Cover Journal of the Atmospheric Sciences
Journal of the Atmospheric Sciences

  • Barker, H. W., 1996: A parameterization for computing grid-averaged solar fluxes for inhomogeneous marine boundary layer clouds. Part I: Methodology and homogeneous biases. J. Atmos. Sci, 53 , 2289–2303.

    • Search Google Scholar
    • Export Citation

  • Barker, H. W., B. A. Wielicki, and L. Parker, 1996: A parameterization for computing grid-averaged solar fluxes for inhomogeneous marine boundary layer clouds. Part II: Validation using satellite data. J. Atmos. Sci, 53 , 2304–2316.

    • Search Google Scholar
    • Export Citation

  • Barker, H. W., J-J. Morcrette, and G. D. Alexander, 1998: Broadband solar fluxes and heating rates for atmospheres with 3D broken clouds. Quart. J. Roy. Meteor. Soc, 124 , 1245–1271.

    • Search Google Scholar
    • Export Citation

  • Cahalan, R. F., W. Ridgway, W. J. Wiscombe, S. Gollmer, and Harshvardhan, 1994: Independent pixel and Monte Carlo estimates of stratocumulus albedo. J. Atmos. Sci, 51 , 3776–3790.

    • Search Google Scholar
    • Export Citation

  • Chandrasekhar, S., 1960: Radiative Transfer. Dover, 391 pp.

  • Fu, Q., and K-N. Liou, 1993: Parameterization of the radiative properties of cirrus clouds. J. Atmos. Sci, 50 , 2008–2025.

  • Geleyn, J-F., and A. Hollingsworth, 1979: An economical analytical method for the computation of the interaction between scattering and line absorption of radiation. Contrib. Atmos. Phys, 52 , 1–16.

    • Search Google Scholar
    • Export Citation

  • Kato, S., T. P. Ackerman, J. H. Mather, and E. E. Clothiaux, 1999: The k-distribution method and correlated-k approximation for a shortwave radiative transfer model. J. Quant. Spectrosc. Radiat. Transfer, 62 , 109–121.

    • Search Google Scholar
    • Export Citation

  • Liou, K-N., 1975: Applications of the discrete-ordinate method for radiative transfer to inhomogeneous aerosol atmospheres. J. Geophys. Res, 80 , 3434–3440.

    • Search Google Scholar
    • Export Citation

  • McClatchey, R. A., R. W. Fenn, J. E. A. Selby, F. E. Volz, and J. S. Garing, 1972: Optical properties of the atmosphere. 3d ed. Environmental Research Paper 411, Air Force Cambridge Research Laboratories, Bedford, MA, 110 pp.

    • Search Google Scholar
    • Export Citation

  • Meador, W. E., and W. R. Weaver, 1980: Two-stream approximations to radiative transfer in planetary atmospheres: A unified description of existing methods and a new improvement. J. Atmos. Sci, 37 , 630–643.

    • Search Google Scholar
    • Export Citation

  • Oreopoulos, L., and H. W. Barker, 1999: Accounting for subgrid-scale cloud variability in a multi-layer 1D solar radiative transfer algorithm. Quart. J. Roy. Meteor. Soc, 125 , 301–330.

    • Search Google Scholar
    • Export Citation

  • Shettle, E. P., and J. A. Weinman, 1970: The transfer of solar irradiance through inhomogeneous turbid atmospheres evaluated by Eddington's approximation. J. Atmos. Sci, 27 , 1048–1055.

    • Search Google Scholar
    • Export Citation

  • Stamnes, K., and R. A. Swanson, 1981: A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. J. Atmos. Sci, 38 , 387–399.

    • Search Google Scholar
    • Export Citation

  • Stamnes, K., S-C. Tsay, W. Wiscombe, and K. Jayaweera, 1988: Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media. Appl. Opt, 27 , 2502–2509.

    • Search Google Scholar
    • Export Citation

  • Stephens, G. L., 1988: Radiative transfer through arbitrarily shaped optical media. Part II: Group theory and simple closures. J. Atmos. Sci, 45 , 1837–1848.

    • Search Google Scholar
    • Export Citation

  • Stephens, G. L., P. M. Gabriel, and S-C. Tsay, 1991: Statistical radiative transport in one-dimensional media and its application to the terrestrial atmosphere. Trans. Theor. Stat. Phys, 20 , 139–175.

