1. METO 621 Lesson 13

2. 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

3. Separation of the radiation field into orders of scattering Formal solutions to these equations are a sum of direct (I S ) and diffuse (I d )

4. Separation of the radiation field into orders of scattering

7. 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  is replaced with an appropriate expression. For example, in spherical geometry, with  Ch(   ) where Ch is the Chapman function

8. 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.

9. 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.

10. 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%

11. Two-stream Approximation- Isotropic Scattering Although anisotropic scattering is more realistic, first let’s look at isotropic scattering i.e. p=1 The radiative transfer equations are

12. 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

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

14. Two-stream Approximation The expressions for the source function, flux and heating rate are

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

16. The Mean Inclination If the radiation is isotropic then the average  is equal to 0.5 in both hemispheres. If the intensity distribution is approximately linear in  then the average is 0.666. We could also use the root-mean-square value For an isotropic field the average  is 1/√3. A linear variation yields a value of 0.71. Quite often, the value used is the result of trial and error.