Radiative Transfer in Coupled Atmosphere and Ocean Systems: the Discrete Ordinate Method
Following from: Radiative transfer in coupled atmosphere and ocean systems: Overview
Leading to: Radiative transfer in coupled atmosphere and ocean systems: Successive Orders of Scattering Method
For simplicity, consider a coupled atmosphere-ocean system consisting of two adjacent slabs separated by a plane, horizontal interface across which the complex refractive index changes abruptly from a value m_{1} ≡ n_{1} + in_{1}^{'} in the air (atmosphere) to a value m_{2} ≡ n_{2} + in_{2}^{'} in the ocean. The radiation field can be split into a direct and scattered (diffuse) part:
(1) |
The direct part is given by
(2) |
where F^{s} is the irradiance (normal to the beam) of the incident (solar) beam. In either of the two slabs, the diffuse radiance distribution I_{diff}(τ,μ,φ) can be described by the radiative transfer equation (RTE) (omitting the subscript)
(3) |
where
(4) |
and the source term J(τ,μ,φ) due to multiple scattering and thermal emission is described in preceding article. In the air, the single-scattering source term S^{*}(τ,μ,φ) in Eq. (4) is
(5) |
where τ_{1} ≡ τ_{air} is the vertical optical depth of the atmosphere, r(-μ_{0};m_{1},m_{2}) is the Fresnel reflectance at the air-ocean interface, μ_{0} = cosθ_{0}, and θ_{0} is the solar zenith angle. The first term on the right-hand side of Eq. (5) is due to first-order scattering of the attenuated incident beam of irradiance F^{s}e^{-τ/μ0} (normal to the beam), whereas the second term is due to first-order scattering of the attenuated incident beam that is reflected at the air-ocean interface. In the ocean, the single-scattering source term consists of the attenuated incident beam that is refracted through the interface, i.e.,
(6) |
where t(-μ_{0};m_{1},m_{2}) is the Fresnel transmittance through the interface, and μ_{0n} is the cosine of the polar angle θ_{0n} in the ocean, which is related to θ_{0} = arccosμ_{0} by Snell’s law.
For a coupled atmosphere-ocean system with source terms given by Eqs. (5) and (6), a solution based on the discrete-ordinate method (Stamnes et al. 1988, Stamnes et al. 2000) of the RTE in Eq. (3) subject to appropriate boundary conditions at the top of the atmosphere, at the bottom of the ocean, and at the air-ocean interface, was first developed by Jin and Stamnes (1994) [see also Thomas and Stamnes (1999)].
1. Isolation of azimuth dependence
The azimuth dependence in Eq. (3) may be isolated by expanding the scattering phase function in Legendre polynomials, P_{l}(cosΘ), and making use of the addition theorem for spherical harmonics (Thomas and Stamnes, 1999)
(7) |
where δ_{0,m} is the Kronecker delta function, i.e. δ_{0,m} = 1 for m = 0 and δ_{0,m} = 0 for m ≠ 0, and
(8) |
Here, χ_{l} = (1/2)∫ _{-1}^{1}d(cosΘ)P_{l}(cosΘ)P(cosΘ) is an expansion coefficient and Λ_{l}^{m}(μ) is given by
(9) |
where P_{l}^{m}(μ) is an associated Legendre polynomial of order m. Expanding the radiance in a similar way,
(10) |
where φ_{0} is the azimuth angle of the incident light beam, one finds that each Fourier component satisfies the following RTE [see Thomas and Stamnes (1999) for details]
(11) |
where m = 0,1,2,...,2N - 1 and p^{m}(μ,μ') is given by Eq. (8).
1.1. Interface Between the Two Slabs
When a beam of light is incident upon a plane interface between two slabs of different refractive indices, one fraction of the incident light will be reflected and another fraction will be transmitted or refracted. For unpolarized light incident upon the interface between the two slabs, the Fresnel reflectance r is given by
(12) |
where r_{⊥} is the reflectance for light polarized with the electric field perpendicular to the plane of incidence and r_{||} is the reflectance for light polarized with the electric field parallel to the plane of incidence (Thomas and Stamnes, 1999; Born and Wolf, 1980). Thus, one finds
(13) |
where μ_{i} = cosθ_{i}, θ_{i} being the angle of incidence, μ_{t} = cosθ_{t}, θ_{t} being the angle of refraction determined by Snell’s law (n_{1} sinθ_{i} = n_{2} sinθ_{t}), and m_{r} = m_{2}/m_{1}. Similarly, the Fresnel transmittance can be written
(14) |
where m_{rel} = n_{2}/n_{1}.
