  Monodromy in analytic families of smooth complex surfaces yields groups of
isotopy classes of orientation preserving diffeomorphisms for each family
member X. For all deformation classes of minimal elliptic surfaces with
p_g>q=0, we determine the monodromy group of a representative X, i.e. the group
of isometries of the intersection lattice L_X:=H_2/torsion generated by the
monodromy action of all families containing X. To this end we construct
families such that any isometry is in the group generated by their monodromies
or does not respect the invariance of the canonical class or the spinor norm.
