In the following we give expressions for the covariance matrix $Z$ (for details see ^{1}), assuming that the (dominant) photon noise follows a Poisson distribution. Display Formula
$cov(\Delta A^,\Delta A^)k,l=2Ntot,k\delta k,l$(12)
Display Formula$cov(\Delta T^,\Delta T^)k,l=2VkNtot,k\delta k,l$(13)
Display Formula$cov(\Delta V^,\Delta V^)k,l=2m4,kc\u2212Vk2Ntot,k\delta k,l$(14)
Display Formula$cov(\Delta A^,\Delta T^)k,l=cov(\Delta T^,\Delta A^)k,l=0$(15)
Display Formula$cov(\Delta A^,\Delta V^)k,l=cov(\Delta V^,\Delta A^)k,l=0$(16)
Display Formula$cov(\Delta T^,\Delta V^)k,l=cov(\Delta V^,\Delta T^)k,l=2m3,kcNtot,k\delta k,l,$(17)
where $k$ and $l$ denote the indices of the matrix elements running from 1 to $M$. The quantities on the right side represent data derived from measured DTOFs, $m3c=\u2211(ti\u2212m1)3Ni/m0$ and $m4c=\u2211(ti\u2212m1)4Ni/m0$ are the third and fourth centralized moments, respectively. The expressions for those submatrices of $Z$ given in Eqs. (12)–(14) and (17) indicate that the respective moments are statistically dependent in case they are derived from the same DTOF, i.e., from the same distance $r(k=l)$. The covariances between $\Delta A^$ and $\Delta T^$ as well as between $\Delta A^$ and $\Delta V^$ turn out to be zero even for $k=l$ [Eqs. (15) and (16)]. However, they do not vanish in general as a property of the moments, but rather because of the specific property of the Poisson distribution $var(Ni)\u2248Ni$ (see ^{1}).