## 1. Introduction

[2] Physical Optics (PO) [*Silver*, 1949] is one of the high frequency asymptotic techniques and is widely used for the diffraction analysis. In PO, the currents induced on the scatterer are approximated by Geometrical Optics (GO) and integrated over the surface to give the scattering fields. The surface integration, called radiation integral hereafter, has highly oscillating integrand in higher frequency. The surface-to-line integral reduction is important for reducing the computational load and extracting the mechanism of diffraction. To this end, in addition to a mathematical method related with Stokes' theorem, the concept of equivalent edge currents (EECs) have been established in asymptotic approaches based on the method of stationary phase (SP) for high frequency. The radiation integral could be asymptotically decomposed into two components in general, GO contributions and diffraction. It has been widely understood that the GO components are separately obtained as the contribution from stationary phase point (SPP) inside of the integration area while the diffraction is extracted from the radiation integral in terms of line integration of EECs along the periphery of the scatterer. The GO components are analytically derived using ray geometries alternatively. In the long history of high frequency diffraction analysis, various types of definitions of EECs have been proposed asymptotically and the accuracy for the surface-to-line integral reduction has been compared intensively [*Michaeli*, 1984; *Ufimtsev*, 1991; *Murasaki and Ando*, 1992; *Johansen and Breinbjerg*, 1995; *Albani and Maci*, 2002; *Albani*, 2011]. The EECs were defined along the edge and discussions were focused upon their accuracy in predicting the diffraction fields near the geometrical boundaries such as the shadow (SB) and the reflection boundaries (RB), for which inner SPP traverses the periphery of the integration area. In this background with asymptotic techniques, the EECs have been considered to have nothing to do with the GO components, the remaining contribution in the radiation integral.

[3] The Modified Edge Representation (MER) is a unique concept for defining the EECs for surface-to-line integral reduction of radiation integral [*Gokan et al.*, 1989; *Murasaki and Ando*, 1992]. MER-EEC is based upon not only asymptotic approximation but also Stokes' theorem. It eliminates the singularities of fields on geometrical boundaries. Furthermore, MER-EEC is defined independently from the orientation of the actual edge and therefore, it could be available not only at the periphery but also at arbitrary points over the scatterer except the SPP where MER-EEC becomes infinite. It has been verified that the surface integral of PO currents is uniformly reduced into a MER-EEC line integration, if no inner SPP [*Sakina and Ando*, 2001]. The superiority to other EECs in terms of accuracy and applicability in surface-to-line integral reduction was fully demonstrated [*Murasaki and Ando*, 1991; *Sakina and Ando*, 2001; *Rodriguez and Ando*, 2005]. It implies that, if SPP exists inside, the surface integration is reduced into two line integrations along the periphery and the infinitesimally small indentation integration around the inner SPP, the singularity in the integrands [*Miyamoto and Wolf*, 1962] as in Figure 1. If the former, named MER-Periphery, is regarded as the diffraction as it was suggested by*Young* [1802], it comes that the latter coming from the MER-EEC indentation integral at inner SPP should be identical with the GO constituent. This poses the new discussion about the relation between EECs and the GO component, which has never opened until the proposal of MER-EEC, which is applicable even for the small and inner integration area. The accuracy of MER-Periphery in diffraction analysis and the comparison of MER-EEC with the GO are two related but separate problems for MER-EEC. This contrasts strongly with the conventional EECs where the line integration along periphery sometimes contains not only diffraction but also the GO [*Johansen and Breinbjerg*, 1995; *Albani and Maci*, 2002].

[4] The equivalence of MER(SPP) and Scattering Geometrical Optics (SGO) was proved rigorously for the planar surface [*Rodriguez and Ando*, 2005] and supported numerically for curved surfaces provided that the wavefront of the reflected ray is spherical [*Rodriguez et al.*, 2007]. However, the difference between MER(SPP) and SGO appears in the general case where the reflected wave is non-spherical and aberration exists.

[5] This paper discusses the difference between MER(SPP) at inner SPP and SGO intensively for the ellipsoid [*Lu and Ando*, 2011a, 2011b] and summarizes the dependence upon the parameters, such as the ray geometry and the local shape of the scatterer. Finally the explicit and analytical expression is derived empirically. Now, the GO term is given with two independent expressions, analytical one from ray geometry and line integration of MER-EEC with analytical correction factor.