METO 621Separation of the radiation field into orders of scatteringSlide 3Slide 4Slide 5Slide 6Slide 7Slide 8Lambda IterationSingle-scattered contribution from ground reflectionTwo-stream Approximation- Isotropic ScatteringSlide 12Slide 13Two-stream ApproximationThe Mean InclinationSlide 16METO 621Lesson 13Separation 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 approximationSeparation of the radiation field into orders of scattering),,(),,(),,(),,(),,(),,(by given are sintensitie range half thegeometry, slabIn φμτφμττφμτμφμτφμττφμτμ−−−+++−=−−=SIddISIddI Formal solutions to these equations are a sum of direct (IS) and diffuse (Id)Separation of the radiation field into orders of scattering)(),0,()( ),,'(')( ),*,(),,()( ),,'(')( ),,0(),,(/)'(*/)*(/)'(0/τφμτφμτμτφμτφμτφμτμτφμφμτμττττμττμτττμτSIISedIeIIISedIeIIdSdS==+=+=++−−+−−++−−−−−−−−∫∫Separation of the radiation field into orders of scattering][)(4),;,( ]1[)1(),,(][)(4),;,( ]1[)1(),,(get wefunctions source thesengsubstituti),;,(4)1(),,'( tosimplifies source thecase scattering single In the]/*/)*[(/0000/)*(//0000//'00000μτμττμτμττμτμτμτμτμμπφμφμμφμτμμπφμφμμφμτφμφμπφμτ+−−−−−+−−−−−±−++−+−−=−−−−+−−=±−+−=eepFaeBaIeepFaeBaIepaFBaSSdSdSSeparation of the radiation field into orders of scattering),,( ),,(),*,(),,(),,()()( ),,(),,0(),,(solution analytican get Then we absorber).perfect a isboundary lower that theassume we(if 0),*,( and)()(),,0( erms,boundary t twointroducemust weintensity totalobtain the To/)*(00//00φμτφμτφμτφμτφμτφφδμμδφμτφμφμτφμτφφδμμδφμμττμτμτ++−−++−−−−−−+−=+=+−−=+==−−=dddSdSIIeIIIeFIeIIIFISeparation 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 is replaced with an appropriate expression. For example, in spherical geometry, with Ch() where Ch is the Chapman functionSeparation 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.Lambda Iteration•Assume isotropic scattering in a homogeneous atmosphere. Then we can writextnxnnetdtedxESEdaSBaS/110/2*01)( where)'(|)'(|'2)(*)()1()(−∞−−∫∫∫==−++−=μτμμτττττττ This integral forms the basis for an iterative solution, in which the first order scattering function is used first for S.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%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 areBaIdaIdaIddI)1()',('2 )',('2),(),(1010−−−−=−+++∫∫μτμμτμμττμτμTwo-stream Approximation- Isotropic ScatteringBaIdaIdaIddI)1()',('2 )',('2),(),(1010−−−−=−−+−−∫∫μτμμτμμττμτμ• 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 equationsTwo-stream Approximation- Isotropic ScatteringBaIaIaIddIBaIaIaIddI)1()(2)(2)()()1()(2)(2)()(−−−−=−−−−−=−+−+−−++++τττττμτττττμIf 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 hemispheresTwo-stream Approximation•The expressions for the source function, flux and heating rate areBIIzFIIIIdFBaIIaBaIIdaSπαμτμτπαδδμτμμτμπμτμτμμπτμτμτμτμτμτ4)],(),([2and )],(),([2)],(),([2)()1()],(),([2 )1()],(),([2)(1010−+≈=+≈+=−++≈−++=−+−−++−+−+−+∫∫HThe Mean Inclination±±±±±==∫∫IFIdIdπμτμπμτμμπμμ2),(2),(2mean weightedintensity theas defined be could n,inclinatiomean the,1010But, 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.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∫∫==≡101022),(),()(μτμμτμμμμμIdIdrms• 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