2. Discrete-Ordinate Solution of the RTE
To solve Eq. (11) for a coupled (atmosphere-ocean) system, one must take into account the boundary conditions at the top of the atmosphere and at the bottom of the ocean as well as the reflection and transmission at the air-ocean interface. In addition, the radiation field must be continuous across interfaces between horizontal layers with different inherent optical properties (IOPs) within each of the two slabs (with constant refractive index). Such horizontal layers are introduced to resolve vertical variations in the IOPs within each slab.
The integrodifferential RTE [Eq. (11)] may be transformed into a system of coupled, ordinary differential equations by using the discrete-ordinate approximation to replace the integral in Eq. (11) by a quadrature sum consisting of 2N_{1} terms in the atmosphere and 2N_{2} terms in the ocean, where N_{1} terms are used to represent the radiance in the downward hemisphere in the atmosphere that refracts through the interface into the ocean. In the ocean, N_{2} terms are used to represent the radiance in the downward hemisphere. Note that N_{2} > N_{1} because additional terms are needed in the ocean with the real part of the refractive index n_{2} > n_{1} to represent the downward radiance in the region of total internal reflection.
Seeking solutions to the discrete ordinate approximation of Eq. (11), one obtains the Fourier component of the radiance at any vertical position, both in the atmosphere and the ocean. The solution for the pth layer of the atmosphere is given by (Thomas and Stamnes, 1999)
(15) |
where i = 1,...,N_{1} and p is less than or equal to the number of layers in the air. The solution for the qth layer in the ocean is given by (Thomas and Stamnes, 1999)
(16) |
where i = 1,...,N_{2}. The superscripts a and o are used to denote air and ocean parameters, respectively, the plus (minus) sign is used for radiances streaming upward (downward), and k_{jp}^{a},g_{jp}^{a},k_{jq}^{o}, and g_{jq}^{o} are eigenvalues and eigenvectors determined by the solution of an algebraic eigenvalue problem, which results when one seeks a solution of the homogeneous version of Eq. (11) (with S^{*m}(τ,μ) = 0) in the discrete-ordinate approximation. The terms U_{p}(±μ_{i}^{a}) and U_{q}(±μ_{i}^{o}) are the particular solutions. The coefficients C_{±jp} and C_{±jq} are determined by boundary conditions at the top of the atmosphere and at the bottom of the ocean, the continuity of the basic radiance (the radiance divided by the square of the real part of the refractive index) at each interface between internal layers in each of each of the two slabs, and Fresnel’s equations at the air-ocean interface.
The numerical code C-DISORT (Jin and Stamnes, 1994) computes radiances at any optical depth, polar, and azimuth angle by solving the RTE in Eq. (11) for each layer of the two slabs by using the discrete-ordinate method to convert the integrodifferential RTE into a system of coupled ordinary differential equations. The C-DISORT method can be summarized as follows:
- The atmosphere and ocean media are separated by a plane interface at which the complex refractive index changes from m_{1} = n_{1} + in_{1}^{'} in the air to m_{2} = n_{2} + in_{2}^{'} in the ocean.
- Each of the two slabs is divided into a sufficiently large number of homogenous horizontal layers to adequately resolve the vertical variation in its IOPs.
- Fresnel’s equations for the reflectance and transmittance are applied at the air-water interface, in addition to the law of reflection and Snell’s Law to determine the directions of the reflected and refracted rays as well as the relations between the diffuse radiance fields across the interface (Jin and Stamnes, 1994; Thomas and Stamnes, 1999) described by the interaction principles.
- Discrete-ordinate solutions to the RTE are computed separately for each layer in the two slabs [see Eqs. (15)-(16)].
- Finally, boundary conditions at the top of the atmosphere and the bottom of the ocean are applied, in addition to continuity conditions at layer interfaces within each of the two slabs.
Fourier components of the radiances at any vertical location given by the pth layer in the air or the qth layer in the ocean are computed from Eqs. (15)-(16), and the azimuth-dependent diffuse radiance distribution from Eq. (10). Upward and downward hemispherical irradiances and mean intensities (scalar irradiances) are calculated by integrating the m = 0 (azimuthally-averaged) Fourier component <I_{p}(τ,+μ_{i},φ)> = I_{p}^{m=0}(τ,+μ_{i}) or <I_{p}(τ,-μ_{i},φ)> = I_{p}^{m=0}(τ,-μ_{i}) over polar angles.
The downward irradiance in the air consists of a direct component
(17) |
and a diffuse component
(18) |
where <I_{d}^{a}(τ,μ)> is the azimuthally averaged diffuse downward radiance at optical depth τ ≤ τ_{1} in the air. Similarly, the upward diffuse irradiance E_{u,diff}^{a} in the air and the average diffuse radiance (mean intensity) are given by
(19) |
(20) |
In the ocean, the downward (direct and diffuse), upward, and average diffuse irradiances become:
(21) |
(22) |
(23) |
(24) |
where the downward and upward azimuthally averaged radiances in the ocean are given by <I_{d}^{o}(τ,μ)> and <I_{u}^{o}(τ,μ)>.
