Canopy Reflectance, Photosynthesis, and Transpiration. III

REMOTE SENS ENVIRON 42 187-216 (1992)

Canopy Reflectance, Photosynthesis, and Transpiration. III. A Reanalysis Using Improved Leaf Models and a New Canopy Integration Scheme.

P. J. Sellers,* J. A. Berry, ~ G. J. Collatz, t C. B. Field, t and E G. Hall* *NASA/Goddard Space Fhght Center, Greenbelt and tCarneg~e Institution, Stanford, Cahfornm

1 h e theoretwal analyses of Sellers (1985, 1987), which hnked canopy spectral reflectance properties to (unstressed) photosynthetw rates and conductances, are cmtwally remewed and significant shortcomings are ~dent~fied These are addressed m thin article principally through the incorporation of a more sophmtwated and reahstw treatment of leaf phystological processes within a new canopy integration scheme It m assumed, based on eco- physiological observations and arguments, that leaf physiologwal properttes vary throughout the plant canopy m response to the radmtion-we~ghted time- mean profile of photosynthetwally actwe radmtion (PAR) These modificatzons yield a s~mpler and more robust theoretical relationship between canopy b~ophysical rates (photosynthesm, conductance) and spectral vegetation radices (SVI) The results indicate that area-averaged SVI, as obtained from coarse resolution satelhte sensors, may gwe good estimates of the area-integrals of photosynthesm and conductance even for spatially heterogenous (though phys~ologwally umform) vegetation c o v e r s

Address correspondence to Piers Sellers, NASA / GSPC, Code 923, Greenbelt, MD 20771

Recesved 6 May 1991, rewsed 28 March 1992

INTRODUCTION

Over the last 10 years, there have been some important advances m our understanding of how leaves asslmdate carbon and control the simulta- neous loss of water vapor through their stomata Farquhar et al (1980) showed how a biochemical model of leaf COz asslmdatlon based on rate constants calculated from a consideration of the enzyme kinetics and electron transport properties of chloroplasts could yield a reahstlc description of photosynthesis for C3 plants Collatz et al (1991), following Ball (1988), used this work and observahons of stomatal conductance to construct a robust semlempmcal model of leaf stomatal function which can reproduce the response of leaf conductance to changes m ambient temperature, humidity, CO2 concentration, and asslmllahon rate A number of mveshgators (Field, 1983, Ter- ashlma and Inoue, 1985, Hlrose and Werger, 1987, Gutschlck and Wlegel, 1988, Farquhar, 1989, Evans, 1989a) explored consequences of the chstnbulaon of photosynthelac capacity in leaves and canopies with respect to light and developed criteria for identifying the distribution of any fixed total capacity that maximizes photosynthetic CO2 asslmdatlon

It would be useful to extend this knowledge

0034-4257 / 92 / $5 O0 ©Elsemer Scwnce Pubhshmg Co lnc, 1992 655 Avenue of the Amerwas, New York, NY 10010 187

1 8 8 Sellers et al

of leaf-level processes up to the canopy scale (meters and kdometers) Among other apphca- tmns, this could lead to the calculatmn of carbon fluxes and evapotransp~ratmn rates on scales con- slstent with global bmgeochemmal cycle studies, see, for example, Tans et al (1990) To do this, it is necessary to quantify the relataonshlps between canopy functmn and spectral signatures because satelhte remote sensing offers the only prachcal means of continuously and consistently momtor- mg bmsphenc processes on a global scale

Sellers (1985, 1987) mveshgated methods of integrating simple leaf-level models of light scat- termg, hght absorphon, photosynthesis, and stomatal conductance over the depth of vegetation canopms His analysis explored a theorehcal basis for analyzing the empmcal connections between spectral vegetation mdmes (SVI) and important functional relationships that regulate canopy photosynthesis and transplratmn A key result showed that for horizontally uniform (plane-parallel) canopies, there is a strong mechamshc basis for a correlatmn between the fractmn of photosynthetically actwe radlatmn absorbed by the vegetation canopy (FPAR) and the assocmted simple ratm vegetatmn index (SR) (near-mfrared reflectance dwlded by wslble reflectance) The analysis also showed that the bulk canopy photosynthehc capacity and the maximum canopy conductance were near-hnearly related to the SR However, the leaf physmlogmal models used m the analysis of Sellers (1985, 1987) suffered from a number of shortcomings

