Changeset 304 in ETALON for reconstruction


Ignore:
Timestamp:
Nov 29, 2015, 4:00:37 PM (9 years ago)
Author:
hodnevuc
Message:

correcter version

Location:
reconstruction/long_paper3
Files:
5 edited

Legend:

Unmodified
Added
Removed

  • reconstruction/long_paper3/phase_reconstruction_paper.aux

    r299 r304  
    1313\citation{E203prstab}
    1414\citation{E203prstab}
    15 \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Example of profiles giving very different $\chi ^2$ despite being relatively similar.   $\chi ^2_{\unhbox \voidb@x \hbox {sine noise}}=3.8219e-08, \chi ^2_{\unhbox \voidb@x \hbox {offset}}=7.2661e-08$; For profile with sine noise: FW0.1M=0.0241, FW0.2M=0.044 FWHM=0.0621 FW0.8M=0.1849 FW0.9M=0.3619. For offset profile all FWXM=0. \relax }}{2}}
     15\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Example of profiles giving very different $\chi ^2$ despite being relatively similar.   $\chi ^2_{\unhbox \voidb@x \hbox {sine noise}}=3.8219\times 10^{-8}, \chi ^2_{\unhbox \voidb@x \hbox {offset}}=7.2661\times 10^{-8}$; For profile with sine noise: FW0.1M=0.0241, FW0.2M=0.044 FWHM=0.0621 FW0.8M=0.1849 FW0.9M=0.3619. As FWXM calculated from top of profile, for all profiles FWXM=0. \relax }}{2}}
    1616\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
    1717\newlabel{Offsine}{{1}{2}}
    18 \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Effect of scaling the constraints on the parameters $\sigma _i$ (top) and $\mu _i$ (bottom) on the $\chi ^2$. For each point 1000 simulations was made.\relax }}{2}}
     18\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The constraints on the parameters $\sigma _i$ (top) and $\mu _i$ (bottom) due to effect of scaling in terms of the $\chi ^2$ and $Delta$FWHM ratio. For each point 1000 simulations were made.\relax }}{2}}
    1919\newlabel{sigma_chi2}{{2}{2}}
    2020\newlabel{eq:lamb}{{7}{3}}
    … …  
    2626\citation{VBthesis}
    2727\citation{VBthesis,DESYthesis}
    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 Ts is Triple sine sampling; mx mean the reconstruction use the maximum number of detectors (blue and red dots on figure \ref  {lin12}).\relax }}{4}}
     28\citation{LaiS}
     29\@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 the maximum number of detectors (blue and red dots in figure \ref  {lin12}).\relax }}{4}}
    2930\newlabel{biglin}{{5}{4}}
    3031\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Effect of the sampling frequencies on the $\chi ^2$ (top) and $\Delta _{FWHM}$ (bottom). \relax }}{4}}
    3132\newlabel{sampling_chi2}{{6}{4}}
    32 \citation{LaiS}
    33 \@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}}
     33\@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 the Hilbert method of phase recovery and $A\omega ^B$ high frequency extrapolation were used.\relax }}{5}}
    3434\newlabel{lf}{{7}{5}}
    3535\@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}}
    3636\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 }}{6}}
     37\@writefile{toc}{\contentsline {section}{Study of the reconstruction performance}{5}}
     38\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces Comparison of different HF extrapolations~: example of spectrum (top) and profile (bottom). For these simulations the Hilbert reconstruction method of phase recovery and Gaussian LF extrapolations were used.\relax }}{6}}
    3839\newlabel{hf}{{9}{6}}
    39 \@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}}
     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}}
    4041\newlabel{hf2}{{10}{6}}
    41 \@writefile{toc}{\contentsline {section}{Study of the reconstruction performance}{6}}
    4242\@writefile{lof}{\contentsline {figure}{\numberline {11}{\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}}
    4343\newlabel{good_profiles}{{11}{6}}
     44\citation{Pelliccia:2014vba}
    4445\@writefile{lof}{\contentsline {figure}{\numberline {12}{\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}}
    4546\newlabel{bad_profiles}{{12}{7}}
    4647\@writefile{lof}{\contentsline {figure}{\numberline {13}{\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). }\relax }}{7}}
    4748\newlabel{profiles_stats_hilbert}{{13}{7}}
    48 \@writefile{lof}{\contentsline {figure}{\numberline {14}{\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}}
     49\@writefile{lof}{\contentsline {figure}{\numberline {14}{\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 in figure~\ref  {profiles_stats_hilbert}.\relax }}{7}}
    4950\newlabel{expKK}{{14}{7}}
    50 \citation{Pelliccia:2014vba}
    5151\@writefile{lof}{\contentsline {figure}{\numberline {15}{\ignorespaces $\Delta _{FWXM}$ for 1000 profiles with both methods.\relax }}{8}}
    5252\newlabel{fwxm}{{15}{8}}
    5353\@writefile{lof}{\contentsline {figure}{\numberline {16}{\ignorespaces Original and reconstructed profile and their difference for bad profile (top) and good profile (bottom).\relax }}{8}}
    5454\newlabel{mod}{{16}{8}}
    55 \@writefile{lof}{\contentsline {figure}{\numberline {17}{\ignorespaces Distribution of the $\chi ^2$ in the case of a lorenzian distribution. \relax }}{8}}
     55\@writefile{lof}{\contentsline {figure}{\numberline {17}{\ignorespaces Distribution of the $\chi ^2$ in the case of a Lorenzian distribution. \relax }}{8}}
    5656\newlabel{lorenz}{{17}{8}}
    5757\@writefile{toc}{\contentsline {section}{Discussion}{8}}