    • Search Google Scholar
    • Export Citation

  • Toon, O. B., C. P. Mckay, T. P. Ackerman, and K. Santhanam, 1989:: Rapid calculation of radiative heating rates and photodissociation rates in inhomogeneous multiple scattering atmospheres. J. Geophys. Res, 94 , 16287–16301.

    • Search Google Scholar
    • Export Citation

  • Wilks, D. S., 1995: Statistical Methods in the Atmospheric Sciences. Academic Press, 467 pp.

  • Wiscombe, W. J., 1977: The delta-Eddington approximation for a vertically inhomogeneous atmosphere. Tech. Note TN-121+STR, National Center for Atmospheric Research, Boulder, CO, 66 pp.

    • Search Google Scholar
    • Export Citation
  • View in gallery

    (a) Probability density functions of cloud optical thickness for gamma distributions with mean optical thickness of 15 and ν = 1.0, 1.5, and 2.0. (b), (c), and (d) show broadband top-of-the-atmosphere reflectance and top-of-the-atmosphere to surface transmittance as functions of mean optical thickness for distributions shown in (a) as computed by the algorithm using analytical integration and by numerical integration of the homogeneous solution weighted by the distributions in (a). Values for homogeneous clouds computed by the standard two-stream approximation are also shown for reference

  • View in gallery

    Difference of the atmospheric absorptance for homogeneous cloud and inhomogeneous cloud shown as a function of mean optical thickness for distributions shown in Fig. 1a

(a) Probability density functions of cloud optical thickness for gamma distributions with mean optical thickness of 15 and ν = 1.0, 1.5, and 2.0. (b), (c), and (d) show broadband top-of-the-atmosphere reflectance and top-of-the-atmosphere to surface transmittance as functions of mean optical thickness for distributions shown in (a) as computed by the algorithm using analytical integration and by numerical integration of the homogeneous solution weighted by the distributions in (a). Values for homogeneous clouds computed by the standard two-stream approximation are also shown for reference

Difference of the atmospheric absorptance for homogeneous cloud and inhomogeneous cloud shown as a function of mean optical thickness for distributions shown in Fig. 1a
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 3 3 3
PDF Downloads 3 3 3
Article Type:
Research Article

Gamma-Weighted Discrete Ordinate Two-Stream Approximation for Computation of Domain-Averaged Solar Irradiance

Seiji Kato1, G. Louis Smith2, and Howard W. Barker3
View More View Less
  • 1 Center for Atmospheric Sciences, Hampton University, Hampton, Virginia
  • | 2 Virginia Polytechnic Institute and State University, Blacksburg, Virginia
  • | 3 Environment Canada, Downsview, Ontario, Canada
Print Publication:
01 Dec 2001
DOI:
https://doi.org/10.1175/1520-0469(2001)058<3797:GWDOTS>2.0.CO;2
Page(s):
3797–3803
Full access

Abstract

An algorithm is developed for the gamma-weighted discrete ordinate two-stream approximation that computes profiles of domain-averaged shortwave irradiances for horizontally inhomogeneous cloudy atmospheres. The algorithm assumes that frequency distributions of cloud optical depth at unresolved scales can be represented by a gamma distribution though it neglects net horizontal transport of radiation. This algorithm is an alternative to the one used in earlier studies that adopted the adding method. At present, only overcast cloudy layers are permitted.

Corresponding author address: Dr. Seiji Kato, Mail Stop 420, 21 Langley Blvd., B1250, NASA Langley Research Center, Hampton, VA 23681-2199. Email: s.kato@larc.nasa.gov

Abstract

An algorithm is developed for the gamma-weighted discrete ordinate two-stream approximation that computes profiles of domain-averaged shortwave irradiances for horizontally inhomogeneous cloudy atmospheres. The algorithm assumes that frequency distributions of cloud optical depth at unresolved scales can be represented by a gamma distribution though it neglects net horizontal transport of radiation. This algorithm is an alternative to the one used in earlier studies that adopted the adding method. At present, only overcast cloudy layers are permitted.

Corresponding author address: Dr. Seiji Kato, Mail Stop 420, 21 Langley Blvd., B1250, NASA Langley Research Center, Hampton, VA 23681-2199. Email: s.kato@larc.nasa.gov

1. Introduction

It has been demonstrated that accurate estimates of domain-averaged solar irradiances for marine boundary layer clouds can be obtained by applying a two-stream approximation to each subgrid cell and averaging the resulting irradiances (the independent pixel or column approximation; Stephens et al. 1991; Cahalan et al. 1994). Barker et al. (1998) showed that the same can be done using entire columns for large domains containing convective clouds. Since this approximation neglects net horizontal fluxes, these results suggest that if the probability density function of cloud optical thickness is known, domain-averaged irradiances can be computed as a function of a few parameters expressing the probability density function.

