This report is in two parts. First is the report of the analysis concerning the hole ice issue, and secondly analysis looking at possible variations in the bulk ice as a function of depth within the detector. Note, if you want a print out of all the figures and tables in the report, click here to download the gzipped postscript files of these figures. (Note, you must click Save As under File to download. And to untar and unzip the file on UNIX machine do gtar -xzf psfigures.tar.gz . A directory called figures will be created, containing the 45 postscript files of the figures and the seperate tables from this report. You'll at least 45 Mbyte of memory to successfully download).
Scott Young, John Kim, and Pat Mock have been trying to understand and quantify the impact of the bubbly-hole ice on muon reconstruction. We have used the AB coincidence events from 1997 to address this issue. Thus far, the analysis can rule out several of the hole-ice photon tables created by Albrecht - the no-hole ice and the 10cm hole ice at the 2 sigma level. Our best fits suggest the 50 cm model. The results are summarized in figure. 1 , with the details tabulated in seperate tables. (Real Data), (MC, no hole ice), (MC, hole ice l_scatt. = 1m), (MC, hole ice l_scatt. = 0.5 m),(MC, hole ice l_scatt. = 0.3 m), (MC, hole ice l_scatt. = 0.1 m). The plot below uses photon tables having the effective scattering length of 24 m, and absorption factor of f096. The uncertainties in the seperate tables are inclosed by paranthesis. The mean hit probability difference of each string is the average of the four hit probability differences of the downfacing OMs from the upfacing OM. The error of the mean is taken from the RMS value of these 4 hit probability differences. Then the final mean is the average of the 3 hit probability differences for each string and the error of the mean is again the RMS of these 3 averaged values.
One additional point about the hole ice results is doing this analysis on MC with photon tables of scattering length of 28 m with the same absorption factor we find an increase of about 0.04 in the hit probability differences. (Fig. 2) This isn't to surprising since in ice with less scattering we expect the hit probability difference between the upfacing and downfacing OMs to be greater. Note that the result agrees with the above plot in the difference between the no hole ice case and the hole ice case with a scattering length of 50 cm. This result raises the issue that with this analysis, which choice of hole ice parameters that best matches the data, also depends on the choice of the bulk ice parameters. In Fig. 2, for example the choice would probably be hole ice scattering length of 30 cm.
There were several ideas guiding our analysis:
1) Vertical tracks are relatively simple to simulate reliably. Large statistical samples can be obtained for a large variety of ice conditions so a thorough probe of the parameter space can be done quickly. Also, we can check the results in many details. We have used the 64-node AENEAS supercomputer at UCI (after struggling with bugs in the upgraded SIEGMUND package, and working through DEC UNIX/LINUX differences) to reconstruct events using RECOOS and BASIEV/AMASIM2 to simulate the events
2) The analysis of hole ice relies on a difference between the hit probabilities between up-facing OMs (OM 10, 30 and 70) and their neighboring downfacing OMs. The asymmetry between and upfacing OM and downfacing OM to downward traveling nearly vertical muons should provide a sensitive probe of the effects of hole-ice. MC simulations suggest that the difference in up-down hit probability should range between 5-20%, so a measurement of 3% can distinguish between the models.
3) We selected muons that pass within 20m of an OM for this test. At larger distances, the bulk ice scattering makes it difficult to disentangle and extract the effects due to hole ice scattering. In fact, ideally we would like to use tracks that pass as close to the OM as possible. However, we had to use all distances within 20m, due to the error in the distance measurement. Simulations confirm that the uncertainty in distance is about 25m (Fig 3) due to two effects - after reconstruction, multiple muons are at least 50% of the data sample (Fig. 4), so the reconstructed trajectory is roughtly the center of mass of muon bundle. - Even if we could identify events with single muons then the distance error is still about 16m (Fig. 5), due to the fact that the perpendicular distance to nearly vertical tracks are governed by amplitude information, which is a relatively poorly measured quantity. One thing to mention is that the uncertainty in the distance depends on which string that module is on. The mean of the distance uncertainty is about 18 m for the outer strings, 20 m for strings 1, 2, and 3, and 25 m for string 4. These differences should have little effect on the hole ice analysis because of this range selection of 20 m. Checks on the distance uncertainties is further explored in the bulk ice analysis. Since we are using 20 m as the range to look at the hit probability, there would be some effect of the bulk ice parameters on the absolute value of the hit probability difference between the upfacing and downfacing modules, which is observed in Fig. 2.
