Section 3. Radiative Transfer

Transcription

1 Secton 3. Radatve Tanfe Refeence Kdde and Vonde Haa: chapte 3 Stephen: chapte, pp ; chapte 3, pp. 8-87, 99- Lou: chapte ; chapte, pp. 38-4; chapte 3, pp , 6-63; chapte 4, pp Lenoble: pat of chapte,3,5,6,7 "All of the nfomaton eceved by a atellte about the Eath and t atmophee come n the fom of electomagnetc adaton." Theefoe, we need to know how th adaton geneated how t nteact wth the atmophee 3. Electomagnetc Radaton Electomagnetc (EM wave ae geneated by ocllatng electc chage whch geneate an ocllatng electc feld. Th poduce a magnetc feld, whch futhe poduce an electc feld. Thee feld thu popagate outwad fom the chage, ceatng each othe. E H decton of popagaton. ELECTRIC FIELD E DIRECTION OF PROPAGATION MAGNETIC FIELD H EM adaton uually pecfed by wavelength ( dtance between cet fequency (ν numbe of ocllaton pe econd c/ wavenumbe ( ν o κ numbe of cet pe unt length (uually cm - / PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-

2 Can uually appoxmate the peed of lght n the atmophee by that n a vacuum. Howeve, the change n denty and humdty wth heght can caue gnfcant efacton (bendng of the EM ay whch mut be taken nto account n atellte pontng. The EM pectum can be dvded nto dffeent pectal egon, whch ae ueful n emote oundng: ultavolet (UV ~ to 4 nm vble ~ 4 to 7 nm nfaed ~ 7 nm to mm mcowave ~ mm to m ado ~ nclude mcowave to > m See fgue (.3 - the electomagnetc pectum. EM adaton cae enegy that can be detected by eno. The enegy pe unt aea pe unt tme flowng pependculaly nto a uface gven by the Poyntng vecto S: S c ε oe H unt of W/m ENERGY S (W / m whee ε vacuum pemttvty o E electc feld H magnetc feld E, H, and S ocllate apdly, o S dffcult to meaue ntantaneouly. Uually meaue the aveage magntude ove ome tme nteval: F S adant flux denty (W/m A m The adant flux denty edefned baed on the decton of enegy tavel: adant extance (M adant flux denty emegng fom an aea adance (E adant flux denty ncdent on an aea RADIANT EXITANCE, M IRRADIANCE, E PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-

3 Radaton alo a functon of decton. Th accounted fo by ung the old angle. old angle (Ω ght acenon! the aea of the pojecton onto a unt phee of an object, whee lne ae dawn fom the cente of the phee to evey pont on the uface of the object z OBJECT (AREA A PROJECTION ONTO UNIT SPHERE SOLID ANGLE Ω y x UNIT SPHERE AREA 4 π SPHERE OF RADIUS AREA 4 π Example: old angle of an object that completely uound a pont: Ω 4 π teadan ( old angle of an nfnte plane: Ω π old angle of an object of co ectonal aea A at dtance fom a pont: Ω A / Ω A da (ung factonal aea o d Ω 4π 4π Dffeental element of old angle gven by: whee µ co θ θ zenth angle 9 elevaton angle φ azmuth angle dω n θ dθ dφ dµ dϕ See fgue (K&VH mathematcal epeentaton of old angle. We can now defne adance o ntenty (I o L adant flux denty pe unt old angle S I W m - - d Ω PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-3

4 Th the mot mpotant of the adometc tem that we have defned. Stctly, the adance epeent the EM adaton leavng o ncdent upon an aea pependcula to the beam. Fo othe decton, t mut be weghted by co θ. What the adant extance, M, (.e., the total amount of adaton leavng the uface fo a mall uface aea emttng adance I? π ππ/ π M I( θ, ϕcoθdω I( θ, ϕcoθnθdθdϕ I( µ, ϕ µ dµ dϕ.e., The adant extance M obtaned by ntegatng the adance I ove a hemphee Fo I ndependent of decton (otopc: M ππ/ I co θ n θ dθ dϕ π I. The adaton may be wavelength dependent. pefx the enegy-dependent tem by "monochomatc" o "pectal". Symbol ae ubcpted accodngly, e.g., I, I ν, I ν efe to I pe unt, ν, ν. Unt: e.g. monochomatc adance: I n W m- - nm - Iν n W m- - Hz - Iν n W m- - (cm - - The total adance then the ntegal ove, ν, o ν of the monochomatc adance. I I ν d Iνd I ν dν o dν ν I Iν d c I ν I ν I ν ν I ν ν I ν.e., The adaton pe unt nteval the ame a adaton pe unt ν o ν nteval, f the ze of the nteval ae accounted fo. Radance a ueful quantty fo atellte meauement becaue t ndependent of dtance fom an object a long a the vewng angle and the amount of ntevenng matte do not change. (Both the adance and the old angle deceae wth o I eman contant. See table (K&VH 3. - adaton ymbol and unt. PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-4