Satellite observations (Barker et al. 1996) and cloud-resolving models (e.g., Barker et al. 1998; Oreopoulos and Barker 1999) indicate that distributions of cloud optical thickness for domains the size of cells in typical weather and climate models can be approximated by gamma distributions. Barker (1996) used a two-stream approximation with boundary conditions of no diffuse irradiance from above and below the cloud layer to derive an analytical solution for domain-averaged reflectance and transmittance for an inhomogeneous cloud layer. Oreopoulos and Barker (1999) used the principles of invariance (Chandrasekhar 1960, 161–166) combined with Barker's (1996) analytical solution to compute irradiance at each level of a multilayered atmosphere. They used the adding method, which differs from discrete ordinate two-stream multilayered algorithms that solve a matrix equation set up by irradiance boundary conditions (e.g., Shettle and Weinman 1970; Liou 1975; Stamnes and Swanson 1981; Stamnes et al. 1988; Toon et al. 1989, Fu and Liou 1993). Since discrete ordinate two-stream algorithms are widely used to compute irradiance for multilayered atmospheres, it is worthwhile incorporating the gamma-weighted two-stream approximation into the discrete ordinate framework.

The main purpose of this paper is to derive analytical solutions for reflected and transmitted irradiance given gamma distributed cloud optical thickness that can be used with a discrete ordinate two-stream model. This provides an alternative to the adding method and entails solving a tridiagonal matrix built from boundary conditions for radiative transfer computations for multilayer atmospheres.

2. Algorithm development

a. Homogeneous cloud layer

The general two-stream solution of the radiative transfer equation for a plane-parallel, homogeneous layer (Toon et al. 1989) is
i1520-0469-58-24-3797-e1
where F+ and F+ are upward and downward irradiance, respectively; ξ is the fraction of cloud depth measured from ξ = 0 at cloud top to ξ = 1 at cloud base, τm is optical thickness of the layer at ξ = 1,
i1520-0469-58-24-3797-e3
For solar radiation,
i1520-0469-58-24-3797-e5
where πFsμ0 is the solar irradiance incident at cloud top; μ0 is the cosine of the solar zenith angle; ω0 is the single-scattering albedo; and γ1, γ2, γ3, and γ4 are coefficients that depend on the form of the two-stream approximation (Meador and Weaver 1980).

b. Inhomogeneous cloud layer

Consider a single-layer overcast cloud for which optical thickness varies over a large horizontal domain. We assume that the optical thickness probability density function of the layer follows a gamma distribution defined as
i1520-0469-58-24-3797-e7
where ν and τm are the shape parameter and mean optical thickness of the layer at ξ = 1, respectively (Wilks 1995, p. 86). The shape parameter can be estimated from moments such as ν = (τm/σ)2 where σ is the standard deviation of τm. For ν < 10, however, it is recommended that the maximum likelihood method be used to estimate ν (Wilks 1995, 86–90). Domain-averaged irradiances are then computed via the independent column approximation as
i1520-0469-58-24-3797-e8
where we assume that ν is independent of ξ.
The unknowns in (1) and (2), k1 and k2, are determined by boundary conditions,
i1520-0469-58-24-3797-e9
Assuming no downward and upward diffuse irradiance at the top and bottom of the cloud, respectively, [i.e., F−(0) = F+(τm) = 0] and expanding the denominators of k1 and k2 by the binomial theorem yields
i1520-0469-58-24-3797-e10
Therefore, using (10) and (11) and substituting (1) and (2) into (8), domain-averaged upward and downward irradiances at level ξ are
i1520-0469-58-24-3797-e12
where
i1520-0469-58-24-3797-e14
Following the notation of Barker (1996), G1 and G2 at layer boundaries are
i1520-0469-58-24-3797-e18
where
i1520-0469-58-24-3797-eq1
Domain-averaged direct irradiance at ξ is given by
i1520-0469-58-24-3797-e22
Note that at the homogeneous limit when ν goes to infinity (i.e., σ goes to zero),
i1520-0469-58-24-3797-e23a

