# A Brief Review of Mueller Matrix Calculations Associated with Oceanic Particles
## 3 General Scattering Method for Suspended Particles
### Numerically Exact Methods  
As the name implies, numerically exact solutions use numerical methods to directly solve [PERSON]'s equations or the volume or surface integral equations derived from [PERSON]'s equations. The computational precision depends on the numerical resolution. The finite difference time-domain method (FDTD) is based on the discretization of [PERSON]'s equation Equations (1) and (2) both in time and space [27]. The FDTD method uses the Yee grid to discretize the space, which was developed by [PERSON] [28] and reviewed by [PERSON] [27], and [PERSON] and [PERSON] [29; 30]. Since the computational space has to be confined to a finite region, a perfectly matched layer is used to absorb all of the electromagnetic waves in the computational boundary and avoid any artificially reflected electromagnetic waves back into the computational region [31]. The computational region usually with cuboid shape has to encompass the scattering particle so the computational region is always larger than the scattering particle in the FDTD application. The electromagnetic fields on the grids are updated with the advance of time so the FDTD is an initial value problem. For a time-harmonic field or a field in the frequency domain, [PERSON]'s equations become the vector Helmholtz equations that are given in Equations (5) and (6). The vector Helmholtz Equations (5) and (6) can be discretized in space while using the finite-element method (FEM) [32]. The boundary condition on the particle surface and the continuity condition on the neighboring grid give a series of linear equations. The FEM is a boundary value problem. A major challenge for applying the FEM to light scattering is choosing the finite region covering the scattering particle so that the field in the computational region satisfies the radiation condition in the far field [33]. Like the FDTD method, the computational region for the FEM is also larger than the region that is occupied by the scattering particle, which can constrain the application regime of the FEM. Using the vector Green function, the differential equations become the volume-integral or the surface-integral equations given by Equations (7) and (8). Even though the volume-integral and the surface-integral methods are equivalent, the volume-integral method is numerically more stable than the surface-integral method because the volume-integral equation in Equation (7) is a Fredholm integral equation of the second kind whose matrix equation is usually diagonally dominant [34]. The discrete-dipole approximation (DDA) method is a typical volume-integral method. The DDA was first proposed by [PERSON] and [PERSON] [35] and it was reviewed by [PERSON] [36; 37] and by [PERSON] and [PERSON] [38]. In the DDA method, the particle volume is discretized into usually cubic cells, as shown in Figure 2. Each cubic cell is represented while using an electric dipole and the excitedfield at that cell is composed of the original incident field and the field from all other cells but excluding the cell itself. The dipoles generate a series of linear equations and the fields with respect to all dipoles can be obtained by solving the corresponding linear equations. Once the total field with respect to each cell is obtained, the amplitude scattering matrix and Mueller matrix can straightforwardly be computed. It is evident from Equation (7) that the computational region is equal to the volume of a scattering particle. That makes the DDA method computationally efficient when compared to other numerically exact methods. Two numerical implementations of the DDA are a FORTRAN implementation referred to as DDSCAT by [PERSON] and [PERSON] [39] and a C implementation referred to as ADDA by [PERSON] and [PERSON] [40]. The DDSCAT is parallelized for scattering only in different orientations so the memory requirement might restrict the computational capability for large particles. The ADDA is parallelized by distributing grids (dipoles) into different CPUs so the ADDA can handle particles of large sizes.