5 3. Blackbody Radaton All matte emt adaton f t at a tempeatue > abolute zeo. A blackbody a pefect emtte - t emt the maxmum poble amount of adaton at each wavelength. A blackbody alo a pefect abobe, abobng at all wavelength of adaton ncdent on t. Theefoe, t look black. Planck' Blackbody Functon No eal mateal ae pefect blackbode. Howeve, the adaton nde a cavty (whoe wall ae opaque to all adaton the adaton that would be emtted by a hypothetcal blackbody at the ame tempeatue. The cavty wall emt, abob, and eflect adaton untl equlbum eached. Planck potulated that atom ocllatng n the wall of the cavty have dcete enege gven by E n h ν whee n ntege (quantum numbe h Planck' contant ν fequency A quantum of enegy emtted when an atom change t enegy tate then E h ν ( n. Ung thee two aumpton, Planck deved the blackbody functon, decbng the adance emtted by a blackbody 5 hc B (T hc exp kt whee B monochomatc adance (W m - - µm - k Boltzmann' contant T abolute tempeatue 5 c Th can be wtten a: B (T c exp T whee c ft adaton contant (.9-6 W m - - c econd adaton contant ( m K PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-5

6 The adance emtted by a blackbody depend only on and T. B (T nceae wth tempeatue the of maxmum B (T deceae wth tempeatue B Sketch B v. T > T > T > T 3 4 cuve fo: Seveal othe elaton can be deved fom Planck' blackbody functon. ( Wen' Dplacement Law The wavelength at whch B (T a maxmum detemned ung B (T Th gve m µm, fo T n K th Wen' dplacement Law. T The hotte the object, the hote the of t maxmum ntenty (e.g., element on a tove. Th law can be ued to detemne the T of a blackbody fom the poton of the maxmum monochomatc adance. ( Stefan Boltzmann Law The monochomatc adant extance mply M (T π B (T becaue the blackbody adance otopc (ndependent of decton. The total adant extance fom a blackbody then 5 π c 4 M(T M (Td πb (Td... T 4 5 c M BB (T 4 σt o 4 σt B (T th the Stefan Boltzmann Law. π whee σ Stefan Boltzmann contant W m - K -4 Th tate that the total amount of adaton emtted fom a gven uface aea popotonal to T 4. PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-6

7 (3 Raylegh Jean Appoxmaton At longe wavelength, the Planck blackbody functon can be mplfed. Fo n the mcowave egon, c / T << fo T elevant to Eath. c c c T exp +, gvng B (T 4 T T c th the Raylegh Jean Appoxmaton. It tate that n the mcowave egon, the adance mply lnealy popotonal to T. Often the adance wll be caled by (c /c -4 to get the bghtne tempeatue, T b, whch adance expeed n unt of tempeatue. T b the tempeatue equed to match the meaued ntenty to the Planck blackbody functon at a gven. T b alo ued n the nfaed (efeed to a the equvalent bghtne tempeatue, but t mut be deved fom the blackbody functon. (4 Kchoff' Radaton Law Thu fa, we have been dcung only blackbody adance. In ode to quantfy how cloely a mateal appoxmate a blackbody, the emttance o emvty of the mateal defned. emtted adance at I emttance ε blackbody adance at B (T Fo a blackbody, ε. Fo othe mateal, called gey bode, ε <. Can alo defne: aboptance α abobed adance at / ncdent adance at eflectance R, ρ eflected adance at / ncdent adance at tanmttance τ tanmtted adance at / ncdent adance at Thee thee quantte decbe the thee poblte fo ncdent adaton. All have value between and. By the conevaton of enegy, α + R + τ. Kchoff' adaton Law tate that fo a body n local themodynamc equlbum (havng a ngle tempeatue tue n atmophee below ~4 km (Lou; km K&VH α ε PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-7

8 Smple Applcaton to Atmophec Remote Soundng See table (Stephen. - popete of blackbody emon. Conde thee wavelength utable fo emote oundng of tempeatue: 4.3 µm n CO abopton band 5 µm n CO abopton band 5 mm n O abopton band Relatve emon of enegy: 5 µm appea to be the bet fo emote oundng at both K and 3 K Sentvty of the emon to a mall change n tempeatue (elated to db /dt: 4.3 µm bette fo meaung wame T, but 5 µm tll bet fo lowe T So fa, the 5 mm egon emt the leat enegy and leat entve to tempeatue. Howeve, the optmum fo emote oundng of tempeatue alo detemned by facto othe than the avalable enegy and the ntument entvty, uch a the atmophec tanmon. Typcal tanmon popete of cloud: cloud tongly abob adaton at 4.3 and 5 µm, but ae almot tanpaent at 5 mm at both K and 3 K (See Smth, Bull. Am. Met. Soc., 53, 74, 97. PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-8

