  We study amoebas of exponential sums as functions of the support set $A$. To
any amoeba, we associate a set of approximating sections of amoebas, which we
call caissons. We show that a bounded modular lattice of subspaces of a certain
vector space induces a lattice structure on the set of caissons. Our results
unifies the theories of lopsided amoebas and amoebas of exponential sums. As an
application, we show that our theory of caissons yields improved certificates
for existence of certain components of the complement of an amoeba.
