this lower-dimensional representation can ultimately be mapped to the full problem by using Green function identities (see [@GCT15]) for a detailed discussion of propagation of a correlation function from a straight boundary in this context).

Furthermore, an explicit quantitative connection with phase space densities can be made by presenting the correlation function as a Wigner function. The Wigner distribution function (WDF) approach has been developed in the context of quantum mechanics [@Hillery1984], but has found widespread applications also for microwaves [@Sto99] and in optics [@Dragoman_1997; @torre2005; @Alonso_2011]. The method introduced below exploits a connection between the field-field *correlation function* (CF) and the WDF [@Littlejohn_1993; @Winston_2006; @Dittrich_2009]. Both quantities have been studied intensively in the physics and optics literature. For wave chaotic systems, Berry’s conjecture postulates a universal CF equivalent to correlations in Gaussian random fields [@Berry01JPA_1977; @Hemmady2012; @GG14]. Non-universal corrections can be retrieved by making a link between the CF and the Green function of the system after suitable averaging [@Creagh_1997; @Hortikar_1998; @Weaver_2001; @Urbina_2006]. The Wigner function formalism can be used to derive the ray-tracing limit for propagating wave fields, see for example [@GCT15]. Higher-order (Airy-function) correction to a ray-tracing approach as well as treating evanescent contributions have been discussed 