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$\begin{figure} \par\includegraphics[width=11.7cm,clip]{6853fig1.eps}\end{figure}$ Figure 1: The projection factor decomposition ( $p=p_{{\rm o}} f_{{\rm grad}} f_{{\rm o-g}}$ ) in the case of the Fe I 4896.439 Å spectral line: $\Delta RV_{{\rm c}}$ ( $\triangle$ ), $\Delta V_{{\rm l}}^{{\rm g}}$ ( $\triangleleft$ ), $\Delta V_{{\rm p}}^{{\rm g}}$ ( $\triangledown$ ) and $\Delta V_{{\rm p}}^{{\rm o}}$ ( $\Box$ ). Reference values p=1.33 and p=1.27 are indicated for $\delta$ Cep and $\ell$ Car, respectively. The proposed method of the $f_{{\rm grad}}$ determination is to first derive $\Delta RV_{{\rm c}}$ as a function of the line depth for all spectral lines (lower part, black points). Then, when translating (D, $\Delta RV_{{\rm c}}$ ) points into (D, $p_{{\rm o}} \Delta RV_{{\rm c}}$ ), a new linear relation is found (upper part of the figure, black points). This new relation is (1) coherent with the pulsation velocity gradient in the atmosphere (open circles) and (2) its zero-point is consistent with the pulsation velocity corresponding to the gaseous layer of the photosphere: $\Delta V_{{\rm p}}^{{\rm g}}$ ( $\triangledown$ ).
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In the text

$\begin{figure} \par\includegraphics[width=15.5cm,clip]{6853fig2.eps}\end{figure}$ Figure 2: $\Delta V_{{\rm l}}^{{\rm g}}$ (open circles) as a function of the line depth estimators D( $\phi =R_{{\rm max}}$ ), $\langle D \rangle$ , and D( $\phi =R_{{\rm min}}$ ). Black points correspond to the $p_{{\rm o}} \Delta RV_{{\rm c}}$ quantity. The symbol $\triangledown$ indicates $\Delta V_{{\rm p}}^{{\rm g}}$ ; i.e., the amplitude of the photospheric pulsation velocity associated to the gas. We find good agreement between $\Delta V_{{\rm l}}^{{\rm g}}$ and $p_{{\rm o}} \Delta RV_{{\rm c}}$ , which both present an interesting linear relation. Through extrapolation (dashed line), we find that the best estimator of the line-forming region is the line depth associated with the minimum radius, a result confirmed for $\delta$ Cep and $\ell$ Car hydrodynamic models.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{6853fig3.eps}\end{figure}$ Figure 3: $f_{\rm o-g}$ as a function of the period.
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In the text

$\begin{figure} \par\includegraphics[width=15.3cm,clip]{6853fig4.eps}\end{figure}$ Figure 4: $\Delta RV_{{\rm c}}$ as a function of the depth Dof the spectral line considered. Uncertainties are indicated. Stars are presented with increasing period. The $\delta$ Cep model is also indicated for comparison. Linear correlations are derived for all stars (see Table 3). The $f_{{\rm grad}}$ quantity is derived from these relations.
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{6853fig5.eps}\end{figure}$ Figure 5: Slope and zero-point of the Period- $f_{{\rm grad}}$ relation ( $f_{{\rm grad}}=S(D) \log P + ZP(D)$ ) as a function of the line depth. The reduced $\chi ^2$ is also indicated.
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{6853fig6.eps}\end{figure}$ Figure 6: $f_{{\rm grad}}$ in the case of the Fe I 4896.439 spectral line is represented as a function of the period. Open circles corresponds to the hydrodynamic models. The slope and the zero-point of the relation depend on the line depth (see Fig. 5).
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{6853fig7.eps}\end{figure}$ Figure 7: The projection factor ( $p=p_{{\rm o}} f_{{\rm grad}} f_{{\rm o-g}}$ ) as a function of the logarithm of the period (diag. b), together with $p_{{\rm o}}$ (diag. a). Black points and open circles correspond to observations and models.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{6853fig3.eps}\end{figure}$	Figure 3: $f_{\rm o-g}$ as a function of the period.
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$\begin{figure} \par\includegraphics[width=15.3cm,clip]{6853fig4.eps}\end{figure}$	Figure 4: $\Delta RV_{{\rm c}}$ as a function of the depth Dof the spectral line considered. Uncertainties are indicated. Stars are presented with increasing period. The $\delta$ Cep model is also indicated for comparison. Linear correlations are derived for all stars (see Table 3). The $f_{{\rm grad}}$ quantity is derived from these relations.
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$\begin{figure} \par\includegraphics[width=7.8cm,clip]{6853fig5.eps}\end{figure}$	Figure 5: Slope and zero-point of the Period- $f_{{\rm grad}}$ relation ( $f_{{\rm grad}}=S(D) \log P + ZP(D)$ ) as a function of the line depth. The reduced $\chi ^2$ is also indicated.
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$\begin{figure} \par\includegraphics[width=7.8cm,clip]{6853fig6.eps}\end{figure}$	Figure 6: $f_{{\rm grad}}$ in the case of the Fe I 4896.439 spectral line is represented as a function of the period. Open circles corresponds to the hydrodynamic models. The slope and the zero-point of the relation depend on the line depth (see Fig. 5).
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$\begin{figure} \par\includegraphics[width=7.8cm,clip]{6853fig7.eps}\end{figure}$	Figure 7: The projection factor ( $p=p_{{\rm o}} f_{{\rm grad}} f_{{\rm o-g}}$ ) as a function of the logarithm of the period (diag. b), together with $p_{{\rm o}}$ (diag. a). Black points and open circles correspond to observations and models.
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