Image parameterization

In the first step of the analysis, the charge in each pixel is determined by a trace analysis, an image cleaning is performed to reduce the noise contribution and the remaining image is parameterized.

Trace analysis

Two typical FADC traces, as recorded by the VERITAS cameras, are shown in Figure 3.8. The left part of the figure corresponds to a signal from Cherenkov photons. It shows the typical pulse shape of a PMT and is characterized by a fast rise to a maximum value (recorded as negative values in digital counts) and a slightly slower decay back to a baseline value, so-called pedestal. The arrival time T0 is defined as the time where the trace rises to half its maximum value. It

is corrected for relative time differences in the camera (see Section 3.1.2). The right part of Figure 3.8 shows the absence of Cherenkov light in the PMT, fluctuating around the pedestal value. The pedestal is an artifical negative offset which is added to the analog PMT signal while the fluctuations arise mainly from NSB photons and electronic noise. To determine the pedestal

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CARE (CAmera and REadout) is a simulation package developed by Nepomuk Otte; git clone http://www.gtlib.gatech.edu/pub/IACT/CARE.git.

value, the externally triggered pedestal events (see Section 3.1.2) are analyzed by summing up the digital counts in each pixel. The mean of this charge distribution is the pedestal and is approximately 16 digital counts (dc). The charge in each pixel is finally determined by summing up the signal of each FADC trace within a given time window, subtracting its pedestal value and applying the relative gain corrections (Section 3.1.2).

pedestal value FADC trace summation window summation window T 0 pedestal

value FADC _trace

T

0

Figure 3.8.: Two typical FADC traces with a sampling rate of 500 Mega-samples per second (i.e. 2 ns). (Left) Temporal pulse profile in response to Cherenkov photons hitting the PMT. (Right) The FADC trace in the absence of Cherenkov photons shows only fluctuations around the pedestal value (note the different ranges on the y-axes). The horizontal lines represent the pedestal value while the vertical dashed lines indicate the arrival time T₀. The grey shadow indicates the summation window used in the analysis for the charge extraction.

To minimize the contribution from NSB photons6 which are continueously recorded and dig- itized, a short summation window, covering the pulse from the Cherenkov light only, is used. This is typically done in two stages, so-called double-pass method (Holder et al., 2005). In the first stage, a wide integration window (typically 18 samples) is applied to each FADC trace and is used to calculate the charge and the arrival times. The resulting images are then cleaned and parameterized (as described in the next sections). In a second stage, a smaller integration win- dow (typically 7 or 12 samples) is positioned upon every trace in the camera. To determine the optimal position of the short integration window, the temporal development of the air shower has to be taken into account. Therefore, the arrival times as a function of the PMT position

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For dark-sky observations of a Crab Nebula-like FoV, approximately 0.13 photo-electrons/ns from the NSB are recorded. This number can increase dramatically when the Moon is above the horizon.

3.2. VERITAS data analysis

along the major axis (determined from the first-stage image parameterization) are fitted by a straight line. The slope of this fit is the time gradient and its value is used to define the start position of the integration window in the second stage. This method allows to use short integra- tion windows with optimal signal-to-noise ratios for small pulses and at the same time prevents signal losses due to significant time gradients in large, far distant showers.

Image cleaning

After the charge in each pixel is determined, an image cleaning is performed. It removes random background fluctuations from the compact shower image by applying tail cuts. These tail cuts can be either fixed or are based on the signal-to-noise ratio in each pixel7_{. The main motivation}

to use variable tail cuts is that the NSB conditions during a typical observation run can change, especially when operating under moonlight conditions. Therefore, every pedestal event within a given time slice (typically three minutes) is selected and the RMS of the pedestal distribution (its pedvar ) is determined. This pedvar is proportional to the NSB (noise) level as the pedestal events characterize the pixel behavior in the absence of Cherenkov light signals.

To determine whether a given pixel belongs to the shower image is done in two stages. In the first stage, each pixel with a charge greater then 5 times its pedvar is selected and forms an image pixel. In a second stage, a lower cut value of 2.5 times the pedvar is used with the condition that at least one of the neighboring pixels is already identified as an image pixel. If this cut is passed, the pixel belongs to the shower image and forms a border pixel. In case, an image pixel does not have any neighbor pixel surviving the cleaning procedure, it is removed as well. The remaining image and border pixels then define the image of the Cherenkov shower.

Hillas parameterization

Once the shower images have been calibrated and cleaned, each image is parameterized. The parameterization is based on the so-called Hillas parameters (Hillas, 1985). They are derived by the zeroth order (amplitude or size), first order (center of gravity) and second order (length, width, orientation) moments of the elliptical image of the shower and are listed in Table 3.2. These parameters describe the shape and the orientation of the image in the camera and are calculated using the formulas given in Fegan (1997).

As images at the edge of the cameras are difficult to reconstruct with this method, they are usually excluded from the analysis by cuts on the distance between the camera center and the image centroid or the loss parameter (the fraction of image size contained in edge pixels).

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In VERITAS, the signal-to-noise ratio is used while for the CTA analysis (see Section 4.3.2) fixed cleaning thresholds are applied.

Parameter Short description

size Total intensity of the image = sum of the integrated charge of all pixels (relates to the energy of the primary particle).

width The rms spread of charge deposit along the minor axis of the image

(relates to the projected lateral development of the cascade).

length The rms spread of charge deposit along the major axis of the image

(relates to the projected longitudinal development of the cascade).

centroid Coordinates of the center of gravity of the image in the camera (θx, θy).

distance The distance from the centroid of the image to the center of the field of view of

the camera.

fui The fraction of image size under the image ellipse. (can be used to remove less compact images)

loss The fraction of image size contained in edge pixels.

(can be used to remove strongly truncated images at the camera edge)

Table 3.2.: Image parameters describing the shape and the orientation of the image in the camera.

These quality cuts result in improved energy and angular resolution at the expense of less reconstructed gamma-ray events at multi-TeV energies. To recover those truncated images at the edge of the camera, a simple log-likelihood fitting algorithm is applied (Maier, 2010). The underlying assumption of the fitting method is that the image of a gamma-ray shower can be described by a two-dimensional normal distribution. Asymmetry is ignored to limit the number of fit parameters which are image centroid position, image width, image length and image size. Noise is estimated by assuming Poisson fluctuations. The starting values for the fit parameters are obtained from the standard image analysis. The following expression is minimized using the MIGRAD method of MINUIT (James, 1998):

L = −(niln Si− Si− niln ni+ ni) (3.1)

where n_i is the measured integrated charge for pixel i, and S_iis the estimated charge from the fit function. In general, this log-likelihood reconstruction method works well for loss values smaller than 50%. It results in an increase of effective area at high energies compared to the standard analysis.

3.2. VERITAS data analysis

3.2.3. Event reconstruction