Chapter 5. Canopy Spectral Invariants

Transcription

1 Chape 5 Canopy Specal Invaians. Inoducion.... Physical Pinciples of Specal Invaians RT Theoy of Specal Invaians Scaling Popeies of Specal Invaians... 6 Poble Ses Refeences Inoducion The showave adiaion budge descibes how he facion of adiaion absobed by o scaeed ou fo he canopy o he undelying bacgound o bac o space ae elaed o he sucual and opical popeies of canopy and bacgound. Opeaional eoe sensing o cliae applicaions of he odel of showave adiaion budge naually equie ha such odel should build upon a canopy epesenaion wih only a sall se of basic paaees which goven he adiaion budge wih sufficien accuacy. The ineacion of sola adiaion wih he vegeaion canopy is fully descibed by he heediensional (3D) RT equaion. The scale of he eleenay volue of he equaion (scale of leaves, banches, wigs, ec.) is lage copaed o he wavelengh of sola adiaion, and, accoding o pinciples of physics, he phoon fee pah beween wo successive ineacions is independen of he wavelengh. Naely, while he scaeing and absopion pocesses ae diffeen a diffeen wavelenghs, he ineacion pobabiliies fo phoons in vegeaion edia (ineacion coss-secion o exincion coefficien) ae deeined by he sucue of he canopy ahe han phoon wavelengh o he opics of he canopy. This feaue of he RT equaion allowed foulaion of he concep of canopy specal invaians. This concep saes ha siple algebaic cobinaions of leaf and canopy specal ansiance and eflecance becoe wavelengh independen and deeine a sall se of canopy sucue specific vaiables. The se of sucual vaiables includes canopy inecepance, ecollision and escape pobabiliies. These vaiables specify he specal esponse of a vegeaion canopy o he inciden sola adiaion and allow fo a siple and accuae paaeeizaion of he paiioning of he incoing adiaion ino canopy ansission, eflecion and absopion a any wavelengh in he sola specu. In addiion o he specal invaiance popey, hese vaiables poses fundaenal scaling popey, allowing o scale RT paaees ove full ange of

2 landscape scales fo leaf inenals, hough leaf, shoos, cowns o whole canopy. Thus, he specal invaian appoach povides a copac alenaive o he full 3D RT equaion fo opeaional eoe sensing o cliae applicaions. This chape is oganized as follows. We sa by inoducing basic physical pinciples of he specal invaians and suppoing field easueens. Nex, we pesen igoous aheaical foulaion of he specal invaians based on he ehod Successive Odes of Scaeing appoxiaions (SOSA) o Neuann seies and eigenvalue/eigenveco heoy of funcional analysis. Finally, we discuss he scaling popeies of specal invaians and illusae his fundaenal feaue wih wo case sudies: (a) scaling fo needles o shoos in he needle leaf canopies and (b) scaling fo leaf inenals o leaf.. Physical Pinciples of Specal Invaians Radiaion Fluxes a eaf and Canopy Scale: The 3D RT equaion can be inepeed as he lin beween leaf and canopy scale adiaion fluxes (Chape 3 and 4). A he leaf scale he adiaion fluxes ae descibed in es of specal leaf ansiance and eflecance. The leaf ansiance (eflecance) is he poion of adiaion flux densiy inciden on he leaf suface ha he leaf ansis (eflecs) (Chape 3). The leaf albedo, ω (, is he su of he leaf eflecance, ρ ( λ ), and ansiance, τ (, ω = ρ ( + τ (. () ( The fluxes a he canopy scale ae descibed in es of specal canopy inecepance, absopance, eflecance and ansiance. The canopy inecepance (absopance) is he aio of he ean flux densiy ineceped (absobed) by canopy leaves o he downwad adiaion flux densiy above he canopy. Siilaly, canopy ansiance (eflecance) is he aio of he ean downwad adiaion flux densiy a he canopy boo (ean upwad adiaion flux densiy a he canopy op) o he downwad adiaion flux densiy above he canopy. Accoding o RT heoy (Chape 3), he canopy inecepance, i( λ ), absopance, a( λ ), eflecance, ( λ ), and ansiance, (, ae defined as follows: i( λ ) d dω σ(, Ω) I( λ,, Ω), (a) V 4π a( λ ) d dωσ I( λ,, Ω), (b) V 4π a ( λ ) dωμ( Ω)I( λ, =, Ω), (c) π+

3 ( λ ) dωμ( Ω)I( λ, = H, Ω). (d) π In he above, I( λ,, Ω) is he adiaion inensiy a wavelengh λ, spaial locaion, and in diecion Ω, σ(, Ω) is he ineacion coss-secion, and σ a is he absopion cosssecion. The ineacion coss-secion is eaed as wavelengh independen consideing he size of he scaeing eleens (leaves, banches, wigs, ec.) elaive o he wavelengh of sola adiaion [Ross, 98]. The ineacion coss-secion, σ (, Ω), consis of absopion, σ a and scaeing, σ, coss-secions (cf. Chape 3): s whee σ, Ω) = σ + σ, (3a) ( a s σ = [ ω( ] σ(, Ω), (3b) a σ s dω σ 4π s (, Ω Ω ) = ω( σ(, Ω). (3c) In he above σs (, Ω Ω ) is he diffeenial scaeing coss-secion. Thus, cobining Eqs. () and (3) we have, a( = [ ω( ]i(. (4) (a) (b) Figue. Needle (Panel a) and canopy (Panel b) specal eflecance (veical axis on he lef side) and ansiance (veical axis on he igh side) fo a Noway spuce (Picea abies (.) Kas) sand a Flaaliden sie in Sweden. Aows show needle and canopy absopance. The effecive AI a he sie was The needle ansiance, τ, and albedo, ω, follow he egession line τ =.47ω.. The canopy absopance, eflecance and ansiance ae he hee coponens of he showave enegy consevaion law which descibes canopy specal esponse o inciden sola 3

4 adiaion a he canopy scale. If eflecance of he gound below he vegeaion is zeo (blac soil, cf. Chape 4), he poion of adiaion absobed, a( λ ), ansied, ( λ ), o efleced, ( λ ), by he canopy oals o uniy, i.e., ( + ( + a( =. (5) The ey efeence daa se fo his Chape will be field daa colleced in Flaaliden sie in Sweden [Huang e al., 7]. Ohe ancillay souces will be enioned as appopiae. Canopy specal ansiance and eflecance, soil and undesoey eflecance speca, needle opical popeies, shoo sucue and AI wee colleced in six 5x5 plos coposed of Noway spuce (Picea abies (.) Kas) locaed a Flaaliden eseach aea ( 64 4 N, 9 46 E), opeaed by he Swedish Univesiy of Agiculual Sudies. The specal easueens of canopy and needles wee aen by ASD Field spec Po handheld specoadioee. Needle specal eflecances and ansiances of an aveage needle wee obained by aveaging 5 easued speca wih highes weigh given o he -yea-old needles (8%) and equal weighs o he cuen (%) and -yea (%) needles. Figue shows needle and canopy ansiance, eflecance and absopance speca a he Flaaliden sie. Mechanis of Scaeing: Conside he following basic schee of he elaionship beween leaf and canopy scales adiaion fluxes. We assue ha canopy is illuinaed fo he op by onodiecional uni flux. Canopy boo is assued o be absoluely absobing, such ha phoons hi bacgound will no e-ene, bu exi canopy. The inciden uni flux undego uliple ineacions wih phyoeleens and uliaely is paiioned ino absobed, a( λ ), ansied, ( λ ), and efleced, (, poions. To analyze uliple scaeing, we sepaae he adiaion flux inciden on vegeaion canopy ino wo coponens (Fig. ): diecly ineceped by leaves and available fo fuue ineacion evens (zeo-ode inecepance, i ), and diecly ansied o he canopy boo wihou hiing a leaf (zeo-ode ansiance, ): + = i. (6a) The ineceped phoons, i, will paicipae in he uliple scaeing and uliaely will be eihe absobed, a( λ ), o scaeed ouside of canopy, s( : i = a( + s(. (6b) Thus, = a( + s( +. (6c) Noe, i and ae zeo-ode scaeing quaniies, while a( λ ) and s( λ ) ae oal quaniies, accuulaed ove uliple evens of scaeing. While a( λ ) and s( ae wavelengh dependen, 4