9 3.3 Radatve Tanfe Equaton Now we can dcu the tanfe of EM adaton though the atmophee. Conde the adaton ncdent on a dffeental volume of mateal (atmophee. d I I + d I Fou pocee wll change the ntenty of the EM adaton a t pae though the volume: A abopton fom the beam (depleton tem B emon by the mateal (ouce tem C catteng out of the beam (depleton tem D catteng nto the beam (ouce tem The change of ntenty wth dtance : di d A + B + C + D. Let' conde each of thee fou tem. (A The abopton tem gven by the Bee-Bougue-Lambet Law whch tate that the change n ntenty : di σa( I d ρk a( I d [.e., tem A σa ( I ρk a( I ] whee σ ( volume abopton coeffcent (m - a k a ( ma abopton coeffcent o co ecton (m kg - o cm molecule - ρ denty of the abobng mateal (kg m -3 o molecule cm -3 Integatng ove a fnte dtance a to b: b I (b I (a exp σa( d I (a τ (a, b a whee τ (a, b tanmttance between a and b α, wth no catteng. PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-9

10 (B The emon tem gven by Kchoff' Law, becaue a mateal a good an emtte a t an abobe. tem B σ B (T ρk ( B (T a ( a (C The catteng nk tem alo follow the fom of Bee' Law. tem C σ ( I ρk ( I whee σ ( volume catteng coeffcent (m - k ( ma catteng coeffcent o co ecton (m kg - o cm molecule - (D The catteng ouce tem moe complex becaue adaton catteed nto the beam fom all decton. ππ σ ( tem D + I( θ', φ' p( ψ n θ' dθ' dφ' 4π whee (θ',φ' epeent the decton of the ncomng adaton (θ,φ epeent the decton of the ognal beam ψ catteng angle angle between (θ,φ and (θ',φ' p( ψ catteng phae functon, whch pecfe what poton of the adaton fom decton (θ',φ' catteed nto decton (θ,φ Note: ψ fo catteng n the fowad decton 8 ψ fo catteng n the backwad decton SCATTERING ANGLE ππ p( ψ fo an otopc cattee and p( ψ n θ' dθ' dφ' 4 π Can egad tem D a the poduct of σ ( and a dectonally weghted aveage of I,.e., tem D σ ( I'. Now thee fou tem can be combned to get the adatve tanfe equaton (RTE fo unpolazed adaton: di σa( I ( θ, φ + σa( B (T σ( I ( θ, φ d σ( + 4π ππ σ ( I ( θ, φ + σ ( B a I ( θ', φ' p( ψ n θ' dθ' dφ' a (T σ ( I ( θ, φ + σ ( I' [RTE] PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-

11 Reaangng di σa( d [ B (T I ( θ, φ ] + σ ( [ I' I ( θ, φ ] [RTE] Let' apply th expeon to a beam of adaton popagatng upwad though a thn laye of atmophee towad a atellte ntument. The ft tem σ ( [ B (T I ( θ, φ ] a epeent the effect of abopton and emon. f σ a (, then the laye tanpaent wth epect to adaton f σ a (, then the tempeatue of the laye and the ncdent ntenty detemne the change of ntenty aco the laye The econd tem σ ( [ I' I ( θ, φ ] epeent the effect of catteng. f σ (, then no catteng patcle ae peent o no catteng occu f σ (, then the ntenty of the beam nceae fo I' I ( θ, φ > deceae fo I' I ( θ, φ The full RTE can be fomally olved, but we wll mplfy t fo ue n emote oundng applcaton. < Reaange RTE: di [ σa( + σ( ] I ( θ, φ + σa( B (T + σ d σ ( I ( θ, φ + σ ( B (T + σ ( I' e whee σ σ ( + σ ( volume extncton coeffcent e( a Smlaly, k ( k ( + k (. e a a ( I' [RTE3] Futhe ntoducng ~ σa ( α abopton numbe σ ( e ~ σ ( ω ngle catte albedo σ ( σe( ρk e( and d dz / co θ dz / µ yeld di I ( θ, φ + α~ B (T + ω~ I' σ ( d e µ di ρk ( dz e I ( θ, φ + α~ B (T + ω~ I' e [RTE4] PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-