We increased the value of Ndir, in an attempt to reduce the uncertainty in the trajectory, which it does (Fig. 6) This cut only made things worse (by reducing statistics, expecially in the Monte Carlo). Also the Ndir cut actually increases the fraction of multi-muons in the sample (Fig. 7). No Ndir cut was used in the analysis presented in the first paragraph.
4) Systematic difficulties are minimized. The variation of bulk ice with depth should not be a big factor since the range of depths is typically less than 80m, and we can average to eliminate linear variations. The difference measurement should not depend on poorly known quantities such as absolute quantum efficiency, collection efficiency, 1pe detection effeciency as long as they are comparable for all OMs used in the test. Three independent strings can be used to search for systematic problems (if hole ice is responsible for these effects, then all strings should respond the same). It should be possible to achieve a few percent accuracy, which is required to differentiate between the different hole ice models.
Therefore, AB events have the possibility of producing a measureable difference between up and down facing OMs, with good statistics and limited sensitivity to systematic problems.
The analysis procedure:
1) True AB coincidence events are identified. The raw sample is ~2/3 random coincidences between A and B. Pat has carefully matched AB events in time, and selected only those events which agree to better than 1 us, which occurs for ~600k events. Am-B10 events had a software cut applied which required 30 OMs to be hit. Simulations show that approximately 35% of the events have multiplicities between 16 and 30 OMs.
2) The B10 events were analyzed using the latest release of RECOOS and the old Pandel functions (ie, we are not using the improved Pandel functions based on the new photon tables).
3) Reconstructions were done using RECOOS. The first guess was cheated by assuming a zenith angle of 0 degrees as the input to the maximum likelihood. This should be fine, since the AB events are only spread in zenith by about 10 degrees. Events that reconstruct within 10 degrees of zenith were selected. The angular distributions obtained after RECOOS reconstruction were compared to MC and good agreement is seen (Fig. 8). No Phit cuts were used, no filtering was done. As mentioned above, no Ndir cut was used in the final analysis, but several values of this parameter were tested. Empirically, if Ndir was much greater than 6, then the statistics of data was dramatically reduced, and it would be very hard to generate the required MC statistics.
4) Muon trajectories were computed, ASSUMING ALL MUONS TRAVERSE THE ENTIRE ARRAY. This is obviously wrong since muons will range out, but this effect is small in the analysis we do since the change in vertical depth of the OMs on a given string is usually less than 80m. For all muons which pass within d < 20m (perpendicular distance, d, between track and OM; although since all trajectories are within 10 degrees of vertical, it is also approximately true that the distance is just the perpendicular distance from the OM in the xy plane) of an OM, we record if the OM was hit or not. The hit probability is the ratio between a hit OM and the number of times a muon passed within 20m of the OM, assuming the fitted muon trajectory travels the entire length of the detector. There should be a monotonic decrease in the hit probability due to muon decays in flight and energy loss, and this was observed in MC simulation. However, this effect was mitigated by averaging the hit probability of two downward facing OMs that are positioned above and below the upward facing OM of interest. The systematic variation of hit probability with depth was also observed in the data, however, it was observed that the depth dependence of the hit probability was larger than expected if we only consider the variation in the vertical muon flux. Thus, it seems that there is some additional effect due to systematic variation in 1pe efficiency of OMs. It also should be noted that the variation in hit probability due to the changing vertical muon flux introduces only small errrors in the up-down difference of hit probabilities (see the error in the MC data).
5) We only used upward facing OMs in the middle of the array to avoid ambiguities related to edge effects at the top and bottom of the array.
The simulations:
1) We used the recent release of BASIEV for muon generation. We simulated atm muons over a zenith angle of 0-10 degrees in B10 to account for the rather narrow range of zenith angles in AB coincidence events. The older version of Basiev had a bug which gave incorrect muon trajectories. We have verified the muon trajectories by hand in the new version of Basiev. We do not expect that the lack of Fe nuclei in the primary flux will impact the multimuon content at the primary energies that generate a trigger.
2) We used the latest release geometry/TO/noise rates from OMDB (as of Aug 98).