5 i and don depend on wavelengh, ha is, hey ae funcion of oveall canopy sucue/achiecue and illuinaion geoey, bu no leaf opical popeies. ( λ ) a( λ ) ( λ ) i a( λ ) s( λ ) Figue. Paiioning of he incoing flux beween canopy absopance, a( λ ), ansiance, (, and eflecance, ( λ ), (lef panel) as he esul of he scaeing pocess (igh panel). The incoing flux is ineceped by canopy, (zeo-ode inecepance, ) o diecly ansied hough canopy (zeo-ode ansiance, ). The ineseped flux paicipaes in uliple scaeing and is fuhe subdivided beween absopance, a( λ ), and scaeing ou of canopy, s(. i The schee of uliple scaeing is as follows. A each individual even of ineacion in he sequence of uliple scaeing, he ω ( poion of ineceped phoons is scaeed and ω( poion is absobed (Fig. 3). In un, he scaeed poion, ω (, can be fuhe subdivided ino wo pas: wih pobabiliy p (ecollision pobabiliy) phoon will fuhe paicipae in uliple scaeing and will hi new leaf again, while wih he pobabiliy -p he phoon will be eoved fo canopy. Thus, he hee coponens of he adiaion budge fo he singe phoonphyoeleen ineacion even ae (Fig. 3): [Absobed] + [Re-scaeed] + [Scaeed ou of canopy] = [ ω( ] + [ pω( ] + [( p) ω( ]. = (7) Single Ineacion [oal poions = ] ω( Absobed [ p] ω( Scaeed ou of canopy p ω ( Re-scaeed Toal Ineacion [oal inensiy = i( λ ) ] a( λ ) Absobed s( λ ) Scaeed ou of canopy s RE ( λ ) Re-scaeed Figue 3. Paiioning of enegy beween absobed, scaeed ou of canopy and escaeed wihin canopy poions is peseved hough he individual scaeing evens (lef panel). This leads o he sae popoions fo he oal adiaion egie (igh panel). Conside he sequence of scaeing evens geneaing he oal obseved adiaion egie as deailed in Fig. 4. Pa of he incoing inensiy in he aoun of is diecly ansied o he canopy boo and will no paicipae in he pocess of uliple scaeing. The eaining pa 5

6 of he incoing adiaion, i, will be ineceped by canopy and becoe he souce of he fis ineacion even esuling in he fis-ode absopance, a = [ ω( ] i, scaeing ou of canopy, s = [ p] ω( i and e-scaeing, s RE, = ω( p i, all ae in popoions as shown in Fig. 3. The e-scaeed poion will seve as a souce fo he second-ode ineacion evens, and so on. Refeing o Fig. 4, he canopy oal inecepance, i( λ ), absopance, a(, scaeing, s(, and escaeing, s RE ( ae calculaed as he su of conibuions of individual scaeing evens: i( i + i( + i ( +... = i [ ω( p] = i n eff ( i, (8a) = pω( ω( a( = i [ ω( ] = pω( [ ω( p] = i, [ p] ω( s( = i [ p] ω( = pω( [ ω( p] = i, pω( s RE ( = i pω( = pω( [ ω( p] = i. (8b) (8c) (8d) s = [ p] ω i s 3 3 = [ p] ω p i i -d Scaeing a = [ ω] i i = ωpi a = [ ω] ωp i i ω p = i a 3 = [ ω] ω p i i ω p = i -s Scaeing 3-d Scaeing s = [ p] ω pi Figue 4. Quaniaive pesenaion of he scaeing schee shown in Fig.. The sequence of cicles epesens sequence of scaeing evens. The enegy budge fo each individual scaeing even (ineceped, absobed, scaeed ou of canopy and escaeed wihin he canopy poions) is accoding o Fig.3. Refe o Fig. 5, which explains he eaning of he effecive nube of phoon-phyoeleen ineacions, n eff, appeaing in Eq. (8a). Accoding o he definiion, he oal ineceped by phyoeleens adiaion is he infinie su of he aouns of ineceped adiaion of deceasing 6

7 inensiy, coesponding o infinie seies of phoon-phyoeleen ineacions [Eq. (8a)] Alenaively, he conibuion of infinie seies can be epesened by finie nube of phoonphyoeleen ineacions, neff, assuing each has consan inecepance of i. Also noe he following noaions. Soeies, he noalized vesions of absopance and scaeing ae used in he lieaue: canopy absopion (scaeing) coefficien, a( λ ) / i ( s( λ ) / i ), is he poion of ineceped phoons ha canopy absob (escape canopy in upwad and downwad diecions). Ineceped phoons i i = i ω i p = iω p n eff = pω (scaeing ode) Figue 5. Deivaion of he effecive nube of scaeing evens, n eff. The oal canopy inecepance is accuulaed wih he infine nube of scaeing evens,, wih declining inecepance, i. Effecively his can be epesened as n eff he conibuion of finie nube,, of ineacions wih consan inecepance, i. The nue n eff is deived fo he condiion ha aea of ecange n eff i is equal o he aea unde cuve. i Refe o Eq. (8) and Figs. 3 and 6, and noe he following popeies of inecepance, absopance, scaeing ou of canopy and escaeing. Fis, he seies fo inecepance, i( λ ), ae unique in es ha he fis e (zeo-ode inecepance) does no depends on wavelengh. Second, he elaionship beween inecepance, absopance, scaeing ou of canopy and escaeing is i( = i + s RE (, i = a( + s(. i( = a( + s( + s RE (. (9) Thus, he oal canopy inecepance, i( λ ), consis of fixed coponen (absopance and scaeing ou of canopy) and ansi coponen (escaeing). The ansi coponen is volaile, enegy ansfeed fo one scaeing o anohe will uliaely be eihe absobed o scaeed ou of canopy. Thid, he elaionship beween enegy fluxes a leaf and canopy ae esablished wih he following aios: i( a( =, ω ( s( a( [ p] ω( =, ω( s RE ( a( p ω( =. ω( () The physical eaning of aios in Eq. () is as follows [cf. Fig. 3]. In view ha a each individual phoon-phyoeleen ineacion he ineceped enegy is disibued beween 7

