Changeset 215 in ETALON


Ignore:
Timestamp:
Mar 19, 2015, 1:47:21 AM (10 years ago)
Author:
delerue
Message:

Changes to the paper and bug fixed in the analysis

Files:
6 edited

Legend:

Unmodified
Added
Removed

  • SPESO/ana2015/read_date_hour_fun.m

    r214 r215  
    99    plots_basedir='/tmp_mnt/pub/ftp/pub/etalon/data_speso/analysis/';
    1010end %which basedir
     11
     12%Initialize the return functions
     13mySigArrayHour=[];
     14mySignalArrayHour=[];
     15mySPMArrayHour=[];
     16mySigmaArrayHour=[];
     17myTimeArrayHour=[];
     18datameanOverHourLPM=0;
     19datameanOverHourSPM=0;
     20datasigmaOverHourLPM=0;
     21datasigmaOverHourSPM=0;
    1122
    1223plots_dir=[ plots_basedir dateStr '/' hourStr '/' ];

  • reconstruction/long_paper3/phase_reconstruction_paper.aux

    r208 r215  
    1818\newlabel{sigma_chi2}{{2}{2}}
    1919\newlabel{eq:lamb}{{7}{2}}
    20 \citation{pchip}
    21 \citation{VBthesis}
    22 \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Comparison of different samplings with $\chi ^2$ criterium (top) and $\Delta $ FWHM (bottom)\relax }}{3}}
     20\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Comparison of different samplings with $\chi ^2$ criterium (top) and $\Delta _{FWHM}$ (bottom)\relax }}{3}}
    2321\newlabel{samp}{{3}{3}}
    2422\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Detector position for linear sampling with $10^o$ (top) and $5^o$ (bottom) MDD.\relax }}{3}}
    2523\newlabel{lin12}{{4}{3}}
     24\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Comparison of different sampling with number of MDD with $\chi ^2$ criterium (top) and $\Delta _{FWHM}$ (bottom). Ls is linear sampling with $1^o,5^o,10^0$ MDD and Ts is Triple sine sampling; mx mean the reconstruction use th maximum number of detectors (blue and red dots on figure \ref  {lin12})\relax }}{3}}
     25\newlabel{biglin}{{5}{3}}
     26\citation{pchip}
     27\citation{VBthesis}
    2628\citation{VBthesis,DESYthesis}
    2729\citation{LaiS}
    28 \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Comparison of different sampling with number of MDD with $\chi ^2$ criterium (top) and $\Delta $ FWHM (bottom). Ls -- is linear sampling with $1^o,5^o,10^0$ MDD and Triple sine sapmling; mx mean that in reconstruction was maximum number of detectors (blue and red on figure \ref  {lin12})\relax }}{4}}
    29 \newlabel{biglin}{{5}{4}}
    30 \@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Effect of the sampling frequencies on the $\chi ^2$ (top) and $\Delta $ FWHM (bottom). \relax }}{4}}
     30\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Effect of the sampling frequencies on the $\chi ^2$ (top) and $\Delta _{FWHM}$ (bottom). \relax }}{4}}
    3131\newlabel{sampling_chi2}{{6}{4}}
    32 \citation{Pelliccia:2014vba}
    3332\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Comparison of different LF extrapolation: example of spectrum (top) and profile (bottom) and histogram with mean $\chi ^2$ for each method (bottom). Gaussian and Taylorian methods are described in the text. "Real LF spectrum" means that the real LF spectrum is used. For this simulation was used the Hilbert method of phase recovery and $A\omega ^B$ high frequency extrapolation.\relax }}{5}}
    3433\newlabel{lf}{{7}{5}}
    3534\@writefile{toc}{\contentsline {section}{Study of the reconstruction performance}{5}}
    36 \@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces Comparison of different LF extrapolation: histogram with mean $\chi ^2$ for each method (top) and $\Delta $ FWHM (bottom). \relax }}{5}}
     35\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces Comparison of different LF extrapolation: histogram with mean $\chi ^2$ for each method (top) and $\Delta _{FWHM}$ (bottom). \relax }}{5}}
    3736\newlabel{lf2}{{8}{5}}
     37\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces Comparison of different HF extrapolation~: example of spectrum (top) and profile (bottom). For these simulation was used the Hilbert reconstruction method of phase recovery and Gaussian LF extrapolations.\relax }}{5}}
     38\newlabel{hf}{{9}{5}}
     39\citation{Pelliccia:2014vba}
     40\@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces Comparison of different HF extrapolation for Gaussian~:histogram with mean $\chi ^2$ (top) and $\Delta _{FWHM}$ (bottom).\relax }}{6}}
     41\newlabel{hf2}{{10}{6}}
     42\@writefile{lof}{\contentsline {figure}{\numberline {11}{\ignorespaces Comparison of different HF extrapolation for Lorenzians~:histogram with mean $\chi ^2$ (top) and $\Delta _{FWHM}$ (bottom).\relax }}{6}}
     43\newlabel{hf3}{{11}{6}}
     44\@writefile{lof}{\contentsline {figure}{\numberline {12}{\ignorespaces Examples of well reconstructed profile. The original profile is in blue and the profiles reconstructed with the Hilbert transform and the full Kramers-Kronig procedures are in red and black respectively.\relax }}{6}}
