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<!-- DOI: 10.1051/0004-6361/200911727  -->

<h2 class="sec">Online Material</h2>

<p>

<h2 class="sec"><a name="SECTION000100000000000000000"></a>&#160; <A NAME="appa"></A>
Appendix A: Examples of inverted spectra
</h2>

<p>

<p>
<div class="inset-old">
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<tr><td><!-- init Label --><A NAME="specs1">&#160;</A><!-- end Label--><A NAME="1633"></A><A NAME="figure1302"
 HREF="img93.png"><IMG
 WIDTH="208" HEIGHT="211" SRC="Timg93.png"
 ALT="\begin{figure}
\par\includegraphics[width=18.4cm,clip]{11727fa1.eps}
\end{figure}"></A><!-- HTML Figure number: 14 --></td>
<td class="img-txt"><span class="bold">Figure A.1:</span><p>
Examples of observed (<EM> black line</EM>) and best-fit spectra (<EM>
    red</EM>) in Op.&nbsp;001 (<EM> 1st and 3rd row</EM>). The <EM> dash-dotted horizontal</EM>
  lines in <I>QUV</I> indicate three times the rms noise level, and the <EM> solid horizontal</EM> line the zero level. The <EM> vertical solid</EM> line denotes the rest wavelength. The <EM> 2nd and 4th row</EM> show the corresponding temperature stratifications of the magnetic component (<EM> solid</EM>), the field-free component (<EM> dash-dotted</EM>), and the HSRA atmosphere that is used as initial model (<EM> dashed</EM>). Field strength and LOS inclination and their respective errors are given in the plot of Stokes <I>I</I> of 1564.8 nm (<EM> upper left in each panel</EM>); the number of each profile is given in the <EM> upper left</EM> corner of each panel.</p></td>
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 ALT="\begin{figure}
\par\includegraphics[width=18.4cm,clip]{11727fa2.eps}
\end{figure}"></A><!-- HTML Figure number: 15 --></td>
<td class="img-txt"><span class="bold">Figure A.2:</span><p>
Same as Fig.&nbsp;<a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#specs1">A.1</a> for TIP Op.&nbsp;005.</p></td>
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<tr><td><!-- init Label --><A NAME="p_exams">&#160;</A><!-- end Label--><A NAME="1635"></A><A NAME="figure1325"
 HREF="img95.png"><IMG
 WIDTH="208" HEIGHT="210" SRC="Timg95.png"
 ALT="\begin{figure}
\par\includegraphics[width=18.4cm,clip]{11727fa3.eps}
\end{figure}"></A><!-- HTML Figure number: 16 --></td>
<td class="img-txt"><span class="bold">Figure A.3:</span><p>
Polarization degree of 1564.8 nm for the profiles shown in Figs.&nbsp;<a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#specs1">A.1</a> (<EM> upper two rows</EM>) and <a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#specs2">A.2</a> (<EM> lower two rows</EM>). The <EM> dash-dotted</EM> and <EM> solid</EM> horizontal lines denote the inversion and final rejection threshold, respectively.</p></td>
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<p>
Figures <a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#specs1">A.1</a> and <a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#specs2">A.2</a> show several profiles taken from the
first and second long-integration observation of 2008 May 21 (Op.&nbsp;001 and
Op.&nbsp;005), respectively. The positions of the profiles are marked by 
consecutive numbers in Fig.&nbsp;<a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#fig1">1</a>. The profiles shown were selected to
have a small polarization degree that in some cases was barely sufficient to
meet the inversion threshold (e.g., profiles Nos.&nbsp;6 and 7). Below the spectra,
the temperature stratifications used in the generation of the
best-fit spectra are shown. With 3&nbsp;nodes in temperature, the SIR code can use
a parabola for changing the stratification; the parabola shape appears quite
prominent for many of the locations. We note, however, that the IR lines at
1.56&nbsp;<IMG
 WIDTH="10" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img16.png"
 ALT="$\mu$">m are not sensitive to the temperature in the atmosphere above log