8 absopance, scaeing ou of canopy and escaeing in he consan popoion, independen on scaeing ode, his sae popoion will be peseved a he whole canopy scale. Fo insance, he scaeing ou of canopy consiue [ p] ω( poion, while absopance consiues ω poion, and his holds ue boh a phyoeleen and canopy scales. i i( ω ) = pω a( ω ) = p ω ω i i ωp ( ω ) = pω RE i [ p] ω s( ω ) = pω s i i p i ω Figue 6. Funcional dependance of canopy inecepance, i( ω ), absopance, a( ω), scaeing ou of canopy, s( ω ), and escaeing wihin canopy, s RE ( ω ), on single scaeing albedo, ω. Hee, i is he zeo-ode canopy inecepance and p is he ecollision pobabiliy. Nex, conside funcional dependence of inecepance, absopance, and wo scaeing quaniies on single scaeing albedo, ω [cf. Fig. 6]. Noe, zeo-ode quaniies and he sae quaniies a ω = convey a disinc eaning and should no be used inechangeably. In he case of inecepance, i and i ( ω = ) coincide, bu he definiion of i does no equie ω equal zeo, as i is consan fo all values of ω. In conas, in he case of absopance, a and a ( ω = ) ae diffeen, and a depends on ω. As ω inceases, oal inecepance inceases saing fo i due o conibuion of uliple scaeing. In conas, oal absopance is highes fo blac leaves ( ω = ) and deceases wih ω, because uliple scaeing eoves enegy ou of canopy. The wo scaeing quaniies (scaeing ou of canopy and escaeing) ae equal o zeo a ω =, bu incease wih inceasing ω. The oveall funcional dependence and specific liis of all above quaniies ae shown in Fig. 6. Finally, noe he consisency beween seies of scaeing foulaion [Eq. ()] and RT equaion [Eq. (4)]. Canopy Specal Invaian fo Inecepance: Conside oal canopy inecepance a wo independen wavelenghs, i( λ ) = i /[ pω( ] and i( λ ) = i /[ pω( λ )] fo λ λ [cf. Eq. (8a)]. The syse of he above wo equaions can be solved fo he ecollision pobabiliy: i( i( λ ) p =. () i( λ ) ω( λ ) i( λ ) ω( λ ) This equaion expesses he pinciple of specal invaiance wih espec o canopy inecepance. Recall [cf. Eqs. (7) and (9)] he oal canopy inecepance, i( λ ), is paiioned beween oal canopy absopance, a( = [ ω( ]i(, and oal canopy scaeing, s( + s RE ( = ω( i(. The pinciple of specal invaiance saes ha he aio beween diffeence in he aoun of 8

9 ineceped phoons, i( i( λ ), and hose of scaeed phoons, ω ( i( ω( λ )i( λ ), is specally invaian wih espec o any wavelengh λ and λ, and is equal o he ecollision pobabiliy. Figue 7 shows aoun of phoons ineceped, i( λ ), and scaeed, ω ( i(, by canopy as funcion of wavelenghs deived fo easueens a Flaaliden sie. Also shown is he fequency of values of he ecollision pobabiliy, p, coesponding o all cobinaions of λ and λ. The shap pea of he disibuion suggess ha he ecollision pobabiliy, p, is invaian wih espec o he wavelengh wih sufficienly high accuacy. The ino spead of he disibuion is due o easueens eos and ignoing suface conibuion. (a) (b) Figue 7. Reieval of he ecollision pobabiliy, p, fo Flaaliden field daa (Fig. ). Panel (a) shows oal canopy inecepance, i( (solid line), and oal canopy scaeing, s( λ ) + s RE ( = ω( i( (dashed line). Panel (b) shows fequency of values of he ecollision pobabiliy deived accoding o Eq. (). The Equaion () can be eaanged o a diffeen fo, which we use o deive ( = i ( ω = ) ) fo field daa, naely, p and i p ( ) i( = ) i ω λ. λ i If he ecipocal of he oal canopy inecepance calculaed fo easued canopy absopion and needle albedo is ploed vesus easued needle albedo, a linea elaionship is obained (Fig. 8). The ecollision pobabiliy, p, and canopy inecepance, i, can be infeed fo he slope and inecep. 9

10 Figue 8. Recipocal of i( and (b) ω ( /[i( i ] vesus leaf albedo ω( deived fo Flaaliden field daa (Fig. ). The ecollision pobabiliy, p =.9, and canopy inecepance, i =.9, ae deived fo he slope and inecep of he line. The ey popeies of he ecollision pobabiliy ae as follows. The ecollision pobabiliy esablishes he lin beween leaf and canopy scales, and hus i is a scaling paaee in RT heoy fo vegeaion. This paaee accouns fo he effec of he canopy sucue on RT egie acoss ange of scales. The paaee is wavelengh independen. Mone Calo siulaions [Solande and Senbeg, 5] sugges ha he ecollision pobabiliy is inially sensiive o ahe lage changes in he diecion of he inciden bea. Howeve, ohe nueical siulaions [ewis and Disney, 7] indicae ha he ecollision pobabiliy depends on scaeing ode and AI (Fig. 9). Thus, one should disciinae beween he acual ecollision pobabiliy, p acual, which is funcion of scaeing ode, is asypoic value, p inf, a plaeau, eached unde condiion of infinie scaeing, and effecive value, p efff, evaluaed ove scaeing evens. (a) (b) Figue 9. Recollision pobabiliy, AI, 5,. Infinie scaeing ode ecollision pobabiliy ( p efff p acual, as a funcion of scaeing ode calculaed fo canopies wih ) and effecive ecollision pobabiliy ( ) as a funcion of AI. Mone-Calo siulaions ae pefoed fo canopies coposed of andoly locaed non-ovelapping diss wih a spheical leaf angle disibuion (fo ewis and Disney, [7]). p inf

11 Canopy Specal Invaian fo Reflecance and Tansiance: The oal canopy scaeing consis of escaeing beween phyoeleens, s RE ( λ ), and scaeing ou of canopy, s( λ ). The escaeing e, s RE (, is chaaceized by ecollision pobabiliy, p. The scaeing ou of canopy e, s(, can be subdivided fuhe ino upwad and downwad coponens o deive eflecance and ansiance. The pobabiliy ha scaeed phoon will escape he vegeaion canopy hough he uppe (o lowe) bounday is called escape pobabiliies ρ and τ, especively. A each individual even of phoon-phyoeleen ineacion, all possible oucoes of scaeing ae liied o phoon escaping canopy in upwad, o downwad diecion, o colliding anohe phyoeleen, hus, ρ + τ + p =. () As in he case of ecollision pobabiliy, p, he escape pobabiliies, ρ and τ, depend, in geneal, on scaeing ode, bu each consan value (plaeau) afe seveal ieaions. The nube of ineacion evens befoe his plaeau is eached depends on he canopy sucue and he needle ansiance-albedo aio. Mone Calo siulaions sugges ha he ecollision and escape pobabiliies sauae afe wo-hee phoon-canopy ineacions fo low o odeae AI canopies [ewis and Disney, 7]. This esul undelies he appoxiaion o he canopy eflecance, (, poposed by Disney and ewis [5], ω( R ( λ ) = ω( R +, (3a) p ω( whee coefficiens R, R and p ae deeined by fiing Eq. (3a) o easued eflecance specu. Unde assupion ha he ecollision, p, and escape pobabiliy, τ, eains consan in successive ineacions, R ρ, pi, p p. (3b) i R ρ The fis e evaluaes he poion of phoons fo he ineceped flux, ha escape he vegeaion canopy in upwad diecions as a esul of one ineacion wih phyoeleens. The second e accouns fo phoons ha have undegone wo and oe ineacions. Violaion of he above condiion esuls in a ansfoaion of ρ i, ρ pi, and p o soe effecive values R, R and p as he esul of he fiing pocedue. The diffeence beween acual and effecive values of he escape pobabiliies depends on is speed of convegence as he nube of ineacions inceases. A deailed analysis of his effec will be pesened in Secion 3. A siplified expession, R = pr, can also be used, wih a educion in accuacy of he appoxiaion [Disney and ewis, 5]. i

