Dimension-variable invariant imbedding (DVIIM) T-matrix computational method for the light scattering simulation of atmospheric nonspherical particles

Hu Shuai; Hu Shuai; Hu Shuai; Li Shulei; Li Shulei; Zeng Qingwei; Liu Lei; Liu Lei

doi:10.1364/OE.472809

1. Introduction

To improve the simulation accuracy of atmospheric radiative transfer, the light scattering properties of aerosols and ice crystals should be accurately simulated [1–7]. However, due to their irregular shapes and inhomogeneous compositions [8–10], their light scattering properties haven’t been adequately understood, thus causing substantial uncertainties in radiative transfer simulation [11,12]. In recent years, with the development of polarized remote sensing, more and more researchers have found that the nonspherical shapes of aerosol and ice crystals have a significant influence on the polarized components of the radiation [2,12,13]. As a result, how to obtain the light scattering properties have become an important issue in the field of atmospheric remote sensing [6,14–16].

To obtain the scattering properties of nonspherical particles, many scattering models are developed, such as GOA/IGOA (Geometrical Optics Approximation/Improved Geometrical Optics Approximation) [17–19], PGHOM (Physical Geometric Optics Hybrid Method, a geometric optical model using a beam-splitting technique) [20,21], T-matrix method [22–25], DDA (Discrete Dipole Approximation) [26–28] and time domain methods (FDTD, PSTD and MRTD, etc.) [29–35]. The semi-empirical geometric-optics method GOA/IGOA is suitable to the scattering simulation of large particles but it fails for small particles because the “ray” concept is not valid when the light wavelength is similar to particle size [20]. DDA and the time domain models are applicable to particles with moderate size parameters However, they only can calculate the light scattering properties of the particle with specific size and orientation in one wave-particle interaction [11]. Since the scattering parameters in the Radiative Transfer Models (RTMs) are volume-averaged values (namely, the scattering properties should be integrated over their size distributions and orientations [3,11]), in order to obtain the scattering parameters of the randomly oriented particles, the computational process should be repeated many times for particles with different orientations, and the process will become very time consuming.

In contrast, T matrix method has the incomparable advantage. Once the T matrix of the nonspherical particle is obtained, the scattering properties of randomly oriented particles can be calculated analytically [36,37]. Thus, it will be much more precise and efficient than other computational models. The T matrix can be calculated by the surface integral and volume integral scheme [38]. The surface integral scheme, which is also called EBCM (Extended Boundary Condition Method), is firstly proposed by P. C. Waterman [39,40]. After the improvement of Mishchenko and other scholars [23,37,41–44], it has become a well-tested scattering model and widely applied in the fields of atmospheric radiation, bio-optics, astrophysics, etc. [38,45]. However, due to the difficulty in solving the surface integral equations, the EBCM T-matrix method is mainly good at the light scattering simulation by rotationally symmetric particles (like spheroidal and cylindrical particles, etc.) [42]. To extend the application scope of the T-matrix methods, the invariant imbedding (IIM) technique is also introduced into the calculation of the T matrix [38,46–48]. Because the IIM T-matrix method is essentially derived from the Helmholtz volume integral equation, it can be applied to particles with arbitrary shapes and inhomogeneous compositions. After the efficient implementation of Bi L. and Yang P. [38,47–49], the IIM T-matrix method has become a powerful tool for the light scattering simulation of nonspherical particles. In recently years, referring to the work of Bi L., we further derived the symmetrical properties in T-matrix method and proposed an efficient realization scheme of this model [50–52]. Based on those works, a FORTRAN code of the IIM T-matrix method has also developed by our team [50,52–54].

In principle, with the increasing of dimension of T-matrix, the higher accuracy can be achieved in the light scattering simulation. Generally, to guarantee the information content of the T matrix, the dimensions of T matrix and other matrices involved are fixed as l_max=k·N_max(N_max+2) (N_max is the expansion order of the electric fields, which is dependent on the size of the nonspherical particle, and k=2 or 3) [38,53,55]. Therefore, for particles with large sizes, all the matrices involved in the computations are supermatrices, and most of the time is consumed on their inversion and multiplication processes. Besides, the T matrix should be updated step by step along the radial direction, all these operations of large matrices should be repeated in every iteration. Thus its computational burden is much heavier than that of EBCM T-matrix method [48]. Owning to this reason, how to improve the modeling efficiency has become an important issue of the IIM T-matrix method. From the principle of IIM T-matrix model, it can be found that in the early iterations, only a part of the particle (the portion in the nth spherical shell) is used. That is to say, the effective size for computation is gradually increased with the iteration performed. According to the T matrix theory, the smaller the particle size is, the smaller dimension of T matrix is needed. Therefore, we consider whether the dimension of the matrices can be reduced in early iterations. From this idea, a dimension-variable invariant imbedding T-Matrix method is proposed in this paper. In this model, the dimension of T matrix is gradually increasing with the increasing of the radius of spherical shell rather than fixed as specific value, i.e., 2N_max (N_max+2). In this way, the computational amount can be cut down notably, especially for particles with large size and aspect ratio.

