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\begin{titlepage}

\begin{large}
\hspace*{14cm}
\parbox[t]{50mm}{
20 August 1999\\
}
\end{large}
\vspace{1.5cm}

\begin{center}
{\Large\bf Analysis of the X-Ray Tomograph Data}

\vspace{0.3cm}
{\Large\bf of the BOS MDT Prototype Chamber}
\vspace{1cm}

{\Large H.~Kroha, A.~Manz and A.~Ostapchuk}

\vspace{0.3cm}

{\Large Max-Planck-Institut f\"ur Physik, Munich}

\end{center}
\vspace{4.0cm}


\begin{abstract}
The complete X-ray tomograph data of the BOS MDT prototype chamber for the
ATLAS muon spectrometer have been analyzed. 
The geometrical parameters of the chamber determined from the
average of all X-ray scans are in good agreement with the design values.
The spread of the measured wire positions with respect to the fitted wire
grid is $17.0~\mu$m ($19.0~\mu$m) RMS in the $z$ coordinate and $18.2~\mu$m ($19.3~\mu$m) RMS
in $y$ on the readout (high voltage) side. The errors include the average measuring
uncertainty of the X-ray tomograph estimated to be $4~\mu$m RMS in $z$ and
$6~\mu$m RMS in $y$ from the reproducibility of repeated measurements in the
same chamber position.

\end{abstract}
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\section{ Introduction}
%
The complete X-ray tomograph \cite{XTOMO}
data of the full-scale prototype of a BOS MDT chamber
constructed in February 1998 at MPI Munich \cite{BOS}
have been analyzed. The chamber has a tube length of 3800~mm
and a width of 2160~mm, the largest chamber width
in the barrel part of the ATLAS muon spectrometer. It contains 432 drift
tubes, 72 in
each of the 6 tube layers. The chamber has been constructed according to the
procedures described in the Technical Design Report for the ATLAS muon
spectrometer \cite{TDR}.

21 X-ray scans of the chamber have been performed at CERN. After analysis of
the X-ray shadowgrams, the wire-coordinate data
for 9 scans of the HV side of the chamber, 3 scans of the RO side
and 3 scans of the middle part of the chamber have been made available for
analysis on the X-ray tomograph web page.

The 9 scans of the HV side were performed in 4 different
chamber orientations (see Figure~\ref{fig:scans}):

\begin{itemize}
\item two scans (numbers 7 and 8 in the sequential scan numbering)
in the initial orientation (``r1'') ,
\item two scans (9 and 11) after rotation of the chamber by $180^{o}$ around the
x-axis (orientation ``r2''),
\item two scans (12 and 14) after rotation of the chamber by $180^{o}$ around
the y-axis with respect to
orientation r2 (orientation ``r3''),
\item and three scans (15, 20 and 21) after rotation of the chamber
by $180^{o}$ around the x-axis with respect to orientation r3
(orientation ``r4'').
\end{itemize}

The three scans of the RO side are the following:
\begin{itemize}
\item one scan (10) in orientation r2,
\item one scan (13) in orientation r3 and
\item one scan (17) in orientation r4.
\end{itemize}

Out of the 432 wires of the chamber, 320--330 were reconstructed for the different scans.
Measurement of the remaining wires was obscured by the long beams.
No wire coordinates provided by the X-ray tomograph have been discarded in the
analysis.

Analysis of the X-ray measurements allows to determine the geometrical parameters
of the prototype chamber as well as the intrinsic resolution of
the X-ray tomograph. Comparison
of the wire coordinates measured by the X-ray tomograph with the
measurements of wire positions in individual tubes
before assembly of the chamber \cite{BOS} gives information about the
contributions of drift tube and chamber assembly to the
mechanical accuracy of the chamber.
\clearpage

\section{Comparison with the Ideal Wire Grid}
%
The geometrical parameters of the chamber are determined by the assembly jigging, the
combs and sphere towers, which are set up on a flat granite table and have been measured
using 3D metrology (see \cite{BOS}). Accordingly, the design parameters of the prototype chamber
at $20^\circ$~C are:

$y$- and $z$-pitch on high voltage (HV) and readout (RO) side:

\begin{eqnarray*}
y{\mathrm -pitch} &=& (26054\pm 10)~\mu{\mathrm m}, \\
z{\mathrm -pitch} &=& (30036\pm 0.6)~\mu{\mathrm m};
\end{eqnarray*}

$z$-shift and $y$-separation (defined between the nearest wire planes) between the two multilayers (ML):

\begin{eqnarray*}
\Delta_{y}^{HV} &=& (347.040\pm 0.010)~{\mathrm mm}\;\; ({\mathrm HV\; side}),  \\
\Delta_{y}^{RO} &=& (347.072\pm 0.010)~{\mathrm mm}\;\; ({\mathrm RO\; side}),  \\
\Delta_{z}^{HV} &=& 0~{\mathrm mm}\;\; ({\mathrm HV\; side}), \\
\Delta_{z}^{RO} &=& 0~{\mathrm mm}\;\; ({\mathrm RO\; side}). \\
\end{eqnarray*}

The vertical $y$-pitch was intentionally increased by $(40\pm 10)~\mu$m compared to the nominal value
of 26.011~mm for the construction of the prototype chamber to avoid touching of tube layers
during glueing.
On the readout side, the $y$-separation between multilayers is increased by a common offset of
$32~\mu$m of the wire positions in all drift tubes of the second multilayer known precisely from
the X-ray measurements of wire positions in individual tubes (see \cite{BOS}).

For the wide BOS chambers, correct compensation of the
gravitational sag of the cross plates during glueing of each tube layer
is an essential part of the assembly process. The sag of the outer cross plates of the prototype
chamber was controlled with the force compensation system and optical cross plate sag monitors
to be below $10~\mu$m for each tube layer glued (see \cite{BOS}).

The wire positions measured in the X-ray scans were fitted with an ideal grid with
the expected geometrical parameters of the chamber.
The $y$- and $z$-coordinate of the origin of the grid and the angular
orientation of the grid around the $x$-axis with respect to the X-ray tomograph coordinate
system are not relevant for the chamber accuracy and were treated as three free parameters
in the fit.

The mechanical accuracy of the chamber can be characterised by the RMS widths
of the residual distributions of the measured wire positions in $y$ and $z$ with respect to their
nominal values. The results of the fit for the whole chamber and for each multilayer
separately are shown in Table~\ref{tab:gr_fit_3p} for the scans on the high-voltage (HV) and
on readout (RO) side and their averages.

\begin{table}[htb]
\caption{\small RMS widths $[\mu$m$]$ of the distributions of the residuals $\delta y$ and $\delta z$ 
of the wire coordinates from the X-ray measurements
with respect to the expected ideal grid with the
fixed geometrical parameters: $y$-pitch~=~26.054~mm, $z$-pitch~=~30.036~mm,
$\Delta_{y}^{HV} = 347.040$~mm, $\Delta_{y}^{RO} = 347.072$~mm,
$\Delta_{z}^{HV} = 0$, $\Delta_{z}^{RO} = 0$, sag of chamber ends~=~0.
The RMS values are also given separately for the two multilayers ML1 and ML2 numbered
in the assembly sequence.}

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c||} \hline
{Scan}&{$RMS^{\mathrm tot}_{y}$}&{$RMS^{\mathrm tot}_{z}$}&
{$RMS^{ML1}_{y}$}&{$RMS^{ML1}_{z}$}&
{$RMS^{ML2}_{y}$}&{$RMS^{ML2}_{z}$}\\
\hline\hline
{HV-r1-07}&{26.4}&{22.6}&{28.8}&{17.8}&{23.9}&{22.5}\\
{HV-r1-08}&{24.5}&{20.6}&{25.0}&{17.2}&{23.7}&{22.4}\\
{HV-r2-09}&{20.4}&{19.5}&{22.4}&{15.6}&{18.2}&{22.2}\\
{HV-r2-11}&{19.4}&{19.8}&{20.7}&{14.2}&{18.0}&{22.2}\\
{HV-r3-12}&{26.0}&{20.1}&{22.4}&{16.4}&{20.6}&{22.5}\\
{HV-r3-14}&{21.7}&{21.0}&{22.3}&{16.6}&{20.8}&{23.4}\\
{HV-r4-15}&{23.4}&{20.2}&{23.4}&{16.4}&{22.3}&{20.9}\\
{HV-r4-20}&{22.5}&{20.3}&{22.5}&{16.2}&{21.7}&{20.6}\\
{HV-r4-21}&{22.1}&{19.6}&{22.4}&{16.3}&{21.3}&{20.3}\\
\hline
{Average}&{23.0}&{20.4}&{23.3}&{16.3}&{21.2}&{21.9}\\
{HV-side}&{$\pm 2.1$}&{$\pm 1.1$}&{$\pm 2.1$}&{$\pm 0.9$}&{$\pm 2.0$}&{$\pm 2.4$}\\ \hline
\hline
{RO-r2-10}&{18.6}&{17.7}&{15.1}&{12.4}&{21.7}&{20.5}\\
{RO-r3-13}&{22.9}&{20.0}&{16.5}&{13.6}&{19.1}&{21.8}\\
{RO-r4-17}&{22.6}&{21.4}&{16.0}&{15.2}&{20.1}&{22.7}\\
\hline
{Average}&{21.4}&{19.7}&{15.9}&{13.7}&{20.3}&{21.7}\\
{RO-side}&{$\pm 1.6$}&{$\pm 1.5$}&{$\pm 0.6$}&{$\pm 1.1$}&{$\pm 1.1$}&{$\pm
0.9$}\\ \hline
\hline
\end{tabular}
\end{center}
\label{tab:gr_fit_3p}
\end{table}

The average RMS values are close to the specified $20~\mu$m \cite{TDR}, but
their variations from scan to scan are large.
The value of $RMS^{tot}_{\delta y}$ 
varies from $19.4~\mu m$ up to $26.4~\mu m$ 
for the 9 scans of the HV side.
This wide range of variation is not compatible with the expected X-ray tomograph
resolution. Possible reasons for the large variations of the X-ray tomograph results
have been investigated.