3. Comparisons of C-DISORT and C-Monte Carlo Results
Gjerstad et al. (2003) compared irradiances obtained from a Monte Carlo (MC) model for the coupled atmosphere-ocean system (C-MC) to those obtained from a discrete-ordinate method (C-DISORT). By treating the scattering and absorption processes in the two slabs in the same manner in both models, Gjerstad et al. (2003) were able to provide a more detailed and quantitative comparison than those previously reported (Mobley et al., 1993). Figure 1 shows a comparison of direct and diffuse downward irradiances computed with C-MC and C-DISORT codes, demonstrating that when precisely the same IOPs are used in the two models, computed irradiances agree to within 1% throughout the coupled atmosphere-ocean system.
Figure 1. Comparison of irradiance results obtained with C-DISORT and a C-MC code for radiative transfer in a coupled atmosphere-ocean system [see Gjerstad et al., (2003), for details].
3.1. Extension to a Non-Flat Atmosphere-Ocean Interface
One shortcoming of the results discussed above is that the interface between the two media with different refractive indices was taken to be flat. This flat-surface assumption limits the applications of C-DISORT, because a wind-roughened ocean surface is a randomly scattering object. In fact, if the ocean surface were flat, a perfect image of the Sun’s disk would be observed in the specular direction. The effect of surface roughness is to spread the specular reflection over a range of angles referred to as the sun-glint region. If the surface is characterized by a Gaussian random height distribution and the tangent plane approximation is invoked (Beckmann and Spizzichino, 1963), the bidirectional reflection distribution function (BRDF) for an isotropic Gaussian surface can be expressed as (Beckmann and Spizzichino, 1963; Tsang et al., 1985; Gordon, 1997)
(25) |
if one ignores the effects of shadowing and multiple reflections due to surface facets. Here Δφ = φ^{'}- φ, r(m_{1},m_{2},μ,μ',Δφ) is the Fresnel reflectance, μ_{n} = cosθ_{n}, θ_{n} is the tilt angle between the vertical and the normal to the tangent plane (see Fig. 1), σ^{2} = 0.003 + 0.00512 WS, and WS is the wind speed in meters per second (Cox and Munk, 1954; Jin et al., 2006; Zhai et al., 2010) [see Eqs. (15)-(20)]. A corresponding expression for the transmittance B_{t} [see Eq. (21)] can be derived as well (Beckmann and Spizzichino, 1963; Tsang et al., 1985).
In addition to affecting the backscattered radiation, the surface roughness significantly affects the directional character of the radiation transmitted through the air-water interface. To deal with the surface roughness in the discrete-ordinate method, Gjerstad et al. (2003) proposed an ad hoc method of mimicking the irradiances obtained from a C-MC model by adjusting the refractive index in C-DISORT. The limitations incurred by this ad hoc method were removed by Jin et al. (2006), who presented a consistent solution of the discrete ordinate radiative transfer problem in a coupled atmosphere-ocean system with a rough surface interface having a Gaussian wave slope distribution [Eq. (25)]. Jin et al. (2006) concluded that the ocean surface roughness has significant effects on the upward radiation in the atmosphere and the downward radiation in the ocean. As the wind speed increases, the angular domain of the sunglint broadens, the surface albedo decreases, and the transmission of radiation through the air-water interface into the ocean increases. The transmitted radiance just below a flat ocean surface is highly anisotropic, but this anisotropy decreases rapidly as the surface wind increases. Also, the anisotropy will decrease as the water depth increases because multiple scattering in the ocean interior eventually will tend to reduce the anisotropy more than the surface roughness. The effects of surface roughness on the radiation field depend on both the wavelength and angle of incidence (i.e., solar elevation). Although their model predictions of the impact of surface roughness agreed reasonably well with observations, Jin et al. (2006) cautioned that the original Cox-Munk surface roughness model (Cox and Munk, 1954) adopted in the simulations [Eq. (25)] may be inadequate for high wind speeds.
REFERENCES
Beckmann, P. and Spizzichino, A., The Scattering of Electromagnetic Waves from Rough Surfaces, MacMillan, New York, 1963.
Born, M. and Wolf, E., Principles of Optics, Cambridge University Press, Cambridge, England, 1980.
Cox, C. and Munk, W., Measurement of the roughness of the sea surface from photographs of the sun’s glitter, J. Opt. Soc. Am., vol. 44, pp. 838-850, 1954.
Gjerstad, K. I., Stamnes, J. J., Hamre, B., Lotsberg, J. K., Yan, B., and Stamnes, K., Monte Carlo and discrete-ordinate simulations of irradiances in the coupled atmosphere-ocean system, Appl. Opt., vol. 42, pp. 2609-2622, 2003.