This paper is organized as follows: in section 2, the basic principle of the Dimension-Variable IIM (DVIIM) T-Matrix method is introduced. Then, to optimally determine the dimension of the supermatrices in each iterative step, a spheroid-equivalent scheme is proposed in section 3. Further, the modeling accuracy is further validated for particles with different shapes and sizes, and computational efficiency of the DVIIM and IIM T-matrix methods are compared quantitatively. Section 7 is a summary of our work.

2. Basic principles of DVIIM T-matrix method

In the DVIIM T-matrix method, the nonspherical particle is also regarded as an inhomogeneous sphere (as shown is Fig. 1) and is discretized into several inhomogeneous spherical shells in the spherical coordinate system. Then, the T Matrix of the nonspherical particle is solved by combining the invariant imbedding technique with the Lorenz-Mie theory or EBCM, where the T matrix of inscribed sphere of the nonspherical particles is firstly calculated by Lorenz-Mie theory or EBCM (the yellow part in Fig. 1(a)), and then, in the spherical mantle region, the principle of the IIM is applied to obtain the T-matrix of a larger sphere of n layers based on the T-matrix of a smaller sphere with n–1 layers (see Fig. 1(c)). The iterative equation for the T matrix computation can be written as

(1)$$\overline{\overline{\mathbf{T}}} ({r_n}) = \overline{\overline{\mathbf{Q}}} _{11}({r_n}) + (\overline{\overline{\mathbf{I}}} + \overline{\overline{\mathbf{Q}}} _{12}({r_n})){[{\overline{\overline{\mathbf{I}}} - \overline{\overline{\mathbf{T}}} ({r_{n - 1}}){\overline{\overline{\mathbf{Q}} }_{22}}({r_n})} ]^{ - 1}}\overline{\overline{\mathbf{T}}} ({r_{n - 1}})[{\overline{\overline{\mathbf{I}}} + {\overline{\overline{\mathbf{Q}} }_{21}}({r_n})} ],$$

where R₀ is the radius of the inscribed sphere, $\overline{\overline{\mathbf{T}}} ({r_n})$ $\overline{\overline{\mathbf{T}}} ({r_n})$ and $\overline{\overline{\mathbf{T}}} ({r_{n - 1}})$ are the T matrices of the spheres with n layers and n-1 layers, $\overline{\overline{\mathbf{Q}}}_{11}\left(r_n\right), \overline{\overline{\mathbf{Q}}}_{12}\left(r_n\right), \overline{\overline{\mathbf{Q}}}_{21}\left(r_n\right)$ \;\text{and}\; $\overline{\overline{\mathbf{Q}}}_{22}\left(r_n\right)$ are the optical matrices of the nth spherical shell.

Fig. 1. A schematic diagram of the discretization of nonspherical particles and the computational regions for IIM technique and Lorenz-Mie theory (or EBCM).

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In principle, with the increasing of particle size, there will be more scattering information existed in the T matrix, and the dimension of matrices should be increased as well. Therefore, in traditional IIM T-matrix method, the dimensions of the T and Q_ij (i, j=1,2) are fixed as l_max=2·N_max(N_max+2) in all iterations to guarantee the convergence of the computation, where, N_max is the expansion order of the electric fields, a parameter dependent on the size parameter of the nonspherical particle. According to the work of Wiscombe [56], N_max generally can be obtained by Eq. (2) [41]

(2)$${N_{\max }}(x) = x\textrm{ + }4.05{x^{1/3}} + 8;$$

where x=2πr/λ is size parameter of the nonspherical particle (λ is the light wavelength, r is the radius of the circumscribed sphere).

However, the effective size of the particle in the initial iterative steps is not as large as that of the last iteration (the effective size for computation is become larger and larger with the iteration process carried out, as shown in Fig. 2). Therefore, according to the T matrix theory, the expansion order of the electric field is not needed to be taken as large as N_max in early iterations, and the dimensions of the T matrix can be reduced in the initial iterative process. Referring to this idea, a dimension-variable IIM T-matrix method is proposed. The basic principle of the method is shown in the Fig. 2. In the computation, the expansion order of the electric field is gradually increasing as effective size of the inner sphere becomes larger. By using this method, both the computational amount and memory consumption can be saved. In the implementation of this model, the expansion order of the electric field in the ith iteration, i.e., N_i, is firstly determined by the Spheroidal Equivalent Scheme (SES, see section 3 for detail), then the dimension of T, U, Q and Q_ij (i, j=1, 2) can be defined as

(3)$$\textrm{For }\mathbf{T}\,\textrm{and}\,{\mathbf{Q}_{ij}}\textrm{matrices}:{l_i} = 2{N_i} \cdot ({N_i} + 2);$$

(4)$$\textrm{For}\,\mathbf{U}\,\textrm{and}\,\mathbf{Q}\,\textrm{matrices}\,\textrm{:}\,{l_i} = 3{N_i} \cdot ({N_i} + 2). $$

Fig. 2. The basic principle of DVIIM T-Matrix method.