12 whee I' 4π ππ I ( θ', φ' p( ψ n θ' dθ' dφ' 4π π I ( µ ', φ' p( ψ dµ' dφ' z SATELLITE z dz θ d dz / co θ dz / µ z Some textbook expe the RTE n tem of the optcal depth δ athe than the vetcal dtance z, whee δ δ lant vetcal (, (z, z z z σ (, d e σ (, zdz e µ δ lant (, Let δ (, z.e., optcal depth fom the uface to heght z. δ vetcal Then dδ σ ( dz µσ ( d ρk ( dz µρk ( d. e e e e We can now ewte the RTE a di µ I ( µ, φ + α~ B (T + ω~ I' dδ whee (agan π I' I( µ ', φ' p( ψ dµ' dφ' 4π [RTE5] Th a geneal fom of the adatve tanfe equaton. It can be mplfed n eveal way to gve veon that ae vey ueful n emote oundng. PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-

13 Smplfed Cae #: The RTE wth No Scatteng The aumpton of neglgble catteng n the atmophee vald at nfaed wavelength when no cloud ae avalable (aumng local themodynamc equlbum. No catteng: ωc αc di The RTE become µ I ( µ, φ + B (T [RTE6] dδ Th can alo be wtten ung ρ ( dz o σ ( dz ntead of d δ. k e e Th Schwazchld' Equaton, a fundamental elaton fo adatve tanfe n the atmophee. It a ft ode, lnea, odnay dffeental equaton whch can be ntegated to calculate the nfaed adance een by a atellte ntument. Schwazchld' Equaton can alo be deved fom a condeaton of the tanfe of enegy between two tate n a molecule. Th eult n a moe geneal fom of the equaton, whee the blackbody tem B eplaced by a moe geneal emon tem J. The ubttuton of B fo J vald n the lowe atmophee, whee Local Themodynamc Equlbum hold. The ft tem on the RHS decbe the deceae n adance due to abopton, and the econd tem on the RHS decbe the nceae n adance due to blackbody emon. z I ( at δ at z at at SATELLITE δ + dδ δ z + dz z + d z,, δ θ I ( SURFACE Let olve Schwazchld Equaton. Each textbook ha t own appoach, a the path fom the gound to the atellte can be epeented by, z/µ, δ lant, o δ vetcal /µ. The followng baed on Kdde & Vonde Haa. PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-3

14 Ft multply both de of the equaton by exp [ δ / µ ]/ µ, and eaange. Keep n mnd that δ δ vetcal (, z the vetcal optcal depth fom the uface to heght z, wth δ deceang a z nceae. δ di δ δ exp + exp I ( µ, φ exp B (T µ dδ µ µ µ µ Th can be wtten a: d δ exp I( µ, φ dδ µ δ exp µ µ B (T Integate th fom the uface ( δ δ (, to the atellte ( δ z δ z (, zat δat δat Th gve: d dδ, ung δ a the vetcal coodnate: δat δ δ dδ exp I ( µ, φ dδ exp B (T µ µ µ δ exp µ I ( µ, φ δ at δ δ at δ exp µ at δat δ exp Iat Io exp µ µ whee I o adance leavng the uface I at adance eachng the atellte B B dδ (T µ dδ (T µ Reaangng to olve fo I at : I at I o δ exp µ at + δ at B (T exp δ at δ µ dδ µ [RTE7a] Altenatve oluton n tem of z and ae: I ( at I I ρk ( exp ( exp e ( z µ at + at ρk e( B µ (T exp ρk e( z dz µ at ( ρk e( at + ρk e( B (T exp[ ρk e( ] z d [RTE7b] Intoduce the vetcal tanmon between two optcal depth: δ, δ exp( δ τ ( δ PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-4

15 Then defne τ τ ( δ, δat.e., τ the vetcal tanmon of the atmophee fom level δ to the atellte, wth τat τ (, δat a the vetcal tanmon fom the uface to the atellte. (Note: th can be confung becaue ometme τ defned a τ δ e. Subttutng th nto RTE7 gve the followng oluton to Schwazchld Equaton: dτ I ( µ µ at Io τat + B (T τ [RTE8] µ τat Fo ovehead vewng (θ, µ, th mplfe to: dτ I at Ioτat + B (T [RTE9] µ τ at Now the phycal ntepetaton become obvou. ft tem uface adance atmophec tanmttance to the atellte econd tem adance emtted by each laye tanmttance fom laye to atellte Th equaton fundamental to the ntepetaton of atellte meauement of adance n tem of atmophec tempeatue and compoton. Smplfed Cae #: The RTE wth Scatteng but No Emon The ntepetaton and ue of the RTE fo the catteng cae moe complex. Howeve, n the ultavolet, vble and nea nfaed, nethe the Eath no t atmophee emt gnfcant adaton, allowng the RTE to be mplfed by doppng the blackbody emon tem. π di ω~ Thu: µ I ( µ, φ + ω~ I' I( µ ', φ' p( ψ dµ' dφ' dδ 4π In ode to olve th equaton, need to fomulate an expeon fo elevant catteng pocee. Fo moe detal of th devaton, ee Kdde & Vonde Haa, Secton ω ~ baed on the Now want to look at the paamete that ae needed n ode to apply the RTE: abopton and catteng popete of the atmophee eflecton popete of the Eath' uface chaactetc of the Sun' adaton PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-5

