  For 2n-stream radiation transfer theory, a stack of m clouds can be
represented as an equivalent cloud. Individual clouds, indexed by c = 1, 2, 3,
..., m are characterized by 2n x 2n scattering matrices S^{c}, that describe
how the cloud interacts with 2n streams of axially symmetric incoming
radiation, propagating in upward and downward Gauss-Legendre sample directions.
Some of the radiation is transmitted, some is absorbed and converted to heat
and some is scattered into 2n outgoing streams along the directions of the
incoming streams. The clouds are also characterized by 2n x 1 thermal source
vectors |J^dot^{c}} that describe the thermal emission of radiation along the
stream directions by cloud particulates and gas molecules. The 2n x 2n
scattering matrix of the equivalent cloud, S^{ev} = S^{m}*S^{m-1}
*...*S^{2}*S^{1}, is the Redheffer star product of the scattering matrices
S^{c} of the individual clouds. The 2n x 1 thermal source vector for the
equivalent cloud |J^dot^{ev}}=G^[m,m}|J^dot^{m}} + G^[m,m-1}|J^dot^{m,m-1}} +
... + G^[m,1}|J^dot^{1}} is a linear combination of the thermal source vectors
of the individual clouds. The 2n x 2n discrete Green's matrices G^[m,c} can be
constructed from the scattering matrices S^{c} of individual clouds. The
equivalent scattering matrix S^{ev} and the equivalent thermal source vector
|J^dot}^{ev}} are analogous to the equivalent resistance and the equivalent
electromotive force of Thevenin's theorem for a network of electrical circuits.
Illustrative numerical examples are given for single clouds, 3-cloud stacks and
10-cloud stacks. These methods are useful for modeling radiation transfer in
Earth's atmosphere, which can be represented by layers of invisible clouds,
consisting of clear air and greenhouse gases, or visible clouds which also
include condensed water, smog, etc.
