The canonical problem of electromagnetic scattering by a buried dielectric or metallic sphere is studied in this paper. Two expansions are presented, of which one is the expansion of the vector plane waves into the vector spherical waves. The other is its converse, the expansion of the vector spherical waves into the vector plane waves. The use of two expansions is to facilitate matching the boundary conditions at the planar ground surface and the sphere surface. A general full-wave analytic solution is given in which not only the high-order multipoles but also the multiple reflections between the sphere and the ground surface are considered. Oblique plane wave and vertical magnetic dipole are considered as sources, though the illumination can be of arbitrary form with the use of Weyl identity.
 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.
 To our knowledge, it was D'Yakonov  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  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  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.
 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.
 The key to the new solution is the transformation between the vector plane waves and the vector spherical waves. Stratton  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 . Jackson  discussed a similar problem in which the plane wave is circularly polarized and incident along z direction.
 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 . 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.
 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.
2. Transformations Between Vector Plane Waves and Vector Spherical Waves
2.1. Expansion of Vector Plane Waves Into Vector Spherical Waves
then the expansion of eik·r is obtained by taking the curl of both sides of equation (9)
 From their definitions, it is seen that eik·r and eik·r represent two unit orthogonal components of one vector field. As far as the electric field is concerned, eik·r refers to the transverse electric (TE) mode, while eik·r refers to the transverse magnetic (TM) mode, both relative to the z axis. If, however, the field is magnetic field, then eik·r will refer to the TM mode instead of the TE mode, and eik·r the TE mode instead of the TM mode accordingly.
 We will refer to eik·r as the horizontally polarized vector plane wave (HPVP), and eik·r the vertically (obliquely) polarized vector plane wave (VPVP). The latter definition is not very clear physically since an obliquely polarized field is not necessarily related to the TE mode or the TM mode exclusively as defined in the above. In the context of this paper, however, its meaning is evident and specific. It refers to either the TE mode or the TM mode, but not both simultaneously.
2.2. Expansion of Vector Spherical Waves Into Vector Plane Waves
 According to Wittmann , the integral representation of the spherical vector wave functions of the third kind are respectively
where X's are the vector spherical harmonics as defined by Jackson . In Cartesian coordinate, they are of the form
 Dot-multiplying both sides of equation (16) by , respectively, and after some algebraic operations, we have
3. Formal Solution to the Scattering by a Buried Sphere
 With the above two expansions in mind, we now proceed to derive the general full-wave solution to the scattering by a buried sphere, as shown in Figure 1. The incident wave in region 3, or the air region, excited by an arbitrary source is expressible as
with the following conventions used
where the plus sign represents the upgoing wave, and the minus sign, the downgoing wave. We are interested in the field between the source and the ground surface, so that the minus sign is chosen in equation (21).
 Considering ·r3− and ·r3− represent respectively the TE mode and TM mode, it is straightforward to write down the transmitted field in region 2, or the Earth region, as
T32TE is the transmission coefficient for the TE mode, and T32TM, the transmission coefficient for the TM mode, from the air to Earth. The occurrence of the phase factor in the two coefficients is due to the phase shift from the origin to the ground surface. By using the expansions (9) and (12), the two vector plane waves in the integrand, namely, ·r2− and ·r2− are expanded into vector spherical waves, then briefly
Matrix and vector notations are used so that the transmitted field is written in compact form. Double indices n and m are used to denote either row number or column number, which are also the indices for vector spherical waves. The single-dot product denotes the inner product with a finite summation, while the double-dot product denotes the inner product with an infinite summation [Chew and Gürel, 1988; Gürel and Chew, 1988; Chew et al., 1991]. Specifically, the single-dot product denotes two-dimensional inner product throughout the paper, with the two dimensions corresponding to HPVP and VPVP respectively. The infinite dimensional row vector (1) represents standing waves. The arrow sign overhead means its entries are physical vectors. Notice that (1) is outside the integral because it is only the function of field point while the integration is over the propagation direction. is an infinitely long two-column matrix. Its first column is composed of the expansion coefficients of HPVP in terms of vector spherical waves, while the second column the expansion coefficients of VPVP in terms of vector spherical waves.