\section{Chamber Deformations During X-Ray Measurements}
%
Both from the readings of the RASNIK cross plate sag monitors (see \cite{BOS})
and from the X-ray tomograph data,
{\sl global} deformations of the high voltage and readout ends of the chamber
were found which exceed the expectations from the Bessel support
of the cross plates
and which vary for the different orientations of the chamber 
in the tomograph (see Figures~\ref{fig:sag_hv} and \ref{fig:sag_ro}).
The deformations observed can be described by a parabola.
One concludes that the chamber in its different orientations
was not supported free of external moments in the tomograph as required. 

For the evaluation of the accuracy of the chamber,
the global deformations of the chamber ends in the tomograph setup have to be taken into
account.

The fits of the measured wire coordinates with a grid were repeated with the same geometrical
parameters as used for Table~\ref{tab:gr_fit_3p} but with a common parabolic
sag of both multilayers as additional free parameter. As expected, the $RMS_y$ values improve
(see Table~\ref{tab:gr_fit_4p}).

\begin{table}[htb]
\caption{\small RMS widths $[\mu$m$]$ of the distributions of the residuals $\delta y$ and $\delta z$
of the wire coordinates from the X-ray measurements
with respect to the expected ideal grid with the
fixed geometrical parameters: $y$-pitch~=~26.054~mm, $z$-pitch~=~30.036~mm,
$\Delta_{y}^{HV} = 347.040$~mm, $\Delta_{y}^{RO} = 347.072$~mm,
$\Delta_{z}^{HV} = 0$, $\Delta_{z}^{RO} = 0$, and the parabolic sag of the 
chamber ends as free parameter.
The RMS values are also given separately for the two multilayers ML1 and ML2 numbered
in the assembly sequence.}

\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c||} \hline
{Scan}&{Sag $[\mu$m$]$}&{$RMS^{\mathrm tot}_{y}$}&{$RMS^{\mathrm tot}_{z}$}&
{$RMS^{ML1}_{y}$}&{$RMS^{ML1}_{z}$}&
{$RMS^{ML2}_{y}$}&{$RMS^{ML2}_{z}$}\\
\hline\hline
{HV-r1-07}&{-53}&{19.2}&{22.8}&{21.4}&{17.8}&{16.6}&{22.6}\\
{HV-r1-08}&{-49}&{19.2}&{20.5}&{19.6}&{17.2}&{17.3}&{22.4}\\
{HV-r2-09}&{-23}&{19.4}&{19.5}&{20.2}&{14.3}&{17.1}&{22.2}\\
{HV-r2-11}&{-18}&{18.5}&{19.8}&{20.8}&{15.6}&{16.7}&{22.2}\\
{HV-r3-12}&{ 28}&{24.2}&{20.1}&{20.4}&{16.4}&{18.5}&{22.5}\\
{HV-r3-14}&{ 24}&{20.1}&{21.0}&{20.4}&{16.6}&{19.4}&{23.4}\\
{HV-r4-15}&{ 29}&{21.3}&{20.2}&{21.2}&{16.4}&{20.3}&{20.9}\\
{HV-r4-20}&{ 28}&{20.2}&{20.2}&{20.0}&{16.2}&{19.6}&{20.6}\\
{HV-r4-21}&{ 26}&{20.3}&{19.6}&{20.3}&{16.3}&{19.6}&{20.3}\\
\hline
{Average}&{}&{20.3}&{20.4}&{20.5}&{16.3}&{18.3}&{21.9}\\
{HV-side}&{}&{$\pm 1.6$}&{$\pm 1.1$}&{$\pm 0.4$}&{$\pm 1.1$}&{$\pm
1.9$}&{$\pm 1.0$}\\
\hline
\hline
{RO-r2-10}&{17}&{17.7}&{17.7}&{21.0}&{20.5}&{14.0}&{12.4}\\
{RO-r3-13}&{-1}&{22.9}&{20.0}&{19.1}&{21.7}&{16.3}&{13.6}\\
{RO-r4-17}&{-7}&{22.3}&{21.6}&{19.7}&{22.7}&{15.7}&{15.2}\\
\hline
{Average}&{}&{21.0}&{19.8}&{19.9}&{21.6}&{15.3}&{13.7}\\
{RO-side}&{}&{$\pm 2.3$}&{$\pm 1.5$}&{$\pm 1.4$}&{$\pm 1.5$}&{$\pm
1.0$}&{$\pm 1.5$}\\
\hline
\hline
\end{tabular}
\end{center}
\label{tab:gr_fit_4p}
\end{table}

As expected from the data in Figures~\ref{fig:sag_hv} and \ref{fig:sag_ro},
the largest fitted sag of approximately $50~\mu$m was found
on the HV-side in orientation r1. Consequently, the most significant
improvement of the RMS value of the $y$-residuals occurs for scans 7 and 8.
Still the variations of the $RMS_{\delta y}$ values for the different scans are
larger than expected: from $18.5~\mu$m to $24.2~\mu$m on the HV side and from
$17.7~\mu$m to $22.3~\mu$m on the RO side.

\section{Reproducibility of the X-Ray Tomograph Measurements}
%
In order to understand the variations in the results of the X-ray tomograph scans,
the reproducibility of the measurements from repeated scans of the chamber 
in the same position has been evaluated.
The differences between two subsequent measurements $i$ and $j$ in the same position,
i.e.\ the mean and RMS of the distributions of the differences 

\begin{eqnarray*}
\Delta y &=& \delta y_i - \delta y_j\;\; {\mathrm and}  \\
\Delta z &=& \delta z_i - \delta z_j
\end{eqnarray*}

of the residuals $\delta y$ and $\delta z$ of the wire coordinates with respect to the expected 
grid with sag (see previous section),
are given in Table~\ref{tab:repr}.

\begin{table}[htb]
\caption{\small
Reproducibility of the X-ray tomograph measurements:
mean and RMS $[\mu$m$]$ of the differences $\Delta y$ and $\Delta z$
of the residuals of the measured wire coordinates with respect to the expected grid with sag
for two subsequent scans of the chamber on the high voltage side in the same position
(top and bottom multilayer separately and combined)}

\begin{center}
\begin{tabular}{|c||c|c||c|c||c|c||} \hline
{Scans}&{$<\Delta y>^{\mathrm tot}$}&{$RMS^{\mathrm tot}_{\Delta y}$}&
{$<\Delta y>^{\mathrm top}$}&{$RMS^{\mathrm top}_{\Delta y}$}&
{$<\Delta y>^{\mathrm bot}$}&{$RMS^{\mathrm bot}_{\Delta y}$}\\
\hline\hline
{\hphantom{0}7--\hphantom{0}8}&{$-1.6$}&{11.4}&\hphantom{1}{5.2}&{7.6}&{$-10.4$}&{9.4}\\
{\hphantom{0}9--11}&{$-1.3$}&\hphantom{1}{9.0}&\hphantom{1}{2.7}&{6.6}&{$-\hphantom{1}6.6$}&\hphantom{1}{9.1}\\
{12--14}&\hphantom{$-$}{2.0}&{14.3}&{11.0}&{6.8}&{$-11.4$}&{10.5}\\
{15--20}&{-0.4}&{\hphantom{1}9.7}&\hphantom{1}{0.1}&{6.5}&{$-\hphantom{1}0.9$}&{12.1}\\
{15--21}&{-0.5}&{11.4}&\hphantom{1}{7.2}&{7.1}&{$-\hphantom{1}8.6$}&\hphantom{1}{9.2}\\
\hline
{Average}&{$-0.4$}&{11.2}&{\hphantom{1}5.2}&{6.9}&{$-\hphantom{1}7.6$}&{10.1}\\
\hline\hline

{Scans}&{$<\Delta z>^{\mathrm tot}$}&{$RMS^{\mathrm tot}_{\Delta z}$}&
{$<\Delta z>^{\mathrm top}$}&{$RMS^{\mathrm top}_{\Delta z}$}&
{$<\Delta z>^{\mathrm bot}$}&{$RMS^{\mathrm bot}_{\Delta z}$}\\
\hline\hline
{\hphantom{0}7--\hphantom{0}8}&{$-0.7$}&{7.3}&{$-5.5$}&{4.2}&\hphantom{$-$}{5.7}&{5.4}\\
{\hphantom{0}9--11}&{$-1.4$}&{5.0}&{$-3.0$}&{4.0}&\hphantom{$-$}{7.5}&{5.3}\\
{12--14}&\hphantom{$-$}{0.4}&{5.5}&\hphantom{$-$}{1.0}&{4.5}&{$-0.3$}&{6.4}\\
{15--20}&{$-0.2$}&{5.6}&\hphantom{$-$}{0.6}&{3.9}&{$-1.2$}&{6.8}\\
{15--21}&\hphantom{$-$}{0.3}&{5.1}&\hphantom{$-$}{0.1}&{4.2}&\hphantom{$-$}{0.5}&{5.8}\\
\hline
{Average}&{$-0.3$}&{5.7}&{$-1.4$}&{4.2}&\hphantom{$-$}{2.4}&{5.9}\\
\hline\hline
\end{tabular}
\end{center}
\label{tab:repr}
\end{table}

The RMS widths of the differences of the residuals of the wire coordinates vary
considerably for the different chamber orientations in the tomograph.
The average RMS widths are $4~\mu$m for the $z$-coordinate and $7~\mu$m for the
$y$-coordinate in the multilayer which was on top during the scan. 
The reproducibility for the bottom
multilayer is 1.5 times worse, $6~\mu$m in $z$ and $10~\mu$m in
$y$, respectively, which can be explained by the larger multiple scattering of the X-rays.
The ratio of the widths for the $z$- and for the $y$-coordinate is about
0.6 for both multilayers which is in agreement
with the expected value of $\tan 30^\circ = 0.58$ due to the $30^\circ$ inclination
of the X-ray beams with respect to the the $y$ axis.

