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Solution

From Eq.6.19, the criterion of the true coordinate basis $(\eta,\zeta)$ emerges naturally: it is the basis which makes the observable $\,d^2N/\,d\eta\,d\zeta$ independent of $\zeta$. Such a criterion can be formulated quantitatively; then, the problem becomes that of a formal minimization, treatable numerically. This is done in the following way. Let $i$ and $j$ index pads. We approximate acceptance of a pad $i$ by a quadrangle and calculate its area on the $\eta,\zeta$ plane $S_i$. Then $\,d^2N/\,d\eta\,d\zeta$ at the pad $i$ is approximated as $N_i/S_i$, where $N_i$ is the pad's mean occupancy. We denote its statistical errorbar (based on the propagation of the fitting error estimates obtained in the fitting procedure described in 6.3) as $\sigma(N_i/S_i)$. We seek an offset such that $N_i/S_i$ and $N_j/S_j$ with $i$ and $j$ at different $\zeta$, but similar $\eta$, be minimally different. In practice, comparison of the $N/S$ quantities must be limited to pads with a finite $\eta$ difference, which is small enough so that the only reason for the difference of the $N/S$ may be the geometrical offset. Then, the quantity to minimize is

$$\frac{ \sum_{i,j \mbox{\ with small $\Delta_{i,j}$}} \frac{(N_i/S... ...S_i)^2+\sigma(N_j/S_j)^2} } { \sum_{i,j \mbox{\ with small $\Delta_{i,j}$}} 1 }$$ (73)

where

$$\Delta_{i,j} = \eta_i - \eta_j$$ (74)

is the maximal separation in $\eta$ allowed. Too small a value of $\Delta _{i,j}$ will result in too few channel pairs to compare. To calculate the function, a geometry transformation is required to find displaced coordinates $(\eta^\prime,\zeta^\prime)$ for every $(\Delta x, \Delta y)$ displacement of the detector in the vertical plane. Displacement along the $z$ axis and rotations were not considered because the problem does not seem sensitive to them. GEANT simulation package [78] was used to calculate the geometrical transformations, and MINUIT minimization package [79] - to search for the minimum of the function given by formula 6.20. At first, I was using $\Delta_{i,j}=0.0225$, with a non-gradient (SIMPLEX [79]) minimization. Then I realized that the function 6.20, not everywhere differentiable, could be made suitable for gradient minimization by smoothing it. The smoothing was done by replacing the sharp cutoff at $\Delta _{i,j}$ by a smoothly decaying weight function:

$$w(i,j) = \frac{1}{1+\exp(2\frac{\Delta_{i,j}-\Delta_{cut}}{\Delta_{cut}})},$$ (75)

where $\Delta_{cut}$ was set at 0.01125. That is, the binary ``yes/no'' decision making on the inclusion of a term was replaced by a weight, varying smoothly between 0 and 1.

Figure 6.4: Alignment results for run 3192. The axes show detector's offsets in $X$ and $Y$ in cm. MIGRAD (see [79]) minimization converged at point $(X,Y) = (0.110 \pm 0.019, 0.031 \pm 0.009)$ cm. The dotted lines cross at the estimated minimum. The contour and the errorbar estimates quoted correspond to the unit deviation of the function from the minimum.

A typical result of the minimization is shown on Figure 6.4. The offsets we find are within the tolerance of the detector/beam positions. The $(\eta,\zeta)$ transformation so found was used in the Monte Carlo detector response simulation to compare the measured data with the event generators.