Since the single-scattering albedo ω0 and asymmetry parameter g are mean values weighted by the optical thickness of constituents, they depend on cloud optical thickness. For the integration of (8), we assume that ω0 and g, as well as λ, which is a function of ω0 and g, are constant throughout the layer.

c. Multilayer atmosphere

For real clouds, downward irradiance at cloud top and upward irradiance at cloud base are not zero [i.e., F−(0) and F+(τm) ≠ 0 in (11)]. Thus, assuming that when G1 and G2 are factored out of the unknowns, that are determined by boundary conditions, the remainder is independent of τm, and domain-averaged upward and downward irradiance for nonzero F−(0) and F+(τm) are
i1520-0469-58-24-3797-e25
where k′1 and k′2 are unknown and are determined by boundary conditions. Substituting (23) and (24) into (25) and (26) it is easy to verify that in the limit as ν goes to infinity, (25) and (26) revert to (1) and (2), except that eλτm is factored out and included in k1.

When an inhomogeneous cloud layer is placed in a vertically inhomogeneous atmosphere consisting of n homogeneous layers, the boundary conditions are set up as a matrix equation that can be solved for n − 1 unknowns of k1 and k2 in (1) and (2) and k′1 and k′2 in (25) and (26). Toon et al. (1989) suggest that a tridiagonal matrix inversion is computationally fast and numerically stable. Therefore, a description of the elements of the matrix when an inhomogeneous cloud layer is included in the atmosphere is given in the appendix.

3. Results

Broadband, domain-averaged irradiances computed by (25) and (26) are compared to corresponding values obtained by numerically integrating (1) and (2) weighted by the appropriate optical thickness distribution [i.e., Eq. (7)]. For the test, a cloud layer of τm = 15 was placed between 1 and 2 km in the midlatitude summer standard atmosphere (McClatchey et al. 1972). We used a spectrally invariant surface albedo of 0.2 and a solar zenith angle of 60°. The k-distribution tables used here in estimating gaseous absorptions are those given by Kato et al. (1999). For numerical integration, lower and upper limits of optical thickness were 0.001 and 500, respectively. The population outside this range is negligible, less than 0.1% for the cases considered here.

Figure 1 shows the top-of-the-atmosphere reflectance and top-of-the-atmosphere to surface transmittance for the analytic model and the numerical integration. Differences between broadband reflectances and transmittances are less than 0.015, 0.01, and 0.007 for ν = 1.0, 1.5, and 2.0, respectively. As mentioned by Barker et al. (1996), since the algorithm's values result from integration over τm from 0 to infinity, differences between algorithmic and numerically integrated irradiances can be caused, in part, by numerical integration over a finite range. As a consequence, differences between estimated irradiances increase as ν and τm decrease; the population outside the range of integration increases to more than 1% when ν = 0.5 and τm = 5. In this case, differences in broadband reflectance and transmittance are 0.006 and 0.023, respectively.

This algorithm runs into difficulty when ω0 approaches unity because of the binomial expansion of the denominator in (9) and because of the singularity in the two-stream approximation. Therefore, a maximum single-scattering albedo is set to avoid the problem (Wiscombe 1977). For a maximum single-scattering albedo of 0.99999 and use of 100 terms to compute G1 and G2 by (14) and (15), the algorithm reproduces conservative-scattering reflectances given by Barker et al. (1996). For example, for four overcast scenes in Barker et al. (1996), the sums of transmittance and reflectance are greater than 0.9995 for all solar zenith angles.

4. Discussion and summary

The result shown in the previous section demonstrates that this algorithm can compute accurate domain-averaged irradiances for multilayered atmospheres with a single-layer inhomogeneous cloud. In such cases, trivial boundary conditions are sufficient for straightforward computation of domain-averaged irradiance. If the cloud layer were to be divided into many layers, trivial boundary conditions are no longer appropriate because irradiance is horizontally variable; columns locating below columns that are optically thinner (thicker) than the mean optical thickness receive more (less) radiation than the domain-averaged irradiance at the top boundary (Stephens 1988; Oreopoulos and Barker 1999). One could, however, resort to an adjustment of optical properties similar to that proposed by Oreopoulos and Barker (1999). When overcast cloud layers are separated by a clear layer, the trivial boundary conditions can be applied once again if one is content to accept the assumption that for multilayered systems, fluctuations in cloud optical properties and irradiance are uncorrelated. Although the discrete ordinate formulation allows for overcast clouds only, the algorithm presented here could be used in a cloud overlap scheme such as that developed by Geleyn and Hollingsworth (1979).