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After the dimension of these matrices is determined, the T matrix can be updated layer by layer by using Eq. (1).

3. Optimal determination of the T-matrix dimension and Gaussian integral points by spheroidal equivalent scheme (SES)

From section 2, we can know that it is very important to determine N_i in each iteration since it is the parameter to determine the dimension of the T, U, Q and Q_ij matrices. However, not only that, N_i is also an important parameter to determine the number of Gaussian discrete points of the azimuth and zenith integrals in the U matrix calculation, which will influence the modeling accuracy of the DVIIM T-matrix method. Due to these reasons, the expansion order N_i should be optimally determined considering both the computational efficiency and accuracy.

In the DVIIM T-matrix method, the computational accuracy of U matrix is very critical since the irregular shape of particles and their optical properties is manifested by the matrix, the expression of U matrix can be written as:

(5)$$\scalebox{0.93}{$\displaystyle\begin{array}{l} \overline{\overline{\mathbf{U}}} _{mn{m^{\prime}}{n^{\prime}}}^{}(r) = \\ = {k^2}{r^2}{( - 1)^{m + {m^{\prime}}}}{\left[ {\frac{{2n + 1}}{{4\pi n(n + 1)}}} \right]^{1/2}}{\left[ {\frac{{2{n^{\prime}} + 1}}{{4\pi {n^{\prime}}({n^{\prime}} + 1)}}} \right]^{1/2}}\int\limits_0^{2\pi } {d\varphi \int\limits_0^\pi {d\theta \sin \theta \exp [{ - i(m - {m^{\prime}})\varphi } ]} } [{{\varepsilon_r}(r,\theta ,\varphi ) - 1} ]\\ \times \left( {\begin{array}{ccc} {{\pi_{mn}}(\theta ){\pi_{{m^{\prime}}{n^{\prime}}}}(\theta ) + {\tau_{mn}}(\theta ){\tau_{{m^{\prime}}{n^{\prime}}}}(\theta )}&{ - i[{{\pi_{mn}}(\theta ){\tau_{{m^{\prime}}{n^{\prime}}}}(\theta ) + {\tau_{mn}}(\theta ){\pi_{{m^{\prime}}{n^{\prime}}}}(\theta )} ]}&0\\ {i[{{\pi_{mn}}(\theta ){\tau_{{m^{\prime}}{n^{\prime}}}}(\theta ) + {\tau_{mn}}(\theta ){\pi_{{m^{\prime}}{n^{\prime}}}}(\theta )} ]}&{{\pi_{mn}}(\theta ){\pi_{{m^{\prime}}{n^{\prime}}}}(\theta ) + {\tau_{mn}}(\theta ){\tau_{{m^{\prime}}{n^{\prime}}}}(\theta )}&0\\ 0&0&{\frac{{\sqrt {n(n + 1){n^{\prime}}({n^{\prime}} + 1)} d_{0m}^nd_{0{m^{\prime}}}^{{n^{\prime}}}}}{{{\varepsilon_r}(r,\theta ,\varphi )}}} \end{array}} \right) \end{array}$}$$

where r is radius of the nth spherical layer, k is wavenumber, $\theta$ and $\varphi$ are the zenith angle and azimuth angle respectively, ${\varepsilon _r}(r,\theta ,\varphi )$ is the permittivity of the medium; $d_{0m}^n$ is the Wigner-d function, ${\pi _{mn}}(\theta )$ and ${\tau _{mn}}(\theta )$ are the angular functions. To guarantee both the simulation accuracy and efficiency of the computation, similar to the way adopted by Mishchenko, the number of the Gaussian discrete points for the integral over the zenith and azimuth angles are taken as [41,42]

(6)$${N_{ZEN}} = 2 \cdot NDGS \cdot {N_i};$$

(7)$${N_{AZI}} = 2 \cdot {N_{ZEN}} = 4 \cdot NDGS \cdot {N_i};$$

where ${N_{ZEN}}$ and ${N_{AZI}}$ are the number of the discrete points for zenith and azimuth integrals (since the variation range of the azimuth angle is twice that of zenith angle, therefore ${N_{AZI}}$ is set as $2 \cdot {N_{ZEN}}$ in our IIM T-Matrix method). From the equations above, on the premise of ensuring the simulation accuracy, not only the expansion order of the electric field should be optimally determined, but also NDGS need to be determined properly.