1 The leaf COz asslmdahon and stomatal conductance models used simple empmcal func- hons which are hard to parametenze

n There was no hnkage between stomatal funchon and leaf CO2 asslmilatmn

in Leaf physmlogmal properhes (photosynthetic capacity, etc ) were assumed to be mvariant through the depth of the canopy

These issues are addressed m the analysis presented m this paper We demonstrate that the mcorporatmn of a more soph~shcated treatment of physmlogacal processes results in a simpler and more robust relatmnshlp between canopy baophyslcal rates (photosynthesis, conductance) and spectral vegetation radices (SVI)

THEORETICAL BACKGROUND

Summary of the Analysis of Sellers (1985; 1987) Many researchers have uhllzed combinations of spectral radiance observahons acquired over vegetated surfaces by satelhte or a~rcraft-mounted sensors as indicators of the density, health, or blomass of the vegetahon These empirical apph- cations of remote sensmg take advantage of the large difference between the hght scattering prop- ertles of green leaves m the visible and near- infrared wavelength intervals

Sellers (1985, 1987) used a two-stream approxlmahon model to describe radlahve transfer wRhln vegetation canopies The equahons obtained from the two-stream method may be used to calculate the hemispheric reflectance of a plant canopy as a function of a) the relative spectral response function of the sensor, b) the radlatmn field incident on the canopy, c) the sod or background reflectance, d) the scattering coefficmnts and geometric arrangement of the leaf elements, and e) the amount of vegetahon present, as specified by the leaf area index, for example A related procedure was used to calculate the profiles of radlahon absorbed by leaves as a function of canopy depth

Sellers (1987) used these equations to show that for 1deal condit ions-uniform green canopy, dark underlying sur face- the spectral vegetation indices (SVI) should be proporhonal to the near- infrared reflectance, a~, and to FPAR [referred to as APAR in Sellers (1985, 1987)] The most commonly used SVI are the simple raho (SR) and the normalized difference (ND) vegetahon mdmes, defined as

"" (la) SIR=--, a~

ND = a, - a~ (lb) a.~ + a~

where aN,a~ = hemispheric canopy reflectances for

near-infrared and wslble wavelength intervals, respectively (sensor-dependent)

Sellers (1987) showed that this useful relahon- ship between the SVI and FPAR holds because the broad-band scattering coefficients of green leaves In the near-infrared (coN) and wslble (mQ

Canopy Reflectance, Photosyntheszs, and Transp~ratura 189

wavelength intervals are very different (Table 1), this difference is such that

aaNoc a(FPAR), for all values of LT, (2) a L~ a L~

where L~ = total leaf area index

when the extinction coefficient for the flux of PAR or visible radiation (k) is roughly double the extinction coefficient for diffuse near-infrared flux (hN) within the canopy, that is, when

k = 2hN, (3a) which may be reexpressed as

C(u) (1 - o v) = 2( - o N) ( a b )

where Ogv, 09, = leaf scattering coefficients in the

visible, near-infrared wavelength intervals, respectively (sensor- dependent),

k = extinction coefficient for direct (solar) beam flux within the canopy

= I t ( u ) / u ] ( 1 - o J 4 h. = extinction coefficient for diffuse

near-infrared flux within the canopy = (1 - ('ON) 1/2,

Table 1 P a r a m e t e r s U s e d T o C a l c u l a t e L e a f P h o t o s y n t h e s i s a n d C o n d u c t a n c e for t h e M o d e l s U s e d m Sel le rs (1985, 1987)

Parameter Umts Value

Photosynthesis al ~tmol m -2 s -1 52 0 bl /*mol m -z s l, W m -2 13800, 3 0 0 0

Conductance a2 /*mol mol - l , J m -3 1268 5, 13966 0 b2 /tmol m -2 s - l , W m -2 046 , 0 1 c2 (mol m -2 s - l ) - l , s m -1 055 , 2 8 0

Leaf propertms G(u) - 0 5 w~ -- 0 2 to~ - - 0 95

Sod reflectance to, -- O1

Solar angle /~ cos- 1 0 5

Adapted from Charles-Edwards and Ludwig (1974) and Jarvls (1976), see Eq (1)

These parameters were obtained by curve fits to the PAR response functmns as gtven by the Farquhar et al (1980) and Collatz et al (1991) models for stress-free (relatwe humidi ty = 1) con&tmns, see Figure 6e, f

