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But the expression in Eq. 50 does not correspond to a properly normalized probability distribution.
Integrating the formula over all ’s yields the normalization:
¢
© ¡ ¢
£
¦
and we get the properly normalized probability distribution function:
©
© £
¦
¡
©
¢
¡ ¢ ¢©
Another way of reaching this result is by noticing that the derivation of the p.d.f. in Eq. 50 assumes
that we actually observe a hit. Dividing it by the ©
probability to have at least one
hit, we get the conditional p.d.f. in Eq. 52.
If the number of photons is Poisson distributed, then the number of photoelectrons is Poisson
too [60]. In reality, the PMs record photoelectrons, so their expectation value is the number of
photons hitting the cathode, times the quantum efficiency ¢ of the PM. Thus, the formula in
Eq. 52 can be used, replacing by
and replacing by the Green’s function given
¦
by Eq. 45.
This distribution has been tested by fitting time distributions taken with high intensity
pulses [23], yielding absorption and scattering lengths compatible with previous analyses. It is
useful where there are no reliable measurements of hit intensities and the only data available are
the hit times. An analysis of electron neutrino induced cascades in AMANDA-A has been made,
making use of a more sophisticated version of Eq. 52 in a reconstructing algorithm of cascades.
It resulted in limits put on the ¡ flux at energies greater than ¡ GeV [61].
5.2 The ice at AMANDA-B depths
Calibration-data from AMANDA-B4 have also been analyzed, yielding much narrower time distributions
than AMANDA-A (see Fig. 28). This is in agreement with the increase in scattering
length with depth measured in AMANDA-A. Indeed, by extrapolating the curve in Fig. 27 to
0, the disappearance ¥ of air bubbles can be predicted below 1200 m, whereas the shallowest
AMANDA-B4 modules is located at 1520 m depth. The remaining time delays in the distributions
are believed to be caused by scattering on dust particles. Preliminary Monte Carlo simulation
comparisons indicated a scattering length so long that the assumption of diffusion could
not hold anymore. A Green’s function could not be derived
for
©
the range of optical parameters,
¡ ¡
where (i.e. of the order of the distance between
strings) and . Instead of
relying on an analytical formula, the optical parameters of the ice between 1520-2000 m had to be
extracted from the data by fitting distributions with different absorption and scattering lengths
to it. Two statistical methods were used: the
Kolmogorov-Smirnov
test
and
a
©
straightforward
¡ ¡
-minimization. The first yielded and using both calibration
data
taken with the 337 -laser module from UCB at 1830 m depth and light sent from a YAG-
laser from the surface through optical fibers ending at different
£
nm¤
levels, assuming an isotropic light
source in both cases. In the latter, pulses of light in the range 475-515 nm could be produced in
situ.
The second method was applied on the -laser module data, with the improvement that
the source was simulated more correctly, as a distribution covering §
sr and pointing upward
[63]. The orientation of the receiving OM was also taken into account. An example of such
¦
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40
(51)
(52)