In this disclosure we describe a technique for imbedding a full-wave source model in the traditional integral equation formulations of acoustic and electromagnetic scattering and inverse scattering. This method directly links the material properties of the scatterers to measurement quantities. Traditionally, we model the physics of wave scattering due to inhomogeneous objects with a surface or volume integral equation over the object. That is, given the object materials and the fields throughout the object, along with a kernel describing wave propagation, one can predict the fields anywhere away from the object. These predicted fields are used for general study or in inverse scattering algorithms to reconstruct the object and form images. In experiment, one can never measure fields directly. We measure a voltage response at the output of an antenna or acoustic transducer. In microwave measurements, we often measure the set of S-parameters, or other voltage ratio. Similar quantities can be measured in acoustics.