  An Okounkov body is a convex subset of Euclidean space associated to a
divisor on a smooth projective variety with respect to an admissible flag. In
this paper, we recover the asymptotic base loci from the Okounkov bodies by
studying various asymptotic invariants such as the asymptotic valuations and
the moving Seshadri constants. Consequently, we obtain the nefness and
ampleness criteria of divisors in terms of the Okounkov bodies. Furthermore, we
compute the divisorial Zariski decomposition by the Okounkov bodies, and find
upper and lower bounds for moving Seshadri constants given by the size of
simplexes contained in the Okounkov bodies.