12 Figue shows coelaion beween easued and evaluaed accoding o Eq. (3) canopy eflecance ove Flaaliden sie. Oveall close ageeen suppos he appoxiaion of Disney and ewis. In his exaple he seleced values fo R and p give he bes fi o he easued eflecance specu. These coefficiens can also be evaluaed fo he slope and inecep of he egession line, deived fo values of he needle albedo, ω (, and he ecipocal of ( λ ) / ω( a wavelenghs [7-75 n]. A hose wavelenghs values of ω ( ae unifoly disibued in he ineval [.,.9] and he canopy eflecance exhibis a song coelaion wih ω(. These feaues allow educing he ipac of gound eflecance and easueen unceainies on he specificaion of R and p fo he egession line. Figue. Coelaion beween easued canopy eflecance and canopy eflecance evaluaed using Eq. (3) wih R =.5, p =.59, and R = p R =.9 fo he specal ineval 4 λ 9 n. The aow indicaes a ange of eflecance values coesponding o ω.9. Field daa ae fo Flaaliden sie (Fig. ). Analogous o Eq. (3) fo canopy eflecance, a siila elaionship can be esablished beween canopy ansiance and phyoeleens albedo, naely T ω( ( λ ) = + p ω( λ, (4a) ) whee he values of coefficiens, T and p ae chosen by fiing Eq. (4a) o he easued specu of canopy ansiance. Analogous o Eq. (3b), he coefficiens T and p ae effecive values and unde assupion ha he ecollision, p, and escape pobabiliy, τ, eains consan in successive ineacions, T τ, p p. (4b) i Unde he above assupion, he value of conveges o zeo-ode ansiance [cf. Eq. (6)]. Figue shows coelaion beween easued and evaluaed accoding o Eq. (4) canopy ansiance ove Flaaliden sie. A heoeical analysis of his appoxiaion will be pesened in Secion 3. I should be noed ha canopy ansiance is sensiive o he needle ansiance

13 o albedo aio τ( / ω( [Panfeov e al., ]. This ay ibue wavelengh dependence o he escape pobabiliies fo low ode phoon scaeing. Figue. Coelaion beween easued canopy ansiance and canopy ansiance siulaed using Eq. (4) wih =.6, T =.7 and p =.94. Enegy consevaions fo i and is peseved wih good accuacy, i.e., i + =.9+.6=.98. Field daa ae fo Flaaliden sie (Fig. ). Ipac of Soil Reflecance: The oal canopy ansiance, ( λ ), eflecance, ( λ ), and absopance, a(, fo he geneal RT poble of canopy above soil bacgound can be epesened by conibuion of blac-soil and soil sub-pobles as follows (cf. Chape 4): BS( ( λ ) = = BS( + ( ρsoil( S (, (4a) ρ ( ( soil S ( λ ) = ( + ( ρ ( (, (4b) BS soil S a( λ ) = a ( + ( ρ ( a (. (4c) BS Hee, ρ is he heispheical eflecance of he canopy gound. Vaiables and ; soil BS S BS and S ; a BS and a S denoe canopy eflecance, ansiance, and absopance calculaed fo a vegeaion canopy () illuinaed fo above by he inciden adiaion and bounded fo below by a non eflecing suface (subscip BS, fo blac soil); and () illuinaed fo he boo by noalized isoopic souces and bounded fo above by a non-eflecing bounday (subscip S ). These vaiables ae elaed via he enegy consevaion law, i.e., a ( λ ) + ( + (, i=bs o S-poble. i i i = The canopy specal invaians ae foulaed fo BS, BS and abs. The easued specal ansiance,, and eflecance,, ae aen as esiaes of BS, BS. The absopance abs is appoxiaed using Eq. (5). I follows fo Eq. (4) ha he elaive eos, Δa, Δ and Δ, and in a BS, BS, and BS due o he neglecing of suface eflecion can be esiaed in es of easued, and as: ρ soil soil S 3

14 a BS a Δ a = ρssoil ( S S ) ρsoil, a (43a) BS Δ = ρsoil S ρsoil, (43b) BS Δ = ρsoil S ρsoil, (43c) Figue. Uppe liis of he elaive eos Δ, Δ, and Δ a in he esiaes of BS, BS and abs, aising due o he effec of non-blac soil eflecance. Refeence field daa ae fo Flaaliden sie in Sweden (Fig. ). Thus, neglecing conibuion of soil, esuls in oveesiaion of eflecance and ansiances and undeesiaion of absopances. Figue shows uppe liis of he elaive eos Δ, Δ,and Δa.as a funcion of he wavelengh fo Flaaliden daa. I follows fo he above analysis ha easued canopy absopance appoxiaes a BS wih an accuacy of abou 5%. Deviaions of easued canopy ansiance and eflecance fo BS and BS in he ineval 4 λ 7 n do no exceed 5%, howeve hey incease subsanially in he ineval 7 λ 9 n. Majo Assupions fo Specal Invaians: We suaize ey assupions of he heoy of specal invaians along hee caegoies. () Bounday condiions assupions: a vegeaion canopy is illuinaed fo above by a wavelengh independen paallel bea and bounded fo below by a non-eflecing (blac) suface. The las assupion is equied o avoid e-enance of phoons exiing hough bacgound. () Phyoeleens scaeing popeies assupions: he ineacion coss-secion, σ (, Ω), is eaed as wavelengh independen consideing he size of he scaeing eleens (leaves, banches, wigs, ec.) elaive o he wavelengh of sola adiaion. (3) Effecive values assupions: he ecollision and escape pobabiliies ( p, ρ, τ ) ae geneally dependan on he scaeing ode, bu end o each plaeau and could be eplaced wih coesponding effecive values. The unceainies of eievals of inecepance (o absopance) ae elaively low, as hose vaiables depend on ecollision pobabiliy only, while unceainies 4

15 fo ansiance and eflecance ae highe as hose vaiables depend boh on ecollision and escape pobabiliies. 3. RT Theoy of Specal Invaians Successive Odes of Scaeing Appoxiaion: Below we foulae he igoous aheaical basis undelying he pinciple of specal invaiance, inoduced in he pevious secion. We adop funcional analysis foulaion of he anspo equaion (Vladiiov [963], Machu e al [98]). e V and δ V be he doain whee adiaive ansfe occus and is bounday, especively. The doain V can be a shoo, ee cown, ee sand, ec. e and be he seaing-collision and scaeing linea opeaos (Chape ), S λ Iλ Ω Iλ + σ(, Ω)Iλ, (5a) SλIλ σs, λ (, Ω Ω)Iλ (, Ω )dω, (5b) 4π whee I λ is he adiaion inensiy a wavelengh λ, spaial locaion and diecion Ω ; σ and σ S ae exincion and diffeenial scaeing coefficiens, especively. In he following we assue ha single scaeing albedo, ω (, does no depend on and Ω. The 3D adiaive ansfe equaion (Chape 4) can be foulaed in opeao noaions as follows I = λ SλIλ. (6a) The bounday condiions include canopy op o be illuinaed fo above by uni bea in diecion Ω and canopy boo o be absoluely absobing, ( op, Ω) = δ( Ω Ω ), op δv, μ( Ω) <, (6b) Iλ ( boo, Ω) =, boo δv, μ( Ω) >. (6c) I λ We solve Eq. (6) wih ehod of successive ode of scaeing appoxiaions (SOSA). The oal adiaion inensiy is epesened as he su of uncollided, Q, and collided, coponens (cf. Chape ), dif I λ = Q + Iλ (, ). (7) Ω By definiion, Q is he adiaion inensiy of phoons in he inciden flux ha will aive a along diecion Ω wihou suffeing a collision. This is a wavelengh independen paaee. saisfies he equaion Q Q =, (8) 5