The total widths for the whole chamber,
$RMS^{\mathrm tot}_{\Delta z} \simeq 6\mu$m and $RMS^{\mathrm tot}_{\Delta y} \simeq 11\mu$m,
differ from the average of the values for top and bottom multilayers,
obviously due to systematic shifts of the measured
wire coordinates in the two multilayers from one scan to the other.

Information about the origin of the systematic effects can be
obtained by studying the difference between
repeated wire coordinate measurements as function of $z$, i.e.\ along the scan direction.
The differences of the $y$- and $z$-residuals for the two subsequent scans no.\ 7 and 8 in the same position
of the chamber,

\[ \Delta y = \delta y_7 - \delta y_8\;\; {\mathrm and}\;\; \Delta z = \delta z_7 - \delta z_8, \]

are shown as a function of $z$-coordinate of the wires
in Figure~\ref{fig:7-8_yz} (for the top multilayer) and
Figure~\ref{fig:7-8_yz_b} (for the bottom multilayer).

Significant systematic effects are visible: large fluctuations near the locations
of the long beams which absorb and scatter the X-rays, average offsets from zero and slopes in scan direction.
The effects are studied further by projecting the $y$- and $z$-coordinates of the wires
in the directions perpendicular to and measured directly by the two X-ray beams:

\begin{eqnarray*}
\Delta_{\mathrm left} &=& \cos(30^\circ)\cdot\Delta z + \sin(30^\circ)\cdot\Delta y,  \\
\Delta_{\mathrm right} &=& cos(30^\circ)\cdot\Delta z - \sin(30^\circ)\cdot\Delta y.
\end{eqnarray*}

The variables $\Delta_{\mathrm left}$ and $\Delta_{\mathrm right}$ describe the reproducibility of the wire
position measurements with respect to the directions of the left and right X-ray beam in scan direction, respectively.
The $z$-dependence of the new variables is shown in Figure~\ref{fig:7-8_lr} for the top multilayer
and in Figure~\ref{fig:7-8_lr_b} for the bottom multilayer.

The analysis leads to the following conclusions on the systematics of the X-ray measurements:

\begin{enumerate}

\item Large fluctuations are observed in both coordinates in the
regions close to the shadows of the long-beams which have not been corrected.

\item The wire coordinates measured by the right X-ray beam are
relatively stable along the scan direction with average offsets of $\Delta_{\mathrm right}$
from zero of $\sim -10~\mu$m for the top multilayer and $\sim +10~\mu$m for the bottom multilayer.
The offsets can be explained by a change of the right X-ray direction
between the two scans.

\item The difference of the wire coordinates measured by the left X-ray beam show a systematic trend
across the width of the chamber which can be explained by a drift of the direction of the
left X-ray beam during one of the two scans.
\end{enumerate}

Similar systematic effects were found in the comparison of the other repeated scans in the same position.
They broaden the residual distributions of the measured wire coordinates and explain the
large variations of $RMS_y$ and $RMS_z$ from scan to scan (see Table~\ref{tab:gr_fit_4p}).
In particular, they deteriorate the accuracy in the measurement of the multilayer
separations in $y$ and $z$.
According to Table~\ref{tab:repr}, the
reproducibility of the measurement of the multilayer separations 

\[ <\Delta y>^{\mathrm top} - <\Delta y>^{\mathrm bot}\;\; {\mathrm and}\;\;
<\Delta z>^{\mathrm top} - <\Delta z>^{\mathrm bot} \]

varies in the range 1 to $22.4~\mu$m for $y$ and $-11.2$ to $-0.4~\mu$m for $z$.

To get rid of the systematic uncertanities in the X-ray measurement of the multilayer
separations in $y$ and $z$ and the $y$- and $z$-pitches for the evaluation of the chamber accuracy,
these parameters have to be left free in the grid fit.
The remaining uncertainty in the wire coordinate measurements is the intrinsic
resolution of the X-ray tomograph which can be estimated from the
reproducibility of repeated scans by dividing the average $RMS_{\Delta y}$ and $RMS_{\Delta z}$
in Table~\ref{tab:repr} by $\sqrt{2}$ since both scans are affected by the measurement errors.

The $y$-resolution of the X-ray tomograph for the measurement of the BOS prototype chamber
is therefore $5~\mu$ for the top multilayer and $7~\mu$m for the bottom multilayer.
The $z$-resolution is $3~\mu$m and $4~\mu$m, respectively, for the top and the bottom multilayer.
The average X-ray tomograph resolutions for the whole chamber are 

\begin{eqnarray}
\sigma^{\mathrm tomo}_y &\simeq& 6\hphantom{.5}~\mu{\mathrm m}\;\; {\mathrm and} \\
\sigma^{\mathrm tomo}_z &\simeq& 3.5~\mu{\mathrm m}.
\end{eqnarray}

\section{Global Fit of the Wire Grid}
%
Given the mechanical tolerances of the jigging \cite{BOS}, \cite{TDR} (see Table~\ref{tab:ch_par})
and the systematic uncertainties of the X-ray tomograph measurements, 
the geometrical parameters and the accuracy of the
chamber are best evaluated by fitting a wire grid to the wire coordinates measured in each scan
with the $y$- and $z$-pitch, the $y$- and $z$-separation between multilayers and
the common sagitta of all layers in $y$-direction as free parameters.
The results for the different scans and their average are given in Table~\ref{tab:gr_fit_8p}.
Included are the rms widths of the residual distributions
and their standard deviations (from fit of a Gaussian) which can be used to evaluate 
the overall accuracy of the chamber assembly. The wire coordinate errors contain
a non-negligible contribution from the resolution of the X-ray tomograph (see previous section).
The residual distributions for two typical scans are shown in
Figures~\ref{fig:res_20} and \ref{fig:res_13}.

\begin{table}[t]
\caption{\small
Results from the global fit of a wire grid to the data from the X-ray tomograph
scans on the high voltage (HV) and readout (RO) side and at intermediate $x$-positions
(see Figure~\ref{fig:scans}).
Repeated scans for the same chamber orientation are grouped together.
For the different scans, the average sagitta of all layers,
$y$- and $z$-pitch and the separations $\Delta_y$ and
$\Delta_z$ between multilayers are determined as parameters in the fit.
The RMS widths and standard deviations (from fit of a Gaussian) of the residual distributions
in $y$ and $z$ are also shown.}

\begin{center}
\begin{tabular}{|c||c|c|c|c|c||c|c||c|c||} \hline
{Scan}&{Sag}&{$y$-pitch}&{$z$-pitch}&{$\Delta_{y}$}&{$\Delta_{z}$}&
{$RMS^{\mathrm tot}_y$}&{$RMS^{\mathrm tot}_z$}&{$\sigma^{\mathrm tot}_y$}&{$\sigma^{\mathrm tot}_z$}\\

{}&{[$\mu m$]}&{[$mm$]}&{[$mm$]}&{[$mm$]}&{[$\mu m$]}&{[$\mu m$]}&{[$\mu
m$]}&{[$\mu m$]}&{[$\mu m$]}\\
\hline\hline

{HV-r1-07}&{$-53$}&{26.057}&{30.0361}&{347.038}&{$-22$}&{18.6}&{20.5}&{18.5}&{18.7}\\
{HV-r1-08}&{$-51$}&{26.054}&{30.0362}&{347.028}&{$-\hphantom{2}5$}&{18.3}&{20.3}&{17.6}&{18.5}\\
\hline
{HV-r2-09}&{$-23$}&{26.058}&{30.0364}&{347.041}&{$-\hphantom{2}6$}&{18.4}&{18.6}&{15.9}&{16.6}\\
{HV-r2-11}&{$-19$}&{26.057}&{30.0362}&{347.036}&{$-11$}&{18.3}&{19.0}&{17.1}&{15.9}\\
\hline
{HV-r3-12}&{$+27$}&{26.059}&{30.0363}&{347.058}&{$+10$}&{19.0}&{19.4}&{16.8}&{18.1}\\
{HV-r3-14}&{$+23$}&{26.059}&{30.0362}&{347.062}&{$+13$}&{19.5}&{20.3}&{17.5}&{18.0}\\
\hline
{HV-r4-15}&{$+28$}&{26.058}&{30.0362}&{347.041}&{$-17$}&{20.3}&{18.8}&{18.8}&{16.3}\\
{HV-r4-20}&{$+28$}&{26.059}&{30.0361}&{347.039}&{$-18$}&{19.3}&{18.5}&{18.2}&{17.3}\\
{HV-r4-21}&{$+25$}&{26.058}&{30.0360}&{347.026}&{$-15$}&{19.7}&{18.4}&{18.0}&{16.4}\\
\hline
{Average}&{}&{26.058}&{30.0362}&{347.041}&{$-\hphantom{2}8$}&{19.0}&{19.3}&{17.6}&{17.3}\\
{HV-side}&{}&{$\pm 0.0005$}&{$\pm 0.0001$}&{$\pm 0.011$}&{$\pm 11$}&{$\pm
0.8$}&{$\pm 0.8$}&{$\pm 0.9$}&{$\pm 1.2$}\\
\hline
\hline
\hphantom{H}{2-r4-19}&{$+25$}&{26.058}&{30.0361}&{347.045}&{$-17$}&{17.7}&{17.3}&{15.9}&{14.2}\\
\hphantom{H}{3-r4-16}&{$+\hphantom{2}4$}&{26.058}&{30.0361}&{347.050}&{$-28$}&{15.0}&{16.1}&{12.5}&{15.5}\\
\hphantom{H}{4-r4-18}&{$-\hphantom{2}4$}&{26.060}&{30.0359}&{347.081}&{$-20$}&{15.5}&{16.7}&{13.7}&{16.4}\\
\hline
\hline                                                                                                                       
{RO-r2-10}&{$+16$}&{26.058}&{30.0362}&{347.061}&{$+12$}&{17.3}&{16.6}&{15.0}&{14.0}\\
\hline                                                                                                                       
{RO-r3-13}&{$-14$}&{26.061}&{30.0362}&{347.086}&{$+17$}&{16.7}&{18.3}&{14.8}&{16.4}\\
\hline                                                                                                                       
{RO-r4-17}&{$-\hphantom{1}8$}&{26.060}&{30.0361}&{347.087}&{$-20$}&{17.0}&{19.7}&{14.6}&{17.2}\\
\hline
{Average}&{}&{26.060}&{30.0362}&{347.078}&{$+\hphantom{2}3$}&{17.0}&{18.2}&{14.8}&{15.9}\\
{RO-side}&{}&{$\pm 0.0013$}&{$\pm 0.00005$}&{$\pm
0.012$}&{$\pm 16$}&{$\pm 0.2$}&{$\pm 1.3$}&{$\pm 0.2$}&{$\pm 0.9$}\\
\hline
\hline
\end{tabular}
\end{center}
\label{tab:gr_fit_8p}
\end{table}


