Three Dimensional Continuum Radiative Transfer

Author(s):  
Jürgen Steinacker
2015 ◽  
Vol 15 (14) ◽  
pp. 8147-8163 ◽  
Author(s):  
M. Schäfer ◽  
E. Bierwirth ◽  
A. Ehrlich ◽  
E. Jäkel ◽  
M. Wendisch

Abstract. Based on airborne spectral imaging observations, three-dimensional (3-D) radiative effects between Arctic boundary layer clouds and highly variable Arctic surfaces were identified and quantified. A method is presented to discriminate between sea ice and open water under cloudy conditions based on airborne nadir reflectivity γλ measurements in the visible spectral range. In cloudy cases the transition of γλ from open water to sea ice is not instantaneous but horizontally smoothed. In general, clouds reduce γλ above bright surfaces in the vicinity of open water, while γλ above open sea is enhanced. With the help of observations and 3-D radiative transfer simulations, this effect was quantified to range between 0 and 2200 m distance to the sea ice edge (for a dark-ocean albedo of αwater = 0.042 and a sea-ice albedo of αice = 0.91 at 645 nm wavelength). The affected distance Δ L was found to depend on both cloud and sea ice properties. For a low-level cloud at 0–200 m altitude, as observed during the Arctic field campaign VERtical Distribution of Ice in Arctic clouds (VERDI) in 2012, an increase in the cloud optical thickness τ from 1 to 10 leads to a decrease in Δ L from 600 to 250 m. An increase in the cloud base altitude or cloud geometrical thickness results in an increase in Δ L; for τ = 1/10 Δ L = 2200 m/1250 m in case of a cloud at 500–1000 m altitude. To quantify the effect for different shapes and sizes of ice floes, radiative transfer simulations were performed with various albedo fields (infinitely long straight ice edge, circular ice floes, squares, realistic ice floe field). The simulations show that Δ L increases with increasing radius of the ice floe and reaches maximum values for ice floes with radii larger than 6 km (500–1000 m cloud altitude), which matches the results found for an infinitely long, straight ice edge. Furthermore, the influence of these 3-D radiative effects on the retrieved cloud optical properties was investigated. The enhanced brightness of a dark pixel next to an ice edge results in uncertainties of up to 90 and 30 % in retrievals of τ and effective radius reff, respectively. With the help of Δ L, an estimate of the distance to the ice edge is given, where the retrieval uncertainties due to 3-D radiative effects are negligible.


Author(s):  
Gisela Widmer

The stationary monochromatic radiative transfer equation (RTE) is posed in five dimensions, with the intensity depending on both a position in a three-dimensional domain as well as a direction. For non-scattering radiative transfer, sparse finite elements [1, 2] have been shown to be an efficient discretization strategy if the intensity function is sufficiently smooth. Compared to the discrete ordinates method, they make it possible to significantly reduce the number of degrees of freedom N in the discretization with almost no loss of accuracy. However, using a direct solver to solve the resulting linear system requires O(N3) operations. In this paper, an efficient solver based on the conjugate gradient method (CG) with a subspace correction preconditioner is presented. Numerical experiments show that the linear system can be solved at computational costs that are nearly proportional to the number of degrees of freedom N in the discretization.


2007 ◽  
Vol 64 (11) ◽  
pp. 4098-4112 ◽  
Author(s):  
Haruma Ishida ◽  
Shoji Asano

Abstract A new calculation scheme is proposed for the explicitly discretized solution of the three-dimensional (3D) radiation transfer equation (RTE) for inhomogeneous atmospheres. To separate the independent variables involved in the 3D RTE approach, the spherical harmonic series expansion was used to discretize the terms, depending on the direction of the radiance, and the finite-volume method was applied to discretize the terms, depending on the spatial coordinates. A bidirectional upwind difference scheme, which is a specialized scheme for the discretization of the partial differential terms in the spherical harmonic-transformed RTE, was developed to make the equation determinate. The 3D RTE can be formulated as a simultaneous linear equation, which is expressed in the form of a vector–matrix equation with a sparse matrix. The successive overrelaxation method was applied to solve this equation. Radiative transfer calculations of the solar radiation in two-dimensional cloud models have shown that this method can properly simulate the radiation field in inhomogeneous clouds. A comparison of the results obtained using this method with those using the Monte Carlo method shows reasonable agreement for the upward flux, the total downward flux, and the intensities of radiance.


2009 ◽  
Vol 48 (16) ◽  
pp. 3019 ◽  
Author(s):  
Yu You ◽  
Peng-Wang Zhai ◽  
George W. Kattawar ◽  
Ping Yang

Sign in / Sign up

Export Citation Format

Share Document