     45\newlabel{good_profiles}{{12}{6}}
     46\@writefile{lof}{\contentsline {figure}{\numberline {13}{\ignorespaces Example of poorly reconstructed profile. The original profile is in blue and the profiles reconstructed with the Hilbert transform and the full Kramers-Kronig procedures are in red and black respectively.\relax }}{7}}
     47\newlabel{bad_profiles}{{13}{7}}
     48\@writefile{lof}{\contentsline {figure}{\numberline {14}{\ignorespaces {$\Delta _{FWHM}$ (top) and $\chi ^2$ (bottom) distribution of 1000 simulations reconstructed using the Hilbert transform method (black line) and Kramers-Kronig reconstruction method (red line). XXX If the top figure is delta FWHM, then the title should say so XXX }\relax }}{7}}
     49\newlabel{profiles_stats_hilbert}{{14}{7}}
     50\@writefile{lof}{\contentsline {figure}{\numberline {15}{\ignorespaces Example of reconstructed profile with zooms on the peak and tails. One can see that the profile reconstructed using the Kramers-Kronig method has a negative component. This will dominate the final $\chi ^2$ and explains why the $\chi ^2$ obtained by this method is higher as shown on figure~\ref  {profiles_stats_hilbert}.\relax }}{7}}
     51\newlabel{expKK}{{15}{7}}
     52\@writefile{lof}{\contentsline {figure}{\numberline {16}{\ignorespaces $\Delta _{FWXM}$ for 1000 profiles with both methods.\relax }}{7}}
     53\newlabel{fwxm}{{16}{7}}
    3854\bibcite{OTR_LURE}{1}
    3955\bibcite{ODR_Cianchi}{2}
    4056\bibcite{Doucas_ESB}{3}
    4157\bibcite{KK}{4}
    42 \@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces Comparison of different HF extrapolation~: example of spectrum (top) and profile (bottom). For these simulation was used the Hilbert reconstruction method of phase recovery and Gaussian LF extrapolations.\relax }}{6}}
    43 \newlabel{hf}{{9}{6}}
    44 \@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces Comparison of different HF extrapolation for Gaussian~:histogram with mean $\chi ^2$ (top) and $\Delta $ FWHM (bottom).\relax }}{6}}
    45 \newlabel{hf2}{{10}{6}}
    46 \@writefile{lof}{\contentsline {figure}{\numberline {11}{\ignorespaces Comparison of different HF extrapolation for Lorenzians~:histogram with mean $\chi ^2$ (top) and $\Delta $ FWHM (bottom).\relax }}{6}}
    47 \newlabel{hf3}{{11}{6}}
    48 \@writefile{toc}{\contentsline {section}{Discussion}{6}}
    4958\bibcite{E203prstab}{5}
    5059\bibcite{Pelliccia:2014vba}{6}
    … …  
    5261\bibcite{pchip}{8}
    5362\bibcite{LaiS}{9}
    54 \bibcite{DESYthesis}{10}
    55 \@writefile{lof}{\contentsline {figure}{\numberline {12}{\ignorespaces Examples of well reconstructed profile. The original profile is in blue and the profiles reconstructed with the Hilbert transform and the full Kramers-Kronig procedures are in red and black respectively.\relax }}{7}}
    56 \newlabel{good_profiles}{{12}{7}}
    57 \@writefile{lof}{\contentsline {figure}{\numberline {13}{\ignorespaces Example of poorly reconstructed profile. The original profile is in blue and the profiles reconstructed with the Hilbert transform and the full Kramers-Kronig procedures are in red and black respectively.\relax }}{7}}
    58 \newlabel{bad_profiles}{{13}{7}}
    59 \@writefile{lof}{\contentsline {figure}{\numberline {14}{\ignorespaces {$\Delta _{FWHM}$ (top) and $\chi ^2$ (bottom) distribution of 1000 simulations reconstructed using the Hilbert transform method (black line) and Kramers-Kronig reconstruction method (red line). XXX If the top figure is delta FWHM, then the title should say so XXX }\relax }}{8}}
    60 \newlabel{profiles_stats_hilbert}{{14}{8}}
    61 \@writefile{lof}{\contentsline {figure}{\numberline {15}{\ignorespaces Explanation of $\chi ^2$ distribution\relax }}{8}}
    62 \newlabel{expKK}{{15}{8}}
    63 \@writefile{lof}{\contentsline {figure}{\numberline {16}{\ignorespaces $\Delta _{FWXM}$ for 1000 profiles with both methods.\relax }}{8}}
    64 \newlabel{fwxm}{{16}{8}}
    6563\@writefile{lof}{\contentsline {figure}{\numberline {17}{\ignorespaces Original and reconstructed profile and their difference for bad profile (top) and good profile (bottom).\relax }}{8}}
    6664\newlabel{mod}{{17}{8}}
    6765\@writefile{lof}{\contentsline {figure}{\numberline {18}{\ignorespaces Distribution of the $\chi ^2$ in the case of a lorenzian distribution. XXX Can you add delta FWHM? XXX\relax }}{8}}
    6866\newlabel{lorenz}{{18}{8}}
    69 \@writefile{lof}{\contentsline {figure}{\numberline {19}{\ignorespaces Mean $\chi ^2$ as function of noise amplitude. XXX Can you add delta FWHM? XXX\relax }}{9}}
    70 \newlabel{noise}{{19}{9}}
     67\@writefile{lof}{\contentsline {figure}{\numberline {19}{\ignorespaces Mean $\chi ^2$ as function of noise amplitude. XXX Can you add delta FWHM? XXX\relax }}{8}}
     68\newlabel{noise}{{19}{8}}
     69\@writefile{toc}{\contentsline {section}{Discussion}{8}}
     70\bibcite{DESYthesis}{10}