<!-- MATH: $\tau\sim -1.5$ -->
<IMG
 WIDTH="52" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img96.png"
 ALT="$\tau\sim -1.5$">
(<a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#cabrera+bellot+iniesta2005">Cabrera Solana et&nbsp;al.  2005</a>). Only one profile
corresponds to a kG field (Fig.&nbsp;<a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#specs2">A.2</a>, <EM>top middle</EM>, No.&nbsp;8). Figure&nbsp;<a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#p_exams">A.3</a> shows the polarization degree of 1564.8&nbsp;nm for all profiles of
the previous figures. Profile No.&nbsp;9 exceeds the inversion threshold of 0.001
of <IMG
 WIDTH="14" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img1.png"
 ALT="$I_{\rm c}$">
near +750&nbsp;m<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img97.png"
 ALT="${\AA}$">
with a spike that is presumably  not of solar
origin, but is instead noise in the Stokes <I>U</I> profile. The final rejection threshold of 0.0014 is, however, only reached by signals clearly related to the Zeeman effect (multiple double or triple lobes).

<p>

<h4 class="sec3"><a name="SECTION000100010000000000000"></a>
Reliability of the inversion results.
</h4>
As discussed in Sect. 3, we used a constant value for the magnetic field
strength (<I>B</I>), inclination (<IMG
 WIDTH="10" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img12.png"
 ALT="$\gamma $">), azimuth (<IMG
 WIDTH="11" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img56.png"
 ALT="$\psi$">), and the LOS
velocity. This inversion setup cannot reproduce the antisymmetric Stokes
<I>Q</I> or <I>U</I> or symmetric Stokes <I>V</I> profiles, which would require
gradients in the magnetic field strength and the velocities along the LOS. The
inversion was initialized with the same model atmosphere on all pixels
(<I>B</I>=0.9 kG, <IMG
 WIDTH="42" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img98.png"
 ALT="$\psi=65$">
deg), only the inclination being modified to 
10 or 170 deg depending on the polarity. In the inversion process, the equal
weight used for <I>QUV</I> in the calculation of <IMG
 WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img99.png"
 ALT="$\chi^2$">
naturally favors the
component of higher polarization signal. For example, in profile no.&nbsp;5 in Fig.&nbsp;<a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#specs1">A.1</a> the <I>Q</I> and <I>U</I> signals are larger than the <I>V</I> signal by almost an order of magnitude, leading to a better fit quality for <I>Q</I> and <I>U</I> than for <I>V</I>. In polarimetric data of low S/N, a difference of this order usually implies that the weaker signal is not seen at all.

<p>
SIR calculates an error estimate for the free fit parameters using the
diagonal elements of the covariance matrix, expressed by the response
functions (<A NAME="aaref6"></A><A NAME="tex2html78"
 HREF="#bellotrubio+etal2000">Bellot Rubio et&nbsp;al.  2000</A>; <A NAME="aaref4"></A><A NAME="tex2html79"
 HREF="#bellotrubio2003">Bellot Rubio  2003</A>). The error estimate
depends on the number of degrees of freedom in each variable; for parameters
constant with optical depth thus a single value is returned. The error
estimate, however, provides only  information about the
  reliability of the best-fit solution for the corresponding
<IMG
 WIDTH="17" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img99.png"
 ALT="$\chi^2$">-minimum inside the chosen inversion setup. The estimated errors 
in the inversion of the profiles shown in Figs.&nbsp;4, <a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#specs1">A.1</a>, and&nbsp;<a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#specs2">A.2</a> are noted on the
Stokes <I>I</I> panel for the Fe&nbsp;I&nbsp;1565.2&nbsp;nm line. The average uncertainties in the calculated magnetic field strength and inclination angle given by SIR are <IMG
 WIDTH="29" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img100.png"
 ALT="$\pm 50~$">G and <IMG
 WIDTH="27" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img101.png"
 ALT="$\pm 10$">&nbsp;deg, respectively. The values agree with a previous error estimate in <a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#beck2006">Beck (2006)</a> derived from a direct analysis of the profile shape of the 1.56&nbsp;<IMG
 WIDTH="10" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img16.png"
 ALT="$\mu$">m lines (Table&nbsp;3.2 on p.&nbsp;47; 
<!-- MATH: $\delta B \sim 50$ -->
<IMG
 WIDTH="49" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img102.png"
 ALT="$\delta B \sim 50$">&nbsp;G and 
<!-- MATH: $\delta \theta \sim 5$ -->
<IMG
 WIDTH="40" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img103.png"
 ALT="$\delta \theta \sim 5$">&nbsp;deg).

