Uniformization of WKB functions by Wigner transform

We develop a technique for uniformizing WKB functions which fail to correctly represent wave fields on caustics due to geometric singularities of ray fields. The uniformization technique is based on appropriate asymptotic surgery of the Wigner transform of the WKB functions, in different regions of the phase space. We present the details of the computation for the model example of the semiclassical Airy equation and we explain how the method can be applied to higher dimensional WKB functions for fold caustics