 Using equation (22), the scattered wave due to the transmitted wave E2t impinging on the sphere is found easily, which is
(3) denotes traveling waves, obtained by replacing the spherical Bessel functions in (1) with the spherical Hankel functions. s is an infinite dimensional diagonal matrix composed of the reflection coefficients of the sphere
Equations (23a) and (23b) are the same as those obtained by Stratton  for the special case in which the TE plane wave is vertically incident. They are the reflection coefficients of TE to r mode and TM to r mode respectively, where r is the radial coordinate of a spherical frame with the origin at the center of the sphere. The superscripts TEr and TMr are used to denote the TE to r and TM to r modes respectively.
 When the scattered wave E2ts goes up to the ground surface, it is also reflected and transmitted. To treat the reflection and transmission by the planar surface, use is made of the expansions (19) and (20) to transform the spherical waves back into the plane waves. To this end
is an infinitely wide two-row matrix, the first column of which consists of the HPVP-related expansion coefficients of vector spherical waves, and the second column of which the VPVP-related expansion coefficients of vector spherical waves. From the definition of and it is readily known that their product is not diagonal. The existence of off-diagonal entries implies a coupling between the TE mode and the TM mode in the reflection of spherical surface, an essential difference than that of planar surface. Using equation (24), the reflected field and the transmitted field are readily obtained
 The definition of 23 is similar to that of 32, with the subscripts 2 and 3 replaced by one another. 23 is defined as
 When the ground-surface-reflected wave E2tsr goes down to the sphere surface, it is reflected again, and hence the above process is repeated, i.e., reflected first by the sphere, then by the ground surface. Such process repeats itself indefinitely, and finally leads to an infinite series accounting for the multiple reflections between the sphere and the ground surface. With equation (25) at hand, we are able to include the multiple reflections into the final solution to arrive at a full-wave solution.
 Letting E3 be the total sphere-scattered field above the ground surface, following the physical process of multiple reflections, we finally have
where the first term on the right hand side of equation (27) is the zeroth-order solution, i.e., the solution neglecting the multiple reflections between the sphere and the ground surface; the second term is the first-order solution in which one reflection is included; the third term is the second solution representing the contribution after two reflections; and the following terms accordingly account for the high-order solutions where high-order multiple reflections are considered. So this infinite series represents a full-wave solution to the scattering by a buried sphere.
Owing to the use of matrix-vector notations, the circle-dot symbol appears more often than the circle symbol alone. In this case, matrix-vector operation is manipulated first, and the circle product is implemented subsequently on each entry of the resulting matrix or vector.
 Formally the above series can be summed as
Concisely, it can further be written as
stands for the zeroth-order sphere-scattered wave above the ground surface, i.e., the wave that propagates from the buried sphere surface into the air.
corresponds to two reflections, one is for the reflection by the ground surface, and the other is for the reflection by the sphere. The inverse of [ − ] thus accounts for the multiple reflections between the ground surface and the buried sphere. Note that is not diagonal, implying a coupling between the multipoles of different order due to the reflection by the ground surface.
refers to the incident wave impinging from the air onto the sphere surface.
4. Representation of the Sphere-Scattered Waves, the Reflection Matrix, and the Incident Wave
 The derivation of the sphere-scattered waves , the reflection matrix and the incident wave a involves lengthy but basic algebraic manipulation. We skip the details and focus mainly on the final result. The representations of the sphere-scattered wave are as follows
The superscripts TEr and TMr implies that the indexed quantities are closely related to TE to r vector spherical waves or TM to r vector spherical waves.
 and are respectively
where α2, α3 are respectively the polar angles of the upgoing wave and with respect to the z axis. and are of the similar forms
Note that all the terms are of the form of the Sommerfeld integral, and the only difference is that the Bessel functions here are of high order. As defined by Chang and Mei , they are referred to as generalized Sommerfeld integrals. In deriving these integrals, the following identity is used
 Taking into consideration that in the expression of , only and have dependence on β2, the forms of which are respectively and . By using the orthogonality of the Fourier harmonics, the integration over the azimuthal angle vanishes, and we finally have the representation for the reflection matrix as follows,
where α′2 is the polar angle of , the reflected part of the upgoing wave , with respect to the z axis, so
δmν is the Kronecker delta function. Therefore is a very sparse matrix, which is favorable for the practical computation. The elements of are also the integrals, but no Bessel functions occur in the integrand like those for .
