Methods for solving the radiative transfer equation. Part 3: Discreteordinate. 1. Discrete-ordinate method for the case of isotropic scattering.

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1 ecture Metos for sov te rtve trsfer equto. rt 3: Dscreteorte eto. Obectves:. Dscrete-orte eto for te cse of sotropc sctter..geerzto of te screte-orte eto for ooeeous tospere. 3. uerc peetto of te screte-orte eto: DSORT Requre re: : 6. Recoee re Tos G.E. K. Stes Rtve trsfer te tospere oce Cpter Dscrete-orte eto for te cse of sotropc sctter. A screte-orte eto s bee eveope by Crser bout 95 see Crser S. Rtve trsfer 96 Dover ubctos. Rec te rtve trsfer equto for zuty epeet ffuse testy: ' ' ' exp or sotropc sctter te sctter pse fucto s. Hece we ve ' ' exp [.] et s ppy te Guss foru to repce te ter Eq.[.] exp [.] ooeeous prt were - ters re te Guss wets costts re qurture es or pots.

2 Eq.[.] s syste of ooeeous fferet equtos: Souto of Eq.[.] eer souto prtcur souto were te eer souto s souto of te ooeeous prt of te Eq.[.] Deot te eer souto of Eq.[.] c be fou s sert Eq.[.3] to Eq.[.] we obt We c f te for exp [.3] [.] were s costt to be etere. Substtut ts expresso for Eq.[.] we ve Eq.[.5] ves soutos for -. [.5] Tus eer souto s exp [.6] were re costts. Te prtcur souto c be fou s exp [.7] were re costts. sert Eq.[.7] to Eq.[.] we ve [.8] ro Eq.[.8] s fou s γ

3 were γ s etere fro γ { } [.9] A te eer souto Eq.[.6] te prtcur souto Eq.[.7] we ve te souto γ exp exp [.] were re costts to be etere fro te boury cotos. H-fucto s bee trouce by Crser s H... [.] Oe c express γ te H-fucto tt Eq.[.] becoes H H exp exp [.] Eq.[.] ves spe souto for te se-fte sotropc tospere see :6.. H H [.3]. Geerzto of te screte-orte eto for ooeeous tospere. et s coser te tospere wt o-sotropc sctter. We c exp te ffuse testy te cose seres ϕ cos ϕ ϕ 3

4 So we ee to sove exp δ o Te eer souto y be wrtte exp φ φ re coeffcets to be etere. Te prtcur souto y be wrtte exp p Z Z s fucto H H Z ζ Te copete souto of te rtve trsfer s Z exp exp φ [.] - et s eerze te copete souto Eq.[.] of te rtve trsfer for te ooeeous tospere. Te tospere c be ve to te ooeeous yers ec s crcterze by se sctter beo pse fucto optc ept. OTE: f yer s ses erosos or cous oe ees to ccute te effectve optc propertes of ts yer.

5 or -t yer we c wrte te souto us Eq.[.]. To spfy ottos et s coser te zut epeet cse.e. so we ve φ exp Z exp [.5] ow we ee to tc te boury cotuty cotos betwee yers. At te top of te tospere TOA: o owwr ffuse testy [.6] At te yer s boury: upwr owwr testes ust be cotuous [.7] At te botto of te tospere ssu te ert surfce: rsur [ exp ] [.8] Eqs.[.6]-[.8] prove ecessry equtos to f te uow coeffcets. 3. uerc peetto of te screte-orte eto: DSORT DSORT s ORTRA uerc coe bse o te screte-orte eto eveope by Stes Wscobe et. DSORT s opey vbe s oo user-ue. DSORT ppes to te ooeeous ototer pe-pre tospere. A user y set-up y ubers of te pe-pre yers. 3 Ec yer ust be crcterze by te effectve optc ept se sctter beo syetry preter f te Heyey-Greeste pse fucto s use. A user y use y pse fucto by prov te eere poyo expso coeffcets. 5 A user seects uber of stres eep tt te coputto te vres s 3. 6 A ey probe s to obt souto for fuxes for stroy forwr-pee sctter. 7 DSORT ows prect te testy s fucto of te recto posto t y pot te tospere.e. ot oy t te boures of te yers. 5

Methods for solving the radiative transfer equation with multiple scattering. Part 3: Exact methods: Discrete-ordinate and Adding.

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