In this paper we further study the theory of standard bases in filtered structures as introduced by Robbiano. In particular, we study the situation in which a commutative ring A is filtered by commutative ordered monoids D and G in such a way that the G filtration refines the D filtration. We use these results to determine conditions under which a D standard basis can be lifted to a G standard basis. Finally, we interpret these results in three situations where the refinements are given by changing the status of the variables in a polynomial ring, refining a partial order on the monomials in a polynomial ring, and refining a filtration by an I-adic filtration for an ideal I in the ring A.