The design values for the $y$- and $z$-pitch and the $y$- and $z$-separations between the
multilayers (see \cite{BOS}) from measurement of the assembly jigging,
are well reproduced by the averaged X-ray tomograph results
within the measurement errors (see Table~\ref{tab:ch_par}). The measured
$y$- and $z$-separations
between multilayers vary considerably from scan to scan indicating the intrinsic systematic
errors of the X-ray measurements.
The small variations in the average $z$-pitch from scan to scan may have been caused by
temperature variations of the chamber in the X-ray tomograph.

\begin{table}[thb]
\caption{\small Comparison of the average X-ray tomograph measurements of the
geometrical chamber parameters with the design values (at $20^\circ$C)
and average RMS values of the measured wire coordinates
(including the resolution of the X-ray tomograph measurements)}

\begin{center}
\begin{tabular}{|l||c|c||c|c||}
\cline{2-5}
\multicolumn{1}{c||}{\hfill} & \multicolumn{2}{c||}{HV side} & \multicolumn{2}{c||}{RO side} \\
\cline{2-5}

\multicolumn{1}{c||}{\hfill}  & X-ray           & Design value     & X-ray           & Design value     \\
\hline\hline

$y$-pitch $[\mu$m$]$        & $\hphantom{.2}26058\pm 0.5\hphantom{11}$  & $\hphantom{.2}26054\pm 10\hphantom{10.}$  & 
$\hphantom{.2}26060\pm 1.3\hphantom{11}$  & $\hphantom{.2}26054\pm 10\hphantom{10.}$ \\

$z$-pitch $[\mu$m$]$        & $30036.2\pm 0.1\hphantom{11}$ & $30036.8\pm 0.6\hphantom{11}$   &
$30036.2\pm 0.05\hphantom{1}$ & $30036.8\pm 0.6\hphantom{11}$ \\

$y$-sep.\ ML $[$mm$]$          & $347.041\pm 0.011$ & $347.040\pm 0.010$ & $347.078\pm 0.012$ & $347.072\pm 0.010$ \\

$z$-sep.\ ML $[\mu$m$]$        & $\hphantom{347}-8\pm 11\hphantom{0.0}$ & $\hphantom{347.01}0\pm 10\hphantom{0.0}$ &
$\hphantom{347.90}3\pm 16\hphantom{0.0}$ & $\hphantom{347.01}0\pm 10\hphantom{0.0}$ \\
\hline

RMS${}_y$ $[\mu$m$]$        & \multicolumn{2}{|c||}{$19.0\pm 0.8$}   & \multicolumn{2}{c||}{$17.0\pm 0.2$} \\

RMS${}_z$ $[\mu$m$]$        & \multicolumn{2}{|c||}{$19.3\pm 0.8$}   & \multicolumn{2}{c||}{$18.2\pm 1.3$} \\
\hline\hline

\end{tabular}
\end{center}
\label{tab:ch_par}
\end{table}

\vspace{-5mm}

\section{Uniformity of Tube Layers}

For each individual tube layer the $y$-residuals of the wires
with respect to the gobal fit were fitted with a parabola

\[ \delta y (z) = a_{y} + b_{y}\cdot z + c_{y}\cdot z^{2} \]

and the $z$-residuals with a a straight line

\[ \delta z (z) = a_{z} + b_{z}\cdot z.  \]

The average values of the parameters
$a_{y}$, $b_{y}$, $c_{y}$, $a_{z}$ and $b_{z}$ for all scans
as well as the average shifts $<\delta y>$ and
$<\delta z> = a_z$ and the spreads $RMS_{\delta y}$ and $RMS_{\delta z}$ with respect to the
fitted grid are shown for each tube layer in
Table~\ref{tab:layers}. The parameters are given in
the chamber coordinate frame for the initial orientation r1 (see Figure~\ref{fig:scans})
with the $y$-axis pointing upwards and the $z$-axis in scan direction.
They allow to evaluate the uniformity of the chamber parameters for all layers.

With the chamber width $W=2160$~mm, the sagitta of the tube layers
after subtraction of the common sag on the supports in the X-ray tomograph is $c_y\cdot (W/2)^2$.
Beyond the common sag, there are no deformations of individual tube layers significantly above the
specification for the cross plate sag compensation.

\begin{table}[thb]
\caption{\small Geometrical parameters of individual tube layers (average of all scans).
The layers are numbered in the assembly sequence,
from the inside out for each multilayer (ML 1: layers 1--3, ML 2: layers 4--5).}

\begin{center}
\begin{tabular}{|c||c|c|c|c|c||c|c|c|c||} \hline
{Layer}&{$a_{y}$}&{$b_{y}$}&{$c_{y}$}&{$<\delta y>$}&{$RMS_{\delta y}$}&
{$a_{z}$}&{$b_{z}$}&{$<\delta z>$}&{$RMS_{\delta z}$}\\
{}&{[$\mu m$]}&{[$\mu m/m$]}&{[$\mu m/m^{2}$]}&{[$\mu m$]}&{[$\mu m$]}&
{[$\mu m$]}&{[$\mu m/m$]}&{[$\mu m$]}&{[$\mu m$]}\\
\hline
\hline
{HV-1}&{1.1}&{2.1}&{-14.3}&{-4.4}&{16.2}&{-6.2}&{-10.3}&{-6.2}&{19.3}\\
{}&{$\pm 2.4$}&{$\pm 2.3$}&{$\pm 2.0$}&{$\pm 2.1$}&{$\pm
1.1$}&{$\pm 4.9$}&{$\pm 2.5$}&{$\pm 4.9$}&{$\pm 1.0$}\\
\hline
{HV-2}&{-3.4}&{3.9}&{12.9}&{1.6}&{16.1}&{-8.9}&{4.6}&{-8.9}&{20.0}\\
{}&{$\pm 1.4$}&{$\pm 1.8$}&{$\pm 1.8$}&{$\pm 1.0$}&{$\pm
1.0$}&{$\pm 5.0$}&{$\pm 1.5$}&{$\pm 5.0$}&{$\pm 0.9$}\\
\hline
{HV-3}&{3.9}&{-14.0}&{-2.5}&{2.8}&{18.2}&{16.2}&{1.5}&{16.2}&{15.3}\\
{}&{$\pm 3.3$}&{$\pm 3.0$}&{$\pm 2.3$}&{$\pm 2.3$}&{$\pm
1.9$}&{$\pm 2.6$}&{$\pm 1.9$}&{$\pm 2.6$}&{$\pm 1.4$}\\
\hline
{HV-4}&{-10.4}&{6.5}&{17.4}&{-3.8}&{21.0}&{-7.9}&{0.0}&{-7.9}&{14.9}\\
{}&{$\pm 2.6$}&{$\pm 3.9$}&{$\pm 4.1$}&{$\pm 2.0$}&{$\pm
0.8$}&{$\pm 4.7$}&{$\pm 1.7$}&{$\pm 4.7$}&{$\pm 0.8$}\\
\hline
{HV-5}&{3.4}&{3.1}&{-9.7}&{0.3}&{14.1}&{0.2}&{-6.4}&{0.2}&{13.2}\\
{}&{$\pm 1.4$}&{$\pm 3.4$}&{$\pm 2.8$}&{$\pm 1.2$}&{$\pm
1.3$}&{$\pm 3.7$}&{$\pm 2.8$}&{$\pm 3.7$}&{$\pm 0.7$}\\
\hline
{HV-6}&{5.3}&{1.8}&{-5.7}&{3.3}&{24.7}&{10.6}&{-10.7}&{10.6}&{15.8}\\
{}&{$\pm 2.3$}&{$\pm 2.1$}&{$\pm 2.4$}&{$\pm 1.9$}&{$\pm
0.9$}&{$\pm 3.4$}&{$\pm 1.6$}&{$\pm 3.4$}&{$\pm 0.6$}\\
\hline
\hline
{RO-1}&{-2.4}&{-3.6}&{9.4}&{1.3}&{11.6}&{-11.2}&{-8.0}&{-11.2}&{11.9}\\
{}&{$\pm 2.5$}&{$\pm 1.0$}&{$\pm 1.3$}&{$\pm 1.9$}&{$\pm
0.2$}&{$\pm 5.5$}&{$\pm 1.3$}&{$\pm 5.1$}&{$\pm 1.1$}\\
\hline
{RO-2}&{3.5}&{-2.3}&{-11.8}&{-1.1}&{13.2}&{4.7}&{-8.3}&{4.7}&{11.4}\\
{}&{$\pm 1.3$}&{$\pm 0.4$}&{$\pm 4.2$}&{$\pm 1.3$}&{$\pm
0.6$}&{$\pm 3.0$}&{$\pm 2.0$}&{$\pm 3.0$}&{$\pm 0.8$}\\
\hline
{RO-3}&{-1.7}&{3.3}&{3.9}&{-0.1}&{17.6}&{2.8}&{-7.4}&{2.8}&{13.7}\\
{}&{$\pm 2.1$}&{$\pm 0.9$}&{$\pm 2.1$}&{$\pm 1.7$}&{$\pm
1.4$}&{$\pm 2.0$}&{$\pm 2.8$}&{$\pm 2.0$}&{$\pm 1.7$}\\
\hline
{RO-4}&{-2.9}&{-10.8}&{0.1}&{-3.0}&{14.7}&{10.6}&{5.8}&{10.6}&{15.2}\\
{}&{$\pm 1.4$}&{$\pm 1.4$}&{$\pm 0.3$}&{$\pm 1.5$}&{$\pm
0.1$}&{$\pm 3.5$}&{$\pm 1.4$}&{$\pm 3.5$}&{$\pm 1.1$}\\
\hline
{RO-5}&{12.4}&{-4.6}&{-13.0}&{7.3}&{18.7}&{5.9}&{-2.7}&{5.9}&{15.9}\\
{}&{$\pm 0.9$}&{$\pm 2.4$}&{$\pm 4.1$}&{$\pm 0.8$}&{$\pm
0.4$}&{$\pm 3.9$}&{$\pm 1.4$}&{$\pm 3.9$}&{$\pm 0.9$}\\
\hline
{RO-6}&{-8.4}&{9.1}&{10.9}&{-4.4}&{20.8}&{-19.2}&{1.9}&{-19.2}&{18.8}\\
{}&{$\pm 1.5$}&{$\pm 1.5$}&{$\pm 1.6$}&{$\pm 2.0$}&{$\pm
1.4$}&{$\pm 4.3$}&{$\pm 0.6$}&{$\pm 4.3$}&{$\pm 0.7$}\\
\hline
\hline
\end{tabular}
\end{center}
\label{tab:layers}
\end{table}


