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.

A physical retrieval algorithm for cloud microphysical properties with a fully linearized forward radiative transfer model was established in this study. The retrieval efficiency is at least tripled and the retrieval accuracy is significantly improved due to the accurate Jacobian in the linearized forward model. Moreover, direct uncertainty estimation and evaluation parameters based on the optimal estimation theory enhance the selections of observations. The Himawari 8 observation were used to retrieve cloud optical thickness (COT), cloud effective radius (CER) and cloud effective variance (CEV). The retrieved cloud properties are in great agreement with the official products, with correlation coefficients of 0.940 and 0.886 for COT and CER, respectively. More importantly, the unreasonable small CER values around the edge pixels of clouds in the official products were detected, and given smooth edges with relatively large uncertainties. Generally, this linearized physical retrieval algorithm is accurate, flexible and multi-scenario applicable.

The algorithm based on the physical-geometric optics method is developed to compute the linearization of single-scattering properties, such as extinction, absorption and scattering cross-sections, and the scattering phase matrix. The algorithm can be applied to any convex facet particles, where a new beam-splitting technique is employed. With the introduction of the winding number method, beams incident on multiple facets can be precisely divided into independent parts that are incident on single facets. The linearization algorithm is verified by the finite-difference method using the regular hexagonal prism. The sensitivities of single-scattering properties with respect to size, aspect ratio, and refractive index are discussed.

The linearized invariant-imbedding T-matrix method (LIITM) and linearized physical-geometric optics method (LPGOM) were applied on regular hexagonal prisms from small to large sizes to obtain the scattering properties and their partial derivatives. T-matrices and their derivatives from the LIITM are presented and discussed in the expansion order, where the minor diagonal elements are dominant. The simulation results of single-scattering properties and their corresponding linearization from both methods are compared. The mutual agreements can be treated as further verification of both linearized methods. Using extinction efficiency as the criterion, the LPGOM are convergent at the LIITM for the particle size parameter larger than 130 with a relative difference of less than 1%, with errors of about 3% and 5% for particle sizes of 50 and 30, respectively. The capability and convergence of the LIITM and LPGOM are discussed in detail based on linearized properties.

The scalar radiative transfer equation in the presence of thermal radiation source is solved in detail, using the adding-doubling method; Planck functions within any given layer are assumed to possess constant, linear, or exponential parameterizations with optical thickness. The radiance profile in any zenith direction is calculated directly in terms of matrix inversions. The inputs to the model are the inherent optical properties (layer total single-scattering albedos, scattering phase functions, and optical thickness) along with temperature and altitude profiles, and the top of the atmosphere and ground surface boundary conditions. The algorithm is implemented in a state-of-the-art MATLAB program, with the cosmic microwave background as the source at the upper boundary and a Lambertian surface reflection at the lower boundary. The simulations are validated against the VLIDORT discrete ordinate radiative transfer model. Results are compared in detail for cases with linear and exponential Planck function parameterizations.