METO 621

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METO 621

• Lesson 13

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Separation of the radiation field into orders of
scattering

• If the source function is known then we may
  integrate the radiative transfer equation
  directly, e.g. if scattering is ignored and we
  are only dealing with thermal radiation.
• This is also true if we can ignore multiple
  scattering , and consider single scattering
  approximation

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Separation of the radiation field into orders of
scattering
Formal solutions to these equations are a sum of
direct (IS) and diffuse (Id)
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Separation of the radiation field into orders of
scattering
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Separation of the radiation field into orders of
scattering
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Separation of the radiation field into orders of
scattering
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Separation of the radiation field into orders of
scattering

• Favorable aspects of the single scattering
  approximation are
• The solution is valid for any phase function
• It is easily generalized to include polarization
• It applies to any geometry as long as t/m is
  replaced with an appropriate expression. For
  example, in spherical geometry, with tCh(m0)
  where Ch is the Chapman function

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Separation of the radiation field into orders of
scattering

• It is useful when an approximate solution is
  available for the multiple scattering, for
  example from the two-stream approximation. In
  this case the diffuse intensity is given by the
  sum of the single-scattering and the approximate
  multiple-scattering contributions
• It serves as a starting point for expanding the
  radiation field in a sum of contributions from
  first- order, second-order scattering etc.

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Lambda Iteration

• Assume isotropic scattering in a homogeneous
  atmosphere. Then we can write

This integral forms the basis for an iterative
solution, in which the first order scattering
function is used first for S.
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Single-scattered contribution from ground
reflection

• The radiation reflected back from the ground is
  often comparable to the direct solar radiation.
• First order scattering from this source can be
  important
• Effects of ground reflection should always be
  taken into account in any first order scattering
  calculation.
• For small optical depths the ratio of the
  reflected component to the direct component can
  exceed 1.0, even for a surface with a
  reflectivity of 10

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Two-stream Approximation- Isotropic Scattering

• Although anisotropic scattering is more
  realistic, first lets look at isotropic
  scattering i.e. p1
• The radiative transfer equations are

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Two-stream Approximation- Isotropic Scattering

• In the two-stream approximation we replace the
  angular dependent quantities I by their averages
  over each hemisphere. This leads to the following
  pair of coupled differential equations

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Two-stream Approximation- Isotropic Scattering
If the medium is homogeneous then a is constant.
One can now obtain analytic solutions to these
equations. m in the above equations is the cosine
of the average polar angle. It generally differs
in the two hemispheres
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Two-stream Approximation

• The expressions for the source function, flux and
  heating rate are

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The Mean Inclination
But, of course, if we knew how the intensity
varied with t and m , we have already solved the
problem. Unfortunately there is no magic
prescription. In general, the value of the
average m will vary with the optical depth and
have a different value in each hemisphere.
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The Mean Inclination

• If the radiation is isotropic then the average m
  is equal to 0.5 in both hemispheres. If the
  intensity distribution is approximately linear in
  m then the average is 0.666.
• We could also use the root-mean-square value

• For an isotropic field the average m is 1/v3.
  A linear variation yields a value of 0.71. Quite
  often, the value used is the result of trial and
  error.