16 and he oiginal bounday condiions, Eq. (6b-c). I dif λ is he collided (o diffuse) adiaion inensiy, ha is, adiaion geneaed by phoons scaeed one o oe ies. This is a wavelengh dependan paaee. Cobining Eqs. (6)-(8) one can veify ha I dif λ saisfies he following equaion, dif Iλ = SQ + SI dif λ, (9) and zeo bounday condiions a he op and boo of canopy. Finally, by cobining Eq. (7) and (9), he Eq. (6a) can be ewien in he fo of he following inegal adiaive ansfe equaion, Iλ + λ = Q TI, () whee opeao T S. The SOSA ehod saes ha he soluion of Eq. () is given by = I λ = Q, whee = TQ T Q, = [, ]. () Q = One can veify he validiy of soluion given by Eq. () by subsiuing i in Eq. () and aing ino accoun popeies of opeao T. The physical eaning of Eq. () is as follows. Q is he adiaion inensiy of phoons, scaeed ies. The uncollided phoons wih inensiy Q, seve as he souce fo he phoons scaeed one ie wih inensiy Q, which in un seve as a souce of phoons scaeed wo ies, and so on (cf. Fig. 4). In Mone Calo siulaions, opeao T coesponds o a pocedue, which inpus Q, siulaes he scaeing even, calculaes he phoon fee pah and oupus he disibuion, Q, of phoons jus befoe hei nex ineacion wih phyoeleens; he pocedue is epeaed wih he souce of phoons evaluaed as oupu a he pevious sep. Specal Invaian fo Canopy Inecepance: e f be he no of 3D adiaion field f (, Ω ) in he doain V 4π, accoding o noaions of funcional analysis [Vladiiov, 963; and Machu e al., 98], f = d dωσ(, Ω) f, () V 4π In es of hese noaions, he oal canopy inecepance, i( λ ), and -ode canopy inecepance, i, ae Iλ and Q, especively [cf. Eqs. (a) and (8a)]. The disibuion of pobabiliy, e, ha a phoon scaeed ies will aive a along he diecion Ω wihou suffeing a collision can be expessed as he adiaion inensiy of he phoons, scaeed ies, noalized by is no 6

17 e Q (, Ω ), e. Q = (3) The noalizaion is equied, as Q is he adiaion inensiy, whose no, inecepance, i, deceases wih ode of scaeing (cf. Fig. 5), while e (, Ω ) is he disibuion of pobabiliy, whose no is equied o be uniy. The ecollision pobabiliy can be expessed in es of Q. The ecollision pobabiliy,, a he sep of scaeing is he aio of adiaion inensiy escaeed inside of canopy o he oal inensiy of scaeing (i.e., escaeed inside of canopy and escaped canopy) [cf. Figs. 3-4 and Eq. (7)]. The adiaion inensiy scaeed - ies is Q, heefoe, Q will be ineceped and ω Q will be available fo he oal scaeing a he cuen sep of scaeing. The oal adiaion available fo scaeing oiginaes he adiaion inensiy Q a cuen sep of scaeing. The escaeed inensiy a he cuen sep is equal o he ineceped inensiy a he nex sep +, Q. Theefoe, p p Q =. (4) ω Q Fo convenience of he following deivaions we will use γ pω. Taing ino accoun Eqs. (), (3) and (4), he disibuions e and e beween successive odes of scaeing - and ae elaed as Te Q Q Q (, Ω) = T = = γ e. (5) Q Q Q The above equaion explicily saes he naue of opeao T: i conves he pobabiliy disibuion of phoons fo pevious o he nex ode of scaeing and evaluaes ecollision pobabiliy. The se ( γ, e ), = [, ], deived fo opeao T accoding o SOSA ehod poses one fundaenal popey esablished in eigenvalues/eigenvecos heoy of funcional analysis [Vladiiov, 963]. An eigenvalue of he adiaive ansfe equaion is a nube χ such ha hee exis a funcion ψ (, Ω) ha saisfies he equaion Tψ = χψ(, Ω) (6) and zeo bounday condiions. Unde soe geneal condiions [Vladiiov, 963], he se of eigenvalues and eigenvecos ( χ, ψ ) is a discee se. Since he eigenvalue and eigenveco poble is foulaed fo zeo bounday condiions, χ and ψ(, Ω) ae independen 7

18 * on he incoing adiaion. The adiaive ansfe equaion has a unique posiive eigenvalue, χ, * ha coesponds o a unique posiive eigenveco, ψ [Vladiiov, 963], * * * * Tψ = χ ψ, ψ =. I should be ephasized ha se ( χ, ψ ), deived accoding o eigenvalue poble [Eq. (6)] is diffeen fo ( γ, e ), deived accoding SOSA ehod [Eq. (5)]. In geneal, ( γ, e ) vay wih he scaeing ode. Howeve, hey end o convege o plaeaus as he nube of ineacions inceases accoding o nueical esuls [ewis and Disney, 998]. Fuhe, accoding o geneal pinciples of funcional analysis [Riesz and Sz.-Nagy, 99; Vladiiov, 963] he se ( γ, e ) conveges o he unique posiive eigenveco/eigenvalue of opeao T, as nube of scaeing inceases, li γ = χ *, li e = ψ *. (7) Assuing negligible vaiaion in γ and e fo he scaeing ode and highe and accouning fo Eqs. () and (3), he adiaion field, I, can be appoxiaed as follows λ I λ = Q e = Q e + = = = + = = = = λ Q Q e e + + = Q I, + δ. γ Q e + Q + e + + δ γ γ + + e + + δ (8) Tha is, adiaion field, I λ, is appoxiaed wih Iλ,, fo which he conibuion of he fis scaeing odes is calculaed exacly and conibuion of highe ode scaeing is appoxiaed assuing ha γ and e ae consan wih espec o fo +. The eo of his appoxiaion is. δ Following he above appoach, we exaine he accuacy of he appoxiaion of he canopy inecepance as a funcion of he scaeing ode. I follows fo Eq. (8) ha he -h appoxiaion, i ( λ ), o i( is γ + θ+ i λ Ω = + = θ + ( ) Iλ, (, ) Q Q i. (9) = γ + = γ + Hee i Q is he zeo-ode canopy inecepance; θ =, and 8

19 θ Q = Q Q Q... = γ γ... Q Q Q Q γ,. (3) The eo, δi (, in he -h appoxiaion is given by [Huang e al., 7] whee θ δ, (3) + i = i( i ( ε γ,+ s + i γ + γ + + γ + θ ε γ,+ = ax, s + =. (3) γ = θ+ Noe ha li θ + = γ. If is lage enough, i.e. θ+ γ, he aio θ + + / θ+ can be appoxiaed by γ. Subsiuing his elaionship ino Eq. (3) one obains s γ /( γ ). Figue 3. Coelaion beween fis and second scaeing ode ecollision pobabiliies ( and p ) fo a ange of AI. p The RT siulaions wee pefoed wih he sochasic RT odel fo a canopy odeled wih idenical cylindical ees unifoly disibued ove blac bacgound. Cown heigh=, gound cove=vaiable and plan AI=, SZA=3. Two facos deeine he accuacy of he -h appoxiaion of canopy inecepance. The fis is he diffeence beween successive appoxiaions γ + and γ + + ; ha is, he salle his diffeence, he oe accuae he appoxiaion is. The second faco is he conibuion of phoons scaeed + o oe ies o he canopy adiaion field. Thei conibuion is given by + θ + /( γ + ) γ /( γ ) which depends on he ecollision pobabiliy, p, and he single scaeing albedo, ω; ha is, he highe γ = p ω is, he highe ode of appoxiaion is needed o esiae he canopy inecepance. This is illusaed in Fig. 3. The vaiaions in he ecollision pobabiliy as funcion of scaeing ode eaches is axiu a high values of p (hee he ecollision pobabiliies fo he fis and second ode of scaeing wee copaed). The specal invaian can no be deived if p ω = since he Neuann seies (4) do no convege in his case. 9

