  We call a symplectic rational surface $(X,\omega)$ \textit{positive} if
$c_1(X)\cdot[\omega]>0$. The positivity condition of a rational surface is
equivalent to the existence of a divisor $D\subset X$, such that $(X, D)$ is a
log Calabi-Yau surface. The cohomology class of a symplectic form can be
endowed with a \textit{type} using the root system associated to its Lagrangian
spherical classes. In this paper, we prove that the symplectic Torelli group of
a positive rational surface is trivial if it is of type $\mathbb{A}$, and is a
sphere braid group if it is of type $\mathbb{D}$.
  As an application, we answer affirmatively a long-term open question that
Lagrangian spherical Dehn twists generate the symplectic Torelli group
$Symp_h(X)$ when $X$ is a positive rational surface. We also prove that all
symplectic toric surfaces have trivial symplectic Torelli groups. Lastly, we
verify that Chiang-Kessler's symplectic involution is Hamiltonian, answering a
question of Kedra positively.
  Our key new input is the recent study of almost complex subvarieties due to
Li-Zhang and Zhang. Inspired by these works, we define a new \textit{coarse
stratification} for the almost complex structures for positive rational
surfaces. We also combined symplectic field theory and the parametrized
Gromov-Witten theories for our applications.
