectrons using the PMT gains. For each pulse a number of quantities are computed: FWHM, height, mean arrival time, number of PMT coincidences, number of digitizer channels saturating, etc. The XY position of each S2 pulse is computed using two differ- ent position reconstruction techniques (see section V B). A few parameters of the entire waveform are also com- puted: its total area, the average and RMS of the baseline before any pulses, and finally the total S1 and S2 signals. An important feature of the XENON10 detector is its ability to localize events in 3D, enabling background re- duction with fiducial volume cuts and multiple scatter events rejection. Due to the small spread of the drift- ing electrons by diffusion in LXe [47], the proportional scintillation signal is localized to a region with the same XY coordinates of the interaction site. Near the edge of the TPC, field non-uniformities lead to systematic ra- dial displacements towards the center of about 1 mm. Charge focussing effects in the extraction region between gate mesh and anode lead to charge displacements of less than half the mesh pitch, i.e., < 1 mm. The event z - coordinate, along the electric field direction, is inferred from the time difference between the primary and sec- ondary signals, with a resolution which is < 1 mm given the precision of the time measurement. Events with en- ergy depositions at different z positions will produce more than one S2 signal and can thus be readily identified and rejected. To obtain the XY event localization, with a few mm resolution, much smaller than the dimension of the PMTs, the light pattern measured by the top PMT array must be de-convolved. Two light pattern recognition al- gorithms were developed and tested on XENON10 data: a minimum χ 2 algorithm and a neural network ( NN ) based algorithm. The minimum χ 2 algorithm compares the measured light pattern with simulated S2 signals, calculating the values for all simulated XY positions. In this formula S$_{i}$ and s$_{i}$ are the measured and simulated S2 signals (in num- bers of photoelectrons) on the i -th PMT, M is the total number of PMTs (48 in the top array), and σ$_{i}$ takes into account the uncertainties of both the measured and simulated signals. The minimum value of the χ $^{2}$( x, y ) function is used as an estimator for the true position in the XY plane. If the statistical uncertainty in the sim- ulated S2 pattern is small, the main contribution to σ$_{i}$ is the fluctuation in the measured signal, which includes the statistical fluctuation on the photoelectric emission from the PMT photocathode σ$_{pe,i}$ and the fluctuation σ$_{g}$$_{i}$ on the gain g$_{i}$ . If the number of detected photoelec- trons is sufficiently large, σ$_{pe,i}$ can be approximated as σ$_{pe,i}$ ≊ √$_{S}$$_{i}$ while σ$_{g}$$_{i}$ is measured for each PMT through its single photoelectron spectrum σ 2 i = σ 2 pe,i [ 1 +$^{(}$ σ$_{gi}$ g$_{i}$ ) 2 ] = S$_{i}$$^{[}$ 1 +$^{(}$ σ$_{g}$$_{i}$ g$_{i}$ ) 2 ] (3) χ $^{2}$( x, y ) = M ∑ i =1 [ S$_{i}$ − s$_{i}$ ( x, y )] 2 σ 2 i (2)