 When the source is the TE plane wave with the electric field polarized in x direction, the representation for a is
and αi is the polar angle of the incident wave propagating in Earth with respect to the z axis. The meanings of superscripts TEr and TMr here are the same as for the sphere-scattered waves .
 When the source is of other types, Weyl identity needs to be used. For example, if the source is a vertical magnetic dipole with the moment IA, then
where Rs, ϕs and zs define the location of the source. It is seen that they are also generalized Sommerfeld integrals.
 It turns out that the evaluation of the Sommerfeld integral can be a separate paper on its own right. Much effort has been taken on the topic. The resulting method, though effective is often elaborate with the necessity of considering the contribution of poles and branch points. The focus of this paper is not on this point. We are less concerned about the efficiency than the correctness and usefulness, so we directly turn to a practical choice, i.e., deform the integral path below the real axis into the fourth quadrant as has been done by Chang and Mei . The efficiency along the new path is basically affected by how deep the path goes into the fourth quadrant. Generally, the depth is small compared to the truncated distance of the horizontal path. However, it is often a trial and error process to choose an appropriate value.
5. Validation and Simulation Results
5.1. Numerical Validation Against Mie Solution and 3-D CGFFT Solution
 In this section we proceed to validate the formal solution using the Mie scattering solution and the CGFFT solution [Cui and Chew, 1999]. It is stipulated throughout section 5 that the media are assumed nonmagnetic (μ = μ0) without illustration. Figures 2a and 2b shows the comparison between the formal solution result and that of the Mie scattering solution for a plane wave impinging upon a sphere in free space. The incident angle of the plane wave is 30° with the electric field polarized in x direction. The diameter of the sphere is 1 λ(λ is the free space wavelength). The scattered field is calculated along a line parallel to and directly above the x axis, and 0.5 λ above the ground surface. The ground surface is 0.6 λ above the sphere center though there is actually no interface since it is assumed that Earth has the same property as the air so that the Mie solution is applicable. Note that the primary field is excluded from the observation so that the sphere-scattering effect is prominent. The agreement is excellent to within at least three significant figures. In this example, the series is truncated at n = 8. Owing to the absence of the electrical discontinuities across the ground surface, there is no multiple reflections ( = 0 in equation (28)), so the comparison with the Mie solution is not a complete validation. However, from the agreement it can be inferred that the two expansions between vector plane waves and vector spherical waves are valid.
 Then we use a model which was used by Chang and Mei  to validate their method. In this model, the Earth permittivity is 15 + i1.5, the radius of the sphere is 0.2 λ, the ground surface is 0.5 λ above the sphere center. The sphere permittivity is 3. Note that the diameter of the sphere is 1.5 wavelengths in the Earth medium. The vertical magnetic dipole (VMD) instead of the plane wave is used as the illumination source. The expansion of the VMD field is more complicated than that of the plane wave field in that integrals are involved, which is given by equation (31). Moreover, the scattered magnetic field is computed other than the electric field as in the above example. One easily derives the expressions for the magnetic field from equation (30) via ∇ × E = −iωμ0H. In other words, the response of a simple transmitter-receiver (Tx-Rx) configuration will be simulated, which is commonly used in realistic measurements. For all the following results, it is assumed that the Tx-Rx offset is 1 λ, and is 0.5 λ above the ground sphere. The tool axis is along the x axis, and it moves along a measurement line parallel to and directly above the x axis. Besides, the series is truncated at n = 8 to include the contribution from multipoles. Figures 3a and 3b shows the comparison between the formal solution result and the CGFFT result. It is seen that the agreement is quite good.