To reasonably determine N_i and NDGS in each iterative process, a Spheroid-Equivalent Scheme (SES) is proposed. The basic principle of SES is illustrated as follows: the particle in the ith spherical shell is firstly equivalent as a spheroidal particle according to its size and shape. Then the optimal expansion order N_i and NDGS is obtained from the convergent condition obtained by the EBCM T-matrix method for the equivalent spheroid. Taking the ith iterative step as the example, the implementation of SES can be divided into 2 steps, as shown in Fig. 3.

Fig. 3. The basic principle of spheroid-equivalent scheme. The key point of this method is to select a spheroid which is similar to the particle in the spherical shell and determine the expansion order using EBCM.

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Step I: determining the shape parameter of the equivalent spheroidal particle. According to the shape of the particle in the ith spherical layer, the maximum aspect ratio and equivalent size parameter is extracted firstly, where the aspect ratio is regarded as the ratio of radius of the ith spherical layer and that of the inscribed sphere, and the equivalent size parameter is set as that of the inscribed sphere, written as

(8)$${\varepsilon _{eq}} = {r_i}/{R_0};$$

(9)$${X_{eq}} = 2\pi {r_i}/\lambda;$$

where ${\varepsilon _{eq}}$ and ${X_{eq}}$ are the equivalent aspect ratio and size parameter, respectively. ${r_i}$ and ${R_0}$ are the radius of the ith spherical shell and the inscribed sphere.

Step II: the optimal determination of N_i and NDGS. To facilitate the determination of the expansion order of electric field and Gaussian discrete points, a Lookup Table (LUT) of N_i and NDGS is established for spheroidal particles with different size parameters, aspect ratios and refractive indices by using the EBCM T-matrix method provided by Mishchenko. The convergence condition of the EBCM model is set as

(10)$$\frac{{|{{C_{ext}}({N_{\max }},NDGS) - {C_{ext}}({N_{\max }} - 1,NDGS)} |}}{{{C_{ext}}({N_{\max }})}} < 0.001,$$

where the size parameter of the spheroid is calculated by X_s=2πb/λ (b is the half length of long axis), and part of the LUT is presented in Table 1. For spheroid particles with large size and refractive index, the EBCM might not be convergent in the computation. In this case, N_i is taken as N_max, namely, the expansion order determined by Eq. (2).

Table 1. The optimal value of N_max and NDGS for spheroidal particles with different sizes, aspect ratios and refractive indices.

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After the LUT is established, then NDGS for particles with an equivalent size parameter and aspect ratio of ${\varepsilon _{eq}}$ and ${X_{eq}}$ can be obtained by

(11)$$NDGS = \max \{{LUT({\varepsilon_{eq}},{X_{eq}},m),\textrm{ }4} \};$$

where $LUT({\varepsilon _{eq}},{X_{eq}},m)$ denotes the linear interpolation in the established lookup table.

For the expansion order N_i, it is obtained in the following manner, expressed as

(12)$${N_i} = \max \{{LUT({\varepsilon_{eq}},{X_{eq}},m),{N_r}} \};$$

(13)$${N_r}\textrm{ = }\left( {\frac{{2\pi {r_i}}}{\lambda }} \right)\textrm{ + }4.05{\left( {\frac{{2\pi {r_i}}}{\lambda }} \right)^{1/3}} + 8;$$

where ${r_i}$ is the radius of the ith spherical shell, λ is the wavelength, ${N_r}$ is the expansion order for the sphere in the ith iteration using Wiscombe’s rule [56].

4. Implementation of DVIIM T-matrix computational code

Based on the principle proposed above, we have developed the FORTRAN code of the DVIIM T-matrix method. As shown in Fig. 4, the computational process of DVIIM T-matrix method can be concluded as following 3 steps:

Step I: The initialization of the model. This function of this part is to do the necessary initialization of the light scattering simulation. In the part, the refractive index of the nonspherical particle should be firstly converted into the permittivity, and radius of the inscribed and circumscribed sphere of the nonspherical particle should be determined. Then, according to the size of the particle, the expansion order of the electric field and the dimensions of matrices T, U, Q, Q_ij (i, j=1,2) are initially determined.
Step II: T-matrix computational process. In this part, the T matrix of the inscribed sphere is firstly calculated by the Lorenz-Mie theory and EBCM, and then the T matrix of the nonspherical particle is updated layer by layer by Eq. (1). In the computational process, the expansion order of the electric field is firstly determined by the SES technique; and based on the results, the dimensions of the matrices and the Gaussian discrete points for the zenith and azimuth integrals can also be determined (see section 3 for detail). The optical properties of the spherical shells, namely, the U matrix, is firstly calculated based on the particle’s geometry, and then the Q, Q₁₁, Q₁₂, Q₂₁ can be Q₂₂ could be calculated based on U matrix. To save the computer memory, the memory consumed by U, Q, Q_ij and $\mathbf{J}({r_n})$ should be deallocated before the scattering parameter computational process.
Step III: The scattering parameter computational process. This process is to calculate the scattering parameters based on the T-matrix obtained by the DVIIM T-Matrix method. In this process, both the scattering parameter of particles with specified orientation and randomly orientations can be obtained. The scattering parameters calculated in this model includes the extinction, absorption and scattering cross sections, single scattering albedo, asymmetric factor, and scattering phase matrix.