  • reconstruction/long_paper3/phase_reconstruction_paper.log

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    1516 Underfull \hbox (badness 1577) in paragraph at lines 211--214
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  • reconstruction/long_paper3/phase_reconstruction_paper.tex

    r299 r304  
    7676
    7777When it is only possible to measure the amplitude of the complex signal, it is necessary to recover the phase of the available data.
     78We assume that the function of the longitudinal beam density is analytical.%VH question 4
    7879For an analytic function this is easier because the real and imaginary part are not completely independent.
    7980The Kramers-Kronig relations~\cite{KK} helps restore the imaginary part of an analytic function $\varepsilon(\omega)$ from its real part and vice versa.
    … …  
    143144  \includegraphics*[width=70mm]{plots/1.eps}
    144145  \caption{Example of profiles giving very different $\chi^2$ despite being relatively similar. \\
    145   $\chi^2_{\mbox{sine noise}}=3.8219e-08, \chi^2_{\mbox{offset}}=7.2661e-08$; For profile with sine noise: FW0.1M=0.0241, FW0.2M=0.044 FWHM=0.0621 FW0.8M=0.1849 FW0.9M=0.3619. For offset profile all FWXM=0. }% VH change name of picture and unite with other
     146  $\chi^2_{\mbox{sine noise}}=3.8219\times 10^{-8}, \chi^2_{\mbox{offset}}=7.2661\times 10^{-8}$; For profile with sine noise: FW0.1M=0.0241, FW0.2M=0.044 FWHM=0.0621 FW0.8M=0.1849 FW0.9M=0.3619. As FWXM calculated from top of profile, for all profiles  FWXM=0. }% VH quetion 5
    146147   \label{Offsine}
    147148\end{figure}
    … …  
    149150
    150151
    151 To 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}.
     152To ensure that the choice of the parameters $\sigma_i$ and $\mu_i$ for the simulations does not bias significantly the results, their value has been varied and this is shown in figure~\ref{sigma_chi2}.
    152153
    153154\begin{figure}[htbp]
    … …  
    157158  \includegraphics*[width=70mm]{plot2210/23.eps}\\
    158159  \includegraphics*[width=70mm]{plot2210/24.eps}\\
    159   \caption{Effect of scaling the constraints on the parameters $\sigma_i$ (top) and $\mu_i$ (bottom) on the $\chi^2$. For each point 1000 simulations was made.}
     160  \caption{The constraints on the parameters $\sigma_i$ (top) and $\mu_i$ (bottom) due to effect of scaling in terms of the $\chi^2$ and $Delta$FWHM ratio. For each point 1000 simulations were made.}%VH question 8
    160161   \label{sigma_chi2}
    161162\end{figure}
    … …  
    165166
    166167Different distributions have been used for the frequencies $\omega_i$: linear, logarithmic, triple-sine.
    167 In most sampling schemes we used 33 frequencies  to make it comparable with the Triple-sine distribution used in~\cite{E203prstab}. These sampling scheme are defined as follow:
     168In most sampling schemes we used 33 frequencies  to make it comparable with the Triple-sine distribution used in~\cite{E203prstab}. These sampling schemes are defined as follow:
    168169\begin{itemize}
    169170\item \textit{Triple-sine} This sampling matches that of the E-203 experiment at FACET~\cite{E203prstab}. Eleven detectors are located every $10^o$ around the interaction point and 3 different sets of wavelengths are used, giving the following distribution:
    … …  
    185186%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.
    186187\end{itemize}
    187 {The study of the sampling is important, as it shows the  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}.
     188{The study of the sampling is important, as it shows the  best position of the detectors and also  how to optimize the system. Linearly sampled spectrum gives the best result as shown in figure~\ref{samp}.
    188189%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. 
    189190}
    … …  