G(p) = relative projected area of leaves in direction cos - 1/,,

= cosine of solar zenith angle Simply put, Eq (2) holds because the near-

infrared reflectance aN is proportional to double the pathlength of near-infrared radiation in the canopy [e--2hNLr], a s this radiation must enter and leave the canopy, while FPAR is proportional to only the one-way penetration and absorption of PAR through the canopy [e -k~T] The two parameters, aN and FPAR, will be proportional to each other if Eq (3) is satisfied

If the sod or background material underl)ang the canopy is relatively dark, so that

a av 0 ,

we can w r i t e

0 (SR) ~ 0 aN (4)

0 LT O LT It follows then that FPAR is proportional to SR

Figure la shows how the canopy visible and near-infrared reflectances, av and aN, the simple ratio vegetation index SR, and FPAR vary with leaf area index for the (almost ideal) model canopy described in Table 1 Figure lb illustrates how the nonlinear functions of leaf area index, SR and FPAR, shown in Figure la are almost linearly related to each other because of the relationship expressed in Eq (3)

The analysis summarized above provides a theoretical foundation for the correlation between the SR and FPAR established by empirical analysis of remote sensing data, Eq (3) is normally a reasonable approximation for canopies composed of randomly distributed elements with identical reflectance properties overlying dark soils It is important to note, however, that the system of equations given in Sellers (1987) provides a general basis for relating hemispheric reflectance measurements to canopy, leaf, and soil proper- t ies-regardless of whether the above approximation is true

Sellers (1985, 1987) also examined the relationship between canopy reflectance and the use of PAR by leaves for net photosynthesis A. and the regulation of stomatal conductance g. This is a more complex problem since these physiological

1 9 0 Sellers et al

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Ftgure 1 a) Varmtmn of canopy optical parameters with leaf area index as calculated by the two-stream approxlma- tmn model of Sellers (1985, 1987) with parameters taken from Table 1 a., aN = wslble and near infrared (hem]- sphermally-mtegrated) reflectances, SR = simple ratm vegetatmn index, FPAR = fractmn of PAR absorbed bv the canopy b) Simple ratm vegetation index plotted against FPAR, replotted from Figure l(a) Dots on the curve reler to values of leaf area index, 0 1, 0 5, 1 0, 2 0, 4 0. 8 0, readmg from left to right

processes are mf luenced by several o ther vanables ( temperature , water vapor content of the aar, water potential of the leaf, and the p rmr history of the leaves m the canopy) in ad&tmn to the absorbed flux of PAR

In the t rea tment of Sellers (1985, 1987), the models of Charles-Edwards and Ludwig (1974) and Jarvls (1976) were used to descnbe leaf pho-

tosynthes]s and lea| stomatal conductance, respectively

I A,, (Sa) [b, + e n I

[a b z + F n t g ,= [f(T)f(~oOf(ge)], (5b) 2 + b2ce + c2F n

A,, = leaf photosynthesis (/tmol m -z s -t)

- P in Sellers (1985, 1987), a,,bl = specms-dependent constants

(/~mol m -2 s i or W m-2), F = (vector) flux o~ PAR

(/~mol m -2 s -2 s -1 or W m-e), n = v e c t m of leaf normal,

f(T),f(~oO,f(Se ) =ad jus tmen t factors to account for the effects of temperature , leaf water potential, and vapor pressure def io t stress,

g~ = leaf stomatal conductance for water vapor, (mol m -2 s -j or m s- '),

az,b2,c2 = specms-dependent constants (mol mo l - ' or J m ~, mol m - s- or W m -, (mol m-2 s 1)-1 or s m ~)

The formulation used m (5b) is consistent with the nomencla tu re used m Sellers (1985, 1987) It was assumed that all the leaves m the canopy respond ldentmally to F n The constants a~, b~, a2, b2, and e2 can be de t e rmined from curve fits to data (see Table 1) The stress factors f(x) vary from umty, under ophmal condltmns, to zero when photosynthesis and transplratmn are totally suppressed by adverse envaronmental con&tlons (see Jarvls, 1986, Sellers et al , 1989, Collatz et al , 1991)

The combinat ion of the enwronmenta l stress factors was assumed to operate more or less um- formly throughout the canopy so that in estimating canopy photosynthesis and conductance it is only necessary to integrate Eqs (5) with respect to the varmtlon of in te rcepted PAR, F n, down through the canopy The canopy integral forms of (5) may then be wri t ten as