When τm/ν is small, (14) and (15) approach (10) and (11) because the second term in the denominators of (14) and (15) are small and the denominators can be approximated by exponential functions. Therefore, the inhomogeneous solutions (25) and (26) approach the homogeneous solutions (1) and (2), respectively. Two cases that give small τm/ν are τm ≪ 1 and ν ≫ 1. When the layer is optically thin, the transmittance and reflectance are linear functions of the optical thickness so that the mean optical thickness provides the domain averaged transmittance and reflectance. As the mean optical thickness increases, the difference of transmittance and reflectance increases. When ν is large, the layer is close to homogeneous; hence, the inhomogeneous solution should approach the homogeneous solution. The absorptance of the atmosphere with a homogeneous cloud layer is always greater than that with an inhomogeneous cloud layer for these cases (Fig. 2). As the mean optical thickness or ν of the cloud layer increases, the absorptance of the atmosphere with an inhomogeneous cloud layer approaches to that with a homogeneous cloud layer. However, the absorption vertical profile can be different.

Acknowledgments

We thank S. Weckmann for useful discussions. The work for the clouds and the Clouds and the Earth's Radiant Energy System (CERES) project was supported by the NASA Langley Research Center Grant NAG-1-1963 to Hampton University and Cooperative Agreement NCC-1-405 to Virginia Polytechnic Institute and State University.

REFERENCES

  • Barker, H. W., 1996: A parameterization for computing grid-averaged solar fluxes for inhomogeneous marine boundary layer clouds. Part I: Methodology and homogeneous biases. J. Atmos. Sci, 53 , 2289–2303.

    • Search Google Scholar
    • Export Citation

  • Barker, H. W., B. A. Wielicki, and L. Parker, 1996: A parameterization for computing grid-averaged solar fluxes for inhomogeneous marine boundary layer clouds. Part II: Validation using satellite data. J. Atmos. Sci, 53 , 2304–2316.

    • Search Google Scholar
    • Export Citation

  • Barker, H. W., J-J. Morcrette, and G. D. Alexander, 1998: Broadband solar fluxes and heating rates for atmospheres with 3D broken clouds. Quart. J. Roy. Meteor. Soc, 124 , 1245–1271.

    • Search Google Scholar
    • Export Citation

  • Cahalan, R. F., W. Ridgway, W. J. Wiscombe, S. Gollmer, and Harshvardhan, 1994: Independent pixel and Monte Carlo estimates of stratocumulus albedo. J. Atmos. Sci, 51 , 3776–3790.

    • Search Google Scholar
    • Export Citation

  • Chandrasekhar, S., 1960: Radiative Transfer. Dover, 391 pp.

  • Fu, Q., and K-N. Liou, 1993: Parameterization of the radiative properties of cirrus clouds. J. Atmos. Sci, 50 , 2008–2025.

  • Geleyn, J-F., and A. Hollingsworth, 1979: An economical analytical method for the computation of the interaction between scattering and line absorption of radiation. Contrib. Atmos. Phys, 52 , 1–16.

    • Search Google Scholar
    • Export Citation

  • Kato, S., T. P. Ackerman, J. H. Mather, and E. E. Clothiaux, 1999: The k-distribution method and correlated-k approximation for a shortwave radiative transfer model. J. Quant. Spectrosc. Radiat. Transfer, 62 , 109–121.

    • Search Google Scholar
    • Export Citation

  • Liou, K-N., 1975: Applications of the discrete-ordinate method for radiative transfer to inhomogeneous aerosol atmospheres. J. Geophys. Res, 80 , 3434–3440.

    • Search Google Scholar
    • Export Citation

  • McClatchey, R. A., R. W. Fenn, J. E. A. Selby, F. E. Volz, and J. S. Garing, 1972: Optical properties of the atmosphere. 3d ed. Environmental Research Paper 411, Air Force Cambridge Research Laboratories, Bedford, MA, 110 pp.

    • Search Google Scholar
    • Export Citation

  • Meador, W. E., and W. R. Weaver, 1980: Two-stream approximations to radiative transfer in planetary atmospheres: A unified description of existing methods and a new improvement. J. Atmos. Sci, 37 , 630–643.

    • Search Google Scholar
    • Export Citation

  • Oreopoulos, L., and H. W. Barker, 1999: Accounting for subgrid-scale cloud variability in a multi-layer 1D solar radiative transfer algorithm. Quart. J. Roy. Meteor. Soc, 125 , 301–330.