Fig. 4. The computational flowchart of the implementation of DVIIM T-matrix method.

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Different from traditional invariant imbedding T-matrix method, the most important in the whole calculation process is how to update the dimension of the matrix in step II, which will seriously affect the calculation accuracy of DVIIM T-matrix method. And it should be noted that it is not necessary to allocate and deallocate the computational memory for the large matrices (e.g., T, U and Q matrices, etc.) in each iterative step. We can allocate the memory according to the highest order of electric field N_max in model’s initial process; and in the computational process, only the sub-matrix required in computation is applied, while the remaining matrix elements in are set to be zero. In this way, the repeated memory allocation processes can be avoided.

5. Model validation and results analysis

To validate the modeling accuracy of DVIIM T-matrix method, its simulation results (including the scattering phase matrices and integral scattering parameters such as extinction, scattering and absorption cross sections) are compared with those obtained by EBCM T-matrix method [36,57], DDASCAT [27,58], and IIM T-matrix method [51].

5.1 Spheroidal particle case

In this case, the computational results of DVIIM T-matrix method are compared with those of EBCM T-matrix method for three spheroidal particles with different sizes, and the results are presented in Fig. 5 and Fig. 6. Figure 5 shows the scattering phase matrices of particles with specific orientations, and Fig. 6 are the scattering matrices of randomly oriented particles. In this test, the light wavelength is taken as 0.5µm, the refractive index is set as m= 1.20+0.0008i, the half-lengths of the horizontal and rotational axis are set as (a, b)=(1µm, 2µm), (2µm, 4µm) and (5µm, 7µm), respectively.

Fig. 5. The comparisons of scattering phase matrices obtained by EBCM and DVIIM T-matrix methods for the single-oriented spheroidal particles with different sizes (the incident light is parallel to the rotational axis).

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Fig. 6. The comparisons of scattering phase matrices obtained by EBCM and DVIIM T-matrix methods for the randomly oriented spheroidal particles with different sizes.

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As can be seen from these figures, a good consistency is obtained between the results of the EBCM and DVIIM T-matrix method for both the particles with specific orientation and random orientations. For particles with specific orientation, the relative errors of the scattering phase functions are all with 10% in scattering angles ranging from 0° to 90°, and the absolute errors of F₁₂/F₁₁ are generally smaller than 0.1, which indicating that the DVIIM T-matrix method can simulate the scattering parameters of spheroidal particles with a high precision. For the particles with random orientations, the modeling accuracy of DVIIM T-matrix method is much higher than particles with single orientation, where the relative errors of the F₁₁ are smaller than 1% in most scattering directions, and the absolute errors of F₁₂/F₁₁ are generally less than 0.03. On the whole, the simulation accuracy of the DVIIM T-matrix method in forward directions is higher than that in large scattering angle, which similar to that of the traditional IIM T-matrix method.

To further illustrate whether the decreasing of the dimension of T matrix in early iterations will influence the simulation accuracy, the scattering phase matrix calculated by DVIIM T-matrix method is also compared with that of the traditional IIM T-matrix method quantitatively, the results are shown in the Table 2.

Table 2. The scattering phase matrices obtained by DVIIM and IIM T-matrix methods for the spheroidal particle with (a, b) = (2µm, 4µm).

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As can be seen, a good agreement is achieved between the results of DVIIM and IIM T-matrix methods. For the scattering phase function, F₁₁, the relative differences of the results are generally smaller than 1%, the absolute differences of F₁₂/F₁₁, F₃₃/F₁₁ and F₃₄/F₁₁ are generally smaller than 0.01. Therefore, it can be concluded that the decreasing of the T matrix dimension in the early iterative steps will not influence the modeling accuracy of the T matrix model notably for spheroidal particles.

Moreover, the integral parameters calculated by IIM and DVIIM T-matrix methods are also compared with each other, the results are shown in Table 3. From the table, it can be found that the results of DVIIM T-matrix method show a good agreement with those obtained by IIM T-matrix method, indicating that the DVIIM T-matrix method can achieve similar computational accuracy of the IIM T-matrix method for spheroidal particles.

Table 3. The integral scattering parameters obtained by DVIIM and IIM T-matrix methods for the spheroidal particle with different sizes.

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5.2 Cylinder particle case

The modeling accuracy of the DVIIM T-matrix method is also tested for cylindrical particles, the scattering phase matrices calculated by different models are shown in Fig. 7 and Fig. 8. In this simulation, three cylinders with different sizes are used for model validation, their diameters and lengths are set as (D, H)=(2µm, 2µm), (4µm, 4µm) and (8µm, 8µm), respectively.