A~ =f(E) .10 [bl + F dL, (6a)

where

Canopy Reflectance, Photosynthesis, and Transpwatmn 191

] g~ =f(Z) 0 2 + b2c2 + C2 F n dL, (6b)

where A~= canopy photosynthesis (pmol m -2 s-l), g~ = canopy conductance

(mol m -2 s -1 or m s-l), f(Z) = f( T)f( gxl)f( ge)

The PAR flux vertmal component is assumed to be attenuated as it passes down through the canopy followmg the semmmpmcal expressmn of Goudrman (1977)

FL = Foe- k~ (7) where

FL = PAR flux at leaf area index L m the canopy (pmol m -2 s -1 or W m-2),

F0 = PAR flux above the canopy (pmol m -2 s -j or W m-2),

L = cumulatwe leaf area index Insertmn of Eq (7) into (6) allows evaluatmn

of A~ and g~ (see Sellers, 1985, Tables 3 and 4) [Note The combmatmn of (6) and (7) only ac- counts for the vanataon of PAR mtenslty with canopy depth, a mean leaf angle is assumed throughout Sellers (1985) explored the impact of this slmph- ficatmn on the calculatmn of the bulk canopy propertms, A~ and g~ Full integratmns over leaf angle and onentatlon were made pnor to the mtegratmn with canopy depth, the &fference between the results obtmned with the full (leaf angles and onentatmns, canopy depth) and slmph- fled (mean leaf angle/ormntatmn, canopy depth) mtegrataons was found to be prachcally neghgible ]

We may now rewrite Eqs (6) as

A~ = A*f(Z), (8a) g~ = g*f(E), (85)

where A* and g* are the mtegrated kernels of (6) and represent the canopy-scale values of the unstressed photosynthetm rate and conductance

Figures 2a and 2b show how An, A* and g~, g* vary with incident PAR flux Fo, accordmg to Eqs (5) and (6) (Here An and g~ refer to leaves at the very top of the canopy, fully exposed to the ambmnt PAR flux F0 )

Figures 2c and 2d show how A* and g* vary with total leaf area index For any given PAR flux, there Is a diminishing increase m A* and g* for

further increments of leaf area index as more and more of the canopy consists of shaded leaves whmh have progresswely lower rates of An and g~ Now an inspection of the solutions to Eqs (6) or (8) [see Tables 3 and 4 in Sellers (1985)] shows that, for low values of the incident PAR flux Fo, the derivatives of A* and g* with respect to total canopy leaf area index LT are proporhonal to the derwatlve of FPAR with LT and to e-k~

OA* 0g*oc 0 (FPAR) 0I.~' OLT e-~T' OLT ' as F 0 ~ 0 (9)

Equation (9) holds because the PAR flux is used to drwe photosynthesis and conductance as it is progresswely absorbed through the canopy Figures 2e and 2f show A* and g* plotted against FPAR, and it can be seen that, for the lower values of F0, Eq (9) holds resonably well This is because, at low values of Fo, all the leaves m the canopy have approximately the same slope of A or g, agmnst F0 (see Figs 2a and 2b) At higher values of F0, however, the leaves at the top of the canopy are saturated (no change in A or g~ with F0), but the leaves lower down are still below saturation, resulting in an increasingly nonhnear relationship between A*, g*, and FPAR with increasing F0 This effect can be seen more clearly in Figures 2g and 2h, where A*/Fo and g*/Fo are plotted agamst FPAR The hnes corresponding to the lowest values of Fo are the most nearly hnear, degradmg to more nonhnear forms as F0 increases

Comparmg Eqs (2), (4), and (9), we have the cham of relationships

OA* 0 g * 0(FPAR) o: Oa,, OLT' OLT OLT OLr

0 (sn) OLd' as F 0 - ' 0

(10) so that under the specafied condl t lons-umform canopy, dark underlying sod, low levels of Fo-- A* and g* should be proporhonal to SR and other SVI However, on clear days F0 values typically reach 200-400 W m -2 and so this approxlmahon should be vmwed with some caution Addlhonally, further nonhnear effects are reduced when the vegetation within the field ofvmw is clumped into spahally heterogenous umts (see Fig 21 of Sellers (1985)1

This apparent nonhneanty of canopy funchon potenhally degrades the prospects for reahstm estimation of CO~ assimdatlon using remote sens-

