## 1. Introduction

[2] Electromagnetic scattering by buried sphere is not a new problem to the geoscience community. The first time it was addressed is more than forty years ago. It is of special interest to researchers since though both the planar and the spherical surfaces exist, it is believed that they are of simple shapes to warrant an analytic solution using classic analytical method. However, no closed form solution exists that does not require the inversion of a matrix system. With the advent of the modern computer technologies, such a special problem could be solved easily using numerical method, but in this paper we will focus on how to solve it analytically. The analytical method provides a clearer picture for the problem being studied and gives a deep insight into its underlying physics. For some cases it can be more efficient than numerical method. Besides, it can be used to validate three-dimensional (3-D) numerical solutions to the scattering of buried objects.

[3] To our knowledge, it was *D'Yakonov* [1959] that discussed this problem first. He obtained a solution by considering a limiting case of the scattering by two nonconcentric spheres with one embedded in the other, but only zeroth-order azimuthal mode, or the axisymmetric case was discussed. *Hill and Wait* [1973] employed the induced dipole method to attack it. As implied by the name, the high-order multipoles are neglected, and in fact the multiple scattering is also excluded. Experience shows that the dipole model always gives a good result when the buried object is not large electrically. When the buried object is relatively large, however, multipoles need to be included to achieve the accuracy. *Chang and Mei* [1981] applied the multipole expansions in half space to represent the scattered waves. The multipoles include vertical electric multipoles, vertical magnetic multipoles, horizontally rotating electric multipoles, and horizontally rotating magnetic multipoles, which constitute a complete set for expanding the field exterior to buried sphere. The expansion coefficients are then determined by matching the boundary conditions on the sphere surface using weighted least squares method. Another related work is the semianalytic mode matching method proposed by *Morgenthaler and Rappaport* [2000, 2001]. Therein the scattering modes are first decomposed into the direct scattering modes and re-scattering modes due to multiple scattering between the buried object and the ground surface, then the mode coefficients are solved for by enforcing continuity at all allocated points on all boundaries. The coefficients thus obtained best fit the boundary conditions in the least squares sense. It is notable that the ground plane may be rough surface, and the buried object may not be limited to only sphere with the use of least squares analysis.

[4] In this paper, a general full-wave analytic solution is presented in which both the high-order multipoles and multiple scattering are taken into consideration. The final solution is de facto an infinite series. Matrix and vector notations, however, are introduced to yield a compact form. Three types of integrals are involved in the final solution, the two of which are similar to the traditional Sommerfeld integrals. The new solution applies not only to the axisymmetric case, but also to a general case in which the incident wave could be of arbitrary polarization and incident angle.

[5] The key to the new solution is the transformation between the vector plane waves and the vector spherical waves. *Stratton* [1941] presented an expansion of a vector plane wave propagating in *z* direction with the electric field linearly polarized in *x* direction. Though any vector plane wave can be expressed in terms of such an expansion after the coordinate rotation, the formula is not convenient to use systematically. The same problem was also addressed by *Tai* [1994]. *Jackson* [1975] discussed a similar problem in which the plane wave is circularly polarized and incident along *z* direction.

[6] In this paper, a new general expansion is presented for a vector plane wave which can be arbitrarily polarized and propagates along any direction. Its generality lends itself to a systematic formulation for the problem containing both the planar and the spherical surfaces. Conversely, the vector spherical waves can also be expanded into vector plane waves with the use of two known integral representations presented by *Wittmann* [1988]. The two expansions constitute an elegant transform pair which facilitates the formulation of the problem where both the planar and the spherical surfaces coexist, like the problem in this paper.

[7] This paper is organized as follows: first, the transformations between vector plane waves and vector spherical waves are introduced which form the basis of our new solution. To this end, the spherical mode expansion of the dyadic Green's function, the stationary phase technique as well as the plane wave expansion of the spherical modes are employed. Second, a systematic formulation of the problem is presented where a general full-wave solution in terms of operators is given. The formulation is intuitive since it follows exactly the physical process of the multiple scattering between the ground surface and the buried sphere. Third, the new solution is validated against known methods and is used to study, though not fully, the effect of multiple reflections and the response of buried dielectric and metallic spheres. The metallic sphere is virtually a special dielectric sphere whose electric conductivity is extremely large. For high frequencies, it is sufficient to model it as a perfect electric conductor (PEC). For induction frequencies, however, it is demonstrated that PEC model will lead to increasingly noticeable error. The paper finally ends with some concluding remarks. Throughout the paper, the time dependence *e*^{−iωt} is assumed and suppressed.