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We derive rigorously from the water waves equations new irrotational shallow
water models for the propagation of surface waves in the case of uneven
topography in horizontal dimensions one and two. The systems are made to
capture the possible change in the waves' propagation, which can occur in the
case of large amplitude topography. The main contribution of this work is the
construction of new multi-scale shallow water approximations of the
Dirichlet-Neumann operator. We prove that the precision of these approximations
is given at the order $O(\mu \varepsilon)$, $O(\mu\varepsilon +\mu^2\beta^2)$
and $O(\mu^2\varepsilon+\mu \varepsilon \beta+ \mu^2\beta^2)$. Here $\mu$,
$\varepsilon$, and $\beta$ denote respectively the shallow water parameter, the
nonlinear parameter, and the bathymetry parameter. From these approximations,
we derive models with the same precision as the ones above. The model with
precision $O(\mu \varepsilon)$ is coupled with an elliptic problem, while the
other models do not present this inconvenience.