16 3.4 Gaeou Abopton and Spectocopy Abopton and emon pecta povde a mean of dentfyng and meaung the compoton of the atmophee. Inteacton of Radaton wth Gae Thee ae fou pmay way n whch adaton can nteact wth atmophec gae; each eult n a chaactetc abopton pectum. ( Ionzaton - docaton adaton tp an electon fom an atom o molecule, o beak apat a molecule poduce elatvely mooth pecta UV and vble pecta ae chaactezed by the abopton co ecton: k a σa / ρ (cm /molecule mpotant band fo the detecton of ozone (Hatley and Chappu band ( Electonc tanton valence electon jump between enegy level wthn an atom o molecule alo chaactezed by the abopton co ecton, whch ha ome tuctue UV and vble alo mpotant fo the detecton of ozone (Huggn band (3 Vbatonal tanton a molecule change vbatonal enegy level, uually fom the gound tate to the ft excted tate poduce dcete pectal lne nfaed (4 Rotatonal tanton a molecule change otatonal enegy level poduce dcete pectal lne fa nfaed and mcowave (pue otatonal lne In pactce, the IR pectum of many molecule due to a combnaton of vbatonal and otatonal tanton. CO and H O ae the mot mpotant abobe n th egon See fgue - UV-vble abopton co ecton. PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-6

17 Lne-Boadenng Pocee Thee vbatonal-otatonal tanton occu at dcete wavelength and ae boadened by thee pocee: ( Natual lne boadenng due to uncetante n the enegy level. only mpotant n the uppe tatophee and meophee ( Peue (o Loentz boadenng due to collon between molecule whch dtot them and caue abopton at lghtly dffeent fequence. mot elevant to the lowe atmophee below 4 km The ma abopton coeffcent of a Loentz-boadened lne S αl k a ( ν π ( ν νo + αl whee S lne tength, a functon of tempeatue and lowe tate enegy E" ν o cental wavenumbe α L Loentz half-wdth (HW at HM o p To αl (T, p α L (To, po po T whee T o and p o efeence tempeatue and peue (73.5 K, 3.5 mba Evey Loentz-boadened lne can be pecfed by fou paamete: ν o, S, See fgue (K&VH Loentz lne at uface and topopaue. o α L, E". (3 Dopple boadenng due to the moton of molecule. mot elevant to the atmophee above 4 km, becomng compaable to Loentz boadenng at 4 km The ma abopton coeffcent of a Dopple-boadened lne S ( ν ν o k ν a( exp πα D αd whee α D Dopple lne-wdth HWHM / ln (o HWHM αd ln kt νo αd(t M c whee k Boltzmann' contant M molecula ma PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-7

18 Note: Dopple lne ae moe ntene at the cente and weake n the wng than Loentz lne. See fgue - Loentz and Dopple lne. The nfluence of Loentz and Dopple boadenng can be combned n a convoluton functon called the Vogt lnehape. Loentz and Dopple-boadened vbatonal-otatonal IR lne ae the ba fo tempeatue oundng and meauement of the concentaton of many tace gae n the atmophee. e.g., 4.3 µm and 5 µm CO tempeatue oundng 6.3 µm H O wate vapou meauement 9.6 µm ozone total ozone meauement alo NO, N O, NO, HNO 3, HCl, HF, CH 4, CO, and CO Applcaton of Bee' Law Pactcal applcaton of UV-vble abopton co ecton and IR ma abopton coeffcent can be demontated ung Bee' Law. I I I di di I ( ρk a ρk ( I a d ( d I ( exp ρk a( d I ( τ(, Meauement of the adance I ( ncdent and I ( outgong, allow the monochomatc tanmon functon τ, to be detemned. Typcally, n the UV-vble egon: ntoduce optcal ma u( ( u ( ρ, d (unt molecule cm - o τ (, exp k ( du a exp( k a( u(, fo k a ( contant fom to u( meaue k a ( fo a ga n the laboatoy - typcally only weakly dependent on T and p thu the total optcal ma (o lant column abundance of the ga can be detemned PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-8