<p>

<h2 class="sec"><a name="SECTION000110000000000000000"></a>&#160; <A NAME="lin_cal"></A>
Appendix B: Calibration of <I>L<sub>tot</sub></I>
to transversal flux
</h2>

<p>
We tried to follow the procedure described in LI08 to calibrate the
linear polarization signal into a transversal magnetic flux estimate that is independent
of the inversion results. To reduce the influence of noise,  LI08 
first determine the  ``preferred azimuth frame'', where the linear polarization
signal is concentrated in Stokes <I>Q</I>. To achieve this, we determined the
azimuth angle from the ratio of <I>U</I> to <I>Q</I>, and rotated the spectra
correspondingly to maximize the Stokes <I>Q</I> signal. The scatter plot in Fig.&nbsp;<a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#lq_comp">B.1</a> compares the previously used total linear polarization, 
<!-- MATH: $L_{\rm tot} = \int \sqrt{Q^2+U^2}(\lambda) {\rm d}\lambda$ -->
<IMG
 WIDTH="142" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img105.png"
 ALT="$L_{\rm tot} = \int \sqrt{Q^2+U^2}(\lambda) {\rm d}\lambda$">,
with the corresponding 
<!-- MATH: $Q_{\rm tot}({\rm rotated}) = \int |Q(\lambda)|{\rm d}\lambda$ -->
<IMG
 WIDTH="149" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img83.png"
 ALT="$Q_{\rm tot}({\rm rotated}) = \int \vert Q(\lambda)\vert{\rm d}\lambda$">
as a measure of the linear polarization. The rotation of the spectra reduces the noise contribution by a constant amount, but the old and new values otherwise have a linear relationship with a slope close to unity.

<p>
We then averaged the rotated <I>Q</I> spectra over all spatial positions exceeding
the polarization threshold for the inversion. The average Stokes <I>Q</I> spectrum
was used as a spectral mask by LI08, but unfortunately their method
fails for the infrared lines. The wavelengths around the line core have
negative values in the average <I>Q</I> profile (Fig.&nbsp;<a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#qmask">B.2</a>), which prevents
to use it in the same way as in LI08. We thus used 
<!-- MATH: $Q_{\rm tot}({\rm
rotated})$ -->
<IMG
 WIDTH="72" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img106.png"
 ALT="$Q_{\rm tot}({\rm
rotated})$">
as defined above instead, which we  understand to be
equivalent to the approach of LI08 despite not using a (somewhat arbitrary) spectral mask. 

<p>
The plot of 
<!-- MATH: $Q_{\rm tot}({\rm rotated})$ -->
<IMG
 WIDTH="72" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img106.png"
 ALT="$Q_{\rm tot}({\rm
rotated})$">
versus transversal flux (Fig.&nbsp;<a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#fig7">12</a>,
<EM>middle upper panel</EM>) showed considerable scatter that places the
use of a single calibration curve in doubt. We thus not only tried to obtain a
calibration curve, but also to quantify the effect of various parameters on
the obtained relation. The <EM>upper part</EM> of Fig.&nbsp;<a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#qcal">B.3</a> shows
calibration curves of 
<!-- MATH: $Q_{\rm tot}$ -->
<IMG
 WIDTH="25" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img11.png"
 ALT="$Q_{\rm tot}$">
versus field strength for different field
inclinations <IMG
 WIDTH="10" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img12.png"
 ALT="$\gamma $">.
The uppermost curve for 
<!-- MATH: $\gamma = 90^\circ$ -->
<IMG
 WIDTH="47" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img107.png"
 ALT="$\gamma = 90^\circ$">
corresponds
to the one used by LI08. With the assumption that the field inclination does not necessary equal <IMG
 WIDTH="22" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img108.png"
 ALT="$90^\circ$">,
one already finds that one and the
same value of 
<!-- MATH: $Q_{\rm tot}$ -->
<IMG
 WIDTH="25" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img11.png"
 ALT="$Q_{\rm tot}$">
can be obtained for a range of around 200-550&nbsp;G
in field strength. The same effect is shown in the <EM>middle part</EM>, where
the magnetic flux, 
<!-- MATH: $\Phi=B\sin \gamma$ -->
<IMG
 WIDTH="68" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img109.png"
 ALT="$\Phi=B\sin \gamma$">,
was kept constant at 
<!-- MATH: $1.8\times
10^{16}$ -->
<IMG
 WIDTH="60" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img110.png"
 ALT="$1.8\times
10^{16}$">&nbsp;Mx, <I>B</I> was varied, and <IMG
 WIDTH="10" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img12.png"
 ALT="$\gamma $">
was derived accordingly from