192 Sellers et al

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F)gure 2 a,b) Unstressed leaf, A. and g~, and canopy, (A* and g*), photosynthesis and conductance as iunehons of iilel- dent PAR flux, Fo, calculated using Eqs (5) and (6) m text, parameters from Table 1 Total leaf area index for the canopy, LT = 8 c,d) Variation of unstressed canopy photosynthesis, A*, and conductance, g*, wlth leaf area index and PAR flux, Fo, calculated using Eqs (6) Values of F0 are marked on the curves

ing It results from the fact that, in the formulation of Sellers (1985, 1987), leaves at different levels in the canopy are saturating with respect to F n at different levels of F0 If photosynthetic capacity, as represented by the values of ai and bl in (5), is constant throughout the canopy, leaves near the top of the canopy would usually be completely light-saturated while leaves near the base would still be on the hnear portion of the hght-response curve and thus unsaturated This formulation cer- tainly exaggerates any nonhneanty that occurs m

nature, because leaves m shaded habits tend to have lower photosynthetic capacities and saturate at lower hght levels (Bjorkman, 1981)

In the following sections, we extend the analysis of Sellers (1985) by replacing the emplncal leaf models with more general, semlmechamshc models of photosynthesis and stomatal conductance In contrast to the uniform distribution of canopy properties with depth assumed above, we explore depth distributions of photosynthetic capacity and maximum stomatal conductance that

Canopy Reflectance, Photosynthesis, and Transp~ratmn 193

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Figure 2 (continued) e,f) Relationships between A*, g*, and FPAR for a range of PAR fluxes using Eqs (6) and the two- stream approximation model described m Sellers (1985, 1987) g,h) Relationship between A* / F0, g* / Fo and FPAR, A* / Fo and g*//7o can be considered as surrogates for a A*/0/7o and 0 g * / 0 F% The points on the curves refer to values of L~ = 0 1, 0 5, 1 0, 2 0, 4 0, 8 0, working from left to right The numbers on each curve refer to values of F0 ,n W m -2 The canopy is assumed to be uniform and green, overlying a dark background Parameter values are given in Table 1, f(Z) -- 1

parallel the depth distribution of photosynthet,- eally actwe radiation. This arrangement is sup- ported by both an expand, ng base of empirical data and a number of theoretical assessments that identify an optimum depth &stnbutlon of photosynthebc capacity for maximizing canopy photosynthesis The results of this analys]s are simpler and more general than those from the prev, ous studies, and indicate that the relation-

sh]ps among SR, FPAR, Ao and g~ should be almost ,ndependent of the spat]al d, stnbubon of the vegetation

Improved Leaf Physiological Models Farquhar et al (1980) presented a blochem]cal model of leaf photosynthes]s (including photores- plrat]on), subsequently extended by von Caem-

1 9 4 Sellers et al

merer and Farquhar (1985) and other researchers, which descnbes CO2 asslmIlataon, A, as rate-hmlted by enzyme kinetics, specifically the amount and cycle t ime of the carboxylatIng enzyme RubIsco, and electron transport, which is a function of incident PAR and the efficiency of the leaf's light- in tercept ing apparatus (chlorophyll) In a recent version of the model (Collatz et al , 1991), A is given by A = min(w,, We, W,), where w,, w,, w~ are functions which desribe the assimilation rates as l imited by the efficiency of the photosynthet ic enzyme system, the amount of PAR captured b~ the leaf chlorophyll and the capacity of the leaf to export or utilize the products of photosynthesis, respectively

The nbulose-bisphosphate (RuBP) carboxylase- oxygenase enzyme (Rublsco) h m a i n g rate on asslmdatlon, w,, is given by

[ ] w,=V,,, c,+ (fTd2/K,,il (11) where

w. = Rubisco-hmited rate of assimilation (/~mol m-e s- 1),

V,,, = maximum catalytic capacity of RubIsco (gmol m -2 S-')

(7, = concentrat ion of CO2 in leaf interior (Pa),

02 =part ia l pressure Oz in leaf interior (Pa),

F * = CO2 compensat ion point (Pa), K, = Mmhaehs -Men ten constant for

CO2, (Pa), K,,=lnhlbltIOn constant for O2 (Pa)

