Journal of Quantitative Spectroscopy and Radiative Transfer
Volume 111, Issue 6, April 2010, Pages 878-894
Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere–ocean system
Abstract
A vector radiative transfer model has been developed for a coupled atmosphere–ocean system. The radiative transfer scheme is based on the discrete ordinate and matrix operator methods. The reflection/transmission matrices and source vectors are obtained for each atmospheric or oceanic layer through the discrete ordinate solution. The vertically inhomogeneous system is constructed using the matrix operator method, which combines the radiative interaction between the layers. This radiative transfer scheme is flexible for a vertically inhomogeneous system including the oceanic layers as well as the ocean surface. Compared with the benchmark results, the computational error attributable to the radiative transfer scheme has been less than 0.1% in the case of eight discrete ordinate directions. Furthermore, increasing the number of discrete ordinate directions has produced computations with higher accuracy. Based on our radiative transfer scheme, simulations of sun glint radiation have been presented for wavelengths of 670 nm and 1.6 μm. Results of simulations have shown reasonable characteristics of the sun glint radiation such as the strongly peaked, but slightly smoothed radiation by the rough ocean surface and depolarization through multiple scattering by the aerosol-loaded atmosphere. The radiative transfer scheme of this paper has been implemented to the numerical model named Pstar as one of the OpenCLASTR/STAR radiative transfer code systems, which are widely applied to many radiative transfer problems, including the polarization effect.
Introduction
The top of the Earth's atmosphere is irradiated by solar radiation, emitting thermal infrared radiation to space. That radiation interacts with molecules and particles that are present in the terrestrial atmosphere, and reacts also with the ground surface. Especially, the solar radiation is unpolarized at the top of the atmosphere, but the state of polarization and the radiant intensity are changed because of the radiative transfer process. The process, including the polarization effect, follows the radiative transfer equation [1].
The discrete ordinate approach is one technique to solve the radiative transfer equation and obtains the radiation field for given boundary conditions. By discretizing the angular coordinates of the phase function and radiance with respect to the zenith angle, the radiative transfer equation is reduced to a linear differential matrix equation. The ways of solving the matrix equation have been studied for the case of multilayered system with underlying ground surface [2], [3], [4]. Based on the solutions, the atmospheric radiative transfer (RT) models have been developed for several purposes such as the radiation budget and remote sensing studies. The scalar RT code based on the formulations of Nakajima–Tanaka (N–T) [4], [5], [6] is a numerical RT model constructed using the discrete ordinate method and the matrix operator method. In the N–T model, a discrete ordinate solution is obtained for each plane-parallel layer; then the matrix operator method is applied to all layers to obtain the radiation field of the multilayered system. By formulating the ground surface as a pseudo-layer in the matrix operator method, the discrete ordinate solution of the N–T formulation is applicable to several different surface conditions.
The N–T model has been used for several studies of radiative forcing [7], [8] as well as satellite-based and ground-based remote sensing [9], [10], [11]. However, because a new generation of earth observation satellites can conduct polarization measurements, the demand for vector RT model including the polarization effect has grown in recent years. For instance, the greenhouse gases observing satellite (GOSAT, http://www.gosat.nies.go.jp), which was launched for observing greenhouse gases such as carbon dioxide and methane, measures the near infrared radiation of wavelength band at 0.76, 1.6, and 2.0 μm using the polarization-sensitive instrument. Especially, it is noteworthy that GOSAT measures the sun glint radiation to observe greenhouse gases over the ocean. Furthermore, as a part of the global change observation mission (GCOM) of the Japan Aerospace Exploration Agency (JAXA), GCOM-C (climate) satellite is planning to conduct the polarization measurement with the second generation global imager (SGLI) [12]. The polarization measurement of SGLI is used for observing land aerosols and vegetation.
As described in this paper, the problem we particularly examine is extension of the N–T formulation to express the polarized radiation field emerging from and traveling in the multilayered system. Furthermore, we develop a vector RT model for coupled atmosphere–ocean system, which consists of the vertically inhomogeneous atmosphere and ocean, as illuminated by solar radiation from the top of the atmosphere.
Section snippets
Vector radiative transfer equation
The radiation field including the polarization effect is described using the Stokes parameters I, Q, U, and V, which are defined with respect to a certain plane of reference and compose the Stokes vector aswhere superscript T denotes the transpose operation. If we consider a stratified plane-parallel medium, then the Stokes vector of the system follows a vector RT equation [1], [13],where τ is the optical depth
Reflection and transmission model of the ocean surface
When we seek the RT solution for the atmosphere–ocean system, we must formulate the reflection and transmission functions for the ocean surface. Although the scattering of the radiation by a flat water surface can be described by the Snell–Fresnel law, the influence of the wind-generated waves on the radiation transfer is a complicated matter that many authors have studied (e.g., [18], [19], [20], [21], [22], [23]). In this section, we briefly describe a rough ocean surface model that includes
Discrete ordinate equation for a homogeneous layer
We next seek the solution of the vector RT equation by application of the discrete ordinate method to a single homogeneous layer. Given the double-Gaussian quadrature as defined by points ±μi and weights wi over the range of , the vector RT equation for the m-th order Fourier component (Eq. (29)) can be written in discrete ordinate form as
Angular interpolation by source function integration
By the discrete ordinate solutions, the Stokes vector of the diffused radiation field is obtained with respect to the discrete ordinate directions. However, we often require the Stokes vector of a specific emergent angle, especially when the RT model is applied to remote sensing data analysis. The analytical angular interpolation scheme, which is more stable and accurate than standard numerical interpolation schemes, has been used in the scalar RT case of the ordinary discrete ordinate method
Verification of a radiative transfer scheme
Based on the RT formulations described in this paper, we developed the Pstar code: a numerical model for simulations of radiation field in the coupled atmosphere–ocean system. The Pstar consists of a RT code and an initialization code to calculate the input parameters for the RT calculation on physical databases. The RT scheme of Pstar can calculate the full Stokes parameters at any interface between the homogeneous layers as well as the top of the atmosphere, and also at any emergent direction
Concluding remarks
This paper has presented a vector RT model for a coupled atmosphere–ocean system based on the discrete ordinate and matrix operator methods. The reflection and transmission matrices and source vectors are obtained through the discrete ordinate solution for each atmospheric or oceanic layer. A vertically inhomogeneous system is constructed using the matrix operator method, which combines the interaction between layers. This RT formulation is flexible for a vertically inhomogeneous system
Acknowledgements
We would like to thank Hiroshi Murakami (Earth Observation Research Center, Japan Aerospace Exploration Agency) for some helpful feedback about the use of the Pstar code. We also acknowledge Ryoichi Imasu (Center for Climate System Research, The University of Tokyo) for providing a computational system of CCSR that is suitable for development and testing of the Pstar code. This work was conducted as a part of the greenhouse gases observing satellite (GOSAT) project of National Institute for
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