3) We have not been able to recompile the latest release of AMASIM2 to run on the LINUX AENEAS system, so we are running with the older release of AMASIM2, and the various photon tables generated by Albrecht. The options were: - the new default photon tables for bulk ice (absorption factor of f096) with max L_absorption = ~105m and effective scattering length near 24m. No hole ice assumes that the hole ice and bulk ice properties are identical. - default bulk ice, hole ice with 100 cm effective scattering in 60cm hole -default bulk ice, hole ice with 10, 30 and 50 cm effective scattering We hope and expect that none of the bug fixes in AMASIM2 will affect this analysis in any important way.
4) We have used the latest update of RECOOS, but not the latest Pandel functions.
Checks:
1) We have verified that the OMs used in the measurements are functioning well, and have the roughly the same 1pe amplitudes so that the 1pe efficiencies are comparible. A small systematic dependence was observed between the 1pe efficiency and the hit probability. The peak of the 1pe distributions of the downward facing OM varied by a factor of 2, which should affect the 1pe detection efficiency by about 20%, which was observed (Fig.9). However, another contribution seems to be the systematic change of the hit probability with depth (due to stopping muons) (Fig. 10). This figure shows the variation for both the real data and the Monte Carlo. Here you can see a variation of about .05 to .1 in the hit probability over the relevant depths (about 50 to 100 m) of the OMs. The flat distribution between 1500-1700m is likely due to the minimum multiplicity requirement (muons must penetrate to ~1700m depth before they trigger the array, on average). Below 1750m, the hit probability falls off at roughly the same slope as seen in the MC (bottom figure), and due to muons ranging out within the array. However, the decrease in hit probability is about a factor 2 in 200m. This is a larger slope than seen in the vertical muon flux, which only changes by a factor of 1.5 in 200m at these depths. This may effect the results of the variation of bulk ice on the depth dependence, by creating a systematic error in the hit probability versus depth.
2) Comparison between geometrical clustering of reconstructed AB events show good agreement betweeen data and MC and show similar features. (Fig. 11- Data and MC) Note small dots represent the positions of the reconstructed tracks and the large black dots are the positions of the strings in the array. The clustering introduces variations in the distance error to the reconstructed tracks. This is explain further under the checks of the bulk ice part of the report. We expect this clustering to have little effect on the hole ice results.
3) We have checked the trajectories of the muons by hand to verify the coordinate system and other information.
4) Several checks were done using both the true track trajectories and the reconstructed track trajectory to estimate the uncertainties introduced by reconstruction. We have verified that reconstruction accuracy (and therefore the distance measurement) is limited by multi-muon events. (thus, YAG data analysis will be an important cross-check of our analysis since this systematic limitation does not exist for that data sample).
Problems:
1) As mentioned, the hit probability calculation assumes all muons travel completely through the detector. However, the analysis is only done over a rather short depth interval since we only use the upward OMs in the middle of the string. In this case, the averaging procedure should be accurate. Also this effect is simulated by the MC.
2) The extracted AB data had several software cuts, the most important for this analysis was number of OM (Nom)in B event > 30. This cut was not applied to MC events. We did apply this cut at one point, but it reduced statistics by 35% and INCREASED the multimuon fraction. So applying this cut to MC data would not improve the accuracy of the track reconstruction. Obviously, once MC statistics are improved, this cut will be applied.
3) We should use the updated Pandel functions when they are released.
4) The absolute hit probabilities for MC are always higher than the data by 20%. This is indirect evidence that we need to include collection efficiency of the PMT in the photon tables, but this should be checked.
5) 1pe probabilities are not exactly the same for all OMs used in the check.
6) Variations in the bulk ice make it difficult to directly compare the differences for OM 10 and 30, which sit at 1725m, and OM 70 which sits at 1750m. However, the data do not show a dramatically different systematic behavior between the OMs on string 1 and 2, and on the deeper string 4
7) Use the new Pandel functions to obtain a better reconstruction. However, it probably will not greatly change the results due to the multimuon component dominating the position resolution.
Discussion:
Does this result make sense? The visual pictures from the camera naively suggest a rather strong affect by the refrozen bubbles in the hole. However, our results indicate that the effect on hit probability are rather small. There could be several arguments to explain our result.