    • Search Google Scholar
    • Export Citation

  • Shettle, E. P., and J. A. Weinman, 1970: The transfer of solar irradiance through inhomogeneous turbid atmospheres evaluated by Eddington's approximation. J. Atmos. Sci, 27 , 1048–1055.

    • Search Google Scholar
    • Export Citation

  • Stamnes, K., and R. A. Swanson, 1981: A new look at the discrete ordinate method for radiative transfer calculations in anisotropically scattering atmospheres. J. Atmos. Sci, 38 , 387–399.

    • Search Google Scholar
    • Export Citation

  • Stamnes, K., S-C. Tsay, W. Wiscombe, and K. Jayaweera, 1988: Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media. Appl. Opt, 27 , 2502–2509.

    • Search Google Scholar
    • Export Citation

  • Stephens, G. L., 1988: Radiative transfer through arbitrarily shaped optical media. Part II: Group theory and simple closures. J. Atmos. Sci, 45 , 1837–1848.

    • Search Google Scholar
    • Export Citation

  • Stephens, G. L., P. M. Gabriel, and S-C. Tsay, 1991: Statistical radiative transport in one-dimensional media and its application to the terrestrial atmosphere. Trans. Theor. Stat. Phys, 20 , 139–175.

    • Search Google Scholar
    • Export Citation

  • Toon, O. B., C. P. Mckay, T. P. Ackerman, and K. Santhanam, 1989:: Rapid calculation of radiative heating rates and photodissociation rates in inhomogeneous multiple scattering atmospheres. J. Geophys. Res, 94 , 16287–16301.

    • Search Google Scholar
    • Export Citation

  • Wilks, D. S., 1995: Statistical Methods in the Atmospheric Sciences. Academic Press, 467 pp.

  • Wiscombe, W. J., 1977: The delta-Eddington approximation for a vertically inhomogeneous atmosphere. Tech. Note TN-121+STR, National Center for Atmospheric Research, Boulder, CO, 66 pp.

    • Search Google Scholar
    • Export Citation

APPENDIX

Matrix Elements of the Domain-Averaged Irradiance for a Multilayer Atmosphere

Using the notation of Toon et al. (1989), when an inhomogeneous cloud is placed in the jth layer, upward and downward irradiance for the layer are
i1520-0469-58-24-3797-ea1
where
i1520-0469-58-24-3797-ea3
Matching the irradiances at the top and bottom with those of neighboring layers provides 2n equations in 2n unknowns. Each of these equations has four unknowns. Eliminating either Y1 and Y2 for one of the layers results in three unknowns in each equations and, thus, leads to a tridiagonal matrix similar to that given by Toon et al. (1989). Following their notation, the tridiagonal matrix is in the form
i1520-0469-58-24-3797-ea5
The coefficients for l = 1 are
i1520-0469-58-24-3797-eqa1
The coefficients for odd values of l from 3 to 2N − 1 are
i1520-0469-58-24-3797-ea6
while those for even values of l from 2 to 2N − 2 are
i1520-0469-58-24-3797-ea7
For l = 2N the coefficients are
i1520-0469-58-24-3797-ea8
where Rsfc is surface albedo.
Elements for the cloud-free layers are
i1520-0469-58-24-3797-ea9
while for inhomogeneous cloud (jth layer), they are
i1520-0469-58-24-3797-ea13
and C±j(τj) are replaced by (16) and (17).

Fig. 1.
Fig. 1.

(a) Probability density functions of cloud optical thickness for gamma distributions with mean optical thickness of 15 and ν = 1.0, 1.5, and 2.0. (b), (c), and (d) show broadband top-of-the-atmosphere reflectance and top-of-the-atmosphere to surface transmittance as functions of mean optical thickness for distributions shown in (a) as computed by the algorithm using analytical integration and by numerical integration of the homogeneous solution weighted by the distributions in (a). Values for homogeneous clouds computed by the standard two-stream approximation are also shown for reference

Citation: Journal of the Atmospheric Sciences 58, 24; 10.1175/1520-0469(2001)058<3797:GWDOTS>2.0.CO;2

Fig. 2.
Fig. 2.

Difference of the atmospheric absorptance for homogeneous cloud and inhomogeneous cloud shown as a function of mean optical thickness for distributions shown in Fig. 1a

Citation: Journal of the Atmospheric Sciences 58, 24; 10.1175/1520-0469(2001)058<3797:GWDOTS>2.0.CO;2

Save