    199200{However, the linear sampling may not be practical to realize in a the real world. One needs to take into account the spatial size of the detectors (about 10 degrees in the case of E-203) and there is also a limit on the start and end points of detectors location (35-145 degrees for E-203). So linear sampling at a wide range of frequencies is impossible with this number of points. An investigation of how many linear sampling points can be used for a given angle difference between detectors shows that such physical constraints reduce strongly the number of detectors that can be used.
    200201%For angle calculation and applying condition for first and final point was used  formula ~\ref{eq:lamb}.
    201 Figure~\ref{lin12} shows examples of detector positions. 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 the minimum detector distance (MDD) given on top of each plot.}
     202Figure~\ref{lin12} shows examples of detector positions. The position of the red points is calculated using formula~\ref{eq:lamb} and blue are possible detector positions  which don't break the minimum detector distance (MDD) given on top of each plot.}
    202203\begin{figure}[htbp]
    203204 \centering
    … …  
    209210\end{figure}
    210211
    211 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).
     212Figure~\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 number of detectors with linear sampling (Lsmx).
    212213%As the Lsmx configuration is physically impossible,
    213214So 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.
    … …  
    217218  \includegraphics*[width=90mm]{plots/51.eps} \\
    218219  \includegraphics*[width=90mm]{plots/52.eps}
    219   \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 the maximum number of detectors (blue and red dots on figure \ref{lin12}).}%VH add  picture
     220  \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 the maximum number of detectors (blue and red dots in figure \ref{lin12}).}%VH add  picture
    220221   \label{biglin}
    221222\end{figure}
    … …  
    223224
    224225%This section has been moved from elsewhere
    225 The choice of 33 frequencies for the sampling of the spectrum was made to match the current layout used on E-203. However it is important to check if there is an optimum value. To perform this check we used the same simulations and the same simulated spectrum but sampled  with 3 to 140  points. The effect of changing the sampling frequencies on the  $\chi^2$ is shown on figure~\ref{sampling_chi2}. This study uses Triple sine sampling with 1000 profiles for each point and both reconstruction method.
     226The choice of 33 frequencies for the sampling of the spectrum was made to match the current layout used on E-203. However it is important to check if there is an optimum value. To perform this check we used the same simulations and the same simulated spectrum but sampled  with 3 to 140  points. The effect of changing the sampling frequencies on the  $\chi^2$ is shown in figure~\ref{sampling_chi2}. This study uses Triple sine sampling with 1000 profiles for each point and both reconstruction method.
    226227
    227228
    … …  
    276277\end{equation}
    277278
    278 Conditions for the constants A, B and C  are the same. Comparison of different LF extrapolation can be found on figure~\ref{lf} and the performances of these methods on figure~\ref{lf2}.
     279Conditions for the constants A, B and C  are the same. Comparison of different LF extrapolation can be found in figure~\ref{lf} and the performances of these methods in figure~\ref{lf2}.
    279280
    280281\begin{figure}[htbp]
    … …  
    283284  \includegraphics*[width=65mm]{plots/72.eps}\\
    284285
    285     \caption{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.}
     286    \caption{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 the Hilbert method of phase recovery and $A\omega^B$ high frequency extrapolation were used.}
    286287   \label{lf}
    287288\end{figure}
    … …  
    300301
    301302
    302 Several high frequency (HF) extrapolation method were also tested. The most common~\cite{VBthesis,DESYthesis} is :
