  We prove quantitative norm bounds for a family of operators involving
impedance boundary conditions on convex, polygonal domains. A robust numerical
construction of Helmholtz scattering solutions in variable media via the
Dirichlet-to-Neumann operator involves a decomposition of the domain into a
sequence of rectangles of varying scales and constructing
impedance-to-impedance boundary operators on each subdomain. Our estimates in
particular ensure the invertibility, with quantitative bounds in the frequency,
of the merge operators required to reconstruct the original
Dirichlet-to-Neumann operator in terms of these impedance-to-impedance
operators of the sub-domains. A key step in our proof is to obtain Neumann and
Dirichlet boundary trace estimates on solutions of the impedance problem, which
are of independent interest. In addition to the variable media setting, we also
construct bounds for similar merge operators in the obstacle scattering
problem.
