Invariant Imbedding T-matrix Method Research Articles

On the relation between ice-crystal scattering phase function at 180° and particle size: implication to lidar-based remote sensing of cirrus clouds.

The lack of knowledge of the relation between a lidar backscatter signal and particle size makes it challenging to retrieve ice-cloud particle size from spaceborne lidar observations. This study employs a synergistic combination of the state-of-the-art invariant imbedding T-matrix method and the physical geometric-optics method (PGOM) to investigate the relation between the ice-crystal scattering phase function at 180° (P11(180°)) and particle size (L) for typical ice-crystal shapes. In particular, the P11(180°) -L relation is quantitatively analyzed. The dependence of the P11(180°) -L relation on particle shape can be used with spaceborne lidar observations to detect ice-cloud particle shapes.

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

The invariant imbedding (IIM) T-matrix method has shown great potential in light scattering field. However, the T-matrix need to be calculated through the matrix recurrence formula derived from the Helmholtz equation, thus its computational efficiency is much lower than Extended Boundary Condition Method (EBCM). To alleviate this problem, the Dimension-Variable Invariant Imbedding (DVIIM) T-matrix method is presented in this paper. Compared with the traditional IIM T-matrix model, the dimensions of the T-matrix and relevant matrices are gradually increasing as the iteration performed step by step, thus the unnecessary operations of large matrices can be avoided in early iterations. To optimally determine the dimension of these matrices in each iterative calculation, the spheroid-equivalent scheme (SES) is also proposed. The effectiveness of the DVIIM T-matrix method is validated from the modeling accuracy and calculation efficiency. The simulation results show that compared with traditional T-matrix method, its modeling efficiency can be improved notably, especially for the particles with large size and aspect ratio, where for the spheroid with a aspect ratio of 0.5, the computational time is cut down by 25%. Though the dimension of the T matrix is cut down in the early iterations, the computational precision of DVIIM T-matrix model is not decreased notably, and a good agreement is achieved between the calculation results of DVIIM T-matrix method, IIM T-matrix method and other well-validated models (like EBCM and DDACSAT), where the relative errors of the integral scattering parameters (e.g., extinction, absorption, scattering cross sections) are generally less than 1%.

Improvement and application of linearized invariant imbedding T-matrix scattering method

The Jacobians of particle scattering properties with respect to aspect ratio and effective radius (re) using the invariant imbedding T-matrix method (IITM) are significantly improved by circumventing the divergence of derivatives. The effective radius re is used in three definitions, which are rV, the equal-volume spherical radius, rS, the equal-surface-area spherical radius, and rVS, the ratio of three times the volume to surface area, respectively. The improved IITM and their Jacobians are applied to discuss the discrepancies between the three different definitions. The analytical Jacobians with respect to aspect ratios and effective radii associated with three effective radii are verified using the finite-difference method. Jacobians with respect to different effective radii are proportional since the ratios of different effective radii are only related to the aspect ratios. Jacobians with respect to aspect ratios are extremely sensitive to the definitions of effective radius.

Analytical Jacobians of single scattering optical properties using the invariant imbedding T-matrix method

Integrated and differential optical properties of a single particle, such as the scattering, absorption, and extinction cross sections, single scattering albedo, asymmetry factor, and scattering phase matrix, are derived from electromagnetic scattering theory. This process depends on microphysical inputs which include particle shape, refractive index, aspect ratio, and size parameter. In this work, we use the invariant imbedding T-matrix method (IITM) to derive analytic expressions for Jacobians of these optical properties with respect to the input parameters. These IITM-derived Jacobians for spheroids, cylinders, and hexagonal prisms are validated by comparison with results calculated with the extended boundary condition method (EBCM) and further validated using finite-difference estimates. We examine the dependencies of these Jacobians as functions of the input microphysical parameters, focusing again on spheroids, cylinders, and hexagonal prisms.