20 Specal Invaians fo he Canopy Tansiance and Reflecance: e he doain V be a laye z H. The sufaces S, z = and z = H consiue is uppe and lowe boundaies, especively. e f and f be nos of 3D adiaion field f in he doain S π, d + f, = dωf μ z= π f, (33) = d dωf μ z= H π whee he fis inegal is aen ove he op bounday and in upwad diecion, while second is aen ove lowe bounday in downwad diecions. In es of noaions of Eq. (33), he canopy eflecance, ( λ ), and ansiance, ( λ ), ae given by I λ and I λ. Recall, accoding o noaions of Eq. () canopy inecepance, i( λ ), is given by Iλ. The elaionship beween eflecance, ansiance and inecepance fo soe ode of scaeing can be deived as follows. Recall, Q = TQ [cf. Eq. ()], o, in es of opeaos and S [cf. Eq. (5)], Q =. Inegaing he las equaion ove he doain V 4π, we have: SQ 4π dω dvω Q + dω dvσ( Ω)Q = dω dv dω σ S ( Ω Ω)Q (, Ω ), V 4π V 4π V 4π 4π dω dsω Q + dω dvσ( Ω)Q = ω dω dvσ( Ω)Q (, Ω ), S 4π V 4π V Q + Q + Q = ω Q. (34) In he above deivaions we accouned fo definiion of nos [Eqs. () and (33)], elaionship beween exincion coefficien, σ (Ω), and diffeenial scaeing coefficien, σs ( Ω Ω), [Eq. (3)] and Gauss heoe fo conveing volue inegal o he suface inegal fo soe scala funcion. Noalizing Eq. (34) by ω Q we finally have ρ + τ +, (35a) p = whee ρ and τ ae escape pobabiliies fo eflecance ansiance and p is he ecollision pobabiliy fo he -h ode of scaeing, Q Q Q ρ, τ, p. (35b) ω Q ω Q ω Q The physical inepeaion of Eqs. (34) and (35) follows hose given fo Eq. (4). The ecollision pobabiliy, p (escape pobabiliy in upwad, ρ, and downwad, τ, diecions) a he sep

21 of scaeing is he aio of adiaion inensiy escaeed inside of canopy (escaped canopy in upwad and downwad diecions) o he oal inensiy of scaeing. The oal inensiy available fo scaeing a he cuen sep of scaeing is ω Q. This oal inensiy is disibued beween escaeed inensiy a cuen sep, Q, escaped canopy in upwad diecions, Q, and escaped canopy in downwad diecion, Q. This explains Eq. (34). The aio of above quaniies accoding o definiion of ecollision and escape pobabiliies explains Eq. (35b). The escape and ecollision pobabiliies coespond o poions of all possible evens of scaeing (escaped upwad, downwad o escaeed), which explains Eq. (35a). Noe also, he escape pobabiliies vay wih he scaeing ode, bu, as in he case of ecollision pobabiliies, hey each plaeaus ( ρ and τ ) as he nube of ineacions inceases. I follows fo Eq. (8) ha he -h appoxiaion, ( and (, o he canopy eflecance and ansiance ae ( I + λ, = ρθ + ωi, (36a) = γ + θ ρ ( I + λ, = + τθ + ωi. (36b) = γ + θ τ Hee i and ae zeo-ode canopy inecepance and ansiance, especively [cf. Eq. (6)]; and θ is defined by Eq. (3). Eos in he -h appoxiaion of canopy eflecance and ansiance ae given by [Huang e al., 7] δ = ( ( ( ε + ε θ ) ρ s ωi +,+ γ,+,, (37a) γ + δ = ( ( ( ε + ε θ ) τ s ωi +,+ γ,+,. (37b) γ + Hee is defined by Eq. (3) and ε γ, κ + + κ + ε κ,+ = ax, θ+ κ + s κ, =, (37c) κ + = θ κ + whee κ and κ epesen eihe canopy eflecance ( κ =, ( κ =, κ = ). τ κ = ρ ) o canopy ansiance

22 p Figue 4. Recollision pobabiliy,, and escape pobabiliies, τ and ρ, as a funcion of he scaeing ode. Thei liis ae p =.75, τ =.5 and ρ =.5. The elaive diffeence ε γ, + in he ecollision pobabiliy is 3% fo = and.8% fo =. Paaees of he RT siulaions ae he sae as fo Fig., excep gound cove=.6. In addiion o wo facos ha deeine he accuacy in he -h appoxiaion of he canopy inecepance [cf. Eq. (3)], δ and δ also depend on he convegence of wo successive appoxiaions κ + and κ + + o ρ o τ. Thus, he eos in he -h appoxiaions o he canopy eflecance and ansiance esul fo he eos in he ecollision and escape pobabiliies, and fo a conibuion of phoon uliple scaeing o he canopy adiaion egie. The -h appoxiaion o he canopy eflecance and ansiance, heefoe, is less accuae copaed o ha o he canopy inecepance. This is illusaed in Fig. 4. In his exaple, he elaive diffeence γ + + γ + / γ + + is 3% fo = and becoes negligible fo. The zeo and fis ode appoxiaions povide accuae specal invaian elaionships fo he canopy inecepance. The coesponding diffeences in he escape pobabiliies do no exceed 4% fo, indicaing ha wo scaeing odes ae equied o evaluae specal invaians fo canopy ansiance and eflecance wih accuacy copaable o ha given by zeo appoxiaion o he canopy inecepance. Specal Invaian fo Canopy BRF: The -h appoxiaion, BRF I λ, (z =, Ω), μ ( Ω) > o he canopy bidiecional eflecance faco (BRF), is given by Eq. (8). Is eo, δ BRF δi (z =, Ω) is given by [Huang e al., 7] θ+ δ BRF i SBRF,+ γ + e + + (z =, Ω) e + (z =, Ω) γ + + γ + ax + ax +, (38), Ω π e = Ω + (z, ) γ + + whee θ S. + + BRF,+ (z =, Ω) = e + (z =, Ω) = θ+

23 If is lage enough, i.e., + θ+ γ e e and +, he e S BRF,+ can be appoxiaed as S BRF,+ e γ /( γ ). Is values, heefoe, ae ainly deeined by he conibuion of phoons scaeed + and oe ies o he canopy adiaion egie. Accoding o Eq. (38), he accuacy in he -h appoxiaion o he canopy BRF depends on he convegence of γ + and e + o he eigenvalue, γ, and coesponding eigenveco, e, of he opeao T. Convegence of he foe is illusaed in Fig. 5. This figue shows vaiaions in ax Ω π+ {e + (, Ω) / e } and in Ω π+ {e + (, Ω) / e } wih he scaeing ode. In his exaple, he diffeence e + + e+ is negligible fo 4, indicaing ha he foh appoxiaion povides an accuae specal invaian elaionship fo he canopy BRF. Vaiaion in he pobabiliy e wih he scaeing ode should be accouned o evaluae he conibuion of low ode scaeed phoons. e Figue 5. Convegence of o he posiive eigenveco, e, of he opeao T. The uppe bounday of vaiaions of he aio e + / e, axω π{e + / e} (solid line) and coesponding lowe bounday inω π{e + / e} (dashed line) ae shown wih espec o he scaeing ode. Fo 5, hei values fall in he ineval beween.98 and.4. Paaees of he RT siulaions ae he sae as fo Fig. 3. Invese inea Appoxiaion o he Canopy Reflecance and Tansiance: The epiical and heoeical analysis indicaes ha he zeo-ode appoxiaion povides an accuae specal invaian elaionship fo he canopy inecepance; howeve oe ieaions ae equied o achieve copaable accuacy fo he canopy ansiance and eflecance. The epiical analysis also suggess ha zeo-ode appoxiaion ay esul in a good accuacy in case of eflecance and ansiance, if he ecollision pobabiliy is eplaced wih is effecive values. In he following we deive effecive ecollision pobabiliies fo canopy eflecance and ansiance in he zeo-ode appoxiaion fo he fis-ode appoxiaion. Accoding o Eqs. (36a), (37a) and (3), he fis-ode appoxiaion (=) o he canopy specal eflecance is ωp p ρ ωpρ ωδ, ( λ ) = ( + δ = ρ + + S, i = ωρi. p p ω (39a) ω ω p ω 3

