Waveform parameterization and priors

The pysiggen parameters specific to each waveform, as discussed in Section 2.2.2, are position, energy, and charge cloud shape. However, for the purposes of the learning model, we use training waveforms from a monoenergetic photopeak, which allows us to model them with a common energy parameter1. This eases a strong correlation in shape that improves convergence of the charge trapping parameter, and will be described in greater detail in Section 3.2.3.

In order to compare thepysiggenwaveform with digitized data, an additional parameter is required to align the waveform in time with the digitized signal. The parameterization and prior for the time alignment, as well position and charge shape, are discussed below.

Position

Although it would be natural to model the position in cylindrical coordinates, it is impor- tant to recognize that the hole drift time is a principal driver of the overall waveform shape. From the isochrones seen in Figure 2.4, it is apparent that there is a strong anticorrelation in drift time between the radial and axial directions. To reduce the impact of this correlation, we perform the fit using polar coordinates, with the origin located at the point contact.

At present, it is assumed that there is no a priori knowledge of a waveform’s position, so the prior for each of the three positions is uniform over the appropriate range of values. For the radial and polar coordinates, this is anywhere within the detector active volume. The crystal axis dependence on drift velocity, discussed in Section 2.2.2, is periodic with a unique angular span of π/4 radians. We therefore set the azimuthal coordinate prior as uniform in the [0, π/4] quadrant, and recognize that any estimation of this parameter is eight-fold degenerate with respect to the true azimuthal position in the detector.

1_{Within a photopeak, there is a spread in energy caused by the statistical variance of the creation of charge}

carriers, which is a Poisson process. The training waveforms are taken from a narrow energy band around the photopeak maximum to account for this.

Charge cloud

The model uses the generalized gaussian model of Equation 2.5. During fitting, we have seen that waveforms at 2614 keV tend to cluster around σ = 20 ns andp= 2. We therefore set each prior with a wide normal distribution around these values: a standard deviation of 20 ns for σ and 10 for p. The sigma prior is limited by the range (1,40) ns, where the lower bound is chosen to prevent σ of less than one sample. The p parameter range bounds are (1,20), which corresponds to a gaussian distribution, and 20 is a value high enough to be indistinguishable from a uniform distribution.

Time alignment

The Demonstratordigitizer continuously digitizes the signal on each input channel at

100 MHz, storing the values in a temporary buffer. When the signal exceeds a threshold over the flat baseline value, defined by the output of an on-board trapezoidal filter, it triggers the digitizer to save the event to disk. To avoid triggering on noise, it is necessary to set the trigger threshold several standard deviations above the noise amplitude. A waveform with a short drift time will therefore cross the trigger threshold closer in time to its true start time than a longer drifting waveform, as shown in Figure 3.4. There is some additional uncertainty on the exact trigger point due to noise and the inherent 10 ns width of the digitization sample.

To enable high precision time alignment, it is therefore necessary to add a parameter which aligns thepysiggenwaveform with the digitized waveform. Aligning by the start point of the waveform would be most convenient. However, because drift time is highly correlated with position, this would tightly correlate the alignment time and position parameters. It advantageous to instead choose an alignment time point at a high fraction of the waveform rising edge, where all waveforms have relatively similar shapes. In addition to reducing correlation with other parameters, it is easier to make a robust prediction of a time point on the rising edge than the start time, especially for long drift-time waveforms which have very

900 950 1000 1050 1100

Digitizer Sample Number [10s of ns]

0 1000 2000 3000 4000 5000 6000 V oltage [adc value] 950 960 970 980 990 1000

Digitizer Sample Number [10s of ns]

0 20 40 60 80 100 120 140

Figure 3.4: Four digitizedDemonstrator waveforms from the 2614 keV Tl208 photopeak.

The plot at left shows the two microseconds of the signal roughly centered around the rising edge. The same waveforms are shown at right, more tightly windowed around the trigger point. The baseline value is subtracted from each waveform. The point at sample _∼ 985 where the waveforms roughly intersect is where the digitizer is triggered. The waveform with the shortest drift time, in blue, has less elapsed time between signal start and the trigger point than the longer drifting waveforms.

little signal amplitude at their start.

Because the pysiggen waveform is calculated in discrete time steps, aligning at an arbi- trary, continuous-valued time point requires interpolation. The siggencalculation produces an output in 1 ns time steps. To compare with digitized data,pysiggen downsamples to 10 ns increments and performs a piecewise-linear interpolation between the calculated points.

We have chosen to align based on the 95% rise-time of the waveform amplitude as the alignment point, defined as the point where the waveform reaches 95% of its maximum value. The prior is normally distributed around an estimated value from the parameter, which is calculated simply as the sample number where the waveform exceeds 95% of its maximum value. The prior has a standard deviation of one sample (10 ns).