Efficient design of the realization scheme of the invariant imbedding (IIM) T-matrix light scattering model for atmospheric nonspherical particles

The invariant imbedding T-matrix method has been recognized as one of the rigorous computational tools to simulate the light scattering by atmospheric nonspherical particles after the efficient implementation by L. Bi and P. Yang. Compared with the T-matrix model based on the Extended Boundary Condition Method (“EBCM”), it is suitable for the light scattering simulation of particles with any shapes. However, since the T-matrix must be updated in each iterative process, its computational efficiency is relatively lower than EBCM, especially for the particles with large sizes and large aspect ratios. To this end, an efficient realization scheme of the IIM T-matrix method is proposed in this paper. The principle and the structure of our IIM T-matrix code is firstly introduced, and then the symmetric properties of the U-matrix (an important matrix in the IIM T-matrix method) is derived; to simplify the iteration of T-matrix calculation, the implicit matrix inversion scheme is also proposed. To validate the effectiveness of these improvements, the computational efficiency and accuracy is systematically investigated for the particles with different shapes and sizes. The simulation results indicate that the computational time is reduced notably after the model is improved by the symmetry of U-matrix and the implicit matrix inversion scheme, where for the spheroid with (a, b) = (2.5 µm, 3.0 µm) at λ=0.5 µm (the particle is treated as complex geometries without any symmetry), the computational efficiency is improved by nearly 40%. Excellent agreement is also achieved between the scattering parameters obtained by the IIM T-matrix method and the well-test scattering models (such as BECM, DDASCAT, MRTD and IGOA, etc.), indicating that the IIM T-matrix method can simulate the light scattering by nonspherical particles effectively.

ON THE CONVERGENCE OF NUMERICAL COMPUTATIONS FOR BOTH EXACT AND APPROXIMATE SOLUTIONS FOR ELECTROMAGNETIC SCATTERING BY NONSPHERICAL DIELECTRIC PARTICLES (INVITED REVIEW)

We summarize the size parameter range of the applicability of four light-scattering computational methods for nonspherical dielectric particles. These methods include two exact methods - the extended boundary condition method (EBCM) and the invariant imbedding T-matrix method (II-TM) and two approximate approaches - the physical-geometric optics method (PGOM) and the improved geometric optics method (IGOM). For spheroids, the single-scattering properties computed by EBCM and II-TM agree for size parameters up to 150, and the comparison gives us confidence in using IITM as a benchmark for size parameters up to 150 for other geometries (e.g., hexagonal columns) because the applicability of II-TM with respect to particle shape is generic, as demonstrated in our previous studies involving a complex aggregate. This study demonstrates the convergence of the exact II-TM and approximate PGOM solutions for the complete set of single-scattering properties of a nonspherical shape other than spheroids and circular cylinders with particle sizes of ~ 48λ(size parameter ~150), specifically a hexagonal column with a length size parameter of kL = 300 where k = 2π/λ and L is the column length. IGOM is also quite accurate except near the exact 180°backscattering direction. This study demonstrates that a synergetic combination of the numerically-exact II-TM and the approximate PGOM can seamlessly cover the entire size parameter range of practical interest. To demonstrate the applicability of the approach, we compute the optical properties of dust particles and demonstrate a downstream application to the retrieval of dust aerosol optical thickness and effective particle size from satellite polarimetric observations.

Impact of calcification state on the inherent optical properties of Emiliania huxleyi coccoliths and coccolithophores

Understanding the inherent optical properties (IOPs) of coccoliths and coccolithophores is important in oceanic radiative transfer simulations and remote sensing implementations. In this study, the invariant imbedding T-matrix method (II-TM) is employed to investigate the IOPs of coccoliths and coccolithophores. The Emiliania huxleyi (Ehux) coccolith and coccolithophore models are built based on observed biometric parameters including the eccentricity, the number of slits, and the rim width of detached coccoliths. The calcification state that specifies the amount of calcium of a single coccolith is critical in the determination of the size–volume/mass relationship (note, the volume/mass of coccoltihs at different calcification states are different although the diameters are the same). The present results show that the calcification state, namely, under-calcification, normal-calcification, or over-calcification, significantly influences the backscattering cross section and the phase matrix. Furthermore, the linear depolarization ratio of the light scattered by coccoliths is sensitive to the degree of calcification, and provides a potentially valuable parameter for interpreting oceanic remote sensing data. The phase function of an ensemble of randomly oriented coccolithophores has a similar pattern to that of individual coccoliths, but the forward scattering is dominant in the coccolithophores due to the large geometric cross sections. The linear depolarization ratio associated with coccolithophores is found to be larger than that for coccoliths as polarization is more sensitive to multiple scattering than the phase function. The simulated coccolithophore phase matrix numerical results are compared with laboratory measurements. For scattering angles larger than 100°, an increase of the phase function with respect to the scattering angle is confirmed based on the present coccolithophore model while the spherical approximation fails.