1) While the camera's view suggests that the ice properties in the hole are rather poor, the camera sits near the center of the hole. It may be in the highest density region. The work by Allan Halgren confirmed the prediction of Bruce Koci that the bubbles form with a strong radial gradient in a cylindrical hole. This observation was checked by inspecting the video tape. Just after the OMs reached depth, nearly 24 hours after the last drilling was done, there is no evidence of bubbles forming at the hole-bulk ice interface. If bubbles were uniformly distributed in the hole, there should have been an ~6" annulus already formed. There is nothing in the video tape that suggests a high density of bubbles in the annulus. Per Olof has more details on this detailed visual inspection of the video tape.
2) Albrechts results are really a combination of scattering length and assumed diameter of the hole. Our best fit is 50cm scattering for a 60 cm diameter hole. If the High density region is only 1/3 as large in diameter, then the scattering length should be reduced by a factor of three. Albrecht stated on the phone that as long as the multiplication of Hole_diameter*scattering_length was constant, the photon table simulations yield roughly the same results. So Albrechts photon tables should be applicable even if the diameter of bubbles is not known very well.
We have made an assessment of the bulk ice clarity as a function of the depth of the Amanda B detector, using the same new method as mentioned in T.C. Miller's internal report "A New Method for Determining Deep Ice Clarity from SPASE-AMANDA Coincidence Data (and Applications to Cosmic Ray Composition)" with the 1997 AB coincident data. The method is to use the probability of a PMT being hit as a function of distance from a muon track, and make an exponential fit to extract an attenuation length of the ice near that PMT. An example of this procedure is seen in Fig. 12. Note the range used for the exponential fit was 20 to 60 m, and 4 m size bins was chosen. An example using the data is seen in Fig. 13. The idea is to make such fits for all the good modules in the amanda B detector. Then plot the extracted attenuation length (which is the inverse of the slope in these fits) for each of the modules as a function of the absolute depth of the modules. The results of this analysis, including all the modules for Amanda B with the AB Coincident data is shown in Fig.14. The variation in the bulk ice with depth is clearly demonstrated. The peaks and valleys agrees with Tim's SPASE/AMANDA result and Seraps original analysis of hit probability with muon triggers. Yet the overall mean of the attenuation length and the amplitude of the variations are different from Tim's results. Our results varying from 25 to 35 m, and Tim's results vary from 55 to 75 m. We don't yet understand these discrepancies.
At this point we still don't know exactly what the absolute value of the attenuation length means. In the diffusion range we expect the hit probability to drop off as 1/d * exp (-d/a), where the a (attenuation length) would be ( La * Ls / 3 ) ** 1/2 . La is the absorption length and Ls is the scattering length. Yet, the range in which reliable fits can be made (20 to 60 m), is intermediate between unscattered cherenkov light on one extreme and complete diffusion on the other. Plus the light is not being generated from a point source, but a moving point source along a line trajectory. The attenuation length that we extract would contain a combination of absorption and scattering length, yet the exact dependence is not yet clear.
Monte Carlo Comparisons:
We have examined the dependence of hit probability versus perpendicular distance, without restriction. In both MC and data, the decrease is adequately described as expontial, b*exp(-d/a), ( we have tried (1/d) exp(-d/a) and (1/sqrt(d)) exp(-d/a), but the exponential fit the distributions better), where d is the perpendicular distance between closest approach of muon track and a given module. The exponential relation is seen in a plot of the Hit probabiltity as a function of distance (impact parameter), using the generated (true) track of the MC. Here is one example (Fig. 15). These plots were created by using the true track information and selecting only single muon events. The relationship is perfectly exponential. The purely exponential dependence is not understood, but probably a consequence of a line source of photons vs a point source, and the fact that a distance of 20-60m is intermediate between unscattered cherenkov light on one extreme and complete diffusion on the other. Now for the hit probability versus distance dependence using the reconstructed tracks, the exponential dependence is seen but not always as clear. The reconstructed tracks position uncertainty (Fig.3), causes the hit probability dependence to have some different features than when using the true track. The following figure compares hit probability dependence from the true track to the reconstructed track for module 11(Fig. 16). Note that in the reconstruction case it levels off at distance less than 20 m. Plus looking at say module 100 on string 5 (Fig. 17) you see how the hit probability from the reconstructed track slope is clearly less than of the true track. This uncertainty in the reconstructed track seems to introduce some systematic errors into the extracted attenuation lengths. This is discussed further later in the report.
