## 1. Introduction

[2] The significance of canopy scattering cannot be overstated in remote sensing, water resources management or hydrologic modeling. The canopy consists of leaves and branches that are randomly oriented and distributed. The leaves are usually approximated by oblate cylinders (circular disks) [*Eom and Fung*, 1984] while the branches are approximated by prolate dielectric cylinders. [*Bracaglia et al.*, 1995; *Eom and Fung*, 1986; *Tsang et al.*, 1981]. Prevailing vegetation scattering models based on vectorized radiative theory (VRT) [*Chandrasekhar*, 1950; *Ishimaru*, 1999] compute the scattering of the crown layer either by using the Generalized Rayleigh-Gans (GRG) [*Eom and Fung*, 1984; *Karam et al.*, 1988] approximation at the low frequency or by the Physical Optical (PO) approximation at the high frequency [*LeVine et al.*, 1983; *Michaeli*, 1984; *Ufimtsev*, 1957]. The total scattering phase function of the crown layer is usually approximated by averaging the intensity of differently oriented scatterers incoherently. This means that 1) the scattering model has to be run repeatedly over multiple discrete orientations, and 2) interactions between the particles in the crown or a sub layer are excluded. The VRT-based canopy scattering models such as MIMICS [*Ulaby et al.*, 1987], KARAM and Matrix Doubling [*Fung*, 1994; *Ulaby et al.*, 1986], improve the precision of simulation significantly by taking the canopy's crown-trunk-ground structure into account. However, the cross-polarized responses tend to be underestimated [*Shen et al.*, 2010] because the contribution of multiple scattering within a sub-layer is neglected.

[3] The numerical algorithms [*Harrington*, 1987; *Tsang et al.*, 1992; *Yee*, 1966; *Zhang et al.*, 1996] provide better accuracy by solving Maxwell's equations without making any approximation. However, these algorithms are not set to calculate the scattering of the vegetation layer that contains thousands of scatterers over a vast range of statistical parameters in remote sensing applications.

[4] The matrix formulation of the scattering problem, referred to as the transfer matrix (T-matrix), has been elaborated [*Mishchenko*, 1990, 1991, 1993; *Mishchenko*, 2000; *Mishchenko and Travis*, 1998; *Mishchenko et al.*, 2005; *Oguchi*, 1981] for more than three decades since it was initially proposed [*Waterman*, 1965]. Most of these advances have been thoroughly discussed by *Mishchenko et al.* [1996]. The T-matrix solution dominates the scattering problem in radar meteorology because it helps quickly obtain the scattering coefficients of arbitrary directions of the incidence, scattering and particle orientation once the T-matrix of the particle is calculated. Moreover, the T-matrix is an exact solution of Maxwell's equation [*Waterman*, 1965] thus yielding better accuracy than GRG or PO in the intersection of their valid region. In addition, the most attractive property of the T-matrix is its capability to take the entire multiple scattering from a cluster of particles into consideration by employing the Foldy-Lax approximation [*Foldy*, 1945; *Lax*, 1951]. The Foldy-Lax approximation traces the interactions between scatterers by superimposing the scattered waves to the incidence. The Foldy-Lax approximation was solved initially by the iterative method [*Tsang et al.*, 2001] and was later solved with better accuracy using the cluster T-matrix approach to calculate densely distributed spheres [*Lu et al.*, 1993; *Mackowski*, 1994; *Mishchenko et al.*, 1996; *Tsang et al.*, 2001; *Wang and Chew*, 1993]. The advantages of the cluster T-matrix version make it a viable candidate for solving the scattering problem of the crown layer.