     303Several high frequency (HF) extrapolation methods were also tested. The most common~\cite{VBthesis,DESYthesis} is :
    303304\begin{equation}
    304305\rho _ {HF} (\omega)=A\omega^{-4},
    … …  
    306307 where $\rho_ {HF} (\omega)$ is the extrapolated spectrum at high frequency and $A=\rho_f \omega_f^{4} $, where $\rho_f$ is spectrum value of final point $\omega_f$.
    307308
    308 The second method use the same consideration as in Lai and Sievers~\cite{LaiS}:
     309The second method uses the same consideration as in Lai and Sievers~\cite{LaiS}:
    309310Assuming that the bunch size is finite with two end points at $z=0$ and at $z=\sigma_z$ then the longitudinal charge distribution ($S$) must follow $S(0)=S(\sigma_z)=0$.
    310311An integration by parts gives~:
    … …  
    324325\rho_{HF}(\omega)=A\omega^{-2}+B\omega^{-3}
    325326\end{equation}
    326 or  extrapolation with the exponent as free parameter:
     327or  extrapolation with degree of frequency as free parameter:
    327328\begin{equation}
    328329\rho_{HF}(\omega)=A\omega^B
    … …  
    330331
    331332
    332 where the A and B coefficient are calculated from the last data samples and the boundary conditions as follow:
     333where the A and B -- coefficients which are calculated from the last data samples and the boundary conditions as follow:
    333334\begin{itemize}
    334335\item $B={\rho_{HF}'(\omega_{fmax}) \omega_{fmax}}/{\rho(\omega_{fmax})}$
    … …  
    337338
    338339  The requirement of finite bunch size requires $B\le2$, so in the case where the fit gives $B>-2$ we use $B=-2$.
    339 Two other extrapolation methods have also been investigated:
     340Two other extrapolation methods also have been investigated:
    340341\begin{itemize}
    341342\item $\rho_{HF}(\omega_f)= 0 $ for $ \omega_f >  \omega_{fmax}$
    … …  
    343344\end{itemize}
    344345
    345 These HF extrapolation methods are compared on figure~\ref{hf} and~\ref{hf2}.\par
    346 Thus, by virtue of the above arguments and simulations, It's naturally to choose the high-frequency extrapolation by power function.
     346These HF extrapolation methods are compared in figure~\ref{hf} and~\ref{hf2}.\par
     347Thus, by virtue of the above arguments and simulations, it's naturally to choose the high-frequency extrapolation by power function.
    347348 
    348349 
    … …  
    352353  \includegraphics*[width=65mm]{plots2/92.eps}\\
    353354
    354     \caption{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.}
     355    \caption{Comparison of different HF extrapolations~: example of spectrum  (top) and profile (bottom). For these simulations the Hilbert reconstruction method of phase recovery and Gaussian LF extrapolations were used.}
    355356   \label{hf}
    356357\end{figure}
    … …  
    360361        \includegraphics*[width=65mm]{plots/101.eps}\\
    361362
    362     \caption{Comparison of different HF extrapolation for Gaussian~:histogram with mean $\chi^2$ (top) and $\Delta_{FWHM}$ (bottom).}
     363    \caption{Comparison of different HF extrapolation for Gaussian~: histogram with mean $\chi^2$ (top) and $\Delta_{FWHM}$ (bottom).}
    363364   \label{hf2}
    364365\end{figure}
    … …  
    382383\section{Study of the reconstruction performance}
    383384
    384 After applying extrapolation and interpolation, the spectrum recovery is completed. Then we 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}.
     385After applying extrapolation and interpolation, the spectrum recovery is completed. Then we 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 in figure~\ref{good_profiles} and examples of poorly reconstructed profile are shown in figure~\ref{bad_profiles}.
    385386
    386387\begin{figure}[htbp]
    … …  
    405406
    406407
