  Let $\rm E/\rm F$ be an unramified quadratic extension of local non
archimedean fields of characteristic 0. Let $\underline{H}$ be an algebraic
reductive group, defined and split over $\rm F$. We assume that the split
connected component of the center of $\underline{H}$ is trivial. Let $(\tau,V)$
be a $\underline{H}(\rm F)$-distinguished supercuspidal representation of
$\underline{H}(\rm E)$. Using the recent results of C. Zhang, and the geometric
side of a local relative trace formula obtained by P. Delorme, P. Harinck and
S. Souaifi, we describe spherical characters associated to $\underline{H}(\rm
F)$-invariant linear forms on $V$ in terms of weighted orbital integrals of
matrix coefficients of $\tau$.