V., is given by the p roduc t of Vm.. and a t empera tu re -dependen t function [see Appendix of Collatz et al (1991) and Table 2] Vm~. IS a proper ty of the leaf (or chloroplast) and is proportional to the Rublseo reserves of the leaf (or chloroplast) and thus its ni trogen content F*, / ( , and K,, are all functions of t empera ture [see Appendix of Collatz et al (1991) and Table 2 of this article]

The hght- l imlted rate of assimilation, w~, is given bv

C,_-F*.] (12) w, = ( F n ) e ( 1 - to , ) C, + 2r*J

where

w, =light- l imited rate o| assimilation (pmol m -e s- ' ) ,

a = ln t r insm quantum effiemncy for CO2 uptake, (pmol pmol -l or pmol J 1)

A third limiting rate has been defined bv Collatz et al (1991) w, is the capacity for the export or utdizatlon of the products of photosynthesis and is es t imated by Collatz et al (1991) to be

u., = ~;,, / 2 (13)

The simplest way to p roceed is to assume that the assimilation rate is the min imum of w,, u~,, and w, (el Farquhar et al , 1980) However, observations indicate that the transition from one hm- a ing rate to another is not instantaneous and that coupling be tween the three processes leads to smooth curves rather than superposi t ioning of straight lines Collatz et al (1991) describe this effect bv combining the rate terms into two qua- dratm equations, which are then solved for their smaller roots

Owl, - w,,(u., + ~,) + w,w, = O, (14a) t ~ A 2 - A(wp + w,) + w,,w, = 0. (14b)

where A = assimilation rate (pmol m -2 s-l),

O,fl = coupling coefficients, w, ,="smoo thed" min imum ofw~ and

w, (pmol m--' s l)

The coefficients 0 and fl can theoretically range from 1 (no coupling effects) to 0 In nature, these coefficients assume values on the order of 0 8 to 0 99 (see Collatz et al , 1990) Figure 6a shows an example of a sharp transition from w~ to wc (0, fl = 1), Figure 6e shows a more gradual transition (0, fl = 0 8)

Net assimilation A,, ~s then given by

A,,= A - R,I. (15) where

Rd=lea f respiration rate (/.tmol m e S l)

Collatz et al (1991) scaled Ra to the leaf carboxylase content by

Rd = 0 015V,,, (16)

Canopy Reflectance, Photosynthests, and Transp~ratugn 195

Table 2 F o r c i n g Varmbles , Lea f Pa rame te r s , and D e r w e d O u t p u t Varmbles for the C o u p l e d S torna ta l -Photosyn thes l s L e a f Phys io logy Models

A Forcing Variables Symbol Varmble Values Units

Ca COz concentration m CAS t 34 Pa Fo PAR flux above canopy 50, 100, 200, 400 W m -z

230, 460, 920, 1840 /Jmol m -2 s -1 gb Leaf boundary layer conductance (H20) 0 04 m s 1 ha Relatwe humlchty of CAS 0 1, 0 25, 0 5, 0 75 02 02 concentratxon m CAS t 20,900 Pa p Atmospheric pressure t 1 013 x 10 ~ Pa T, Leaf temperature 310 K /~ Cosine of incident angle of PAR flux 0 25, 0 5, 0 75, 1 0

B Leaf Parameters b Minimum stomata] conductance t 0 01 mol m -2 s-1 G(g) Leaf angle &stnbutlon funclaon 0 5 -

(replaces n) K~ Mlehaehs-Menten coefficient for CO2' 30*2 1 Q, Pa Ko Inhibition coettqclent for O2 t 30,000.1 2 Q, Pa Lr Total leaf area index 0 1, 0 5, 1 0, m 2 m- ~

2 0 , 4 0 , 8 0 m Slope parameter t 9 0 - Qt Qlo temperature coefllclent t (T, - 298) / 10 -- S CO2 / 02 speefllelty t 2600*0 570' -- Vm Maxamum leaf catalyhe capactty at T~ Vm~x*2 °' /Jmol m -2 S-l Vm. Maxamum leaf catalyhc capaoty at 298 K 200 /Jmol m - 2 s-