<!-- MATH: $\gamma={\rm arcsin} (\Phi/B)$ -->
<IMG
 WIDTH="96" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img111.png"
 ALT="$\gamma={\rm arcsin} (\Phi/B)$">.
Again a range of around 200-500&nbsp;G in <I>B</I> corresponds to the same value of 
<!-- MATH: $Q_{\rm tot}$ -->
<IMG
 WIDTH="25" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img11.png"
 ALT="$Q_{\rm tot}$">.
As a final test, we chose to
investigate the influence of the temperature stratification on the resulting

<!-- MATH: $Q_{\rm tot}$ -->
<IMG
 WIDTH="25" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img11.png"
 ALT="$Q_{\rm tot}$">-value. We retained the magnetic flux, field strength and field
inclination constant at (
<!-- MATH: $1.8\times 10^{16}$ -->
<IMG
 WIDTH="60" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img110.png"
 ALT="$1.8\times
10^{16}$">&nbsp;Mx, 20&nbsp;G, <IMG
 WIDTH="22" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img112.png"
 ALT="$75^\circ$">), and
synthesized spectra for different temperature stratifications. We used 10&nbsp;000
temperature stratifications that were derived for the magnetic component in
the inversion, and thus can be taken to be an estimate of the range of
temperatures  expected in the quiet Sun. The histogram of the resulting

<!-- MATH: $Q_{\rm tot}$ -->
<IMG
 WIDTH="25" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img11.png"
 ALT="$Q_{\rm tot}$">-values is displayed in the <EM>bottom part</EM> of
Fig.&nbsp;<a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#qcal">B.3</a>. The value of 
<!-- MATH: $\sqrt{Q_{\rm tot}}$ -->
<IMG
 WIDTH="36" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img14.png"
 ALT="$\sqrt {Q_{\rm tot}}$">
ranges from nearly zero up to 0.01, which also
roughly corresponds to the scatter in 
<!-- MATH: $\sqrt{Q_{\rm tot}}$ -->
<IMG
 WIDTH="36" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img14.png"
 ALT="$\sqrt {Q_{\rm tot}}$">
in Fig.&nbsp;<a href="/articles/aa/full_html/2009/30/aa11727-09/aa11727-09.html#fig7">12</a>. We
thus conclude that the largest contribution to the scatter
comes from temperature effects. We remark that we used a magnetic filling
factor of unity in all calculations. Any additional variation in the
filling factor due to unresolved magnetic structures would
increase the scatter in 
<!-- MATH: $Q_{\rm tot}$ -->
<IMG
 WIDTH="25" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img11.png"
 ALT="$Q_{\rm tot}$">
even more.

<p>
We conclude that the usage of a calibration curve for a derivation of transversal magnetic flux from 
<!-- MATH: $L_{\rm tot}$ -->
<IMG
 WIDTH="22" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img4.png"
 ALT="$L_{\rm tot}$">
or 
<!-- MATH: $Q_{\rm tot}$ -->
<IMG
 WIDTH="25" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img11.png"
 ALT="$Q_{\rm tot}$">,
regardless of the exact calculation of the wavelength integrated quantities, is not reliable for a solid estimate, mainly because the strong influence of the thermodynamical state of the atmosphere on the weak polarization signals.