Tables~\ref{tab:layers_z} and \ref{tab:layers_y} show the averaged measured
geometrical parameters
of individual tube layers, $z$-pitch, $z$-shift and average $y$-separation.
The $z$-pitch of a layer is given by $\delta z_0 [\mu{\mathrm m}] + b_z\cdot \delta z_0 [{\mathrm m}]$
where $\delta z_0$ is the average global $z$-pitch of the chamber.
The uniformity of the chamber parameters is very good with respect to the tolerances of the
assembly tooling which fulfill the specifications.

The only significant deviations from the design parameters
of the chamber are global $z$-shifts of $+16~\mu$m and $-19~\mu$m of the two outer tube layers
with respect to the fitted grid, in the
first multilayer on the HV side and in the second multilayer on the RO side.
The same sphere tower interface is responsible for $z$-positioning in both cases and
the two shifts point in the same direction in the coordinate frame of the assembly jigging.
The $z$-shifts are the reason for the larger RMS values for the $z$- compared to the
$y$-coordinate. The foreseen optical sphere monitors allow to detect such shifts
if they should occur during series production.

\section{Intermediate Scans}

For chamber orientation r4, three successful scans have been performed at intermediate $x$-positions
between the chamber ends (see Figure~\ref{fig:scans}).
The chamber parameters from the global fits to these data are given in Table~\ref{tab:gr_fit_8p}.
The RMS widths of the residual distributions for the top and bottom multilayer during the scans
separately and combined are shown in Figures~\ref{fig:middle_tot}--\ref{fig:middle_bot}
and in Table~\ref{tab:middle} as function of $x$.

The wire positions measured at intermediate $x$-positions between the
chambers ends (see Figure~\ref{fig:scans})
show a smaller spread than at the HV and RO ends (see
Table~\ref{tab:middle}) because they are determined by interpolation
between both ends and by the wire sag. Fluctuations in the wire sag of $2~\mu$m RMS
due to the variations of the measured wire tension within $1\%$ RMS (see \cite{BOS})
can be neglected. The expected widths of the residual distributions as function of $x$
are therefore determined by

\begin{equation}
\sigma^{\mathrm res}_{y,z}(x)=\sqrt{\biggl ({x\over L}\biggr )^2\sigma^{{\mathrm res}\, 2}_{y,z}(RO) +
\biggl ({x\over L}-1\biggr )^2\sigma^{{\mathrm res}\, 2}_{y,z}(HV)}
\end{equation}

where $L=3800$~mm is the length of the tubes.

The results in Table~\ref{tab:middle} show that the
measured values of $\sigma_{y,z}$ at intermediate $x$-positions
are always larger than predicted from equation (3).
A possible source of this discrepancy is the contribution of the X-ray
measurement errors $\sigma_{y,z}^{\mathrm tomo}$ which do not interpolate. 
To account for this effect,
the error propagation has to be modified:

\begin{equation}
\sigma^{\mathrm res}_{y,z}(x) =\sqrt{\biggl ({x\over L}\biggr)^2
(\sigma^{{\mathrm res}\, 2}_{y,z}(RO)-\sigma_{y,z}^{{\mathrm tomo}\, 2}) + \biggl ({x\over L}-1\biggr)^2
(\sigma^{{\mathrm res}\, 2}_{y,z}(HV)-\sigma_{y,z}^{{\mathrm tomo}\, 2})}\;\; \oplus\; \sigma_{y,z}^{\mathrm tomo}.
\end{equation}

The predictions for the X-ray tomograph measurement errors which describe the RMS values of the data
with equation (4) are given in Table~\ref{tab:middle}. These predictions are higher than
the estimates from the reproducibility of repeated scans, especially for the $z$-coordinate.

\begin{table}[hbt]

\caption{\small Measured $z$-pitch and $z$-shift (with respect to the fitted wire grid)
for individual tube layers (average of all scans on HV and RO side, respectively).
The layers are numbered in the sequence of glueing during chamber construction and
from the inside out for each multilayer (multilayer 1: layers 1--3, multilayer 2: layers 4--5).
}

\begin{center}
\begin{tabular}{|c||c|c||c|c||c|c||}
\hline
Layer no. & \multicolumn{2}{c||}{$z$-pitch $[\mu$m$]$} & \multicolumn{2}{c||}{$z$-shift $[\mu$m$]$} \\
\hline
& HV side & RO side & HV side & RO side \\

\hline\hline
1 & $30035.9 \pm 0.08$ & $30036.0 \pm 0.04$ & $-\hphantom{1}6\pm 5$ & $+11\pm 6$            \\
2 & $30036.3 \pm 0.05$ & $30036.0 \pm 0.06$ & $-\hphantom{1}9\pm 5$ & $-\hphantom{1}5\pm 3$ \\
3 & $30036.3 \pm 0.06$ & $30036.0 \pm 0.08$ & $+16\pm 3$            & $-\hphantom{1}3\pm 2$ \\
\hline            
4 & $30036.2 \pm 0.05$ & $30036.4 \pm 0.04$ & $-\hphantom{1}8\pm 5$ & $-11\pm 4$            \\
5 & $30036.0 \pm 0.08$ & $30036.1 \pm 0.04$ & $\hphantom{-1}0\pm 4$ & $-\hphantom{1}6\pm 4$ \\
6 & $30035.9 \pm 0.05$ & $30036.3 \pm 0.02$ & $+11\pm 3$            & $+19\pm 4$            \\
\hline
Average & $30036.2 \pm 0.1$ & $30062.0 \pm 0.05$ & & \\
\hline\hline

\end{tabular}
\end{center}
\label{tab:layers_z}
\end{table}


\begin{table}[hbt]

\caption{\small Averaged measured $y$-distances $[\mu$m$]$ between
subsequent tube layers (numbering as in Table~\ref{tab:layers_z})
}

\begin{center}
\begin{tabular}{|c||c|c||c||}
\hline
Layer no. & HV side & RO side & Design value \\

\hline\hline
1--2    & $26064 \pm 2.0$ & $26058 \pm 2$ & $26046 \pm 10$ \\
2--3    & $26059 \pm 3.0$ & $26061 \pm 2$ & $26061 \pm 10$ \\
\hline                           
4--5    & $26054 \pm 2.0$ & $26050 \pm 2$ & $26041 \pm 10$ \\
5--6    & $26055 \pm 2.0$ & $26072 \pm 2$ & $26066 \pm 10$ \\
\hline
Average & $26058 \pm 0.5$ & $26060 \pm 1$ & $26054\pm 10$ \\
\hline\hline

\end{tabular}
\end{center}
\label{tab:layers_y}
\end{table}


\begin{table}[thb]