Vm~ Vm~ for top leaves* 200 /Jmol m -2 s -1 F* CO2 photocompensatlon point* Oz / (2S) Pa e Quantum efllelency for COz uptake t 0 08 mol mol- l ~ Leaf scattenng coefficient for PAR 0 2 -- 0, fl "Linkage ~ coefficients between 1 0, 1 0, -

we, w . w, 0 8, 0 8 C Derived Varmbles

A Ass~mdat~on rate -- /~mol m - ~ s- C, Leaf interior CO2 concentration -- Pa C, Leaf surface CO2 concentration -- Pa E Transp~ralaon rate -- mol m -2 s -~ or m s -~ g, Leaf stomatal conductance (H20) - mol m - 2 s- 1 or m s- 1 h, Leaf surface specific huml&ty -- -- Ptd Respzrataon rate -- /~mol m-2 s-1

* Values of forcing variables and leaf parameters used to test mtegrat~on schemes were taken from Collatz et al (1991) when m&cated by t CAS stands for canopy mr space

Collatz et al (1991) w e n t on to incorporate the above photosynthesis model with the Ball (1988) semlempmca] model for leaf stomata] resistance

g~=mA"h,p+b, (17) C,

w h e r e

g, = stomata] conductance for water vapor (mol m -~ s -1 or m s- l ) ,

m = c o e f f l o e n t from observahons = 9 for Ca plants,

gs

b = coefficient from observahons 0 01 for Ca plants

(mol m -~ s -1 or m s- l ) , relatwe humidity at leaf surface, CO2 concentration at leaf surface (Pa),

p = atmospheric pressure (Pa), ps=s tandard atmospheric pressure

= 1 013 × 105 (Pa), Tf= freezing temperature -- 273 16 K,

(m s -1 )=00244 Tps _~ - - - - g~ (mol m s - 1)

h8

C,=

196 Sellers et al

The leaf surface environment variables, h, and C,, are given by (see Fig 3)

( H , - Ho)g = ( H , - = ( l S a )

We also see f rom F igu re 3 that

h , = ~ ,

(C.-C,) ~, (C , -C , ) g~ =A,,, (18c) p 14 p 16

where 11,, H,, H, = water vapor concentration in the

external air, at the leaf surface and Inside the leaf, respectively (mol mol- l),

gb = leaf boundary layer conductance tor water vapor (mol m -2 s-I or m s 1).

E=transplrat lon rate (tool m--' s-l or m s i).

C., C, = carbon dioxide concentration in the external air and at the leaf surface, respeetwely (Pa),

1 4, 1 6 = factors to account for different dlffUSlVltles of H20 and CO2 in the leaf boundary laver and s tomata pores, respectively

The system is then closed by calculating the CO2 concentration of the leaf interior

1 6A,, (7, = C, -p (19)

g~

F~gure 3 Coupled stomatal-photosvnthesls models as described by Collate et al (1991) for canopy photosynthesis and transp~ratmn System shows path- ways for heat, CO.~ and water vapor flux

I

~ - S t o m a t e - ~ Leaf Boundary ( 1/gs) ' Laye r

~ Ts

(1/g b) H - ~

d v v v v v ~ , • Sensible T a Heat

E _~ Water V v v v ~ N ~ '• Vapor

Ha

.~- A vvvvvvv~ • CO 2

Ca

Figure 3 shows how the fluxes of water vapor, temperature, and heat from a leaf to the external air are assumed to behave in this model Figure 4 shows how the system of equations is solved in practice Table 2 hsts the required forcing vanables (environmental conditions), leaf parameters, and the derived (output) variables gtven by the solution of the equation set described in this section and depleted in Figure 4

The Scaling of Leaf Physiology to Canopy Position Leaves in shaded mlerosltes have lower photosynthetic capacmes and maximum stomatal eonduc- tances than leaves in exposed mlerosltes (Bjork- man, 1981) This extremely general pattern apphes to plants grown in controlled environments with high and low hght (Bjorkman and Holmgren, 1963, Evans, 1989a), to lower and upper leaves on a single plant (Field, 1983, Hlrose and Werger, 1987), and to maxed species assem- blages with shade-tolerant understory species be- neath a canopy of different species (Bazzaz, 1979, Sehlmel et al, 1991) Species &tier greatly in their phenotyplc plasticity (Seemann et al, 1987,

F~gure 4 lteratwe solution method tor the photosvnthesls-stomatal conductance model, alter Collate et al (1991) equation set Equatton

(16)

(11)

(12)

(13)

(14 15)

(17,18)

(19)

V m ~ f(Vma x T)

Rd ~ f(Vm)

K c K o ~ f(T)

F* = f(T, 0 2)

Estimate C~

w c ; f(C,, V m, F* , K o, K c)

w e ; f(C, F°n, £, ~ F* )

