  A variety of problems in acoustic and electromagnetic scattering require the
evaluation of impedance or layered media Green's functions. Given a point
source located in an unbounded half-space or an infinitely extended layer,
Sommerfeld and others showed that Fourier analysis combined with contour
integration provides a systematic and broadly effective approach, leading to
what is generally referred to as the Sommerfeld integral representation. When
either the source or target is at some distance from an infinite boundary, the
number of degrees of freedom needed to resolve the scattering response is very
modest. When both are near an interface, however, the Sommerfeld integral
involves a very large range of integration and its direct application becomes
unwieldy. Historically, three schemes have been employed to overcome this
difficulty: the method of images, contour deformation, and asymptotic methods
of various kinds. None of these methods make use of classical layer potentials
in physical space, despite their advantages in terms of adaptive resolution and
high-order accuracy. The reason for this is simple: layer potentials are
impractical in layered media or half-space geometries since they require the
discretization of an infinite boundary. In this paper, we propose a hybrid
method which combines layer potentials (physical-space) on a finite portion of
the interface together with a Sommerfeld-type (Fourier) correction. We prove
that our method is efficient and rapidly convergent for arbitrarily located
sources and targets, and show that the scheme is particularly effective when
solving scattering problems for objects which are close to the half-space
boundary or even embedded across a layered media interface.