\caption{\small RMS widths of the residual distributions for 5 scans
at different positions along the tubes in the
same orientation r4 of the chamber (see Figure~\ref{fig:scans}).
The expectations according to equation (3)
from interpolation of the wire locations between the fixed ends 
and the predicted X-ray tomograph resolutions $\sigma_{y,z}^{\mathrm tomo}$
are also shown.}

\begin{center}
\begin{tabular}{|c|rl||c|c||c|c||c|c||} \hline
{Scan}&\multicolumn{2}{c||}{$x$-position $[$mm$]$}&
{$RMS^{\mathrm tot}_{y}$ $[\mu$m$]$}&{$RMS^{\mathrm tot}_{z}$ $[\mu$m$]$}&
{$RMS^{\mathrm tot}_{y}$ $[\mu$m$]$}&{$RMS^{\mathrm tot}_{z}$ $[\mu$m$]$}\\

& & & measured & measured & expected & expected \\
\hline\hline
{15}&    0 & (HV side) &{20.3}&{18.8}&{20.3}&{18.8} \\
{19}&  L/4 &           &{17.7}&{17.3}&{15.1}&{14.7} \\
{16}&  L/2 & (Center)  &{15.0}&{16.1}&{12.9}&{13.5} \\
{18}& 3L/4 &           &{15.5}&{16.7}&{13.6}&{15.5} \\
{17}&    L & (RO side) &{17.0}&{19.7}&{17.0}&{19.7} \\
\hline
\multicolumn{5}{c||}{\hfill} &
$\sigma_{y, {\mathrm tot}}^{\mathrm tomo} = 10.0~\mu$m &
$\sigma_{z, {\mathrm tot}}^{\mathrm tomo} = 12.1~\mu$m \\ [3mm]
\cline{6-7}\hline

{Scan}&\multicolumn{2}{c||}{$x$-position $[$mm$]$}&
{$RMS^{\mathrm top}_{y}$ $[\mu$m$]$}&{$RMS^{\mathrm top}_{z}$ $[\mu$m$]$}&
{$RMS^{\mathrm top}_{y}$ $[\mu$m$]$}&{$RMS^{\mathrm top}_{z}$ $[\mu$m$]$}\\

& & & measured & measured & expected & expected \\
\hline\hline
{15}&    0 & (HV side) &{18.5}&{20.7}&{18.5}&{20.7} \\
{19}&  L/4 &           &{16.0}&{18.3}&{14.3}&{16.0} \\
{16}&  L/2 & (Center)  &{12.6}&{13.8}&{11.8}&{13.0} \\
{18}& 3L/4 &           &{12.2}&{13.1}&{11.9}&{12.9} \\
{17}&    L & (RO side) &{14.6}&{15.7}&{14.6}&{15.7} \\
\hline
\multicolumn{5}{c||}{\hfill} &
$\sigma_{y, {\mathrm top}}^{\mathrm tomo} = \hphantom{1}6.3~\mu$m &
$\sigma_{z, {\mathrm top}}^{\mathrm tomo} = \hphantom{1}6.6~\mu$m \\ [3mm]
\cline{6-7}\hline

{Scan}&\multicolumn{2}{c||}{$x$-position $[$mm$]$}&
{$RMS^{\mathrm bot}_{y}$ $[\mu$m$]$}&{$RMS^{\mathrm bot}_{z}$ $[\mu$m$]$}&
{$RMS^{\mathrm bot}_{y}$ $[\mu$m$]$}&{$RMS^{\mathrm bot}_{z}$ $[\mu$m$]$}\\

& & & measured & measured & expected & expected \\
\hline\hline
{15}&    0 & (HV side) &{21.9}&{16.8}&{21.9}&{16.8} \\
{19}&  L/4 &           &{19.1}&{16.3}&{17.1}&{13.8} \\
{16}&  L/2 & (Center)  &{16.9}&{17.9}&{14.5}&{14.1} \\
{18}& 3L/4 &           &{18.0}&{19.5}&{15.2}&{17.5} \\
{17}&    L & (RO side) &{18.9}&{22.7}&{18.9}&{22.7} \\
\hline
\multicolumn{5}{c||}{\hfill} &
$\sigma_{y, {\mathrm bot}}^{\mathrm tomo} = 12.3~\mu$m &
$\sigma_{z, {\mathrm bot}}^{\mathrm tomo} = 15.6~\mu$m \\ [3mm]
\cline{6-7}\cline{6-7}
\end{tabular}
\end{center}
\label{tab:middle}
\end{table}

\section{Comparison with X-ray Data for Individual Tubes}

In Figures~\ref{fig:sita} and \ref{fig:sitb}, the residuals (with respect to the fitted grid)
of the $y$-coordinates of the wires from X-ray tomograph scans on the RO side  
are plotted against the $y$-displacements
of the wires from the center of individual drift tubes measured with X-rays before chamber assembly
(see \cite{BOS}). Similar correlations exist for the $z$-coordinate.

The expected correlations between the two wire coordinate measurements are obvious, independent
of the orientation of the chamber during the X-ray tomograph scans.
Using a simple model with Gaussian errors, this information can be used to roughly disentangle and
estimate the different contributions to the total uncertainty in the wire coordinates:
\clearpage

\begin{enumerate}

\item The positioning accuracy $\sigma^{\mathrm wire}$ of the wires inside individual drift tubes.

\item The resolution $\sigma^{\mathrm X-ray}$ of the X-ray measurements of individual drift tubes.

\item The positioning accuracy $\sigma^{\mathrm tube}$ of drift tubes in the chamber.

\item The average resolution $\sigma^{\mathrm tomo}$ of the X-ray tomograph measurements.

\end{enumerate}

Since the four types of errors are statistically independent, they add quadratically
to give the widths $\sigma^{\mathrm res}$, $\sigma^{\mathrm disp}$ and $\sigma^{\mathrm diff}$
of the distributions of the measured parameters shown in Figures~\ref{fig:sigmas_y} and \ref{fig:sigmas_z},
the residuals of the wire coordinates from the X-ray tomograph, the wire displacements
measured in the individual drift tubes and the difference of the two correlated variables:

\begin{eqnarray}
   \sigma^{\mathrm res}  &=& \sigma^{\mathrm wire}\oplus\sigma^{\mathrm tube}\oplus\sigma^{\mathrm tomo}, \\
   \sigma^{\mathrm disp} &=& \sigma^{\mathrm wire}\oplus\sigma^{\mathrm X-ray}, \\
   \sigma^{\mathrm diff} &=& \sigma^{\mathrm tube}\oplus\sigma^{\mathrm X-ray}\oplus\sigma^{\mathrm tomo}.
\end{eqnarray}

The average resolution $\sigma^{\mathrm tomo}$ of the X-ray tomograph over all chamber orientations
has been estimated in section 4, equations (1) and (2).
Therefore, the three equations (5)--(7) allow
to determine the three unknown errors 
$\sigma^{\mathrm wire}$, $\sigma^{\mathrm tube}$ and $\sigma^{\mathrm X-ray}$.
The standard deviations (from fit of a Gaussian) of the measured distributions,
$\sigma^{\mathrm res}$ averaged over the scans, $\sigma^{\mathrm disp}$ and $\sigma^{\mathrm diff}$
(compare Figures~\ref{fig:sigmas_y} and \ref{fig:sigmas_z} for two scans), and the values for
$\sigma^{\mathrm wire}$, $\sigma^{\mathrm tube}$ and $\sigma^{\mathrm X-ray}$
calculated from them are given in Table~\ref{tab:sigmas} separately
for the HV and RO side and the two multilayers numbered in the sequence of assembly.

The expection of larger $z$-positioning errors $\sigma_z^{\mathrm tube}$ on the HV side of multilayer 1
and on the RO side of multilayer 2 due to global $z$-shifts of the two outer tube layers (see section 6)
is reproduced by the results.

\begin{table}[htb]
\caption{\small
The uncertainties $\sigma^{\mathrm res}$, $\sigma^{\mathrm disp}$ and $\sigma^{\mathrm diff}$
in the wire coordinates measured by the X-ray tomograph (average of scans) and for indidvidual tubes 
and their difference
and the errors $\sigma^{\mathrm wire}$, $\sigma^{\mathrm tube}$ and $\sigma^{\mathrm X-ray}$
determined from them separately for the two chamber ends and the two multilayers ML~1 and ML~2
numbered in the assembly sequence (all numbers in $[\mu$m$]$).}

\begin{center}
\begin{tabular}{|c|c|c||c|c|c||c|c|c||} \hline
{Side}&{Multilayer}&{Coord.}&
{$\sigma^{\mathrm res}$}&{$\sigma^{\mathrm disp}$}&{$\sigma^{\mathrm diff}$}&
{$\sigma^{\mathrm wire}$}&{$\sigma^{\mathrm tube}$}&{$\sigma^{\mathrm X-ray}$}\\
\hline\hline
{HV}&{ML~1}&{$y$}&{16.6}&{14.4}&{16.3}&{10.4}&{11.4}&{10.0}\\
{HV}&{ML~2}&{$y$}&{12.8}&{10.4}&{10.9}&{\hphantom{1}8.8}&{\hphantom{1}7.1}&{\hphantom{1}5.5}\\
\hline
{RO}&{ML~1}&{$y$}&{17.2}&{12.6}&{15.0}&{10.7}&{12.0}&{\hphantom{1}6.7}\\
{RO}&{ML~2}&{$y$}&{16.0}&{11.6}&{11.9}&{11.2}&{\hphantom{1}9.8}&{\hphantom{1}3.2}\\
\hline
\hline
{HV}&{ML~1}&{$z$}&{17.6}&{\hphantom{1}8.6}&{16.5}&{\hphantom{1}7.5}&{15.5}&{\hphantom{1}4.2}\\
{HV}&{ML~2}&{$z$}&{12.5}&{11.1}&{11.0}&{\hphantom{1}8.9}&{\hphantom{1}8.0}&{\hphantom{1}6.6}\\
\hline
{RO}&{ML~1}&{$z$}&{10.5}&{\hphantom{1}9.2}&{\hphantom{1}9.8}&{\hphantom{1}7.0}&{\hphantom{1}7.0}&{\hphantom{1}6.0}\\
{RO}&{ML~2}&{$z$}&{15.3}&{11.5}&{16.0}&{\hphantom{1}7.4}&{12.9}&{\hphantom{1}8.8}\\
\hline
\hline
\end{tabular}
\end{center}
\label{tab:sigmas}
\end{table}

\section{Bad Wire Coordinates}

Tails of the distributions of the residuals of the wire coordinates are caused by a few
wire coordinates far away from the fitted grid. During drift tube testing and chamber assembly it was
verified that such large deviations did not occur (see \cite{BOS}). Therefore these wires most likely
have been wrongly measured by the X-ray tomograph. 
They have, however, not been removed from the data sample for the grid fits.
They are partly responsible for the rather large
variations in the X-ray tomograph resolution for different chamber orientations (see section 4 and
Table~\ref{tab:repr}. 

