## 1. Introduction

[2] The physical optics (PO) is widely used for estimating the high frequency scattering of electromagnetic fields [see *Harrington*, 1961]. PO predicts the scattering fields from conducting surfaces by two well-defined steps. Firstly, the induced currents at the point of interest in the illuminated region are approximated in the sense of the geometrical optics (GO) by replacing the scattering surface with its tangential infinite plane. Secondly, the scattering fields are obtained by integrating the PO currents over the illuminated region of the scattering surface. The resulting PO fields are free of discontinuities and singularities on the geometrical boundaries and the caustics where GO and the geometrical theory of diffraction (GTD) fail as shown by *Yamashita *[1990, and references therein].

[3] The PO surface radiation integral suffers of difficulties in the physical interpretation. Theoretical works by *Miyamoto and Wolf* [1962], *Rubinowicz* [1965] and *Asvestas* [1986], asymptotics by *Ando* [1984], *Michaeli* [1986], *Gokan et al.* [1989], *Murasaki and Ando* [1991], *Sakina et al.* [2001], *Rodriguez and Ando* [2004a, 2004b], and *Shijo et al.* [2004] and exacts by *Johansen and Breinbjerg* [1995] and *Albani and Maci* [2002] on the surface-to-line integral reduction developments have been presented. Some of these works have provided important and effective tools for the extraction and corrections of the PO errors [see, e.g., *Ufimtsev*, 1971; *Ando et al.*, 1991; *Oodo and Ando*, 1994].

[4] For the surface-to-line radiation integral reduction the equivalent edge currents (EEC) method is introduced with two principal streams in the study. The works in the first category are associated with the asymptotic discussions based upon the principle of stationary phase; where the line integration of the equivalent edge currents around the periphery of the scattering surface is regarded as the edge contributions or the diffraction components as shown by *Ando* [1984], *Michaeli* [1986] and *Oodo and Ando* [1994]. This concept is harmonizing with the work by *Young* [1802] and was demonstrated rigorously by *Rubinowicz* [1965] for scalar potentials. In the second category the line integration is derived rigorously based upon the field equivalence theorem by *Asvestas* [1986], *Johansen and Breinbjerg* [1995] and *Albani and Maci* [2002] where the integral reduction for the planar surface was conducted rigorously with the help of the image theory.

[5] The modified edge representation (MER) concept was proposed for defining the equivalent edge currents on the periphery of the scattering surface, reported by *Ando* [1984]. MER is conceptually different from the general EECs in its derivation and have two important advantages, the wider applicability to curved surfaces and for the source close to the scattering surface [*Rodriguez and Ando*, 2007]. The line integration of the MER currents along the periphery of the scattering illuminated region reproduces the PO diffracted fields with remarkably high accuracy [see *Gokan et al.*, 1989; *Murasaki and Ando*, 1991].

[6] For the integration region having no stationary phase point (SPP), *Sakina et al.* [2001] mathematically showed, by applying Stoke theorem identities and the high frequency approximation, the equivalence of the MER line integration around the periphery of the scattering surface to the PO surface integration, reproducing the diffraction components. On the other hand, if the SPP is located in the surface integration area the entity of the MER line integration along the periphery, whether it is only diffraction or it is scattering is not clear, as it is also the case with the general EECs.

[7] This paper presents novel findings obtained as the by-product in the above study of MER for the case where SPP is inside of the integration area. The MER line integration along the infinitesimally small contour at SPP is calculated numerically, showing a good approximation to the scattering geometrical optics (SGO) for large variety of surface curvatures.

[8] This methodology provides a new and alternative way for calculating SGO emanating from curved surfaces, which differs from the stationary phase method and the classical geometrical optics. For the special case of the planar scattering surface, this numerical result was also proved mathematically by *Rodriguez et al.* [2005] and different types of line integral expressions of GO were independently derived as part of the development by *Yukimasa et al.* [2006].

[9] In addition, this numerical result indirectly identifies the entity of the MER line integration along the periphery for the scattering surface with inner SPP and answers aforementioned open question; the MER line integration along the periphery of the scattering surface corresponds only to diffraction irrespectively of the observer position.