    407 The  $\Delta_{FWXM}$ and  $\chi^2$ distribution of the 1000 simulations which were made and then reconstructed using the Hilbert transform method and Kramers-Kornig reconstruction are shown in on figure~\ref{profiles_stats_hilbert}. There is a good agreement in FWHM between the two methods indicating that they are both good at finding the bunch length. However, the Hilbert method gives lower $\chi^2$ indicating that this method is better at reconstruction the bunch profile.
     408The  $\Delta_{FWXM}$ and  $\chi^2$ distribution of the 1000 simulations which were made and then reconstructed using the Hilbert transform method and Kramers-Kornig reconstruction are shown in figure~\ref{profiles_stats_hilbert}. There is a good concordance in FWHM between  two methods indicating that they are both good at finding the bunch length. However, the Hilbert method gives lower $\chi^2$ indicating that this method is better at reconstruction of the bunch profile.
    408409
    409410
    … …  
    420421 \centering
    421422  \includegraphics*[width=100mm]{plots/14.eps} 
    422   \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
     423  \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 in figure~\ref{profiles_stats_hilbert}.}% VH change name of picture and unite with other
    423424   \label{expKK}
    424425\end{figure}
    425426% VH add this block
    426427%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    427 Figure~\ref{fwxm} shows the different values the FWXH for different values of X. This shows that at   different height of the profiles the quality of reconstruction varies: There is a better agreement in the tails (X=10\%) than at the top of the profile (X=90\%).
     428Figure~\ref{fwxm} shows the different FWXH values   for different values of X. This shows that at   different height of the profiles the quality of reconstruction varies: there is a better agreement in the tails (X=10\%) than at the top of the profile (X=90\%).
    428429 Figure~\ref{mod} shows the modulus of the difference between the original and reconstructed profiles. One can see oscillations in the difference between the original and reconstructed profile.
    429430 
    … …  
    447448
    448449
    449 While 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 we 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.
     450While 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 we simulated 1000 Lorenzian profiles and performed a similar study. This is shown in 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.
    450451
    451452\begin{figure}[htbp]
    … …  
    454455  \includegraphics*[width=70mm]{plot1510/172.eps} \\
    455456%  \includegraphics*[width=75mm]{THPME088f10.eps}
    456   \caption{Distribution of the $\chi^2$ in the case of a lorenzian distribution. }
     457  \caption{Distribution of the $\chi^2$ in the case of a Lorenzian distribution. }
    457458   \label{lorenz}
    458459\end{figure}
    … …  
    479480\section{Discussion}
    480481
    481 We performed extensive simulation to estimate the performance of two phase recovery methods in the case of multi-gaussian and lorenzian profiles. In both cases we found that when the sampling frequencies are chosen correctly  we obtained a good agreement between the original and reconstructed profiles (in most cases $\Delta_{FWXM} < 10\%$;  $\chi^2 \sim 10^{-6}$). This confirms that such methods are suitable to reconstruct the longitudinal profiles measured at particle accelerators using radiative methods.
     482We performed extensive simulation to estimate the performance of two phase recovery methods in the case of multi-gaussian and Lorenzian profiles. In both cases we found that when the sampling frequencies are chosen correctly  we obtained a good agreement between the original and reconstructed profiles (in most cases $\Delta_{FWXM} < 10\%$;  $\chi^2 \sim 10^{-6}$). This confirms that such methods are suitable to reconstruct the longitudinal profiles measured at particle accelerators using radiative methods.
    482483
    483484
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