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In laying out the dst format for MC studies, I claimed in an earlier
message that the error matrix used to store the errors on neutral
hit parameters only needs to contain 3 (bcal) or 2 (fcal)
off-diagonal elements. Our argument follows below. The starting
point for this is a single cluster reconstructed in the bcal or
fcal. No vertex hypothesis has yet been applied to this hit, nor
has a momentum vector been assigned to it. It is just a neutral
cluster in a calorimeter.<br>
<br>
<ol>
<li><b>bcal</b> - the set of measured hit parameters which are
most loosely coupled in terms of the raw measurements are
(r,phi,z,E,t). A transformation can be used to convert into any
other preferred set of hit parameters. Correlations are (r,z) ,
(r,E) , and (z,E).<br>
</li>
<ul>
<li>phi is decoupled from all of the others.<br>
</li>
<li>r is directly measured through the radial profile, but
couples to z and E implicitly.<br>
</li>
<li>z is independently measured using timing, but is implicitly
correlated with r through fluctuations in the energy
deposition for oblique showers, and with E through the
position of the shower maximum, and explicitly with E through
the attenuation correction.<br>
</li>
<li>E correlates implicitly to r,z through shower depth
variations, and explicitly to z through the attenuation
correction.<br>
</li>
<li>the t of the shower maximum correlates with r,z (and through
them with E). However this correlation is trivial, and will
be taken out when the time is projected back from the shower
maximum to a reference vertex. Since the speed of light is
independent of E, this correlation of t with r,z is trivial
and need not be carried along with the shower.<br>
</li>
</ul>
<li><b>fcal</b> - the set of measured hit parameters which are
most loosely coupled in terms of the raw measurements are
(x,y,E,t). Note that the depth z of the shower maximum is not
measured in the fcal, so it does not belong in the list, being a
pure function of E. Correlations are (x,y) , (x,E) , and (y,E).<br>
</li>
<ol>
<li>x,y are implicitly correlated with each other by the shape
of shower deposition in the fcal, and with E through shower
depth.<br>
</li>
<li>as for the fcal, the t of the shower maximum is trivially
correlated with E,x,y. I argue that this correlation is
trivial and need not be carried along with the shower object
for the same reasons as given above under the bcal heading.</li>
</ol>
</ol>
Thus, for the bcal we must keep 3 off-diagonal error matrix elements
per shower. For the fcal there are also 3 non-trivial off-diagonal
elements, but two of them are related to each other in a simple way:<br>
<br>
x * sigma(y,E) = y * sigma(x,E)<br>
<br>
So strictly speaking there are only 2 off-diagonal error matrix
elements per shower that must be kept in the fcal.<br>
<br>
-Richard J.<br>
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