Principles of multiple scattering in the atmosphere. Radiative transfer equation with scattering for solar radiation in a plane-parallel atmosphere.

Transcription

1 Lectue 7 incipes of utipe scatteing in the atosphee. Raiative tansfe equation with scatteing fo soa aiation in a pane-paae atosphee. Objectives:. Concepts of the iect an iffuse scattee soa aiation.. Souce function an a aiative tansfe equation fo the iffuse soa aiation. 3. Singe scatteing appoxiation.. Legene poynoia expansion of the scatteing phase function. Requie eaing: L: Appenix E. Concepts of the iect an iffuse soa aiation. The soa aiation fie is taitionay consiee as a su of two istincty iffeent coponents: iect an iffuse: i if Diect soa aiation is a pat of soa aiation fie that has suvive the extinction passing a aye with optica epth an it obeys the Bee-Bougue-Labet extinction aw: i exp / [7.] whee is the soa intensity at a given waveength at the top of the atosphee an is a cosine of the soa zenith ange θ cosθ. θ Ι o The iect soa fux is i i exp / [7.]

2 . Souce function an a aiative tansfe equation fo the iffuse soa aiation. Diffuse aiation aises fo the ight that unegoes one scatteing event singe scatteing o any utipe scatteing. Ω Ω o o Singe scatteing Ω ' ' ' Mutipe scatteing Reca Lectue 3 whee we efine the souce function J j thea j scatteing / β e whee j thea is the thea eission j β B thea a T an j scatteing is the e-aiation fo utipe scatteing. Using the voue scatteing coefficient β s an the phase function we have β j Ω Ω Ω [7.3] s scatteing Ω Ω Ω OTE: Reca the scatteing phase function i.e. the eeent of the scatteing atix epesents the angua istibution of scattee enegy as a function of iection. By the efinition see Lectue3 it is noaize as whee Θ is the scatteing ange Ω cos Θ Ω cosθ cosθ'cosθ sinθ'sinθ cos'- / / cos'-

3 3 Thus the souce function fo iffuse soa aiation ay be witten as two coponents / exp ' ' ' ' ' ' J [7.] whee the is the singe scatteing abeo an is the scatteing phase function. OTE: n Eq.[7.] the fist te on the ight-han sie shows that the phase function eiects the incoing intensity in the iection to the iection an the integas account fo a possibe scatteing events within the soi ange. The souce function fo scatteing Eq.[7.] is oe copicate than a thea souce function: i t invoves conitions thoughout the atosphee whie the thea souce function epens on oca conitions ony ii The phase function ay be a vey copex function of the iections an in genea state of poaization. Reca the aiative tansfe equation efine in Lectue 3 fo a pane-paae atosphee J Thus using the souce function fo scatteing we can wite the aiative tansfe equation fo the iffuse aiation as oitting the subscipt if in / exp ' ' Ω Ω Ω Ω Ω Ω Ω Ω [7.5]

4 OTE: Eq.[7.5] is an intego-iffeentia equation. To sove Eq.[7.5] one nees to know the scatteing coefficient β s absoption coefficient β a an scatteing phase function as a function of waveength in each atospheic aye. Eq.[7.5] can be sipifie if thee is no epenency on the aziuth ange. o aziuthay inepenent case we ay efine the phase function as [7.6] Using Eq.[7.6] we can wite the aziuthay inepenent aiative tansfe equation fo the iffuse aiation ' ' ' exp / [7.7] To fin a soution of the aiative tansfe equation fo iffuse aiation i.e. to sove Eq.[7.5] o [7.7] vaious appoxiate an exact techniques have been eveope: Appoxiate ethos: i Singe scatteing appoxiations this ectue ii Two-stea appoxiations Lectue 8 iii Eington an Deta-Eington appoxiations Lectue 8 Exact ethos: i Discete-oinate technique Lectue ii Aing-oubing technique Lectue iii Monte-Cao technique Lectue

