大气辐射传输模拟中,强各向异性散射相函数的精确计算是关键难点。传统δ-M截断方法在处理液态云、气溶胶及冰晶等具有尖锐前向峰的粒子散射时易产生振荡,导致计算精度下降。δ-M+方法在δ-M基础上通过以高斯加权的δ*函数替代传统狄拉克函数,使高阶勒让德系数平滑衰减,从而有效抑制非物理振荡,但δ-M+方法仅应用于标量辐亮度计算。本文将δ-M+方法推广至矢量辐射传输模式计算,并在 SOSVRT 模式中实现了基于矩阵形式和比值缩放的两类方案。基于球形气溶胶、液态水云滴以及非球形冰晶的散射相矩阵重建应用于全矢量辐射传输仿真试验,结果表明δ-M+方法能在保持偏振精度的同时显著提升模拟效率,与传统δ-M方法相比,在达到 1% 辐亮度误差条件下所需流数约其三分之一。研究表明,在勒让德展开系数满足单调递减条件的前提下,δ-M+方法为强前向散射条件下的矢量辐射传输提供了高效且稳定的数值计算方案。

This study addresses a critical challenge in the theory of atmospheric radiative transfer. The strongly anisotropic scattering of large atmospheric particles, such as aerosols, water clouds and ice crystals, continues to hinder the accurate simulation of radiative transfer. Although the δ-M truncation method is widely used to address the sharp forward scattering peak, it can introduce oscillations in the expansion of higher-order Legendre polynomials and reduce accuracy when the peak of the phase function is extremely sharp. The recently proposed δ-M+ method mitigates this issue by replacing the Dirac delta function with a Gaussian-weighted representation. However, current studies of this method have focused exclusively on scalar radiative transfer, whereas actual scattered radiance often includes polarization. This prevents the method from being applied to polarized simulations, which are essential for high-precision remote sensing. This study extends the δ-M+ method to vector radiative transfer, evaluating its numerical accuracy and efficiency under highly anisotropic scattering conditions. The scalar δ-M+ formulation is generalized to scattering phase matrices using two independent vectorization schemes. The first scheme yields a complete analytical matrix expansion by applying Gaussian-weighted Legendre coefficients to each element of the phase matrix. The second scheme uses a ratio-preserving approach, scaling the remaining matrix elements according to the truncated F11 component. Both schemes are incorporated into the SOSVRT (Successive Orders of Scattering Vector Radiative Transfer) model. To evaluate the accuracy of the reconstructions, spherical aerosol and cloud-droplet phase matrices are generated using Mie theory. Non-spherical ice-crystal matrices are obtained from a high-resolution database. Benchmark solutions are produced using 360-stream polarized radiative transfer simulations. Then, comparisons are performed across a broad range of stream numbers to quantify errors in radiance, degree of linear polarization (DoLP), and computational efficiency. Experiments show that the δ-M+ method significantly improves the reconstruction of the phase matrix of strongly forward-scattering particles" phase functions using a few Legendre moments. δ-M produces pronounced oscillations, discontinuities, and incorrect angular curvature. For both aerosol and cloud scattering cases, the δ-M+ method reduces reconstruction errors across all matrix elements. For highly anisotropic ice crystals, the method preserves the extreme forward peak while avoiding the nonphysical artifacts that arise from the δ-M truncation. In radiative transfer simulations, the δ-M+ method achieves much higher accuracy with far fewer streams. Reaching 1% radiance error and 0.01 absolute DoLP error requires one-third the number of streams of the δ-M method. The analytical δ-M+ formulation maintains stable and accurate Stokes Q, U, and V components. In contrast, the ratio-based approach introduces noticeable biases in the polarized components transmitted through optically thick clouds. CPU time tests further indicate that δ-M+ reduces computational cost in most cases and is markedly faster than δ-M at high stream counts in cloud scenarios. This study clarifies how the δ-M+ truncation behaves in polarized radiative transfer under strongly forward-scattering conditions and offers a feasible approach for handling highly anisotropic scattering in vector models. Future work will incorporate forward-peak angle correction strategies to refine the method further.