Memorandum on the relation between MC weight, the effective area, and
Expected event rate from a primary neutrino flux
at the earth surface can be obtained by
where is energy of the secondary lepton such as ,
is primary energy of neutrinos,
is number of secondary leptons produced
inside the earth and reaching to the IceCube volume,
and is the effective area of the IceCube observatory.
The integral in the equation above accounts for the propagation
effect in the earth and obtained by resolving the relevant
transport equation . This is equivalent to
the secondary lepton flux at the IceCube depth and pre-calculated
by JULIeT class like PropagatingNeutrinoFlux.java or
PropagatingAtmMuonFlux.java. An example is shown in
Fluxes of the EHE particles at the IceCube depth
for a scenario of the neutrino production by the GZK mechanism.
Two cases in the nadir angle are shown in the figure.
The effective area in Eq. 1
can be estimated by either the semi-analytical way 
or the full-brown MC. The semi-analytical method gives
where is energy deposit of the secondary lepton
propagating over 1km inside the IceCube volume. In EHE
the energy deposit takes place mostly in form of a bunch of cascades.
is typically 1 km for the IceCube.
By the full MC, the effective area will be given by
where is number of MC events passing your criteria,
is the MC primary particle spectrum
of the secondary leptons (mostly and for EHE).
is the area of throwing primary particles in MC.
is a bin width.
Putting Equations  and 
together will lead to the proper event weight as
Note that could be given by the GZK neutrino model,
or any other EHE neutrino models as well as by the atmospheric
muon/neutrino background prediction. So the weight given above
could be not unique but various depending on what models
In this context the IceCube ``sensitivity'' for EHE neutrinos
can be obtained from a quasi-differential event rate in
neutrino model independent way. This approach has been widely
used in many other experiments (for example, see ).
The neutrino flux upper-bound with energy of
from non-existence of signals is evaluated by putting
into Eq. 1. We get
where is threshold energy of the secondary leptons,
is number of upper bound
events with 95 % C.L. The Poisson distribution gives 2.3 events