5 5 3. Singe scatteing appoxiation. f ight has been scattee ony once the souce function fo Eq.[7.3] becoes / exp J [7.8] an using the soution eive in Lectue 3 of the aiation tansfe in a pane-paae atosphee boune by on two sies at an : fo upwa intensity efecte J exp exp an ownwa intensity tansitte J exp exp we can wite the soution fo iffuse aiation in a singe scatteing appoxiation as ] [ exp exp [7.9a] ] [ exp exp [7.9b] Assuing that thee is no iffuse ownwa aiation at the top of the atosphee an no upwa iffuse aiation at the suface i.e. no efection fo the suface [7.]

6 6 Then fo Eq.[7.9ab] fo a finite atosphee with optica epth we fin the efecte an tansitte iffuse intensities exp [7.] an fo is OT equae to exp exp [7.] an fo exp [7.3] o the singe scatteing appoxiation the iffuse intensities ae iecty popotiona to the phase function. OTE: The singe scatteing appoxiation is vai fo the opticay thin atosphee i.e. sa optica epth. o << cae the singe scatteing appoxiation in eote sensing Eq.[7.] sipifies to Θ. Legene poynoia expansion of the scatteing phase function. o any pactica appications the phase function ust be nueicay expane in Legene poynoias with a finite nube of tes as Θ Θ cos cos [7.] whee Θ is the scatteing ange cosθ cosθ'cosθ sinθ'sinθ cos'- / / cos'- an is the expansion coefficients expesse as Θ Θ Θ cos cos cos [7.5]

7 OTE: Othogona popeties of the Legene poynoias: cosθ cosθ cos Θ fo k k k cosθ cosθ cos Θ fo k The fist few Legene poynoias ae given by: x x x x 3x 3 3 x 5x 3x x 35x 3x x 63x 7x 5x 8 7

8 8 Rewiting the aiative tansfe equation in tes of associate Legene poynoias: Eq.[7.] can be expesse in the tes of associate Legene poynoias cos [7.6] whee!! δ an δ is the Konecke eta: δ fo an othewise δ. n siia anne we ay expan the iffuse intensity in the cosine seies cos [7.7] Using Eqs.[7.6] an [7.7] an the othogonaity of the associate Legene poynoias the equation of the aiative tansfe fo the iffuse intensity Eq.[7.7] spits into inepenent equations in the fo / exp δ o [7.8] > aziutha inepenent case: o Eq.[7.6] the aziuth-inepenent phase function can be expesse as [7.9] o this case Eq.[7.8] sipifies to oitting the supescipt fo / exp o [7.]

9 Expansion of the Henyey-Geenstein phase function in the Legene poynoias The Henyey-Geenstein scatteing phase function is a oe phase function which is often use in aiative tansfe cacuations: HG cosθ g g g cosθ 3/ g is the asyety paaete. Let s take g.5 epesentative of aeosos Legene expansion: HG cos Θ cos Θ an HG cosθ cosθ cos Θ f > o cosθ cosθ cos Θ HG HG cosθ cos Θ OTE: o is aways because of noaization of the phase function cosθ cos Θ Thus in the case of one te in the expansion we have cos Θ cos Θ HG ot beow shows that using ony one te gives poo appoxiation OTE: n a pots beow the back cuve shows the Henyey-Geenstein phase function an the e cuve is fo the Legene expansion a as a function of cosθ. 9

10 f > o cosθ cosθ cos Θ HG HG cosθ cos Θ cosθ cosθ cos Θ cosθ cosθ cos Θ.5 HG HG Thus in the case of two tes in the expansion we have cos Θ cos Θ.5cos Θ - sti accuacy is not so goo! HG

11 We can continue by incuing oe tes to get a esiabe accuacy

12

13 7- gives goo appoxiation to the Henyey-Geenstein scatteing phase function with g.5. The age g the age nube of tes wi be equie to achieve acceptabe accuacy. 3