  • reconstruction/long_paper3/phase_reconstruction_paper.log

    r208 r215  
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    17271718(epstopdf)             Output file: <rev2/hfsp-eps-converted-to.pdf>
    1728 (epstopdf)                    date: 2015-03-17 16:41:52
     1719(epstopdf)                    date: 2015-03-18 16:27:07
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    17441735(epstopdf)                    size: 24512 bytes
    17451736(epstopdf)             Output file: <rev2/hfpr-eps-converted-to.pdf>
    1746 (epstopdf)                    date: 2015-03-17 16:41:52
     1737(epstopdf)                    date: 2015-03-18 16:27:07
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    17621753(epstopdf)                    size: 10577 bytes
    17631754(epstopdf)             Output file: <new203/HFGauss-eps-converted-to.pdf>
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     1755(epstopdf)                    date: 2015-03-05 12:30:00
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    17801771(epstopdf)                    size: 10457 bytes
    17811772(epstopdf)             Output file: <rev4/hfgas-eps-converted-to.pdf>
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     1773(epstopdf)                    date: 2015-03-18 16:27:07
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    17991790(epstopdf)             Output file: <new203/HFLorenz-eps-converted-to.pdf>
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    18171808(epstopdf)             Output file: <rev4/hflor-eps-converted-to.pdf>
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     1809(epstopdf)                    date: 2015-03-18 16:27:07
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    18351826(epstopdf)             Output file: <rev1/541-eps-converted-to.pdf>
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    18521843(epstopdf)             Output file: <rev1/658-eps-converted-to.pdf>
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  • reconstruction/long_paper3/phase_reconstruction_paper.tex