To estimate the effect of the ``bad'' wire coordinates on the RMS widths of the residual distributions,
they have been fitted with a Gaussian and $\pm 3\sigma$- and $\pm 2\sigma$-cuts were applied which
typically rejected $2\%$ and $10\%$ of the wires, respectively.
The $RMS$ values of the residual distributions after the cuts are shown in
Table~\ref{tab:fit_8p_yz}.

\begin{table}[htb]
\caption{RMS values and standard deviations $\sigma$ (from fit of a Gaussian) 
of the residual distributions for
the global fit to the X-ray tomograph measurements (average of all scans)
before and after elimination of outlying wire coordinate measurements (in $[\mu$m$]$).}

\begin{center}
\begin{tabular}{|c||c|c||c|c||} \hline
{Side}&{$RMS^{\mathrm tot}_{y}$}&{$\sigma^{\mathrm tot}_{y}$}&
{$RMS^{\mathrm tot}_{y}$ ($3\sigma$ cut)}&{$RMS^{\mathrm tot}_{y}$ ($2\sigma$ cut)}\\
\hline\hline

{Average}&{19.0}&{17.6}&{17.5}&{15.4}\\
{HV side}&{$\pm 0.8$}&{$\pm 0.9$}&{$\pm 0.8$}&{$\pm 0.6$}\\
\hline
{Average}&{17.0}&{14.8}&{15.6}&{13.0}\\
{RO side}&{$\pm 0.2$}&{$\pm 0.2$}&{$\pm 0.2$}&{$\pm 0.1$}\\
\hline
\hline

{Side}&{$RMS^{\mathrm tot}_{z}$}&{$\sigma^{\mathrm tot}_{z}$}&
{$RMS^{\mathrm tot}_{z}$ ($3\sigma$ cut)}&{$RMS^{\mathrm tot}_{z}$ ($2\sigma$ cut)}\\
\hline\hline
{Average}&{19.3}&{17.3}&{17.9}&{15.1}\\
{HV side}&{$\pm 0.8$}&{$\pm 1.0$}&{$\pm 0.6$}&{$\pm 0.9$}\\
\hline
{Average}&{18.2}&{15.9}&{16.0}&{13.9}\\
{RO side}&{$\pm 1.3$}&{$\pm 1.3$}&{$\pm 1.2$}&{$\pm 1.0$}\\
\hline
\hline
\end{tabular}
\end{center}
\label{tab:fit_8p_yz}
\end{table}

\section{Conclusions}

The complete X-ray tomograph data of the BOS prototype chamber have been
analyzed. The results lead to the following conclusions:

\begin{enumerate}

\item From repeated scans of the chamber in the same position,
the reproducibility and intrinsic accuracy of the X-ray tomograph
measurements has been evaluated. On top of the systematic variations
in the X-ray measurements of the geometrical chamber parameters 
(minimised by averaging over all scans), the average intrinsic resolution of the X-ray tomograph
was estimated to be $6~\mu$m rms for the $y$ coordinate and $4~\mu$m rms in $z$
from the reproducibility of the measurements of the BOS prototype chamber.


\item 
The geometrical parameters of the prototype chamber
are determined by fitting a wire grid to the measured wire coordinates
with $y$- and $z$-pitch, multilayer (ML) separations in $y$ and $z$ and the
global cross plate sag in the X-ray tomograph supports as free parameters.
After averaging over all scans, the chamber parameters and the RMS widths
of the residual distributions for high voltage and readout side are the following
(without cut against outlying wire coordinate measurements):

\begin{tabular}{|l|c|c|c|c|}

\cline{2-5}
\multicolumn{1}{c}{\hfill} & \multicolumn{2}{|c}{HV side} & \multicolumn{2}{|c|}{RO side} \\
\cline{2-5}

\multicolumn{1}{c|}{\hfill}  & X-ray           & Design value     & X-ray           & Design value     \\
\hline

$y$-pitch $[\mu$m$]$        & $\hphantom{.2}26058\pm 0.5\hphantom{11}$  & $\hphantom{.2}26054\pm 10\hphantom{10.}$  & 
$\hphantom{.2}26060\pm 1.3\hphantom{11}$  & $\hphantom{.2}26054\pm 10\hphantom{10.}$ \\

$z$-pitch $[\mu$m$]$        & $30036.2\pm 0.1\hphantom{11}$ & $30036.8\pm 0.6\hphantom{11}$   &
$30036.2\pm 0.05\hphantom{1}$ & $30036.8\pm 0.6\hphantom{11}$ \\

$y$ sep.\ ML $[$mm$]$          & $347.041\pm 0.011$ & $347.040\pm 0.010$ & $347.078\pm 0.012$ & $347.072\pm 0.010$ \\

$z$-sep.\ ML $[\mu$m$]$        & $\hphantom{347}-8\pm 11\hphantom{0.0}$ & $\hphantom{347.01}0\pm 10\hphantom{0.0}$ &
$\hphantom{347.90}3\pm 16\hphantom{0.0}$ & $\hphantom{347.01}0\pm 10\hphantom{0.0}$ \\
\hline
\multicolumn{5}{|l|}{Including X-ray tomograph resolution:} \\
\hline

RMS${}_y$ $[\mu$m$]$        & \multicolumn{2}{|c}{$19.0\pm 0.8$}   & \multicolumn{2}{|c|}{$17.0\pm 0.2$} \\

RMS${}_z$ $[\mu$m$]$        & \multicolumn{2}{|c}{$19.3\pm 0.8$}   & \multicolumn{2}{|c|}{$18.2\pm 1.3$} \\
\hline
\multicolumn{5}{|l|}{Wire locations with X-ray tomograph resolution subtracted:} \\
\hline

RMS${}_y$ $[\mu$m$]$        & \multicolumn{2}{|c}{$18.0\pm 0.8$}   &
\multicolumn{2}{|c|}{$15.9\pm 0.2$} \\ 
RMS${}_z$ $[\mu$m$]$        & \multicolumn{2}{|c}{$19.0\pm 0.8$}   &
\multicolumn{2}{|c|}{$17.8\pm 1.3$} \\
\hline

\end{tabular}

The fitted parameters are in good agreement with the design values (at $20^\circ$C);
uncertainties in the wire locations are below the required $20~\mu$m rms.

The design parameters of the prototype chamber (at $20^\circ$C) determined by the assembly
jigging are well reproduced by the X-ray tomograph data within the
measurement errors. The wires have been positioned in the whole chamber with an accuracy
which is better than the required $20~\mu$m RMS.

\item A further important achievement for the validation of the chamber assembly procedure
is the good uniformity of the $z$-pitch and of the $y$-separation of all tube layers.

The only significant deviations from the design parameters
of the chamber are global shifts in $z$ of the two outer tube layers 
by $+16~\mu$m and $-19~\mu$m, respectively, with respect to the fitted grid,
layer 3 in the assembly sequence on the high-voltage side and layer 6
on the readout side. The two shifts of equal size and same direction in the coordinate frame
of the assembly jigging can be attributed to the
same sphere tower interface being outside tolerances in the $z$-displacement of $\pm 10\mu$m.
Because of the $z$-shifts,
the RMS width for the $z$-coordinate is larger than for $y$ in spite of the uniform
$z$-pitch.

\item Both with the cross plate sag monitors and from the X-ray tomograph data,
global gravitational sags of the high voltage and readout ends of the prototype chamber
were found which are larger than expected from the Bessel support
of the cross plates and which vary from scan to scan.
Obviously the chamber was not supported free of external moments
in the tomograph. Kinematical mounts
as in ATLAS should be used to support the chambers for the X-ray tomograph measurements.

To evaluate the accuracy of the chamber construction, it is necessary to take the global
deformation into account by fitting a common parabola for all layers.
Beyond this global correction, no systematic deformations of individual tube
layers were found.

\end{enumerate}



\section*{ Acknowledgments}

The authors gratefully acknowledge Hermann Wellenstein for helpful
discussions.

\begin{thebibliography}{99}

\bibitem{XTOMO} D.~Drakulakos et al.,  {\em The High-Precision
X-ray Tomograph for Quality Control of the ATLAS MDT Muon Spectrometer},
CERN-OPEN-97-023, July 1997.

\bibitem{BOS} V.Bartheld et al.,  {\em Construction of the Full-Scale
Prototype of a BOS MDT Chamber}, ATL-MUON-98-256, October 1998.

\bibitem{TDR} {\em ATLAS Muon Spectrometer--Technical Design Report},
CERN/LHCC/97-22, ATLAS TDR 10, 31 May 1997.

