\begin{table}%t3
  %\begin{table}%t4
\par
\section{Data modeling}
\par
As will be illustrated in the following, the observations clearly require a wavelength dependence of apparent diameter.  Further, the visibility data are not compatible with classical limb-darkened disk models.  We have already measured chromatic diameters between the~$K$ and the~$L$~bands (Mennesson et~al.~2002; Chagnon et~al.~2002).  It was possible to account for those observations with a model in which a photosphere was surrounded by an emissive and absorbing layer a few radii above the stellar surface.  The nature of the layer was assumed to be~${\rm H}_{2}{\rm O}$ as this molecule had also been detected spectroscopically.  The narrow band observations reported here will allow us to identify some of the sources of opacity.  Because of the large number of parameters and data we have chosen a simpler model than the one used in \cite{mennesson2002}. This model has an analytical solution for the radiative transfer calculations which makes the parameter search quicker and the computation of error bars possible. The model is spherically symmetric. Although some asymmetries have been found on Mira stars (Quirrenbach et~al.~1992; Lopez et~al.~1997; Hofmann et~al.~2000) they have been measured at wavelengths outside of the maximum of flux emission where sensitivity to the star environment is larger (TiO~bands and dust). Besides, it is clear in our case, as will be demonstrated, that the chromaticity of visibility curves can be explained by the physical properties of the object rather than by a particular symmetry. This is why we have privileged spherical symmetry. We have also preferred to use an ad-hoc geometrical model rather than a model derived from hydrodynamical calculations in order to provide physical parameters that are then useful to hydrodynamical calculations. Other types of models will be discussed in Sect.~\ref{sec:model}.
\par
\subsection{Description of the model}
\par
The model we have chosen is close to the one used in \cite{mennesson2002}. Instead of having an extended layer around the star, the layer is compact. The star is modeled by a stellar surface of diameter~$\O_{\star}$ which emits as a blackbody of temperature~$T_{\star}$.  It is surrounded by a spherical layer of diameter~$\O_{{\rm layer}}$ with no geometrical thickness.  The layer is characterized by its temperature~$T_{{\rm layer}}$ and its optical depth~$\tau_{\lambda}$.  The layer absorbs the radiation emitted by the star and re-emits like a blackbody.  Scattering is neglected for our range of wavelengths.  The space between the layer and the stellar photosphere is empty. Figure~\ref{fig:model} gives a sketch of the star+layer model. Limb darkening of the central star is neglected, but the composite model has interesting brightness variations, more complex than classical limb darkening. 
\begin{figure}%f2
\htmlimage{scale=2.5, external, thumbnail=0.4}
     %\centering
\par
     \includegraphics[angle=90,width=7cm,clip]{1098f2.eps}
      \caption{Model of the star surrounded by a thin molecular layer.~$\theta$ is the angle between the radius vector and the line of sight at the layer surface.\label{fig:model} }           \end{figure}
\par
The analytical expression of the model is given by the equations below:
\begin{eqnarray}
I(\lambda,\theta) & = & B(\lambda,T_{\star})\exp(-\tau(\lambda)/\cos(\theta)) \\ \nonumber
 & & +B(\lambda,T_{{\rm layer}})\left[1-\exp(-\tau(\lambda)/\cos(\theta))\right]\end{eqnarray}for~$\sin(\theta) \leq \O_{\star}/\O_{{\rm layer}}$ and:\begin{equation}I(\lambda,\theta)=B(\lambda,T_{{\rm layer}})\left[1-\exp(-2\tau(\lambda)/\cos(\theta))\right]\end{equation}otherwise, where~$B(\lambda,T)$ is the Planck function and~$\theta$ is the angle between the radius vector and the line between the observer and the center of the star. The intensity distribution is taken at the central wavelength of the filter. The effect of bandwidth has been neglected as the maximum error on the visibility model is less than~0.4\% in the bands we have used. The visibility model is obtained by taking the squared Hankel transform of the circularly symmetric intensity distribution.  This simple model is also used in \cite{scholz2001} to illustrate the impact of the circumstellar environment on the center to limb variation.  Although it is a quite simple view of the atmosphere of an evolved object it is very convenient to use as it only depends on a small number of parameters and allows relatively easy and quick computations. It is important to note that the only physical parameters which vary from one wavelength to another are the optical depths. The temperature and geometrical parameters are the same at all wavelengths. 
\par
 \subsection{Parameter search procedure}
 \label{sec:procedure}
 We have minimized a~$\chi^{2}$ to find the parameters that best fit the data:
 \begin{equation}    \chi^{2}=\sum_{i=1}^{N}\left[\frac{V_{i}^{2}-M(\O_{\star}, \O_{{\rm layer}},     T_{\star}, T_{{\rm layer}}, \tau_{\lambda_{i}} ;S_{i})}{\sigma_{i}}\right]^{2}
 \end{equation}
 where~$M(\O_{\star}, \O_{{\rm layer}}, T_{\star}, T_{{\rm layer}}, \tau_{\lambda_{i}} ;S_{i})$ is the visibility model in which~$S_{i}$ is the spatial frequency. 
\par
 The hyper-surface defined by the~$\chi^{2}$ is not convex in the parameter space and many local minima occur.  Also, the minimum might be badly defined and may be very sensitive to  individual measurements.  