24 Hee [cf. Eq. (37c)] and chaaceize he accuacy of he fis appoxiaion, S, Δ, ρ p θ+ ρ+ ρ+ ρ+ γ + γ S, Δ, = [ + S, ],. = +. (39b) ρ p = θ ρ ρ+ γ + Noe, accoding o zeo-ode appoxiaion [cf. Eq. (4a)] he ecipocal of he canopy specal eflecance noalized by he leaf albedo, ω, vaies linealy wih ω. Based on his obsevaion, we eplace he elaionship beween he ecipocal of /( ω i ρ ) and he leaf albedo ω given by fis-ode appoxiaion [Eq. (39a)] wih is zeo-ode fo, given by a linea egession, Y = α βp ω. The coefficiens R, R and p in he zeo-ode appoxiaion (4a), can be specified fo he slope β and inecep α, naely ω( R iρ β ( = ω( R +, R =, p p ω( α = p, (4a) α α Δ, Δ, = p ω( 3ω) dω, β = 6 p Δ ω ω(ω ) dω. (4b) p Δ ω,, Siilaly, he canopy ansiance is ( T ω = ω p i τ T = β p = p α,,. (4) α τ τ Hee α and β ae given by Eq. (4b) bu fo Δ, and S, which ae calculaed wih,. We e his appoach an invese linea appoxiaion. Noe ha if he escape pobabiliies do no vay wih he scaeing ode ( Δ, = Δ, = ), he slope β = β = p and inecep α = α =, and he invese linea appoxiaion coincides wih he zeo-ode appoxiaion. If vaiaions in he escape pobabiliies becoe negligible fo, ( ε κ,, κ =, ), he effecive pobabiliies p and p ae funcions of p, p, ρ, ρ and, p, p, τ, τ especively. Figue 6 deonsaes he enegy consevaion [Eq. (35a)] fo =. The escape pobabiliies ae calculaed fo Eqs. (4) and (4) as R / i and T / i. I follows fo Fig. 5 ha he ipac of he egession coefficiens α and α on he escape pobabiliies is inial; ha is, deviaion of R / i + T / i + p fo uniy does no exceed 5%. This is no supising because values of ( Δ κ, ) /( Δ κ,p ω) in Eq. (4b) fo α ( κ = ) and α ( κ = ) ae uliplied by he funcion ω( 3ω), inegal of which is zeo. The effecive values of he ecollision pobabiliies, p and p, howeve, depend on β and. β 4

25 Figue 6. Enegy consevaion elaionship ρ + τ + p = as funcion of AI. The escape pobabiliies ρ and τ wee calculaed as he aios of coefficiens and T in he R i invese linea appoxiaions o, i.e., ρ = R / i and τ = T / i. The deviaion of ρ + τ + fo uniy does no exceed 5%. p Paaees of he RT siulaions ae he sae as fo Fig. 3. Since eigenvalues and eigenvecos of he opeao T ae independen fo he inciden adiaion, he liis p, ρ and τ of he ecollision and escape pobabiliies do no vay wih he inciden bea. Solande and Senbeg [5] showed ha he fis and highe odes of appoxiaions o he ecollision pobabiliy ae insensiive o ahe lage changes in he sola zenih angle. Alhough he fis appoxiaions o he escape pobabiliies exhibi a highe sensiiviy (Fig. 7) o he sola zenih angle, hei su, ρ + τ = p, eains alos consan. This is consisen wih he above heoeical esuls, suggesing ha he canopy ineacion coefficien equies less ieaions o each a plaeau copaed o he canopy eflecance and ansiance. The sensiiviy of he effecive ecollision pobabiliies o he sola zenih angle is uch salle copaed o he canopy inecepance. Figue 7. Recollision pobabiliy,, is effecive values, p and p, escape pobabiliies, ρ and τ, and he canopy inecepance, i, as funcions of SZA. Equaion (4) was used o specify ρ and τ. Paaees of he RT siulaions ae he sae as fo Fig. 3. p Figue 8 shows elaive eos in he invese linea appoxiaion and he -h appoxiaions, =, and 3, o he canopy eflecance as a funcion of ω and AI. The eo deceases wih he scaeing ode. Fo a fixed, i inceases wih ω and AI. This is consisen wih he 5

26 heoeical esuls saing ha he convegence depends on he axiu eigenvalue γ = p ω ; ha is, he highe is value is, he highe ode of appoxiaion is needed o esiae he canopy eflecance. In his exaple, he hid and invese linea appoxiaions have he sae accuacy level, i.e., hey ae accuae o wihin 5% if ω. 9. The elaive eo in he canopy ansiance (no shown hee) exhibis siila behavio. Figue 8. Relaive eo in he canopy eflecance as a funcion of ω and AI. Paaees of he RT siulaions ae he sae as fo Fig Scaling Popeies of Specal Invaians The scaling effec, o scale dependence of RT paaees, aises due o phenoena of spaial heeogeneiy (disconinuiy) of canopy opical popeies. Fo insance, single scaeing albedo, ω(λ, V), is a funcion of he scale (volue, V) whee i is defined. Conside sequence of nesed scales epesened by coesponding volues: needle leaf ee sand ( V ), ee cowns ( V ), needle leaf shoos ( V ), and needles ( V3 ) (Fig. 9). Selec a couple of scaeing albedos, ω ( λ,v ) and ω( λ,v ), which quanify he scaeing popeies a he scale of ee cown of volue and consiuen objecs (shoos) of volue V. By definiion, single scaeing of V 6

27 V volue is he aio of enegy scaeed by ha volue o he aoun of he enegy ineceped by he sae volue. Accoding o Eq. (8b) he ee cown single scaeing albedo, ω( λ,v ), can be expessed as s( λ,v ) p(v V ) ω ( λ,v ) = ω( λ,v ). (44a) i (V ) ω( λ, V ) p(v V ) Hee i (V ) and s(v ) ae he poion of phoons ineceped and scaeed by he volue V, and p(v V ) is he ecollision pobabiliy defined as he pobabiliy ha a phoon scaeed by a volue V (shoo) esided in he volue V (ee cown) will hi again anohe volue V (anohe shoo) in V. Is value is deeined by he disibuion of volues V (e.g., shoos) wihin V (cowns). Thus, Eq. (44a) can be inepeed as one ha povides a lin beween vegeaion RT popeies a diffeen scales. V V V V 3 V Figue 9. Scheaic plo of nesing of V scales. Tee sand occupies volue, which consis of individual ees cowns of V volue, which, in un, consis of shoos of volue V, which in un consis of V 3 V V3 needles of volue. The ee volues ae nesed: V V. Boh ω ( λ,v ) and p(v V ) vay wih he scale V. Howeve since he lef-hand side of Eq. (44a) does no depend on V, he algebaic expession on he igh-hand side of his equaion should also be independen on he scale of V. Based on his popey, vaiaion in he leaf single scaeing albedo and he ecollision pobabiliy wih he scale V can be specified as follows. e us ewie Eq. (44a) fo needles ( ) and shoo ( ), V 3 V p(v3 V ) ω ( λ,v ) = ω( λ,v3 ). (44b) ω( λ,v )p(v V ) 3 3 Subsiuing whee ω λ,v ) fo Eq. (44b) ino Eq. (44a) peseves he sucue of Eq. (44a): ( p(v3 V ) ω ( λ,v ) = ω( λ,v 3), (44c) ω( λ,v ) p(v V ) 3 3 [ p(v V )] p(v V ) p(v3 V ) = p(v3 V ) + 3. (45) 7