19 Smla calculaton can be done n the IR egon, except that k a ha a tonge dependence on T and p though Loentz and Dopple boadenng. have τ (, exp u( k a u( (, T, pdu thee ae vaou appoxmaton fo the tanmon along uch an nhomogeneou path e.g., τ (, exp( k (, T, p u ~ a o o whee m p To u~ u p a caled optcal ma o T m contant dependent on the ga T, p tempeatue and peue epeentatve of the path The accuacy of abundance and tempeatue deved n th way clealy depend on the accuacy of the UV-vble abopton co ecton and the IR lne paamete, whch ae typcally only accuate to 5-%. Intumental Condeaton Stctly, the RTE and tanmon functon apply to monochomatc cae. Howeve, typcal ntument meaue ove ome pectal egon, whch may nclude eveal IR lne, fo example. In th cae, the meaued adance become I meaued f(i d whee f nclude the ntument eoluton functon and the pectal epone functon of the ntument ove t bandpa fom to. Such calculaton of adance ae mot accuately done "lne-by-lne", calculatng I at many cloely paced wavelength and then ntegatng ove the ntument bandpa. Thee calculaton ae vey ntenve, and ae theefoe often appoxmated by band model whch ae ued to calculate the tanmon ove an ente band ung ome utable paametezaton. See fgue (K&VH 3., 3., 3.3, tanmttance of gae n Eath' atmophee. PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-9

20 3.5 Atmophec Scatteng Th a complex topc - we wll jut touch on the bac. Radaton catteed fom a patcle depend on: patcle hape patcle ze patcle ndex of efacton wavelength of the ncdent adaton vewng geomety Me (98 appled Maxwell' EM equaton to the cae of a plane EM wave ncdent on a phee, and howed that the catteed adaton fo a phee depend only on: vewng angle ndex of efacton ze paamete χ π /, whee adu of the phee Even Me catteng an appoxmaton fo mot patcle, nce they ae nonphecal. χ povde a mean fo defnng thee catteng egme: ( Me catteng:. < χ < 5 compaable to the ccumfeence of the patcle adaton nteact tongly wth the patcle full Me equaton mut be ued ueful n the detecton of andop ung ada, cloud dop n the IR, and aeool (moke, dut, haze n the vble See fgue (K&VH catteng egme. Seveal tem can be defned n ode to chaacteze the catteng: ( Scatteng effcency Q total catteed adaton / ncdent adaton volume abopton coeffcent / (numbe denty co-ectonal aea of patcle σ k ρπ π See fgue (K&VH 3. - catteng effcency of wate phee. Th plot of the catteng effcency how a ee of maxma and mnma, wth malle pple upempoed. The maxma and mnma ae caued by ntefeence between the lght dffacted and tanmtted by the phee, and the pple due to ay that gaze the phee and deflect enegy n all decton. PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-

21 Note alo that the catteng effcency can be geate than one, and that t tend to ocllate about a value of a χ. Th called the extncton paadox. Q mean that a lage patcle emove twce a much lght fom the ncdent beam a t ntecept. A puely geometcal condeaton of extncton would ugget a lmtng value of detemned by the amount of adaton blocked by the co-ectonal aea of the patcle. Howeve, the edge of the patcle caue dffacton whch concentated n a naow lobe about the fowad decton and contan an equal amount of enegy to that ncdent on the co ecton of the patcle. The lght emoved fom the fowad decton of the ncdent beam thu cont of a dffacted component that pae by the patcle, and a catteed (o blocked component that undegoe eflecton and efacton nde the patcle.. EXTERNAL REFLECTION 4. TWO INTERNAL REFLECTIONS LIGHT RAYS. TWO REFRACTIONS. DIFFRACTION 3. ONE INTERNAL REFLECTION LIGHT RAYS SCATTERED BY A SPHERE Note: fo N( d the numbe of dop of adu to + d pe unt volume, the Me catteng co ecton σ ( π Q N( d PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-

22 ( Complex ndex of efacton account fo both catteng and abopton m n - n' whee n eal pat peed of lght n a vacuum / peed of lght n the medum n' magnay pat whch account fo abopton nde the catteng patcle (mpotant n the IR n' -. (µ m See fgue (K&VH magnay pat of the ndex of efacton of wate and ce. ( Scatteng phae functon detemne the decton n whch the adaton catteed peak n the fowad decton a χ nceae χ χ χ See fgue (K&VH 3. - pola plot of the catteng phae functon of wate dop. ( Geometc (optc catteng: χ > 5 much malle than the patcle adaton can be teated ung geometc optc, tacng the eflecton and efacton of ay at the uface of the phee fo nonabobng phee, Q appoache a χ nceae becaue adaton tkng the phee catteed and adaton appoachng the phee dffacted aound t ued to detect dzzle and andop, to detect cloud dop n the nfaed, and to explan anbow and halo PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-