We have measured the attenuation length ("a") of the MC data taken from the true track and also from the reconstructed track for each PMT, using different photon tables which contain the ice parameter information. We looked at tables having an absorption factor of f096 (corresponding to a max. absorption length ~ 115 m) with scattering lengths of 22 m, 24 m, 28 m, and also an absorption factor of f110 (max. absorption length ~ 105 m) with a scattering length of 24 m. For this analysis the introduction of hole ice doesn't effect the extracted attenuation lengths, so for these comparisons no hole ice was used.
We firstly found that the attenuation length for all the PMTs cluster about 22 m with a scatter of about 1 m (Fig.18) for the true track, and about 25 m with a scatter of about 2.5 m (Fig. 19) for the reconstructed track. Note that there is no variation in the attenuation length as a function of depth in MC as is seen in the data.. This is expected since no depth variation of optical properties are in the photon tables. A closer look at the distribution of the attenuation length for each of the above mentioned ice parameter cases has been done. First result shows that there is a very small change in the attenuation length for the different scattering lengths used, for both the true track (Fig. 20) and the reconstructed track (Fig. 21). Note the use of nomenclature used in the plots goes as follows: s22, s24, or s28 means scattering length of 22m, 24m, or 28m; f096 or f110 means the absorption factor f096 or f110, and hi0 means no hole ice. These plots show only about 3% change in the attenuation length with about a 30% change in the scattering length.This at first was surprising, but considering that the range in which the fit was made are 5 to 61 m for the true track and 20 to 60 m for the reconstructed track, at least 2/3 of this range the photons are already isotropic. Thus the overall effect between 22 m and 28 m in the scattering length wouldn't be to large. This result is confirmed by comparing the photon flux as a function of distance (rho) produced directly from the photon tables of a near infinite muon track for each of the three scattering lengths 22m, 24m and 28m. (Fig.22) This plot is a comparision of histogram ID # 35000, from the hbook files that Albreicht Karle produced with those photon tables. The comparison between the two different absorption factors however do show a significant difference in the means of the attenuation length in both the true track (Fig. 23) and the reconstructed track (Fig. 24). The difference is about 1.6 m for a difference of about 10 m in the absorption length. The fraction difference in this case is more meaningful. Thus about a 10% change in the absorption length of the ice produces about a 10% change in the extracted attenuation length. Again this observed difference is confirmed by this flux vs. rho plot, comparing the effect of different absorption factors used. (Fig. 25) These results point out that this analysis is nearly insensitive to a variation of scattering length in the ice, but the fractional variation in the absorption length will produce roughly the same fractional variation in the measured attenuation length. Hence in the data we see a variation of about 25 to 35 m, then this would approximately correspond to a 30% to 40% variation in the absorption length of the ice. The next step would be to MC events using photon tables with different absorption factors (ie. f080 to f130) to assess this variation more accurately.
A comment about the range of the fit used on these hit probability versus distance plots. For all comparisons, a range of 5 to 61 m was used with the true track. If a range of 20 to 60 m were selected, the mean does increase by about .5 m and the sigma gets a bit larger, about 1.5 m (Fig.26). Hence there is some uncertainty in the selection of the range for these fits. The range 5 to 61 m is used with all true track fits, since it does give a better fit and the extracted attenuation length's sigma is smaller. Note that the hit probability versus distance plot for the reconstructed tracks, level off within a distance of 20 m. This is mainly do to the uncertainty in the position of the reconstructed track as discussed in the hole ice part of this report. This causes the best range for the fits with reconstructed tracks to be 20 to 60 m. This seems to be the most reliable range to extraction the attenuation length with the reconstructed tracks.