Fig. 7. The comparisons of scattering phase matrices obtained by EBCM and DVIIM T-matrix methods for the single-oriented cylindrical particles with different sizes (the incident light is parallel to the rotational axis).

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Fig. 8. The comparisons of scattering phase matrices obtained by EBCM and DVIIM T-matrix methods for the randomly oriented cylindrical particles with different sizes.

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From the results, it can be seen that the curves of the matrix elements obtained by DVIIM T-matrix method is almost coincided with those of EBCM, where for F₁₁, the relative differences are generally smaller than 10%, and for F₁₂/ F₁₁, their absolute errors are with 0.1 for most scattering directions. Similar to the spheroidal particle case, the simulation accuracy of DVIIM is much higher for the randomly oriented particles, and the relative differences of the scattering phase functions are smaller than 2% in most scattering angles. From the discussion above, it can be found that the high modeling accuracy can be achieved by the DVIIM T-matrix method in the light scattering simulation of cylinders.

Similar to the spheroidal particles, the scattering phase matrix calculated by DVIIM T-matrix method is also compared with the traditional IIM T-matrix method, and the results are presented in Table 4. From the results, it can be found that the difference between the results of DVIIM and IIM T-matrix methods is small, where the relative differences of F₁₁ are generally less than 1%, and the absolute differences of F₁₂/F₁₁ are smaller than 0.01 for most scattering angles. From the results, it can be concluded that the spheroid-equivalent scheme will not cause the decreasing of the modeling accuracy remarkably in spite of the decreasing of dimension of the T matrix in the early iterative computations.

Table 4. The scattering phase matrices obtained by DVIIM and IIM T-matrix methods for the cylindrical particle with (D, L) = (2µm, 2µm).

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Besides, the integral scattering parameters of DVIIM and IIM T-matrix methods are also compared with each other, and the results are shown in Table 5. It can be seen that the cross sections calculated by DVIIM T-matrix method show a good agreement with those of the traditional one, where the relative differences of C_ext, C_ab, and C_sc are generally smaller than 1%, which further validates both the modeling accuracy of the DVIIM T-matrix method.

Table 5. The integral scattering parameters obtained by DVIIM and IIM T-matrix methods for the cylindrical particle with different sizes.

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5.3 Hexagonal ice crystals case

To further test the modeling accuracy of the DVIIM T-matrix method for particles with non-rotational symmetrical geometry, the simulation results is also validated against DDASCAT for hexagonal particles, and the results of the random-orientation case are presented in Fig. 9. In this case, the wavelength of the incident light is set as λ=0.5µm, the refractive index of the particles are all set as m=1.22+0.0008i, the shape parameters (a, L) are set as (1µm, 2µm) and (2µm, 4µm), respectively.

Fig. 9. The comparisons of scattering phase matrices obtained by EBCM and DVIIM T-matrix methods for the random oriented hexagonal particles with different sizes.

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It can be found that a good consistency is also achieved between the results of DVIIM T-matrix method and DDASCAT, and the curves of the phase matrix elements obtained by different models are very close to each other. For the F₁₁, their relative differences are generally smaller than 5% in forward scattering directions, though the differences in large scattering angles are slightly larger, but they are with 15% for most directions. The high agreement indicates that the DVIIM T-matrix method can simulate the light scattering process of ice crystals with a high precision.

To further investigate the spheroid-equivalent scheme whether will cause the decreasing of the modeling accuracy of invariant imbedding T-matrix method for hexagonal ice crystals, the scattering phase matrices obtained by DVIIM and IIM T-matrix methods are also compared with each other, the results are illustrated in Table 6.

Table 6. The computational results for the IIM and DVIIM T-Matrix models for ice crystal particle (R = 1µm, L = 2µm).

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As can be seen, the computational results of DVIIM T-matrix method agree with those of IIM T-matrix methods well, where for F₁₂/F₁₁ and F₄₄/F₁₁, their absolute differences are generally smaller than 0.01, which validates the effectiveness of the spheroid-equivalent scheme and the modeling accuracy of the DVIIM T-matrix method.

6. Discussion of model efficiency

To further investigate the modeling efficiency of the DVIIM T-matrix method, the computational time required by IIM and DVIIM T-matrix methods are also compared for particles with different sizes and shapes. In these simulations, the light wavelength is set as 0.5µm, and the scatters are set as spheroidal particles. All the scattering processes are simulated on the same computer (32bit 3.1 GHz, Windows operation system), and the computational time of the IIM and DVIIM T-Matrix codes is presented in Table 7 and Table 8.