\end{thebibliography}
\clearpage

\begin{figure}[p]
\begin{center}
\begin{large}
\mbox{\epsfig{figure=xray_scan.eps,height=0.93\textheight}}
\caption{\normalsize
   The orientations of the prototype chamber (r1, r2, r3, r4) during the
   X-ray tomograph measurements. The two multilayers ML1 and ML2 are numbered
   in the assembly sequence.}
\label{fig:scans}
\end{large}
\end{center}
\end{figure}
\clearpage

\begin{figure}[thb]
\begin{center}
\begin{picture}(400,600)

\put(60,460){\makebox(0,0){\epsfig{figure=sj1yr1.eps,
width=0.55\textwidth}}}

\put(330,460){\makebox(0,0){\epsfig{figure=sk1yr2.eps,
width=0.55\textwidth}}}

\put(60,115){\makebox(0,0){\epsfig{figure=ss1yr3.eps,
width=0.55\textwidth}}}

\put(330,115){\makebox(0,0){\epsfig{figure=st1yr4.eps,
width=0.55\textwidth}}}
\end{picture}
\vspace{15mm}

\caption{\normalsize
Average of the $y$-residuals of the wires in the six tube layers as a function of the $z$-coordinate
for the 4 orientations (r1, r2, r3, r4) of the HV-side of the chamber.}
\label{fig:sag_hv}
\end{center}
\end{figure}
\clearpage

\begin{figure}[thb]
\begin{center}
\begin{picture}(400,600)

\put(60,460){\makebox(0,0){\epsfig{figure=sk5yr2.eps,
width=0.55\textwidth}}}

\put(330,460){\makebox(0,0){\epsfig{figure=ss5yr3.eps,
width=0.55\textwidth}}}

\put(60,115){\makebox(0,0){\epsfig{figure=st5yr4.eps,
width=0.55\textwidth}}}
\end{picture}
\vspace{15mm}

\caption{\normalsize
Average of the $y$-residuals of the wires in the six tube layers as a function of the $z$-coordinate
for the three orientations (r2, r3, r4) of the RO-side of the chamber.}
\label{fig:sag_ro}
\end{center}
\end{figure}
\clearpage

\begin{figure}[p]
\vspace{-15mm}

\begin{center}
\begin{large}
\mbox{\epsfig{figure=wire_7-8_yz.eps,width=0.75\textwidth}}

\vspace{-9mm}
\caption{\normalsize
   Differences of the residuals of the measured wire positions in $y$ and $z$ in the 
   top multilayer for scans no.\ 7 and 8
   as functions of the wire position in $z$.}
\label{fig:7-8_yz}
\end{large}
\end{center}

\vspace{-12mm}
\begin{center}
\begin{large}
\mbox{\epsfig{figure=wire_7-8_yz_b.eps,width=0.75\textwidth}}

\vspace{-9mm}
\caption{\normalsize
   Differences of the residuals of the measured wire positions in $y$ and $z$ in the 
   bottom multilayer for scans no.\ 7 and 8
   as functions of the wire position in $z$.}
\label{fig:7-8_yz_b}
\end{large}
\end{center}
\end{figure}
\clearpage

\begin{figure}[p]
\vspace{-16mm}

\begin{center}
\begin{large}
\mbox{\epsfig{figure=wire_7-8_lr.eps,width=0.75\textwidth}}

\vspace{-10mm}
\caption{\normalsize
   Differences of the residuals of the projected wire coordinates measured with the left and the right                      
   X-ray beam in the top multilayer for scans no.\ 7 and 8 
   as functions of the wire position in $z$.}
\label{fig:7-8_lr}
\end{large}
\end{center}

\vspace{-17mm}
\begin{center}
\begin{large}
\mbox{\epsfig{figure=wire_7-8_lr_b.eps,width=0.75\textwidth}}

\vspace{-10mm}
\caption{\normalsize
   Differences of the residuals of the projected wire coordinates measured with the left and the right                      
   X-ray beam in the bottom multilayer for scans no.\ 7 and 8
   as functions of the wire position in $z$.}
\label{fig:7-8_lr_b}
\end{large}
\end{center}
\end{figure}
\clearpage

\begin{figure}[thb]
\begin{large}
\mbox{\hspace{-20mm}\epsfig{figure=hv_r4_20.eps,width=1.20\textwidth}}
\caption{\normalsize
Distributions of the residuals of the
wire positions measured by the X-ray tomograph on the high voltage (HV) side
(scan no.\ 20) with respect to the fitted wire grid.}
\label{fig:res_20}
\end{large}
\end{figure}
\clearpage

\begin{figure}[thb]
\begin{large}
\mbox{\hspace{-20mm}\epsfig{figure=ro_r3_13.eps,width=1.20\textwidth}}
\caption{\normalsize
Distributions of the residuals of the
wire positions measured by the X-ray tomograph on the high voltage (RO) side
(scan no.\ 13) with respect to the fitted wire grid.}
\label{fig:res_13}
\end{large}
\end{figure}
\clearpage

\begin{figure}[thb]
\begin{center}
\begin{large}
\mbox{\epsfig{figure=middle_tot.eps,width=1.0\textwidth}}
\caption{\normalsize
   The widths $RMS_{y}$ and $RMS_{z}$ of the residual distributions of the global fit
   to the wire coordinates of both multilayers
   for scans no.\ 20, 19, 16, 18 and 17 at increasing $x$ from the HV to the RO side of the chamber 
   in the same orientation r4.}
\label{fig:middle_tot}
\end{large}
\end{center}
\end{figure}
\clearpage

\begin{figure}[thb]
\begin{center}
\begin{large}
\mbox{\epsfig{figure=middle_top.eps,width=1.0\textwidth}}
\caption{\normalsize
   The widths $RMS_{y}$ and $RMS_{z}$ of the residual distributions of the global fit
   to the wire coordinates of the top multilayer
   for scans no.\ 15, 19, 16, 18 and 17 at increasing $x$ from the HV to the RO side of the chamber
   in the same orientation r4.}
\label{fig:middle_top}
\end{large}
\end{center}
\end{figure}
\clearpage

\begin{figure}[thb]
\begin{center}
\begin{large}
\mbox{\epsfig{figure=middle_bot.eps,width=1.0\textwidth}}
\caption{\normalsize
   The widths $RMS_{y}$ and $RMS_{z}$ of the residual distributions of the global fit
   to the wire coordinates of the bottom multilayer
   for scans no.\ 15, 19, 16, 18 and 17 at increasing $x$ from the HV to the RO side of the chamber
   in the same orientation r4.}
\label{fig:middle_bot}
\end{large}
\end{center}
\end{figure}
\clearpage

\begin{figure}[thb]
\begin{center}
\begin{large}
\mbox{\epsfig{figure=sita.eps,width=1.0\textwidth}}
\caption{\normalsize
   Correlations of the residuals of the $y$-coordinates measured by the X-ray tomograph (Xtomo) with the
   $y$-displacements of the wires from the center measured in individual drift tubes with X-rays before assembly (Xray)
   (for the 6 tube layers on the RO side in orientation r4 of the chamber).}
\label{fig:sita}
\end{large}
\end{center}
\end{figure}
\clearpage

\begin{figure}[thb]
\begin{center}
\begin{large}
\mbox{\epsfig{figure=sitb.eps,width=1.0\textwidth}}
\caption{\normalsize
   Correlations of the residuals of the $y$-coordinates measured by the X-ray tomograph (Xtomo) with the
   $y$-displacements of the wires from the center measured in individual drift tubes with X-rays before assembly (Xray)
   (for the 6 tube layers on the RO side in orientation r2 of the chamber).}
\label{fig:sitb}
\end{large}
\end{center}
\end{figure}
\clearpage

\begin{figure}[thb]
\begin{picture}(400,600)

\put(110,440){\makebox(0,0){\epsfig{figure=sib.eps,
width=0.53\textwidth}}}
\put(370,440){\makebox(0,0){\epsfig{figure=sic.eps,
width=0.53\textwidth}}}
\put(110,100){\makebox(0,0){\epsfig{figure=sid.eps,
width=0.53\textwidth}}}
\put(370,100){\makebox(0,0){\epsfig{figure=sie.eps,
width=0.53\textwidth}}}
\end{picture}
\vspace{14mm}

\begin{center}
\caption{\normalsize
   Distributions  of the residuals of the $y$-coordinates of the wires from the
   X-ray tomograph (TOMO), of the
   $y$-displacements of the wires measured in individual tubes (XRAY) and of their difference
   for the RO side and separately
   for the two multilayers (ML1: layers 1--3, ML2: layers 4--6)
   in the orientations r2 and r4 of the chamber.}
\label{fig:sigmas_y}
\end{center}
\end{figure}
\clearpage

\begin{figure}[thb]
\begin{picture}(400,600)

\put(110,440){\makebox(0,0){\epsfig{figure=vz52c.eps,
width=0.53\textwidth}}}
\put(370,440){\makebox(0,0){\epsfig{figure=vz52d.eps,
width=0.53\textwidth}}}
\put(110,100){\makebox(0,0){\epsfig{figure=vz54c.eps,
width=0.53\textwidth}}}
\put(370,100){\makebox(0,0){\epsfig{figure=vz54d.eps,
width=0.53\textwidth}}}
\end{picture}
\vspace{14mm}

\begin{center}
\caption{\normalsize
   Distributions  of the residuals of the $z$-coordinates of the wires from the
   X-ray tomograph (TOMO), of the
   $z$-displacements of the wires measured in individual tubes (XRAY) and of their difference
   for the RO side and separately
   for the two multilayers (ML1: layers 1--3, ML2: layers 4--6)
   in the orientations r2 and r4 of the chamber.}
\label{fig:sigmas_z}
\end{center}
\end{figure}
\clearpage

\end{document}