Gordon, H. R., Atmospheric correction of ocean color imagery in the earth observing observation system era, J. Geophys. Res., vol. 102, pp. 17081-17106, 1997.
Jin, Z. and Stamnes, K., Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system, Appl. Opt., vol. 33, pp. 431-442, 1994.
Jin, Z., Charlock, T. P., Rutledge, K., Stamnes, K., and Wang, Y., An analytical solution of radiative transfer in the coupled atmosphere-ocean system with rough surface, Appl. Opt., vol. 45, pp. 7443-7455, 2006.
Mobley, C. D., Gentili, B., Gordon, H. R., Jin, Z., Kattawar, G. W., Morel, A., Reinersman, P., Stamnes, K., and Stavn, R. H., Comparison of numerical models for computing underwater light fields, Appl. Opt., vol. 32, pp. 7484-7504, 1993.
Stamnes, K., Tsay, S. C., Wiscombe, W. J., and Jayaweera, K., Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media, Appl. Opt., vol. 27, pp. 2502-2509, 1988.
Stamnes, K., Tsay, S. C., Wiscombe, W. J., and Laszlo, I., DISORT, A General-Purpose Fortran Program for Discrete-Ordinate-Method Radiative Transfer in Scattering and Emitting Layered Media: Documentation of Methodology, ftp://climate.gsfc.nasa.gov/pub/wiscombe/Multiple Scatt/, 2000.
Thomas, G. E. and Stamnes, K., Radiative Transfer in the Atmosphere and Ocean, Cambridge University Press, Cambridge, England, 1999; 2nd ed., 2002.
Tsang, L., Kong, J. A., and Shin, R. T., Theory of Microwave Remote Sensing, Wiley, Hoboken, NJ, 1985.
Zhai, P.-W., Hu, Y., Chowdhary, J., Trepte, C. R., Lucker, P. L., and Josset, D. B., A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface, J. Quant. Spectrosc. Radiat. Transfer, vol. 111, pp. 1025-1040, 2007.
Verweise
- Beckmann, P. and Spizzichino, A., The Scattering of Electromagnetic Waves from Rough Surfaces, MacMillan, New York, 1963.
- Born, M. and Wolf, E., Principles of Optics, Cambridge University Press, Cambridge, England, 1980.
- Cox, C. and Munk, W., Measurement of the roughness of the sea surface from photographs of the sun’s glitter, J. Opt. Soc. Am., vol. 44, pp. 838-850, 1954.
- Gjerstad, K. I., Stamnes, J. J., Hamre, B., Lotsberg, J. K., Yan, B., and Stamnes, K., Monte Carlo and discrete-ordinate simulations of irradiances in the coupled atmosphere-ocean system, Appl. Opt., vol. 42, pp. 2609-2622, 2003.
- Gordon, H. R., Atmospheric correction of ocean color imagery in the earth observing observation system era, J. Geophys. Res., vol. 102, pp. 17081-17106, 1997.
- Jin, Z. and Stamnes, K., Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system, Appl. Opt., vol. 33, pp. 431-442, 1994.
- Jin, Z., Charlock, T. P., Rutledge, K., Stamnes, K., and Wang, Y., An analytical solution of radiative transfer in the coupled atmosphere-ocean system with rough surface, Appl. Opt., vol. 45, pp. 7443-7455, 2006.
- Mobley, C. D., Gentili, B., Gordon, H. R., Jin, Z., Kattawar, G. W., Morel, A., Reinersman, P., Stamnes, K., and Stavn, R. H., Comparison of numerical models for computing underwater light fields, Appl. Opt., vol. 32, pp. 7484-7504, 1993.
- Stamnes, K., Tsay, S. C., Wiscombe, W. J., and Jayaweera, K., Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media, Appl. Opt., vol. 27, pp. 2502-2509, 1988.
- Stamnes, K., Tsay, S. C., Wiscombe, W. J., and Laszlo, I., DISORT, A General-Purpose Fortran Program for Discrete-Ordinate-Method Radiative Transfer in Scattering and Emitting Layered Media: Documentation of Methodology, ftp://climate.gsfc.nasa.gov/pub/wiscombe/Multiple Scatt/, 2000.
- Thomas, G. E. and Stamnes, K., Radiative Transfer in the Atmosphere and Ocean, Cambridge University Press, Cambridge, England, 1999; 2nd ed., 2002.
- Tsang, L., Kong, J. A., and Shin, R. T., Theory of Microwave Remote Sensing, Wiley, Hoboken, NJ, 1985.
- Zhai, P.-W., Hu, Y., Chowdhary, J., Trepte, C. R., Lucker, P. L., and Josset, D. B., A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface, J. Quant. Spectrosc. Radiat. Transfer, vol. 111, pp. 1025-1040, 2007.