Checks:
There are systematic affects in the reconstructed tracks that cause the extracted attenuation lengths versus depth to have scatter beyond statitstics. This broadening of the attenuation length seems to be caused by some systematic difference between strings. In particular the attenuation lengths for the modules of the outer strings are in general significantly larger than the modules for the inner strings.For the data (Fig. 27), the extracted attenuation length is up to 10 m greater in the outer strings than that of the inner strings, and for Monte Carlo (Fig. 28) the difference is about 5 m. This shows that the slope of the hit probability versus distance is generally smaller for the outer 6 strings than the 4 inner strings, which is readily seen when looking at the hit probability as a function of distance (Fig. 29). Some possibilities that could cause this difference are variation of ice in the x-y plane, variations in PMT properties (like 1pe efficiency, quantum efficiency, or collection efficiency) from string to string in the data, variations in the distance error of the reconstructed track at different distances from each module, and possible effects from the fact that half of the data are multimuon events. Several checks have been done in attempt to understand this systematic difference.
In looking at the distribution of the attenuation length of the inner strings seperately from the outer strings, we find that the inner strings agree well with the attenuation length obtained from the true track (Fig. 30). While the outer string modules are typically 3 to 4 m greater (Fig. 31). One check that was done with the MC, was to check on what effect does having half of the events being multimuons (Fig. 4) have on the reconstructed events. A cut with the MC was made to select out single muon events. We have mentioned already (in the hole ice part) that this selection decreases the distance error of the reconstructed track by about 7 to 8 m. (Fig. 32) We found that this selection decreases the difference between inner string's attenuation length and the outer string's attenuation length only slightly. (Fig. 33) The main difference is this cut shifts the overall attenuation length's down about 2 m. This shift in the attenuation length may be caused by a decrease in the distance error. We expect that the uncertainty in the distance would spread out and thus tend to flatten out the hit probability curve resulting in the slope to decrease, thus giving a larger attenuation length. Since the distance error has decreased, the slope should then become steeper and thus result in a lower attenuation length. The only problem with this explanation is that in Fig. 29, the inner string's attenuation length decreases to about 20 m, yet the true track gives 21.5m for that particular ice parameters used in the MC. The expectation is the inner string's attenuation length should approach the true track's attenuation length. We don't yet understand why it would go below, but suspiciously is to do still the distance error in the reconstructed tracks. The above cut has been done only with the MC. At this point, we haven't found an effective set of cuts that would reduce the multimuons with the real data.
Another check that was done, was to make a Ndir cut on both the data and MC.We have found that the distance error of the reconstructed track decreases by about 8m. (Fig 34). Note that the Ndir cut used was 6 or more hits with the direct time window of -5 to 25 ns. Despite the decrease in the distance error, the MC shows that the fraction of multimuons increased in making this cut. (Fig. 7) This cut really reduces the statistics of the MC (only 10% of the data remains), hence the scatter of the attenuation length gets to large to be helpful. Applying this cut on the data however, we find that for both the inner string (Fig. 35) and the outer string (Fig. 36) that the measure attenuation length shifts down in general about 2 m. This agrees with the case mentioned above when we made a cut of using single muon events only. Both cases, a decrease in the distance error caused the extracted attenuation length to decrease. When the Ndir cut is applied the hit probability versus distance plot begin to show more odd features, making the fits on each module in general worse. Here is one example (Fig. 37). Note how the no cut case seems to simply smear out the curve, thus loosing the features that are in the Ndir cut case. Making a Ndir cut does nothing to this systematic difference between inner and outer strings. Right now, it is not clear whether making a Ndir cut is benefical or not. This shift in the attenuation lengths is expected, yet MC shows that the inner string's attenuation length shifts even below the expected (True track's) attenuation length. With all these considerations, it currently appears that the attenuation lengths extracted from the inner string modules without making any cuts gives the most reliable results in showing the variation in the attenuation length as a function of depth. Thus a more reliable plot showing the attenuation length variation as a function of depth is given by Fig. 38.