    r208 r215  
    4646
    4747\author{
    48 N. Delerue~\thanks{delerue@lal.in2p3.fr}, J. Barros,  M. Vieille-Grosjean, LAL, Orsay, France\\
    49 Oleg Bezshyyko, Vitalii Khodnevych, Taras Shevchenko National University of Kyiv , Ukraine}
     48N. Delerue~\thanks{delerue@lal.in2p3.fr}, J. Barros,  LAL, Orsay, France\\
     49O. Bezshyyko, V. Khodnevych, Taras Shevchenko National University of Kyiv , Ukraine}
    5050
    5151\maketitle
    … …  
    129129where $\mbox{rset} = \{ 0.1 ; 0.2 ; 0.5 ; 0.8 ; 0.9\}$, $FWXM_{\mbox{orig}}$ and $FWXM_{\mbox{reco}}$ are the FWXM of the original and reconstructed profiles respectively.
    130130
    131 \begin{figure}[!htb]
     131\begin{figure}[htbp]
    132132 \centering
    133133  \includegraphics*[width=70mm]{rev1/chiexp.eps}
    … …  
    138138To ensure that the choice of the parameters $\sigma_i$ and $\mu_i$ for the simulations does not biais significantly the results, their value has been varied and this is shown on figure~\ref{sigma_chi2} XXX can you add FWHM ? XXX.
    139139
    140 \begin{figure}[!htb]
     140\begin{figure}[htbp]
    141141 \centering
    142142  \includegraphics*[width=70mm]{newfigures/chi_sigma.eps}\\
    … …  
    169169%Further we will see, that due to space limitations of detector size, only Triple-sine sapmling is physical, but this study will give us information which sampling and further detector position are preferable.
    170170\end{itemize}
    171 {The study of the sampling is important, as it show  best position of the detectors and also  how to optimize the system. Linearly sampled spectrum gives the best result (see figure (\ref{samp})).
     171{The study of the sampling is important, as it show  best position of the detectors and also  how to optimize the system. Linearly sampled spectrum gives the best result as shown on figure~\ref{samp}.
    172172%XXX WE need to discuss this XXX  This is not surprising, because in the process of profile recovery is present  interpolation procedure  for spectrum, which is well known works best with a uniform sampling. 
    173173}
    174174
    175 \begin{figure}[!htb]
     175\begin{figure}[htbp]
    176176 \centering
    177177  \includegraphics*[width=70mm]{rev1/4.eps} \\
    178178    \includegraphics*[width=70mm]{rev1/5.eps}
    179179  %\includegraphics*[width=70mm]{newFig/lin27e203line203.eps}
    180   \caption{Comparison of different samplings with $\chi^2$ criterium (top) and $\Delta$ FWHM (bottom)}%VH add  picture
     180  \caption{Comparison of different samplings with $\chi^2$ criterium (top) and $\Delta_{FWHM}$ (bottom)}%VH add  picture
    181181   \label{samp}
    182182\end{figure}
    … …  
    184184%For angle calculation and applying condition for first and final point was used  formula ~\ref{eq:lamb}.
    185185On figure~\ref{lin12} examples of detector positions are shown. The position of the red points is calculated using formula ~\ref{eq:lamb} and the blue are the possible positions of detector which does not break minimum detector distance (MDD) XXX This is incomplete XXX.}
    186 \begin{figure}[!htb]
     186\begin{figure}[htbp]
    187187 \centering
    188188  \includegraphics*[width=70mm]{rev1/mmd5.eps} \\
    … …  
    195195\textbf{Figure~\ref{biglin} shows a comparison of the performances achieved with such positioning for different MDD. In each case the triple sine sampling (Ts) is better than the linear sampling (Ls) and close from the maximum linear sampling (Lsmx).
    196196%As the Lsmx configuration is physically impossible,
    197 So Ts configuration is favored and will be used in the rest of this paper. The comparison between Ts1, Ts5 and Ts10 shows that reconstruction performances are limited by the MDD.}\par
    198 \begin{figure}[!htb]