23 (3 Raylegh catteng: χ <. much lage than the patcle th allow the Me equaton to be mplfed the catteng become ndependent of patcle hape a molecule (not patcle catte UV and vble adaton, o Raylegh catteng mut be taken nto account when obevng at thee wavelength (e.g. etevng ozone n the UV The Raylegh catteng co ecton, n cm /molecule, can be calculated a 3 8π [ no( ] σ( f( δ 4 3 N whee n o ( the eal pat of the efactve ndex of tandad a, N the numbe denty n molecule cm -3 f(δ.6 a coecton fo anotopy 4 σ Thu σ ρ Th mpotant becaue t mean that adaton of hote catteed moe. Ade: The ky blue becaue the of blue lght (~45 nm hote than that of ed lght (~65 nm. Thu blue lght catteed 5.5 tme moe than ed lght. The ky ed at unet becaue moe blue lght catteed out of the long atmophec path, leavng ed lght n the dect path to the vewe. See fgue (K&VH 3. - vetcal tanmon though the atmophee due to Raylegh catteng. See fgue ("" - molecula v. aeool catteng. PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-3

24 In moe detal, followng the appoach of Wtt et al. The Raylegh molecula catteng co ecton n cm /molecule gven by Penndof 3 3π [ no( ] 6 + 3ρn( σr( 4 3 N 6 7ρn( o whee ( the efactve ndex of tandad a, baed on Equaton ( of Edlen, and n o N o Lochmdt numbe, cm -3 (fo 5 C and ρ n.35 [Penndof] o o Note that: [ n ( ] [ n ( ] The depolazaton facto : ρ ( n FK ( F ( K The Kng coecton facto fo a gven by (e.g., Bate F whee ν wavenumbe n vacuum (cm -. vac - - K ( (5.38 νvac + (.34 ν 4 vac The Raylegh catteng phae functon calculated ung (Goody and Yung, p P R( θ [ + ρn + ( ρn co θ]. + ρ Refeence n G. Wtt, J.E. Dye, and N. Wlhelm, Rocket-Bone Meauement of Scatteed Sunlght n the Meophee, J. Atmo. Te. Phy., 38, 3 (976. R. Penndof, Table of the Refactve Index fo Standad A and the Raylegh Scatteng Coeffcent fo the Spectal Regon between. and. µ and The Applcaton to Atmophec Optc, J. Opt. Soc. Am., 47 (, 76 (957. B. Edlen, The Dpeon of Standad A, J. Opt. Soc. Am., 43 (5, 339 (953. D.R. Bate, Raylegh Scatteng by A, Planet. Space Sc., 3 (6, 785 (984. R.M. Goody and Y.L. Yung, Atmophec Radaton, Second Edton, Oxfod Unvety Pe, Oxfod ( PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-4

25 Cloud and Scatteng Cloud cont of wate dop and ce cytal of adu µm. cloud dop < dzzle < an dop µm < µm < µm ( mm The catteng behavou of cloud depend on wavelength. Fo a km thck (tatu cloud wth the un ovehead: vble ~.% adaton abobed * (.55 µm ~79.8% catteed out the top of the cloud (geometc optc ~% catteed out the bottom of the cloud * Lqud wate doe not abob much n the vble. Becaue all vble ae catteed equally well, due to the dop ze dtbuton and lage χ, cloud ae whte. nea nfaed ~% abobed (lqud wate abob moe n the nea IR (aveaged ove IR ~73.8% catteed out the top (geometc optc ola pectum 6.% catteed out the bottom nfaed cloud dop ae Me cattee but abob vey tongly ( µm act almot a blackbode mcowave catteng by cloud dop neglgble, abopton mall (τ 9% fo non-anng cloud but anng cloud nteact moe wth adaton, povdng a mean of detectng pecptaton 3.6 Suface Reflecton Remote oundng of the atmophee and uface often ele on meauement of eflected adaton. Analy of th adaton eque knowledge of the eflecton popete of the uface. Seveal tem ae ued to decbe thee eflecton popete. All the followng tem ae dependent (monochomatc. Bdectonal eflectance ( γ o R elate the adaton eflected n decton ( θ, φ to the adaton ncdent n decton θ, φ. ( PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-5