Table 7. The computational time needed for DVIIM and IIM T-Matrix models for the particles with different aspect ratios (b = 3µm)

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Table 8. The computational time needed for the traditional and improved IIM T-Matrix models for particles with different sizes (a/b = 2/3)

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As can been seen, on the whole, the computational time needed for DVIIM T-matrix method is less than that required by the IIM T-matrix method. With the increasing of the aspect ratio (a/b), the computational efficiency of DVIIM T-matrix method becomes higher as well. It can also be found that with the increasing of particle size, the improvement of the computational efficiency becomes much more remarkable. The discussion above indicates that the Spheroid-Equivalent Scheme is effective, and can improve the computational efficiency notably.

7. Conclusion

With the development of the computational technology, the IIM T-matrix method has become one of the most promising models for the light scattering simulation. Because this model is based on volume-integral form of the Helmholtz equation, it is suitable for the particles with arbitrary shapes, which greatly expands the application scope of the T-matrix method. However, since the IIM T-matrix method should update the T-matrix layer by layer, how to improve the computational efficiency and stability has become a key problem to be solved. To further improve the modeling efficiency of the IIM T-matrix method, a Dimension-Variable Invariant Imbedding (DVIIM) T-matrix method is designed in this paper, in which the dimension of the T matrix can be changed according to the actual size and shape in the recurrent iteration. To optimally determine the expansion order of the electric field, the Spheroid-Equivalent Scheme is proposed. To validate the computational accuracy of DVIIM T-matrix method, its calculation results are compared with those of EBCM, traditional IIM T-matrix method and DDASCAT. Moreover, the calculation efficiency of the model is also quantitatively compared with the traditional IIM T-matrix method. The main conclusion of this paper can be drawn as follows:

(1) The calculation results obtained by the DVIIM T-matrix method show a good agreement with the EBCM, DDASCAT and IIM T-matrix method, which indicates that the computational precision of DVIIM T-matrix method is not decreased notably though the dimension of the T matrix is reduced in the early recurrent iterations.
(2) The modeling efficiency of DVIIM T-matrix method is higher than that of traditional IIM–matrix method. With the increasing of particle size and aspect ratio, the computational time consumed can be cut down more notably.

It should also be noted that, the expansion order of the electric field and NDGS is determined from the spheroid-equivalent scheme (SES), namely, we taken the particles in the spherical shells equivalent as spheroid, therefore it mainly suitable for convex particles with smooth surface, while for particles with very complex geometry or sharp edges, its computational accuracy might not be so high as the particles calculated in this paper. Besides, because the spheroid-equivalent scheme is established based on the convergence of the EBCM T-matrix method, the DVIIM T-matrix computational method might not be very good at light scattering of particles with very large axial ratio.

Funding

National Natural Science Foundation of China (42175154, 62105367); Natural Science Foundation of Hunan Province (2020JJ4662, 2021JJ40666).

Acknowledgments

Thanks for the Tmatrix code provided by Prof. M. I. Mishchenko. Thanks for the DDSCAT code provided by Prof. Bruce T. Draine and Piotr J. Flatau.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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θ/deg	Model	F₁₁	F₁₂	F₃₄	F₄₄
0	IIM	2906.8163	0.0000	0.0000	2906.5544
	DVIIM	2904.5799	0.0000	0.0000	2904.3507
	difference	-2.2364	0.0000	0.0000	-2.2037
30	IIM	6.3846	0.1407	-0.1561	6.3441
	DVIIM	6.3892	0.1564	-0.1647	6.3468
	difference	0.0046	0.0157	-0.0086	0.0027
60	IIM	0.4446	0.0149	-0.0240	0.4053
	DVIIM	0.4449	0.0144	-0.0219	0.4449
	difference	0.0003	-0.0005	0.0021	0.0396
90	IIM	0.4228	-0.0277	-0.0928	0.3414
	DVIIM	0.4221	-0.0258	-0.0937	0.3376
	difference	-0.0007	0.0019	-0.0009	-0.0038
120	IIM	0.3610	-0.0613	-0.0505	0.2346
	DVIIM	0.3612	-0.0620	-0.0511	0.2304
	difference	0.0002	-0.0007	-0.0006	-0.0042
150	IIM	0.1567	-0.0082	-0.0082	0.0411
	DVIIM	0.1567	-0.0086	-0.0080	0.0397
	difference	0.0000	-0.0004	0.0002	-0.0014

(a, b)/µm	Model	C_ext	C_ab	C_sc
(1, 2)	IIM	14.73710	0.23213	14.50497
	DVIIM	14.74426	0.23204	14.51221
	difference	0.04857	-0.03627	0.04994
(2, 4)	IIM	52.42747	1.83024	50.59723
	DVIIM	52.40656	1.82043	50.58613
	difference	-0.03988	-0.00980	-0.01110
(5, 7)	IIM	207.73997	18.04309	189.69688
	DVIIM	207.73908	18.07180	189.66728
	difference	-0.00042	0.15910	-0.01560