     197So the Ts configuration is favored and will be used in the rest of this paper. The comparison between Ts1, Ts5 and Ts10 shows that reconstruction performances are limited by the MDD.}
     198
     199\begin{figure}[htbp]
    199200 \centering
    200201  \includegraphics*[width=90mm]{rev1/hist1.eps} \\
    201202  \includegraphics*[width=90mm]{rev1/hist1fw.eps}
    202   \caption{Comparison of different sampling with number of MDD  with $\chi^2$ criterium (top) and $\Delta$ FWHM (bottom). Ls -- is linear sampling with $1^o,5^o,10^0$ MDD and Triple sine sapmling; mx mean that in reconstruction was maximum number of detectors (blue and red on figure \ref{lin12})}%VH add  picture
     203  \caption{Comparison of different sampling with number of MDD  with $\chi^2$ criterium (top) and $\Delta_{FWHM}$ (bottom). Ls is linear sampling with $1^o,5^o,10^0$ MDD and Ts is Triple sine sampling; mx mean the reconstruction use th maximum number of detectors (blue and red dots on figure \ref{lin12})}%VH add  picture
    203204   \label{biglin}
    204205\end{figure}
    … …  
    210211
    211212
    212 \begin{figure}[!htb]
     213\begin{figure}[htbp]
    213214 \centering
    214215  \includegraphics*[width=70mm]{rev4/chN.eps}\\
    215216   \includegraphics*[width=70mm]{rev4/fwN.eps}
    216   \caption{Effect of the sampling frequencies on the $\chi^2$ (top) and $\Delta$ FWHM (bottom). }
     217  \caption{Effect of the sampling frequencies on the $\chi^2$ (top) and $\Delta_{FWHM}$ (bottom). }
    217218   \label{sampling_chi2}
    218219\end{figure}
    … …  
    256257Conditions for A, B and C constants are the same. Comparison of different LF extrapolation can be found on figure~\ref{lf}.
    257258
    258 \begin{figure}[!htb]
     259\begin{figure}[htbp]
    259260 \centering
    260261  \includegraphics*[width=65mm]{rev2/lfsp.eps}\\
    … …  
    265266\end{figure}
    266267
    267 \begin{figure}[!htb]
     268\begin{figure}[htbp]
    268269 \centering
    269270
    270271  \includegraphics*[width=65mm]{new203/LF.eps}
    271272    \includegraphics*[width=65mm]{rev3/lffw.eps}
    272     \caption{Comparison of different LF extrapolation: histogram with mean $\chi^2$ for each method (top) and $\Delta$ FWHM (bottom).  }
     273    \caption{Comparison of different LF extrapolation: histogram with mean $\chi^2$ for each method (top) and $\Delta_{FWHM}$ (bottom).  }
    273274   \label{lf2}
    274275\end{figure}
    … …  
    317318 
    318319 
    319 \begin{figure}[!htb]
     320\begin{figure}[htbp]
    320321 \centering
    321322  \includegraphics*[width=65mm]{rev2/hfsp.eps}\\
    … …  
    325326   \label{hf}
    326327\end{figure}
    327 \begin{figure}[!htb]
     328\begin{figure}[htbp]
    328329 \centering
    329330    \includegraphics*[width=65mm]{new203/HFGauss.eps}\\
    330331        \includegraphics*[width=65mm]{rev4/hfgas.eps}\\
    331332
    332     \caption{Comparison of different HF extrapolation for Gaussian~:histogram with mean $\chi^2$ (top) and $\Delta$ FWHM (bottom).}
     333    \caption{Comparison of different HF extrapolation for Gaussian~:histogram with mean $\chi^2$ (top) and $\Delta_{FWHM}$ (bottom).}
    333334   \label{hf2}
    334335\end{figure}
    335 \begin{figure}[!htb]
     336\begin{figure}[htbp]
    336337 \centering
    337338        \includegraphics*[width=65mm]{new203/HFLorenz.eps}\\
    338339          \includegraphics*[width=65mm]{rev4/hflor.eps}
    339         \caption{Comparison of different HF extrapolation for Lorenzians~:histogram with mean $\chi^2$ (top) and $\Delta$ FWHM (bottom).}
     340        \caption{Comparison of different HF extrapolation for Lorenzians~:histogram with mean $\chi^2$ (top) and $\Delta_{FWHM}$ (bottom).}
    340341   \label{hf3}
    341342\end{figure}
    … …  
