4006 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 10, OCTOBER 2012 Electromagnetic Radiation Fields in Three-Layered Media With Rough Interfaces Samira Tadros Bishay, Osama M. Abo-Seida, and Hanan Shehata Shoeib Abstract—Research related to the radiation and propagation of the electromagnetic fields in half space or stratified media is of interest and is considered by many authors. In this paper, a theoretical study is discussed about the propagation of radio waves in the sea (three-layered media). The variations which occur in the shape of the sea surface and sea bottom are considered. The effects of the roughness exercised onto the electromagnetic field of ar- rangements radiating a pure transverse electric field in the sea are studied by using the perturbation method. Closed-form expression for the far field generated by a vertical magnetic dipole embedded below the sea surface is calculated by using a simple technique to evaluate Sommerfeld integrals with the aid of the complex image theory, which was quite difficult to evaluate previously. The results obtained are compared with those mentioned elsewhere. Index Terms—Far field, radiation in the sea, rough surfaces, stratified media, vertical magnetic dipole. I. I NTRODUCTION C OMMUNICATION of radio waves in the sea is required for human activities that take place, particularly in recent years. It is worthwhile researching the far-region electromag- netic radiation due to a vertical magnetic dipole in the sea because it is often used for the underwater communication. Wait [1], Moore and Blair [2], and Durrani [3], considered the problem, but in their works, the sea was assumed to be a two- layered conducting medium (air and seawater). If the sea is proportionally deep to dissipate radio waves, it is reasonable to assume that it is also semi-infinite downward and the sea bottom need not be included. However, the effect of the sea bottom is important in some situations such as a shallow sea, a low frequency, and the transmitting and receiving points are close to the bottom. Thus, Arutaki and Chiba [4] considered the sea as a three-layered conducting medium (air, seawater, and ground) for communication near the sea bottom. They investigated the sequences of considering the sea to be of a finite depth. The effect of the sea bottom was found to play an essential role in the cases mentioned above. In their treatment, however, the sea Manuscript received April 18, 2011; revised October 24, 2011 and January 16, 2012; accepted January 22, 2012. Date of publication April 3, 2012; date of current version September 21, 2012. S. T. Bishay is with the Department of Mathematics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt (e-mail: stbishay@yahoo.com). O. M. Abo-Seida is with the Department of Mathematics, Faculty of Sci- ence, Kafr El-Sheikh University, Kafr El-Sheik 33516, Egypt. Currently with the Department of Mathematics, Faculty of Science, King Faisal University, Al-Ahsaa 31982, Saudi Arabia (e-mail: aboseida@yahoo.com). H. S. Shoeib is with the Faculty of Science, Ain Shams University, Cairo 11566, Egypt (e-mail: hanan.shehata@yahoo.com). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2012.2188409 surface was taken to be planar. Afterward, Bishay presented different studies, regarding the effect of the sea waves that occur, on the sea surface only (see [5] and [6]) while others on both the sea surface and the sea bottom (see [7] and [8]). The deviation of the two rough sea surface and sea bottom from flat ones is small and has small slopes. Osama et al. [9] analyzed the effect of sea waves that occur on the sea surface and obtained the formal solution of the far field in the seawater. Recently, Franceschetti et al. [10] investigated analytically the scattering from layered structures with one rough interface. Furthermore, Imperatore et al. [11] studied scattering from layered structures with an arbitrary number of rough interfaces. In this paper, as an alternative, with the aid of the perturbation method which Becker [12] and Bishay and Mohammed [13] used, we study the effect of the variations which occur in both the sea surface and bottom due to the sea waves. Moreover, we look into the solution for the case of uniform sea height (flat one), which Arutaki and Chiba [4] and Long et al. [14] derived. The studies [4]–[8] got the formula of the far field in the seawater by resolving the Sommerfeld integrals through the residue and saddle-point methods. These methods involved lengthy algebra and several transformations which were rather complicated and difficult to evaluate. Studies [4]–[7], also did not calculate or include the disturbed field in the sea or their numerical results. Therefore, in this work, we use a simple technique [14] to evaluate Sommerfeld integrals in a few easily remembered steps with the aid of the complex image theory [15], to obtain a closed-form expression for the far field in the region of the seawater due to a vertical magnetic dipole in a sea. We also offer some numerical calculations which show the importance of the bottom existence and discuss the physical meaning. II. STRUCTURE OF THE SPACE UNDER I NVESTIGATION We adopt the following model as shown in Fig. 1. A small electric current loop antenna, whose magnetic moment is IS 0 , is located in the middle layer (i.e., in the sea) at depth d 1 , horizontally. An observing point p(r, z) is located at depth d 2 . We suppose that the thickness of the sea is a, and that air is infinite upward and the ground downward along the z-axis. All regions are infinite sideward and are homogeneous. R is the distance between the source and the observation point P . The material constants are assumed to be as follows. The dielectric constant in the air, sea, and ground are ε 0 , ε 2 , and ε 3 , respectively. The magnetic permeability is taken equal to that of the free space in every layer. The conductivity of the 0196-2892/$31.00 © 2012 IEEE