# A Brief Review of Mueller Matrix Calculations Associated with Oceanic Particles
## 3 General Scattering Method for Suspended Particles
### Semi-Analytical T-Matrix Method  
The T-matrix method was originally proposed by [PERSON] [44; 45]. The incident and scattered fields are expanded in a series of the vector spherical wave functions. The T-matrix connects the incident and scattered expansion coefficients because of the linearity of [PERSON]'s equations. The T-matrix of a particle only depends on the intrinsic properties of the particle, such as the refractive index, morphology, and the orientation of the particle frame of reference and its origin location, but not on the incident state. Corresponding to Equations (7) and (8), the T-matrix can be obtained using the surface-integral and volume-integral methods. The computational method of the T-matrix based on the surface integral is called the extended boundary condition method (EBCM) or the null field method, and it was reviewed by [PERSON] et al. [46], [PERSON] et al. [47; 48], [PERSON] and [PERSON] [49], and [PERSON] et al. [50]. The T-matrix method based on the volume integral equation is called the invariant-imbedding T-matrix method (IITM) and it was originally proposed by [PERSON] [51]. The IITM was reviewed and developed by [PERSON] et al. [52]. For a particle with axial symmetry, the T-matrix is decoupled into a block-diagonal form, so the computation is significantly simplified. The applications of the EBCM on spheroids, cylinders, and Chebyshev shapes are exceptionally effective [49]. However, when a particle has a large size or an aspect ratio far from unity or one that is asymmetric, the matrices in the T-matrix computation are often ill-conditioned. The T-matrix method that is based on the volume integral is much more stable than the EBCM because the volume integral equation in Equation (7) is a Fredholm integral equation of the second kind, which is often less ill-conditioned [34]. The extreme stability of the IITM has been validated by applying the IITM to particles with large sizes, extreme aspect ratios, or asymmetric particles [52; 53]. For instance, the IITM were used to compute the light scattering of oceanic particles, such as _Emiliania huxleyi_ coccoliths and coccolithophores by [PERSON] and [PERSON] [54], and diatoms by [PERSON] et al. [11].  
When compared to the numerically exact methods, the significant advantage of the T-matrix solution is the analytical realization for a particle with a random orientation. The computational time of T-matrix methods is usually shorter than the numerically exact methods because T-matrix methods use matrix inversion instead of iterations. Moreover, in contrast to the relatively large refractive indices of atmospheric particles, such as ice crystals (m \(\sim\)1.33) and aerosols (roughly m \(>\) 1.5), the relative refractive indices of oceanic particles are usually smaller than 1.2. For this reason, using the T-matrix method for oceanic particles normally yields faster convergence and higher computational efficiency than for atmospheric particles. However, the computational time and memory requirements of the T-matrix are strongly related to the radius of the circumscribed sphere of a particle and its morphology. For instance, for a needle particle with small volume but large circumscribed radius or a complex morphology, such as a porous particle, T-matrix methods are not as efficient as the numerically exact methods.  
The numerically exact solutions can provide results for particle size parameters x \(\sim\)100 or less; for the T-matrix solutions EBCM can reach x \(\sim\)180, and the IITM can yield accurate results for x \(\sim\)300. For even larger oceanic particles, approximate solutions must be used.