 The next example is a lossless case. In this case, no energy is absorbed by the media, and there is some resonance inside the sphere when it is electrically large. Though it does not impede the formal solution, our experience shows that it can cause the CGFFT solution to lose some accuracy and converge much slower, especially when the permittivity contrast is high between inside and outside the sphere. To guarantee the accuracy, a 32 × 32 × 32 grid is used for the CGFFT solution. The Earth permittivity is 4, the radius of the sphere is 0.25 λ, and the ground surface is 0.6 λ above the sphere center. The sphere permittivity is 8, meaning the field inside the sphere varies faster than the field in Earth. The diameter of the sphere in this case is 1 λ in the Earth medium. Figures 4a and 4b shows the scattered field computed by both the formal solution and CGFFT for the lossless case. They agree well with each other.
5.2. Multiple Reflections
 The multiple reflections turn out to be of interest as to how it will affect the scattered field. To observe the effect, we leave out the Green's function for reflected field in the CGFFT formulation, which is equivalent to ignoring the influence of multiple reflections. Also the reflection matrix is extracted from equation (28) to be in line with the treatment of the CGFFT formulation. The responses to the above two models are re-calculated and the results are shown in Figures 5a and 5b and Figures 6a and 6b, respectively. The formal solution agrees well with CGFFT, which allows for the formal solution to be used for the analysis of the effect of multiple reflections. We therefore overlap the formal solution results for the two cases and show them in Figures 7a and 7b and Figures 8a and 8b. It is seen that for the first model, though the sphere is electrically larger, there is almost no contribution from the multiple reflections. While for the second model, the multiple reflections effect is appreciable, though small. The reason is that high-order reflected waves travel longer distance than the leading-order reflected waves. The higher the order, the larger the distance. When wave travels in a lossy medium, it is absorbed by the medium. The absorption rate depends on the imaginary part of the dielectric permittivity. Therefore high-order reflected waves are absorbed more than the leading-order waves, and accordingly attenuate much faster than the latter. In addition, in fact it is almost absorbed completely by the ambient medium before it emerges into the air. In lossless media, the wave attenuation only comes from the geometrical decay, and though the contribution from multiple reflections is comparatively small, it is not negligible.
 As is observed from the expression of the reflection matrix , the effect of multiple reflections relies on (1) the permittivity contrast between the air and Earth as well as Earth and the sphere, (2) the buried depth, (3) the size of the sphere. It is desirable that a complete analysis is implemented as to how these factors affect the multiple reflections quantitatively. The objective here is to show some general rules and qualitative properties, so we will confine ourselves to a typical case. In this case, the position of the two-coil system is fixed with its center directly above the sphere. All other parameters are kept the same as above. The permittivity contrast between the sphere, Earth and the air is 5:2:1. d is the buried depth, defined as the distance between the ground surface and the center of the sphere. a is the radius of the sphere. HS,X/Z,Q is the sphere-scattered field including multiple reflections. HS,X/Z,nQ is the sphere-scattered field without multiple reflections. The magnitude ratio of HS,X/Z,Q and HS,X/Z,nQ is plotted against the radius a and shown in Figures 9a and 9b. Note that the unit of the radius is in the wavelength in Earth. Most interesting is that the deviation of the ratio from 1.0 achieves its maximum around a = 0.75 λ ∼ 1.0 λ. Beyond this region, generally the deviation is small, though it slightly increases with the radius. The possible reason is resonant scattering. When the frequency of the incident wave is close to the characteristic frequency of the sphere, the sphere is forced to resonate with the incident field and the radiated field will be strong. One may argue that when the radiated field from the sphere becomes large, HS,X/Z,Q and HS,X/Z,nQ both get large. Why does the deviation of their ratio become large also? One explanation is that when resonance occurs, the coupling between the multipoles, either self coupling or mutual coupling or both, becomes larger than when there is no resonance. This coupling will definitely increase the size of , and therefore enhance the multiple reflections. One also notices that when d becomes large, the ratio deviation becomes small. As implied previously, that is due to the geometrical decay. When the buried depth is large, the reflected waves will travel longer distance, experiencing larger attenuation, hence the effect of multiple reflections becomes weak.
