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Relative Thresholds (1 PE Fast Pulse Amplitudes) from TOTs: Part II
5 September 2001
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Albrecht and I have developed a new method for determining the relative threshold (or the amplitude of the prompt pulse) from TOT information. Here we present the progress made on this technique since the method was first described in May at (http://alizarin.physics.wisc.edu/steele/research/thresh/Thresh.htm)
This method, and the information it provides, would be useful for AMANDA-II simulations. Previously, the only way determine the prompt pulse amplitude was by measuring the ratio of the prompt to delayed amplitude (see Steve Barwick's message on ABS under Data Related -> Delay/undelay calibration 2000 ). If this new method is successful, it can be used on previous years' data for which no direct measurements of prompt vs. delayed amplitude is available. The method can also be used to determine a possible time dependance of relative threshold from the data.
The method depends on the TOT being accurately described by the monte carlo. When tested on monte carlo we have successfully reproduced the 1 PE amplitudes as set in the AMANDA-II geometry file to within 5% Here, we present the method and resulting list of 1 PE fast pulse amplitudes as deduced with this method from 2000 data.
The AMANDA-II geometry file contains several parameters, including the "threshold" and "1 PE/mV". As seen in the following schematic figure for typical AMANDA-II electronics, an optical signal arrives from the OM and is converted into an electrical signal by the fast ORB. The resulting signal is referred to in this report as the "fast pulse." AMANDA-II monte carlo simulations require as input the amplitude of the fast pulse which corresponds to 1 photoelectron (point A in the figure below). Also required is the threshold setting of the DMADD (point B). The threshold is set directly, (and is thus known exactly) whereas the 1 PE fast pulse amplitude is set indirectly by the fast ORB gain.
In this analysis, we define the "relative threshold" as the fixed DMADD threshold setting divided by the 1 PE fast pulse amplitude.
relative threshold = DMADD threshold / 1 PE fast pulse Amplitude
Since the DMADD threshold is fixed, an increase in relative threshold corresponds to a decrease in 1 PE fast pulse amplitude. Obviously, given the DMADD threshold, we can determine the 1 PE fast pulse amplitude from the relative threshold:
1 PE fast pulse Amplitude = DMADD threshold / relative threshold
The Time-Over-Threshold (TOT) is defined as the amount of time which the fast pulse signal exceeds the DMADD threshold. It stands to reason that the lower the relative threshold (the higher the 1 PE fast pulse amplitude), the longer the measured TOT. We seek to use this relationship to extract the relative threshold (and thus the 1 PE fast pulse amplitudes) from TOT data.
In the past, the 1 PE fast pulse amplitude have been determined by correlation studies between the delayed pulse (ADC) and the fast pulse (see Steve Barwick's note "Delay/undelay calibration 2000" posted on the ABS page). We believe this method to be problematic because the delayed ORB gain is tunable independently of the fast pulse gain, so there is no direct relationship between the two. These calibration data are currently what supply the 1PE/mV parameters in the AMANDA-II geometry file. Such calibration data are only available for AMANDA-II channels. The parameters as set for B-10 channels remain rather arbitrary. Hopefully our new method can also be adapted and applied to the old analog channels, and supply better 1PE/mV values for 2001 data, as well as for years for which there is no calibration data at all.
To use this method, we must assume that all standard AMANDA-II pulses (of a given category, i.e. optical) are identical in shape. Furthermore, we assume pulses vary only in their amplitudes, and are not stretched on in the time axis. We believe this to be a reasonable assumption because each electron propagates through the dynode stack of the PMT independently. Should further accuracy be desired, one could compile a waveform database containing an average pulse for each OM.
We start by determining the function Th( TOT ) which returns a relative threshold given a TOT. This was done by using a digitized average pulse of an AMANDA-II optical module obtained from Stephan's AMANDA-II webpage The fast pulse appears on the left in the figure below. To compute Th( TOT ), we simply measure the TOT as we move the relative threshold up the pulse. The resulting Th( TOT ) function (with 5 degree polynomial fit) is shown on the right side of the figure.
The following fifth degree polynomial is a good fit to the numerically computed function in the region 4ns < TOT < 55ns.
Th( TOT ) = 0.907 + 0.03351(TOT) - 0.00546(TOT)2 + 0.0001874 (TOT)3 - 0.000002669(TOT)4 + 0.00000001398(TOT)5
Now we examine TOT distributions for individual channels, a few of which are plotted in the next figure.
The question then becomes, how do we choose a TOT from this distribution which will lead us to the correct 1 PE fast pulse amplitude via our function Th(TOT)?
To test how well our developing method worked, we performed each iteration of the analysis on monte carlo and tried to reproduce the 1 PE amplitude as it was already set in the geometry file.
Our first attempt was to use the TOT corresponding to the peak of the TOT distribution for each channel. The results showed that using this choice of TOT, we could determine the 1 PE fast pulse amplitude to within only 30 to 50 percent.
Next we tried using the median TOT of the distribution. This method got us closer, down to a 25 or 30 percent error.
Finally, we computed the average threshold for each OM by summing over the number of events of each TOT mulitplied by the threshold corresponding to each TOT and dividing by the total number of events. This is actually the worst, resulting in 1PE/mV values that are up to 80% off the true value.
The relative errors of the calculated amplitude vs calculated 1PE amplitude as just discussed appear in the left plot of the figure below. In this figure, black corresponds to the peak TOT method, red to the median method, and blue to the average threshold method.
The figure on the right shows the actual 1PE/mV as set in the monte carlo plotted against the calculated value. We observe that the red curve (median method) comes closests to a straight line, but having a slope less than unity. This observation suggests that instead of simply calculating the threshold and 1 PE amplitude from a TOT read directly from the distribution, we can add one extra step to reduce our error. Once we have calculated a 1 PE amplitude using a given method, we then find another function that returns us the correct 1 PE amplitude (with even smaller error) given the calculated amplitude. This function is exactly what is plotted on the right, above.
The following linear parameterization of the red curve is valid in the range 50 mV < (Calculated 1PE/mV) < 1300 mV:
1PE Amplitude = -13.85 + 0.902 * (Calculated 1PE/mV)
The following figure shows the actual amplitude vs calculated for the median method with the above fit superimposed.
Plotted below are the relative errors vs OM for the final result. We are able to determine monte carlo 1PE fast pulse amplitudes to within 5%.
Here is a link to a complete amasim geometry file containing 1pe/mV values calculated by this method using 2001 data for OM's on strings 14, 15, 16 and 19 having a threshold of 50 mV.
Of course the success of this model depends on how well the TOT is modeled by the monte carlo. As of this writing, the TOT is NOT described well by the simulation, but work is in progress. Therefore, the above geometry file should be used with caution.
In addition, an investigation of the systematic errors due to fast pulse amplitude uncertainty is underway and results will be summarized on this page when complete.
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