The last check done was to look at a possible systematic effect introduced by a distance error of the reconstructed track varying at different positions within the detector. Since these tracks are all nearly vertical, zenith angle less than 10 degrees, a distance to a track is approximately the distance from the module to the track in the x-y plane. The following plot shows the x and y position of the closest distance of the reconstructed track to the detector center (0, 0, 0). (Fig. 39). The small dots represents approximately the x-y position of the reconstructed track at the center of the detector, ie z= 0 m. The large black dots are the 10 strings positions. Note how the reconstructed tracks density is greater near each pair of the outer strings. Then looking at the corresponding true tracks position (Fig. 40), which has uniform density within the detector. An explanation for this difference in the position of the reconstructed track to the true track is since these tracks are coming at near vertical direction, RECOOS in reconstructing these tracks has to rely more on the amplitude of the hits rather than on the timing of the hits, which results in a higher distance error in the fit. Plus RECOOS seems to bias the reconstructed track to certain locations within the detector, where the best possible fits can be done with RECOOS. This biasing causes the distance error to be greater at these three locations where the reconstructed tracks tend to cluster. A look at the distance error to the detector center (0,0,0) versus the reconstructed tracks position clearly show the variations in the distance uncertainty versus position within the detector. (Fig. 41). This plot shows the variations in the mean of the distance error of the reconstructed track versus the approximate x and y positions of the reconstructed track. The maximums in the distance error does correpond to regions where the reconstruction tracks cluster and the mininums to the regions between string 4 and strings 1 to 3. This is confirmed with comparisions of the mean of the distance error in a 10 by 10 m region located where the reconstructed tracks cluster to a location between string 4 and the other 3 inner strings. This comparison shows that the distance error mean is about 3.5 m greater in the cluster of reconstructed tracks region. The following plot shows one example of this difference in the uncertainty error in the region near strings 9 and 10, to the region between string 3 and 4.(Fig. 42). The x and y in Fig. 39 refer to the x and y positions of reconstructed track closest to the detector center. This variation in the distance error would effect the plot of the hit probability versus distance. Presummably an increase of the distance uncertainty would cause the hit probability curve to spread out more, thus making the slope less.
There is a difference between the modules of different strings how this distance error variations depend on distance. The following plot shows a comparison of the distance error as a function of distance between module 11 on string 2 and string 5, module 26 (Fig. 43). The events with hits, is the distance error for events that make a hit in the module, and then all events is the distance error to every event whether it hit that module or not. Since to obtain the hit probability requires the ratio of the number of hits as function of distance to the number for all events as function of distance, both then contribute to the overall error in the distance. Note that the distance error vs. distance for events with hits is less than for all events. This is reasonable because it is more likely for these events to have better reconstructed tracks near that module since it produced a hit there, which is the information used by RECOOS to produce the reconstructed track in the first place. The expectation is an increase in the distance error will cause the hit probability slope to decrease faster, and a decrease in the distance error would cause the slope of the hit probability slope to decrease less. When we compare now this distance error vs. distance to the hit probability vs. distance in a module, we do not see clearly this expected effect. Even though there is a difference in distance error versus distance between inner and outer strings, we are still unable to tell if this difference explains why the outer strings attenuation length is greater than the inner string's attenuation length.
Both the Ndir cut and the single muon cut does decrease the overall distance error, but the distance error variations seen as a function distance to the reconstructed track is still there for both the Ndir cut (Fig. 44) and the single muon cut (Fig. 45). Hence whatever effect the distance error variations have on the extracted attenuation lengths, still would effect the results using an Ndir cut as well. Plus this distance error variations is not just a multi-muon effect, but this is seen also in the single muon events.
Conclusion:
The conclusion with regard to the bulk ice is the results show a depth dependent variation in at least the absorption length of the ice. Based on Monte Carlo comparisions, the variation of 30 to 40% in the measure attenuation length corresponds to approximately a 30 to 40% variation in the absorption length. This analysis shows that it is nearly insensitive to any variations in the scattering length. There is a systematic difference between the attenuation length obtain from the inner string modules to that of the outer strings modules using the reconstructed tracks. This difference we are still unable to adequately explain. According to MC, over half of the events in the data are multiple muons. A cut with the MC, selecting out only the single muons, decreases this difference between outer and inner string only a little, but does shift the overall attenuation lengths down by about 2 m. A simple cut of Ndirect does not reduce this systematic difference, but does cause all attenuation lengths to shift down about 2 m in the data. Both of the above cuts performed decreases the distance error of the reconstructed track by about 8 to 9 m. Hence a decrease in the distance error, causes the extracted attenuation lengths to shift down by about 2 m, in general, for both inner and outer strings. The check on distance error as a function of distance to the reconstructed track finds variations between inner and outer strings, yet it still doesn't adequately explain the systematic difference between the inner and outer strings. Lastly based on these checks and comparsion with MC, right now, the attenuation length obtained from the inner strings while applying no cuts, give the most understandable results in terms of its absolute value and variation that is seen in the data. (Fig. 38).