\par
We have used the following strategy to find the best parameter estimates.  We first have noticed that for a given set of diameter and temperature values the hyper-surface is close to convex leading to a fast convergence of the Levenberg-Marquardt algorithm towards the optimum optical depths.  On the other hand, the temperatures are not well constrained by just the interferometric data.  The simple reason for this is that the visibilities are not absolute flux measurements but are relative to the zero spatial frequency.  The visibility data do not therefore constrain the total flux of the model.  In our case they are just sensitive to the relative flux of the star and of the layer.  As a consequence, the locus for the optimum set of temperatures is not a single low point but a trough.  The temperatures of the star and of the layer are coupled and inaccurate.  We therefore first have constrained the diameters of the photosphere and of the layer by computing a 2D~grid of~$\chi^{2}$~values.  Each~$\chi^{2}$ is the result of the minimization with the parameters left, temperatures and optical depths, using a Levenberg-Marquardt algorithm.  The first guess temperatures are~2000 and 3000~~K for the layer and photosphere respectively.  The first guess optical depths are~0.1, 0.5, 0.5 and~0.1 for the $K$203, $K$215, $K$222 and~$K$239~filters respectively.  The first solution found for the diameters is generally very well defined as the~$\chi^{2}$ is locally convex.  
\par
In a second step, the diameters are set to the values found by this procedure.  A 2D~grid of~$\chi^{2}$~values is built in temperature space, the optimum optical depths being computed with the Levenberg-Marquardt algorithm.  As mentioned, the locus of best parameters is a trough if the number of visibilities is larger than the number of free parameters and if the visibility function is adequately sampled.  The optimum set of temperatures is determined by applying a photometric constraint.  A synthetic $K$~magnitude is computed from the model parameters and solutions which are not compatible with the measured photometry (to within its uncertainty, see Sect.~\ref{sec:photometry}) are excluded.  $L$~band photometry is also used to set the photometric constraint when $L$~band visibility data are available.  The intersection of the temperature trough (defined as the area to within the range~$\chi_{\rm min}^{2}$ to~$\chi_{\rm min}^{2} + 1$ with respect to the bottom of the trough) and of the photometry-compliant area yields the best set of temperatures and their uncertainties.  Eventually the best solution is computed by doing a global fit with the parameters found in the two previous steps as first guess parameters. 
\par
This procedure has proved to converge for all the cases studied here. 
\par
\section{Results of the modeling}
\par
%\begin{figure}%f3
\htmlimage{scale=2.5, external, thumbnail=0.4}
\par
         The visibilities we have obtained for all Miras are clearly wavelength dependent. The general shapes of the visibility functions are in agreement with what we have measured in 1997 on R~Leo (Perrin et~al.~1999). They are very different from a simple limb-darkened disk visibility model and are not Gaussian as is often the case at shorter wavelengths, especially in the visible. We generally find a characteristic plateau or slowly decreasing function around~20\% visibility. As shown in \cite{mennesson2002} this effect can be accounted for by a spherical layer around the star and above the photosphere.
%\begin{table}%t5
\par
\subsection{Comparison of compact layer and extended layer models}\label{sec:comparison}We first have performed the analysis of the \object{R Leo} data of \cite{mennesson2002} with the model proposed in the current paper as a cross-check of the slightly different models to test the influence of the geometric thickness of the layer. In that paper, the~$K$ and $L$~band data were not recorded at the same phase.  Also, the fitting procedure described here really requires narrow band data to disentangle the contributions of the continuum and of the molecular bands in which optical thicknesses hence intensity distributions are quite different -- data from that paper are wide band in both~$K$ and~$L$.  Thus the test must be carried out with caution.  We have not been able to find the solution iteratively, and we have performed a manual adjustment of the parameters such as angular dimensions and temperatures. As shown in Fig.~\ref{fig:RLeowideKL} a set of parameters can be found to account for the data and the quality of the fit is comparable to the one of \cite{mennesson2002}. The parameters derived with the two models are listed in Table~\ref{tab:comparaison_modles}. We find similar characteristic sizes -- the photosphere diameter of 21~mas and the layer diameter of 47.3~mas are intermediate between the inner and outer radii of \cite{mennesson2002}. The temperatures cannot be easily constrained. Choosing a temperature of 2700~K for the star yields a temperature of 1300~K for the layer and optical depths of~1.04 
and~0.81 for the~$K$ and~$L'$~bands respectively. The main difference between the results with the two models is the optical depth which we find larger with the model considered in this paper. Unfortunately we cannot conclude if this is a general discrepancy between the two models. It is to be noticed that our model better fits the highest spatial frequency point. We conclude that the agreement between the two models is satisfactory and that our simpler model can be used as well to interpret the data.
\begin{figure}%f3
\htmlimage{scale=2.5, external, thumbnail=0.4}
%\centering
\par
\includegraphics[angle=-90,width=8.3cm,clip]{1098f3.eps}
      \caption{Fit of the 1997 $K$~band (triangles) and 2000 $L$~band (circles) data of \object{R Leo}.\label{fig:RLeowideKL}}         
         \end{figure}
\par
\begin{table}%t5
%\centering
\par
      \caption{Model fit of the 1997 $K$~band data and 2000 $L'$~band data of R~Leo and comparison       with the results obtained by \cite{mennesson2002}.\label{tab:comparaison_modles}}	          \begin{tabular}{lll}            \hline
            \hline            \noalign{\smallskip}
            Model & \cite{mennesson2002} & This paper  \\             \noalign{\smallskip}            \hline            \noalign{\smallskip}
$R_{\star}$ (mas)                    & 10                    & 10.5   \\
$R_{{\rm layer}}$ (mas) &15--27               & 23.65 \\
$T_{\star}$ (K)                         & 2700                & 2700 \\
$T_{\rm {layer}}$ (K)      &~$< $1730        & 1300 \\
$\tau_{{K}}$	      & ~$\simeq$$ 0.5$ & 1.04 \\
$\tau_{{L'}}$	       & ~$\simeq $$0.5$ & 0.81 \\	\noalign{\smallskip}            \hline         \end{tabular}	
\end{table}