[5] Despite the potential of applying the T-matrix to vegetation remote sensing, the inability to handle large dimensions and extreme shapes whose aspect ratio, the ratio of the radius to the half-length, is far from unity [*Mishchenko*, 1993; *Mishchenko and Travis*, 1998; *Mishchenko et al.*, 1996] prevents the T-matrix method from being widely used in this field. The currently available T-matrix codes require numerical integration of Bessel and Hankel functions over the surface of the particle in its kernel step. Unfortunately, Bessel functions*j*_{n} (*k*_{s}*r*) oscillate quickly with respect to the radius, *r* [*Tsang et al.*, 2000]. When the wave number *k*_{s} is a complex in a lossy medium, the order in magnitude of *j*_{n} (*k*_{s}*r*) grows rapidly when *r* increases. Meanwhile, the Hankel function *h*_{n} (*kr*) increases sharply when *r* approaches the origin, where *k* is the wave number of the free space. In case of extreme shapes, the integral with respect to *r* does not often converge over a wide range. Moreover, even if the integration can be calculated accurately, the matrix that requires inversion is usually singular, especially at highly truncated orders, which is the case of extreme shaped particles. Some studies [*Iskander and Lakhtakia*, 1984; *Lakhtakia et al.*, 1983; *Yan et al.*, 2009] employed the iterative extended boundary condition method (IEBCM) to solve the scattering of extreme shaped particles by dividing them into sub-objects and matching the extended boundary condition iteratively. However, such a strategy does not apply to high-loss dielectric objects [*Iskander and Lakhtakia*, 1984] and incurs substantial changes to the T-matrix algorithm thus increasing the complexity [*Mishchenko et al.*, 1996] of the code significantly. Meanwhile the IEBCM does not include the oblate particle type. *Lakhtakia et al.* [1984]indicates that a modified Gram-Schmidt reinforced orthogonalization can circumvent inverting the singular matrix numerically for the lossless particle. The recursive T-matrix method [*Gürel and Chew*, 1992; *Wang and Chew*, 1992, 1993] combines the MOM and T-matrix to solve the scattering of a 2-D thin conductor. It requires great knowledge of the workflow of MOM in order to recur to the current distribution step in MOM, and also limits the valid region of the recursive T-matrix to that of MOM. The Point Matching Method (PMM) of the T-matrix was initially proposed by*Nieminen et al.* [2003] for optical traps. By using data of the extended precision instead of the double precision [*Mishchenko and Travis*, 1998; *Mishchenko et al.*, 1996], the T-matrix works for aspect ratios as large as 20 according to [*Mishchenko et al.*, 1996]. Such aspect ratios meet the range of hydrometeors such as aerosols, raindrops and even dry snow, but not leaves and branches. Typical dry snow has an aspect ratio of 6, whereas that of a 5-cm radius leaf may reach 200. Additionally, since the T-matrix is unable to handle the opposite extreme, i.e., aspect ratios smaller than unity by several orders of magnitude, it is not an adequate solution to branches. Finally, compared to atmospheric particles, scatterers in canopies are dielectrically too large to make the T-matrix practical.

[6] To utilize the advantages of the T-matrix to calculate the canopy scattering, this paper proposes a matrix inversion approach that circumvents its limitation on extreme shapes. Instead of calculating the T-matrix elements directly, this approach obtains the T-matrix elements by solving two linear sets of equations that use the scattering amplitudes calculated by an existing electromagnetic (EM) model. This paper is organized in the following order.Section 2derives the formulae of the proposed approach. The basic definition of the original T-matrix theory is introduced inSection 2.1 for ease of reference. In Section 2.2, we obtain the kernel formulae of this work as well as describe the computational steps. In Section 3, some numerical issues are discussed, followed by a description of the EM model. We then compare the scattering results of the extreme shapes calculated by the proposed method and the employed EM model. The limitation of the proposed method is analyzed at the end of Section 3. The T-matrix discussed in this paper involves only the traditional localized source (LS) type. In the framework of the LS T-matrix, all the spherical harmonics have one origin that is usually the geometric center of the particle. Therefore, the valid region of the LS T-matrix corresponds to the proposed method, which is outside of the minimum circumscribing sphere of the scatterer.