    351352After applying extrapolation and interpolation, the spectrum recovery is completed. Than was used  different reconstruction techniques to reconstruct the original profile. For each reconstruction method some profiles are very well reconstructed whereas some other are not so well reconstructed. Examples of well reconstructed profiles are shown on figure~\ref{good_profiles} and examples of poorly reconstructed profile are shown on figure~\ref{bad_profiles}.
    352353
    353 \begin{figure}[!htb]
     354\begin{figure}[htbp]
    354355 \centering
    355356%  \includegraphics*[trim=0 0 275 0 ,clip,width=95mm]{plot1000700.png}
    … …  
    361362\end{figure}
    362363
    363 \begin{figure}[!htb]
     364\begin{figure}[htbp]
    364365 \centering
    365366%  \includegraphics*[trim=0 0 275 0 ,clip,width=95mm]{plot1000183.eps}
    … …  
    376377
    377378
    378 \begin{figure}[!htb]
     379\begin{figure}[htbp]
    379380 \centering
    380381  \includegraphics*[width=70mm]{rev1/2.eps} \\
    … …  
    383384   \label{profiles_stats_hilbert}
    384385\end{figure}
    385 \textbf{The fact that phase recovery based on Kramers-Kronig relation work worst than Hilbert method is caused by negative part of tails of profiles. At figure \ref{expKK} is shown example of one of the profiles.}
    386 \begin{figure}[!htb]
    387  \centering
    388   \includegraphics*[width=70mm]{rev3/compFig.eps} 
    389   \caption{Explanation of $\chi^2$ distribution}% VH change name of picture and unite with other
     386\textbf{The fact that the phase recovery method based on the Kramers-Kronig relation gives a worst $\chi^2$  than the method based on Hilbert relation has been investigated. It is caused by the presence of negative components in the tails of the profiles. Figure~\ref{expKK} highlights this issue for one of the profiles.}
     387\begin{figure}[htbp]
     388 \centering
     389  \includegraphics*[width=100mm]{rev3/compFig.eps} 
     390  \caption{Example of reconstructed profile with zooms on the peak and tails. One can see that the profile reconstructed using the Kramers-Kronig method has a negative component. This will dominate the final $\chi^2$ and explains why the $\chi^2$ obtained by this method is higher as shown on figure~\ref{profiles_stats_hilbert}.}% VH change name of picture and unite with other
    390391   \label{expKK}
    391392\end{figure}
    … …  
    397398   
    398399     
    399 \begin{figure}[!htb]
     400\begin{figure}[htbp]
    400401 \centering
    401402  \includegraphics*[width=70mm]{rev1/1.eps}
    … …  
    403404   \label{fwxm}
    404405\end{figure}
    405 \begin{figure}[!htb]
     406
     407\begin{figure}[htbp]
    406408 \centering
    407409  \includegraphics*[width=65mm]{rev4/d1.eps}\\
    … …  
    415417While doing this work we also became aware of the discussion in~\cite{Pelliccia:2014vba} where it is argued that these reconstruction method have more difficulties with lorentzian profiles  than gaussian profiles. Therefore was simulated 1000 lorenzian profiles and performed a similar study. This is shown on figure~\ref{lorenz}. Although the $\chi^2$ is slightly worse in that case than in the case of gaussian profiles there still a good agreement between the original and reconstructed profiles.
    416418
    417 \begin{figure}[!htb]
     419\begin{figure}[htbp]
    418420 \centering
    419421  \includegraphics*[width=70mm]{THPME088f9.eps} \\
    … …  
    432434maximum noise for that simulation (depending on the case this can be 5\%, 10\%, 20\%, 30\%, 40\% or  50\%). This study was done using linear sampling with 33 samples  and 1000 simulated profiles for each noise value. The figure~\ref{noise} shows how the $\chi^2$ is modified when this noise component is added.
    433435 
    434 \begin{figure}[!htb]
     436\begin{figure}[htbp]
    435437 \centering
    436438  \includegraphics*[width=70mm]{newfigures/new2/noise.eps}\\
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