Chapter 11
Calculating Solar Radiative Characteristics
in Clouds with Asymptotic Formulas
of the Radiative Transfer Theory
Abstract Asymptotic formulas of the radiative transfer theory are presented for
thick atmosphere with weak absorption. The applicability region over the optical
thickness, single scattering albedo, and phase function asymmetry parameter is
discussed together with the approach exactness. The simplest model of cloud layer
is proposed for calculation of radiative characteristics.
11.1 The Basic Formulas
Let us consider the model of an extended and horizontally homogeneous cloud of a
big optical thickness t ) 1 as Fig. 11.1 illustrates. Here, the cloud layer is assumed
vertically homogeneous as well and the influence of the clear atmosphere layers
above and below the cloud layer is not taken into account. The volume coefficients
of scattering a and absorption k, linked with the cloud characteristics as k þ a
t 0 /Dz, a o 0 t 0 /Dz, k t 0 (1 À o 0 )/Dz, are used for the cloud description.
The optical properties of the cloud are described by the following parameters:
single scattering albedo (or the probability of photon surviving in a single interaction act) o 0 ; optical thickness t, and mean cosine of the scattering angle (asymmetry factor) g, which characterizes the Henyey-Greenstein phase function (Eq. 1.16,
Chap. 1). From the bottom the cloud layer adjoins the ground surface and its
reflectance is described by ground albedo A. The underlying atmosphere could be
taken into account if albedo A is implying as an albedo of the system “surface þ
atmosphere under the cloud”. Parallel solar flux F 0 is falling on the cloud top at
incident angle arccosm 0 . The reflected and transmitted radiance is observed at
viewing angle arccosm. The reflected radiance (in the units of incident extraterrestrial flux F 0 m 0 ) is expressed with reflection function r(t,m,m 0 ) and the transmitted
radiance (in the same units) is expressed with transmission function s(t,m,m 0 ).
At the sufficiently big optical depth within cloud layer far enough from the top
and bottom boundaries the asymptotic or diffusion regime is installed owing to the
multiple scattering. This regime permits a rather simple mathematical description.
I. Melnikova et al., Remote Sensing of the Environment and Radiation Transfer,
DOI 10.1007/978-3-642-14899-6_11, # Springer-Verlag Berlin Heidelberg 2012
105
Calculating Solar Radiative Characteristics
in Clouds with Asymptotic Formulas
of the Radiative Transfer Theory
Abstract Asymptotic formulas of the radiative transfer theory are presented for
thick atmosphere with weak absorption. The applicability region over the optical
thickness, single scattering albedo, and phase function asymmetry parameter is
discussed together with the approach exactness. The simplest model of cloud layer
is proposed for calculation of radiative characteristics.
11.1 The Basic Formulas
Let us consider the model of an extended and horizontally homogeneous cloud of a
big optical thickness t ) 1 as Fig. 11.1 illustrates. Here, the cloud layer is assumed
vertically homogeneous as well and the influence of the clear atmosphere layers
above and below the cloud layer is not taken into account. The volume coefficients
of scattering a and absorption k, linked with the cloud characteristics as k þ a
t 0 /Dz, a o 0 t 0 /Dz, k t 0 (1 À o 0 )/Dz, are used for the cloud description.
The optical properties of the cloud are described by the following parameters:
single scattering albedo (or the probability of photon surviving in a single interaction act) o 0 ; optical thickness t, and mean cosine of the scattering angle (asymmetry factor) g, which characterizes the Henyey-Greenstein phase function (Eq. 1.16,
Chap. 1). From the bottom the cloud layer adjoins the ground surface and its
reflectance is described by ground albedo A. The underlying atmosphere could be
taken into account if albedo A is implying as an albedo of the system “surface þ
atmosphere under the cloud”. Parallel solar flux F 0 is falling on the cloud top at
incident angle arccosm 0 . The reflected and transmitted radiance is observed at
viewing angle arccosm. The reflected radiance (in the units of incident extraterrestrial flux F 0 m 0 ) is expressed with reflection function r(t,m,m 0 ) and the transmitted
radiance (in the same units) is expressed with transmission function s(t,m,m 0 ).
At the sufficiently big optical depth within cloud layer far enough from the top
and bottom boundaries the asymptotic or diffusion regime is installed owing to the
multiple scattering. This regime permits a rather simple mathematical description.
I. Melnikova et al., Remote Sensing of the Environment and Radiation Transfer,
DOI 10.1007/978-3-642-14899-6_11, # Springer-Verlag Berlin Heidelberg 2012
105