θ/deg	Model	F₁₁	F₁₂	F₃₄	F₄₄
0	IIM	105.0270	0.0000	0.0000	104.9918
	DVIIM	104.6016	0.0000	0.0000	104.5702
	difference	0.4254	0.0000	0.0000	0.4216
30	IIM	2.5217	0.0489	-0.1506	2.5040
	DVIIM	2.5165	0.0517	-0.1587	2.4986
	difference	-0.0052	0.0028	-0.0081	-0.0054
60	IIM	0.2905	-0.0222	-0.0332	0.2807
	DVIIM	0.2907	-0.0214	-0.0366	0.2805
	difference	0.0002	0.0008	-0.0034	-0.0002
90	IIM	0.0906	-0.0309	-0.0183	0.0733
	DVIIM	0.0903	-0.0303	-0.0181	0.0732
	difference	-0.0003	0.0006	0.0002	-0.0001
120	IIM	0.0362	-0.0123	-0.0126	0.0215
	DVIIM	0.0358	-0.0121	-0.0128	0.0217
	difference	-0.0004	0.0002	-0.0002	0.0002
150	IIM	0.0382	-0.0105	-0.0016	0.0046
	DVIIM	0.0376	-0.0101	-0.0019	0.0044
	difference	-0.0006	0.0004	-0.0003	-0.0002

(D, L)/µm	Model	C_ext	C_ab	C_sc
(2, 2)	IIM	9.73318	0.17953	9.55364
	DVIIM	9.71009	0.17915	9.53093
	Difference/%	-0.23730	-0.21516	-0.23771
(4, 4)	IIM	40.47588	1.42837	39.04750
	DVIIM	40.32274	1.42391	38.89882
	Difference/%	-0.37833	-0.00445	-0.14867
(8, 8)	IIM	167.38404	10.87531	156.50873
	DVIIM	166.21868	10.78675	155.43193
	Difference/%	-0.69622	-0.81436	-0.68801

θ/deg	Model	F₁₁	F₁₂	F₃₄	F₄₄
0	IIM	104.1718	0.0000	0.0000	104.1259
	DVIIM	104.0669	0.0000	0.0000	104.0338
	difference	-0.1049	0.0000	0.0000	-0.0921
30	IIM	1.7677	0.1018	-0.1178	1.7344
	DVIIM	1.7648	0.1015	-0.1174	1.7339
	difference	-0.0029	-0.0003	0.0004	-0.0005
60	IIM	0.2182	-0.0029	-0.0315	0.1947
	DVIIM	0.2185	-0.0029	-0.0309	0.1939
	difference	0.0003	0.0000	0.0006	-0.0008
90	IIM	0.0648	-0.0123	-0.0093	0.0502
	DVIIM	0.0646	-0.0122	-0.0093	0.0504
	difference	-0.0002	0.0001	0.0000	0.0002
120	IIM	0.0312	-0.0117	-0.0080	0.0129
	DVIIM	0.0310	-0.0117	-0.0081	0.0127
	difference	-0.0002	0.0000	-0.0001	-0.0002
150	IIM	0.0276	-0.0066	-0.0030	0.0029
	DVIIM	0.0274	-0.0065	-0.0031	0.0029
	difference	-0.0002	0.0001	-0.0001	0.0000

Dimension-variable invariant imbedding (DVIIM) T-matrix computational method for the light scattering simulation of atmospheric nonspherical particles

Abstract

1. Introduction

2. Basic principles of DVIIM T-matrix method

3. Optimal determination of the T-matrix dimension and Gaussian integral points by spheroidal equivalent scheme (SES)

4. Implementation of DVIIM T-matrix computational code

5. Model validation and results analysis

5.1 Spheroidal particle case

5.2 Cylinder particle case

5.3 Hexagonal ice crystals case

6. Discussion of model efficiency

7. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Tables (8)

Equations (12)

Optics Express

m	Size parameter	a/b = 0.4		a/b = 0.5		a/b = 0.7		a/b = 0.9
m	Size parameter	N_max	NDGS	N_max	NDGS	N_max	NDGS	N_max	NDGS
1.1 + 0.0005i	10	23	4	20	4	19	4	19	4
	20	43	4	37	4	32	4	32	4
	30	65	8	56	4	44	4	43	4
	50	80	12	76	8	72	4	65	4
1.3 + 0.0005i	10	23	6	24	4	19	4	19	4
	20	43	8	41	4	35	4	32	4
	30	66	10	57	8	48	4	43	4
	50	89	12	86	10	79	4	65	4
1.5 + 0.0005i	10	32	8	28	4	22	4	19	4
	20	45	8	50	4	39	4	33	4
	30	72	10	62	8	54	4	43	4
	50	94	12	88	10	85	4	66	4

Aspect ratio	Number of the threads	Computational time T₀ of the traditional IIM T-Matrix code /s	Computational time T₁ of the DVIIM T-Matrix code /s	(T₁- T₀)/ T₀× 100%
0.5	4	36221	27053	25.3112
0.7	4	22989	19889	13.4847
0.9	4	7558	7017	7.1579