Assessment of the accuracy of the conventional ray-tracing technique: Implications in remote sensing and radiative transfer involving ice clouds

A fundamental problem in remote sensing and radiative transfer simulations involving ice clouds is the ability to compute accurate optical properties for individual ice particles. While relatively simple and intuitively appealing, the conventional geometric-optics method (CGOM) is used frequently for the solution of light scattering by ice crystals. Due to the approximations in the ray-tracing technique, the CGOM accuracy is not well quantified. The result is that the uncertainties are introduced that can impact many applications. Improvements in the Invariant Imbedding T-matrix method (II-TM) and the Improved Geometric-Optics Method (IGOM) provide a mechanism to assess the aforementioned uncertainties. The results computed by the II-TM+IGOM are considered as a benchmark because the II-TM solves Maxwell׳s equations from first principles and is applicable to particle size parameters ranging into the domain at which the IGOM has reasonable accuracy. To assess the uncertainties with the CGOM in remote sensing and radiative transfer simulations, two independent optical property datasets of hexagonal columns are developed for sensitivity studies by using the CGOM and the II-TM+IGOM, respectively. Ice cloud bulk optical properties obtained from the two datasets are compared and subsequently applied to retrieve the optical thickness and effective diameter from Moderate Resolution Imaging Spectroradiometer (MODIS) measurements. Additionally, the bulk optical properties are tested in broadband radiative transfer (RT) simulations using the general circulation model (GCM) version of the Rapid Radiative Transfer Model (RRTMG) that is adopted in the National Center for Atmospheric Research (NCAR) Community Atmosphere Model (CAM, version 5.1). For MODIS retrievals, the mean bias of uncertainties of applying the CGOM in shortwave bands (0.86 and 2.13μm) can be up to 5% in the optical thickness and as high as 20% in the effective diameter, depending on cloud optical thickness and effective diameter. In the MODIS infrared window bands centered at 8.5, 11, and 12μm, biases in the optical thickness and effective diameter are up to 12% and 10%, respectively. The CGOM-based simulation errors in ice cloud radiative forcing calculations are on the order of 10Wm−2.

Accurate simulation of the optical properties of atmospheric ice crystals with the invariant imbedding T-matrix method

The invariant imbedding T-matrix method (II-TM) is employed to compute the optical properties of randomly oriented ice crystals of various shapes including hexagonal columns, hollow columns, droxtals, bullet rosettes and aggregates. The II-TM is shown to be numerically stable and capable of obtaining the single-scattering properties of hexagonal ice crystals with size parameters up to 150. The 22° and 46° halo peaks in the phase function of compact hexagonal ice crystals begin to emerge at a size parameter of approximately 80 and tend to become insensitive to particle size as the corresponding size parameter approaches 150. Furthermore, the II-TM solutions are shown to be in agreement with their counterparts based on the discrete dipole approximation (DDA) method and the pseudo-spectral time-domain (PSTD) method. In addition, the accuracy of the improved geometric-optics method (IGOM) is examined for randomly oriented hexagonal ice crystal cases over a wide size-parameter range from the resonant to geometric-optics regimes. The II-TM is also used to study the effects of particle surface roughness and internal inclusions on the single-scattering properties of ice particles.

Modeling of light scattering by biconcave and deformed red blood cells with the invariant imbedding T-matrix method

The invariant imbedding T-matrix method (II-TM) is employed to simulate the optical properties of normal biconcave and deformed red blood cells (RBCs). The phase matrix elements of a RBC model computed with the II-TM are compared with their counterparts computed with the discrete-dipole approximation (DDA) method. As expected, the DDA results approach the II-TM results with an increase in the number of dipoles per incident wavelength. Computationally, the II-TM is faster than the DDA when multiple RBC orientations are considered. For a single orientation, the DDA is comparable with or even faster than the II-TM because the DDA efficiently converges for optically soft particles; however, the DDA method demands significantly more computer memory than the II-TM. After the applicability of the II-TM is numerically confirmed, a comparison is conducted of the optical properties of oxygenated and deoxygenated RBCs and of normal and deformed RBCs. The spectral variations of RBCs' optical properties are investigated in the wavelength range from 0.25 to 1.0 μm. Furthermore, the statistically averaged phase matrix of spheres and biconcave RBCs are compared. Conducted numerical simulations suggest the applicability of the II-TM for the inverse light scattering analysis and radiative transfer simulations in blood.