28 One can see ha he pobabiliy p(v 3 V ) ha a phoon scaeed by a volue V3 (e.g., needles) will ineac wihin volue V (e.g., cown) again follows he Bayes foula. Accodingly, Eq. (45) is called nesing of scales. The single scaeing albedo and p-paaee exhibi he following scaling popeies. Refeing o Fig. 9 and aing ino accoun Eqs. (44)- (45), povided ω, p, one can deive ha, ω λ, V ) ω(, V ) and (V V ) p(v V ), if V ; ( λ 3 p 3 3 V p(v3 V ) p(v3 V ), if V V. (46) Conside he second popey shown in Eq. (46), which conveys a fundaenal law. I iplies ha he ecollision pobabiliy inceases wih inceasing coplexiy of canopy achiecue (cf. Fig. ). Naely, accoding o is definiion, p= fo he Big eaf odel, as hee ae no uliple scaeing. As we add oe hieachical levels of sucue, if phoon eached paicula sucue eleens i ge apped on sucual sublevels, which inceases pobabiliy of escaeing and hus value of p-paaee. The ecollision pobabiliy, heefoe, is a scaling paaee ha accouns fo a cuulaive effec of he landscape s uli-level hieachy. a) Big eaf b) Hoogeneous Canopy p inceases Figue. Recolision pobabiliy, p, as funcion of canopy sucual hieachy: (a) Big eaf Canopy; (b) Tubid Mediu; (c) 3D Canopy wih nesed scales of sucue. c) 3D Canopy Case Sudy - Scaling fo Needles o Shoos: The scaling popeies of he p-paaee wee fis deonsaed by Solande and Senbeg [3; 5] in he applicaion fo conifeous canopies. The 3D sucue of he conifeous canopies exhibis foliage cluping a uliple scales, including cluping of needles ino shoos and cluping of shoos ino ee cowns; boh give ise o he scaling effec. In paicula, sall-scale cluping of needles ino shoos esuls in uual shading and uliple scaeing of ligh beween needles of a shoo (Fig. ), which 8

29 uliaely leads o he nown RT effec of conifeous canopies o appea dae han boadleaved canopies. Figue. Scaeing of phoons on individual needles wihin shoo. The scaeing is associaed wih loss of enegy and hus shoo albedo is lowe han albedo of individual needles (fo Solande and Senbeg, 3]. The canopy cluping is descibed in he RT appoach wih spaially vaying foliage volue densiy. Howeve, he scale of vaiaion of foliage densiy is liied by he size of canopy eleenay volue. The eleenay volue us be lage enough o conain sufficien nube of saisically independen foliage eleens fo he foliage volue densiy o be defined. Theefoe, he appoach is ypically ipleened a he lage (landscape) scale, idenifies individual ee cowns and space beween he and defines he eleenay volue o conain uliple leafs o needle shoos. The sall-scale cluping of needles ino shoos equies coplex saisical descipion of disibuion of needles. To ovecoe his poble, sho iself is ypically used as he basic sucual eleen in place of leaf fo boadleaved canopies. The deviaion of opical popeies of eleenay volue fo hose of individual needles, caused by shoo sucue, is ypically accouned fo in he RT equaion by adjusing exincion coefficien wih epiically esiaed cluping index (cf. Chape 3). This ad hoo appoach is deficien in descibing physical pocess of ligh scaeing inside of a shoo, as i ignoes wavelengh dependence of he pocess and aificially couples shoo sucue and needle opics. Solande and Senbeg [3; 5] developed p-paaee based RT faewo o descibe he effec of he sall-scale cluping of needles ino shoos o explain he diffeence in RT egies in boadleaved and conifeous canopies. This sudy was focused only on sall-scale cluping and lage scale cluping was ignoed. To suppo he heoy, ay acing siulaions wee pefoed fo he odel of canopy sucue saisfying inial equieens needed o ee he objecives of he sudy: a) ealisic 3D odel of shoos o epesen sall-scale sucue; b) siple hoogeneous ubid ediu odel fo Poisson canopy o epesen acoscopic sucue. Foliage eleens (shoos o leaves) wee andoly disibued and spheically oiened (G-funcion and phase-funcion fo spheically oiened leaves, cf. Chape 3). Needle eflecances and ansiance wee assued o be siila o hose of leaves, hus he 9

30 diffeence beween he wo canopies eflecances caused solely by shoo sucue. Geoeical odel of Scos Pine (Pinus Sylvesis.) shos was ipleened efeencing field easueens [Senbeg e al., ]. Thee ypes of canopies wee siulaed: ) fla leaves, ) shoos, 3) shoo-lie leaves, coposed of leaves wih he sae G funcion and siila scaeing popeies as shoos. Siulaions wee pefoed o geneae eflecance of hese canopies, assued o be bounded below by blac soil. Recall (Chape 3), shoo scale scaeing in he needle leaf canopies is paaeeized in es shoo silhouee o oal aea aio (STAR). Spheically aveaged STAR is ypically uilized, STAR = SSA( Ω)dΩ, TNA 4π 4π (47) whee SSA( Ω ) is he shoo silhouee aea in diecion Ω and TNA denoes he oal needle aea of he shoo. The STAR paaee is analogous o G-funcion fo leaves (cf. Chape 3). In he case of spheically oiened scaeing eleens, he following holds:.5, fo leaves, G = STAR, fo shoos. Noe, he STAR paaee is elaed o shoo sucual paaee, p( Sh), which can be shown as follows. I follows fo Cauchy s heoe fo of convex, non self-shadowing objecs ha he aio of silhouee o oal aea is ¼,. In conas, needle leaf canopy shoo is a selfshadowing objec due o self-shadowing of needles, and his aio, he STAR paaee, is salle. Theefoe, ( / 4 STAR) / 4 4 STAR quanifies he degee of self-shadowing, o he poion enegy apped inside of objec due o self-shadowing. Fo anohe side, shoo sucual paaee, p( Sh), is defined as he pobabiliy ha a phoon scaeed by needle of he shoo will ineac again wih anohe needle of he sae shoo. Copaing he above wo definiions, we infe ha p( Sh) 4 STAR. (48) Noe he following feaues of Eq. (48). Fis, he eason fo he lac of exac equaliy is ha 4 STAR is defined as he ean ove poins on he suface, while p( Sh) is defined as spaially aveaged ove poins of ineacion. Second, in conas o STAR, p( Sh) is no jus a funcion of shoo geoey bu has soe dependency on needle opical popeies since hey affec he diecional disibuion of scaeed phoons. Thid, p( Sh) is defined based on he assupion ha he pobabiliy of ineacions says consan wih successive ineacions. Equaion (48) was veified wih ay-acing siulaions fo nine pine shoos and esuls ae pesened in Fig.. 3