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     \begin{document}


\begin{equation*}
\langle S_{grain} S^*_{leaf}\rangle= 
\sum_{m=1}^4\sum_{n=1}^4 \langle S_{grain}^m S^{n*}_{leaf}\rangle 
\end{equation*}


\begin{equation*}
S_{grain}^m= S_{grain}^{m\prime}(\bar{s}) \;\Phi^m(\bar{p})
\end{equation*}

\begin{align*}
\bar{s}= \text{ {\bf State variables} inclding: } & \text{moisture} \\
& \text{size} \\ & \text{shape} \\ & \text{radar} \\  & \text{extinction} \\
\end{align*}

\begin{align*}
\bar{p}= \text{ {\bf Propagation variables} inclding: } & \text{soil moisture} \\
& \text{position} \\ & \text{stalk state} \\ & \text{radar} \\ 
 & \text{extinction} \\
\end{align*}

\begin{multline*}
\langle S_{grain} S^*_{leaf}\rangle =
\sum_{m=1}^4\sum_{n=1}^4 \int S_{grain}^{m\prime}(\bar{s_g})\; p(\bar{s_g})
\;d\bar{s_g}\;
 \int S^{n\prime*}_{leaf}(\bar{s_l})\; p(\bar{s_l}) \;d\bar{s_l} \\
\int \Phi^m_{grain}(\bar{p_g}) \Phi^{n}_{leaf}(\bar{p_l})^* p(\bar{p_g},\bar{p_l}) d\bar{p}
\end{multline*}

\begin{equation*} 
\langle S_{stalk} S^*_{leaf}\rangle =
\sum_{m=1}^4\sum_{n=1}^4 \langle S_{stalk}^{m\prime} \Phi_{leaf}^{n*}\rangle
\langle S^{n \prime *}_{leaf} \rangle
\end{equation*}

\end{document}