<p>

<p>

<p>
<div class="inset-old">
<table>
<tr><td><!-- init Label --><A NAME="lq_comp">&#160;</A><!-- end Label--><A NAME="1639"></A><A NAME="figure1389"
 HREF="img113.png"><IMG
 WIDTH="100" HEIGHT="63" SRC="Timg113.png"
 ALT="\begin{figure}
\par\includegraphics[width=8.9cm,clip]{11727fb1.ps}
\end{figure}"></A><!-- HTML Figure number: 17 --></td>
<td class="img-txt"><span class="bold">Figure B.1:</span><p>
Scatter plot of the integrated Stokes <I>Q</I> signal in the preferred reference frame versus the total linear polarization without rotation. <EM> Solid line</EM>: unity slope; <EM> dashed line</EM>: unity slope with an offset of 0.0001.</p></td>
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<tr><td><!-- init Label --><A NAME="qmask">&#160;</A><!-- end Label--><A NAME="1640"></A><A NAME="figure1396"
 HREF="img114.png"><IMG
 WIDTH="98" HEIGHT="63" SRC="Timg114.png"
 ALT="\begin{figure}
\par\includegraphics[width=8.9cm,clip]{11727fb2.ps}
\end{figure}"></A><!-- HTML Figure number: 18 --></td>
<td class="img-txt"><span class="bold">Figure B.2:</span><p>
The average Stokes <I>Q</I> profile.</p></td>
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<tr><td><!-- init Label --><A NAME="qcal">&#160;</A><!-- end Label--><A NAME="1718"></A><A NAME="figure1401"
 HREF="img115.png"><IMG
 WIDTH="99" HEIGHT="153" SRC="Timg115.png"
 ALT="\begin{figure}
\par\includegraphics[width=8.9cm,clip]{11727fb3.ps}
\end{figure}"></A><!-- HTML Figure number: 19 --></td>
<td class="img-txt"><span class="bold">Figure B.3:</span><p>
<EM> Top</EM>: calibration curves from 
<!-- MATH: $Q_{\rm tot}$ -->
<IMG
 WIDTH="25" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img11.png"
 ALT="$Q_{\rm tot}$">
into field strength <I>B</I> for field inclinations <IMG
 WIDTH="10" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img12.png"
 ALT="$\gamma $">
from 10 to 90&nbsp;deg (<EM> bottom to top</EM>). The <EM> horizontal dotted line</EM> is at 
<!-- MATH: $Q_{\rm tot}=0.007$ -->
<IMG
 WIDTH="73" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img13.png"
 ALT="$Q_{\rm tot}=0.007$">;
the <EM> solid part</EM> of it denotes a range in <I>B</I> that gives the same 
<!-- MATH: $Q_{\rm tot}$ -->
<IMG
 WIDTH="25" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img11.png"
 ALT="$Q_{\rm tot}$">
at different <IMG
 WIDTH="10" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="img12.png"
 ALT="$\gamma $">.
<EM> Middle</EM>: 
<!-- MATH: $\sqrt{Q_{\rm tot}}$ -->
<IMG
 WIDTH="36" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img14.png"
 ALT="$\sqrt {Q_{\rm tot}}$">
versus field strength for constant magnetic flux. <EM> Dotted line</EM> and <EM> solid part</EM> as above for 
<!-- MATH: $\sqrt{Q_{\rm tot}}=0.0063$ -->
<IMG
 WIDTH="92" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img15.png"
 ALT="$\sqrt {Q_{\rm tot}}=0.0063$">.
<EM> Bottom</EM>: histogram of 
<!-- MATH: $\sqrt{Q_{\rm tot}}$ -->
<IMG
 WIDTH="36" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="img14.png"
 ALT="$\sqrt {Q_{\rm tot}}$">
for constant flux but varying temperature stratifications <I>T</I>. The <EM> vertical line</EM> denotes the value resulting from the HSRA atmosphere model.</p></td>
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