Figures 10a and 10b is to show how multiple reflections affect the scattered field in the lossy Earth. The parameters are the same as above except loss is added to Earth. The Earth permittivity then becomes 2 + i1. As analyzed before, due to the absorption by Earth, the multiple reflections become weak. The calculated results shown in Figures 10a and 10b agree with the analysis. However, one may infer that when the imaginary part of the permittivity is not large, there will still be multiple reflections though small.
5.3. Response of Buried Metallic Sphere
 If the sphere is a PEC sphere, the reflection coefficients assume simpler forms, i.e.,
One may also be interested in the case when the sphere is made of steel, which is commonly encountered in the detection of unexploded ordnance (UXO). Steel is generally of both high conductivity and magnetic permeability. The typical values are 107 S/m for the conductivity and 150 for the relative magnetic permeability.
 Consider a two-coil system the offset of which is 2 m. It is 0.5 m above the ground surface and the tool axis is pointed along the x axis. As in the last section, fix the system position with its center directly above the sphere. The response to both a PEC and steel sphere is computed and plotted against the frequency.
Figures 11a and 11b shows the response in a lossy whole-space Earth and a half-space Earth. The receiver of the two-coil system is polarized in x direction. It can be seen that the response takes on significant value even when the frequency is beyond 1 MHz. For both the whole-space case and the half-space case, there is a rich amount of information about the buried object encoded in high frequency range as seen from the plots. For the half-space case, it is seen that the response is stronger due to the interaction of the object with the air-Earth interface. For the low frequency range (<10 KHz), there is an increasingly larger discrepancy between the response of the PEC sphere and that of the steel sphere. That means that the skin effect at low frequency is not negligible. The oscillation in the high frequency zone is from the interference of the electromagnetic field with the scatterer due to its wave nature.
Figures 12a and 12b shows the response of the two-coil system the receiver of which is vertically polarized. Similar features are observed. Also it shows that high frequency response is rich in the information about the buried object. It is interesting that the signal of the horizontally polarized receiver is quite different than that of the vertically polarized receiver. It implies that the coil polarization yields new information about the buried target.
 In this paper, two new expansions are presented, of which the first is the expansion of the vector plane waves into the vector spherical waves, and the second is its converse, the expansion of the vector spherical waves into the vector plane waves. The first expansion can be thought of as the generalization of the one presented by Stratton which is specialized for a vertically incident TE plane wave. The second is based on two integral representations [Wittmann, 1988]. The two expansions facilitate the formulation of problems in which both the planar and the spherical surfaces coexist.
 More importantly, a general full wave analytic solution to the scattering by a buried sphere is presented, in which not only the high-order multipoles, but also the multiple reflections are considered. It is capable of dealing with arbitrary sources. In this paper, only formulas for the plane wave illumination and the vertical magnetic dipole source are given, the reader will find it easy to generalize to other sources.
 The method has been validated against the Mie solution and the 3-D CGFFT solution. It has been used to analyze the effect of multiple reflections. It is found that when Earth is conductive, multiple reflections can always be neglected without causing large error. When Earth is lossless, multiple reflections are important only when resonant scattering occurs. It has also been used to analyze the response of buried metallic sphere. It is observed that a lot of information about buried metallic sphere is contained in the frequency beyond the induction range which remains to be explored. When the frequency lowers, there is increasing discrepancy between the response of steel sphere and that of its PEC approximation due to the skin effect.
 The method may open a new way for the analysis of the scattering by buried objects when combined with other methods, such as finite element method, integral equation method, or T-matrix method. Also, for some special cases, it is possible to derive some approximate methods from this general solution which could be more convenient and efficient than the present complete form.
 The authors are grateful to the anonymous reviewers. Their comments has helped improve the manuscript.