Raman scattering in the Earth’s atmosphere is caused predominantly by its most abundant molecular components, N${}_2$ and O${}_2$. Modeling Raman scattering in the atmosphere is challenging for several reasons: the first challenge is the large number of molecular constants required for the computation of the wavelength/wavenumber shifts due to inelastic scattering by a given molecular species, and the corresponding variations in the scattering cross-sections and depolarization ratios depending on whether the scattering is rotational, vibrational or rovibrational. This is compounded by the lack of a consistent notation and uniform formalism across the seminal theoretical and experimental works that form the basis of our modeling effort.

This work unifies the formalisms of Long (1977) and Chandrasekhar (1950), in order to make the quantum theoretical work of (Placzek, 1934) amenable to radiative transfer modeling using familiar quantities like the scattering cross-section and the scattering matrix. We also replace the form of key parameters like mean polarizability, $\alpha$, the polarization anisotropy, $\gamma$, and their derivatives as introduced by Long (1977) for arbitrarily shaped molecules with the simplified formulations of Buldakov et al. (1996, 2000) for diatomic molecular species. The resulting scattering cross-sections and phase matrices for N${}_2$ and O${}_2$ are presented.

In a companion paper, we provide the generalized radiative transfer formulation for the first fully polarized, exact simulations of inelastic scattering using the matrix operator method based RTM vSmartMOM, with optimizations for GPU runs to allow computationally competitive multi-wavelength radiative transfer computations of inelastic scattering. We subsequently use the formalism developed here to simulate key phenomena involving inelastic scattering in the Earth’s atmosphere, viz., the spectral response to a monochromatic light source as in Raman Lidar measurements, the Ring effect, the ghosting of Fraunhofer lines due to vibrational Raman scattering and the spectral impact of inelastic scattering on the O${}_2$ A-band.

地球大气中的拉曼散射主要是由其最丰富的分子成分 N 引起的${}_2$和 O${}_2$. 由于几个原因，对大气中的拉曼散射进行建模具有挑战性：第一个挑战是计算由于给定分子种类的非弹性散射引起的波长/波数偏移所需的大量分子常数，以及散射交叉的相应变化-截面和去极化比取决于散射是旋转的、振动的还是振动的。由于在构成我们建模工作基础的开创性理论和实验工作中缺乏一致的符号和统一的形式主义，这使情况更加复杂。

这项工作统一了 Long (1977) 和 Chandrasekhar (1950) 的形式，以使 (Placzek, 1934) 的量子理论工作能够使用散射截面和散射矩阵等熟悉的量进行辐射传递建模。我们还替换了平均极化率等关键参数的形式，$\alpha$，偏振各向异性，$\gamma$，以及 Long (1977) 引入的用于任意形状分子的衍生物，以及 Buldakov 等人的简化公式。(1996, 2000) 用于双原子分子物种。得到的 N 的散射截面和相位矩阵${}_2$和 O${}_2$被提出。

在一篇配套论文中，我们使用基于 RTM vSmartMOM 的矩阵算子方法为第一次完全极化的非弹性散射精确模拟提供了广义辐射传递公式，并优化了 GPU 运行以允许计算具有竞争力的非弹性散射的多波长辐射传递计算. 我们随后使用此处开发的形式来模拟涉及地球大气中非弹性散射的关键现象，即拉曼激光雷达测量中对单色光源的光谱响应、环形效应、由于振动拉曼散射引起的弗劳恩霍夫线重影以及非弹性散射对 O 的光谱影响${}_2$A波段。