26 SUN I z SATELLITE I θ θ φ φ I d Ω z d Ω I θ θ da PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-6

27 The adance eflected fom a mall element of uface I ( θ, φ ππ/ I ( θ, φ γ ( θ, φ ; θ, φ co θ n θ dθ dφ adaton I co θ fom decton ( θ, φ ncdent on the uface aea da I co θ avalable to be eflected facton γ eflected nto decton ( θ, φ ntegatng ove all ncdent old angle ( dω n θdθdφ gve the total eflected adance I ( θ, φ Note that γ unchanged f the decton of ncomng and outgong adance ae wtched: γ ( θ, φ; θ, φ γ ( θ, φ; θ, φ Th the Helmholtz ecpocty pncple. One of the mot commonly ued eflecton popete the albedo A. M adant extance due to eflecton A (no unt E adance whee A. A ππ/ ππ/ I ( θ, φ co θ I ( θ, φ co θ n θ dθ dφ n θ dθ dφ Conde the mplfed cae of dect-beam ola adaton (come dectly fom the Sun wth no catteng whch come fom a naow ange of angle: ( Iun co θundω E IunΩ un co θun un ncdent adance whee Ω un old angle of the Sun ubtended at Eath ( I Ω co θ γ ( θ, φ ; θ, φ I ( θ, φ eflected ntenty un un un un un Thu: γ ( θ, φ ; θ un, φ un γ I un I E un I ( θ, φ Ω un co θ un PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-7

28 In addton, M I un Ω un coθ un ππ/ γ ( θ, φ ; θ un, φ un coθ nθ dθ dφ A M E ππ/ γ ( θ, φ ; θ un, φ un co θ n θ dθ dφ π γ co θ dω o A a functon of ola decton (though Two lmtng cae fo eflectng uface: ( a Lambetan o otopc eflecto co θ un. eflect adaton unfomly n all decton "pefectly ough" A γ contant fo all decton π π A a all of the ncdent adaton catteed e.g., now, flat whte pant ( a pecula eflecto eflect adaton n one decton only: θ θ and φ φ + π "pefectly mooth" lke a mo e.g., the uface of wate whch peudo-pecula becaue t ough (efeed to a un glnt o un gltte Note: All tem above ae monochomatc (hould have ubcpt. They can be ntegated ove the bandpa of a atellte ntument, but E and M mut be ntegated epaately befoe calculatng A. Typcal value of A ntegated ove the ola pectum: bae ol. -.5 deet and ga foet. -. clean now ea uface (un > 5 above hozon appoache Lambetan <. appoache pecula PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-8

29 One addtonal eflecton tem: the anotopc eflectance facto (ξ : often ued to pecfy the behavou of eal catteng uface can make uface that have ξ fo many wavelength and angle π ξ ( θ, φ ; θ, φ γ ( θ, φ ; θ, φ A Fo a Lambetan uface, ξ ( π / ( π /. Fo a non-lambetan uface, ξ the ato of the uface adance to that of a Lambetan uface havng the ame albedo A. ξ > f the uface eflect moe than a Lambetan uface ξ < f the uface eflect le than a Lambetan uface Fo any ncdent decton θ,, ππ/ ( un φun ξ ( θ, φ ; θ, φ co θ n θ dθ dφ π. un un 3.7 Sola Radaton The ouce of mot adaton ued n emote oundng the Sun. Mot of the ola EM adaton eachng Eath emtted by the photophee whch a gaeou about 5 km thck. It ometme called the uface of the Sun and concde wth t vble dk. The tempeatue n the photophee ange fom 4 to 8 K, and the emtted adaton appoxmate a Planck blackbody cuve at ~6 K. Th adaton tavee the chomophee (eveal km thck, 4 to,, K and the coona, whee gae abob and emt adaton. The ola pectum the ola EM adaton a a functon of whch ncdent on the top of the Eath atmophee. The ola pectum uually pecfed a adance (W/m ntead of adance (W/m / becaue the old angle ubtended by the Sun (6.8-5 at Eath o mall that the ola adaton effectvely come fom the ame decton. PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-9

30 E Eath π ISunΩSun ISun M Sun R R Once the ola adance ha eached the top of the atmophee, t futhe abobed and catteed befoe t eache the uface. See fgue (K&VH 3.3 ola pectal adance. how domnant abobe H O, CO, O 3, O We can compae the ola pectum wth the adaton emtted by Eath. Sola adance peak at vble, nea.48 µm fall off apdly at IR hence known a hotwave adaton Eath adance mla to that of a blackbody at 45-5 K peak at IR, nea µm emt no vble adance known a longwave Sola and teetal adaton ae equal at ~5.7 µm. In ode to mantan the Eath n themal equlbum, the total amount of ncomng ola and emtted teetal enegy mut be equal. See fgue (Hae.3 the tanmon pectum of the atmophee how B o that aea ae equal the ola pectum ha hghe adance but ove a malle ange and old angle than the teetal pectum The ola contant, S un, defned a the annual aveage total adance ncdent on the top of the Eath atmophee. Cuently accepted value, a meaued by atellte adomete: S un 368 W/m. Th equvalent to a blackbody at 5774 K. It vae wth tme, e.g., by ±.6 W/m due to the unpot cycle by ± 3.4% due to the eccentcty of Eath obt. PHY 499S Eath Obevaton fom Space, Spng Tem 5 (K. Stong page 3-3