  The anelastic and pseudo-incompressible equations are two well-known
soundproof approximations of compressible flows useful for both theoretical and
numerical analysis in meteorology, atmospheric science, and ocean studies. In
this paper, we derive and test structure-preserving numerical schemes for these
two systems. The derivations are based on a discrete version of the
Euler-Poincar\'e variational method. This approach relies on a finite
dimensional approximation of the (Lie) group of diffeomorphisms that preserve
weighted-volume forms. These weights describe the background stratification of
the fluid and correspond to the weighed velocity fields for anelastic and
pseudo-incompressible approximations. In particular, we identify to these
discrete Lie group configurations the associated Lie algebras such that
elements of the latter correspond to weighted velocity fields that satisfy the
divergence-free conditions for both systems. Defining discrete Lagrangians in
terms of these Lie algebras, the discrete equations follow by means of
variational principles. Descending from variational principles, the schemes
exhibit further a discrete version of Kelvin circulation theorem, are
applicable to irregular meshes, and show excellent long term energy behavior.
We illustrate the properties of the schemes by performing preliminary test
cases.