Ws ~ f(Vm)

A° i gs hs Cs =

fiw~ We, W s (::), tJ)--R d

Ad ust C

f(A n m b, gb Ca)

New estimate, C, '= tiC s A n , gs)

, Exit

Canopy Reflectance, Photosynthesis, and Transptratton 197

Evans, 1989a), in response to growth m low or high-light habitats (Bjorkman and Holmgren, 1963), and m the extent to which stress factors other than light depress photosynthetic capacity (Mooney, 1972) All of this intrinsic complexity is the raw material from which ecological and evolutionary factors shape plant canopies, with the result that the depth-distribution of photosynthetic capacity m real canopies tends to be quite predictable

Why does the shaping of plant canopies of ecological and evolutionary factors constrain the response of photosynthetic capacity to light avaal- ablhtyP Three factors appear to be critical First, the basic mechanisms of photosynthesis are essentially identical in all C3 plants (more than 85 % of all plant species), and differences in capacity are largely driven by differences in investment in the biochemical machinery of photosynthesis The strong, linear relationship between photosynthetic capacity and leaf nitrogen (Field and Moo- ney, 1986, Evans, 1989b) attests to this hnkage between investment and capacity Second, the investments required to support photosynthesis are expensive Nitrogen nutrition alone can con- sume from 20% to 45% of the carbon fixed m photosynthesis (Chapm et al, 1987) Third, any plant that invests too much photosynthetic capacity in any leaf should be at a competitive disadvan- tage to a plant that matches investments to local resource (especially light) availability For a single plant with leaves distributed through a canopy, the cnterlon for the most efficient chstnbutmn of capacity is gwen by the solution to the lsoperime- tric problem In dynamic control theory (see Intnl- lgator, 1971, Bloom et al , 1985) Specifically, if photosynthetic capacity is limited by some quantity Z that is expensive for the plant to acquire, then the returns on any fixed investment m Z are greatest when

O Amt 0--~-= 2, (20)

w h e r e Alnt lS photosynthesis Integrated over an appropriate spatial or temporal scale and 2 is an undefined Lagranglan multiplier In the analyses to date, Z has been considered as transpiration in a single leaf (Cowan and Farquhar, 1977), leaf nitrogen in a canopy (Field, 1983), leaf mass m a canopy (Gutschlck and Wiegel, 1988), and photosynthetic capacity in a leaf (Farquhar, 1989) Amt

is typically the instantaneous value of photosynthesis for single leaf analyses but the single-leaf dmly integral for canopy analyses If the condition m Eq (20) holds for instantaneous values, it also holds for daily integrals The problem of optimal resource allocation has no general solution when a canopy is composed of many plants or species, but the single plant solution may be approximately correct, as long as all plants tap a common pool of below-ground resources and construct tissues of similar composition and cost

PAR is also expenswe to acqmre, because plants cannot capture it without investing m leaves, stems, and branches Formally, the problem is identical to those already solved As long a s

a2A < 0 (21) 0 t(F-n)(1 - oa~)] 2

for all A, then we can also write

aA = 21, (22a)

a [(F. n)(1 - v)l which leads to

0& = 22, (22b)

a(FPAR) where the value of 2 as &fferent for each quantity (water, mtrogen, mass, or PAR) and also depends on the level of each quantity invested as well as on the status of the environmental vanables that influence photosynthesis and transpiration

Equation (22) Implies that, at all levels m the canopy, leaves begin to saturate with respect to (F n) (1- o~) at the same F0 In fact, the light response curves for leaves at all levels of the canopy should be scaled versions of a single response, w~th a scaling factor proportional to [(F n)(1 - a)~)]/Fo As long as dark respiration Is the same at all levels m the canopy or is proportional to Vmax [as m this analysis, see Eq (16)], then the light saturated photosynthetic rate, or A . . . . should also be proportional to [(F n)] (1 - a~v)] / F0 or to ( F n) (1 - co 0

At any level in the canopy, the value of [(F n ) (1 - o9~)]/Fo changes during the day, on a time scale that changes m Amax are unlikely to track Again, for optimum efficiency, adjustments In Ama~ are hkely to follow a mowng time average of the hght regime An expanding body of empirical data, mcluchng the results of Field (1983),

198 Sellers et al

Waiters and Field (1987), and Pons et al (1990), d~rectly supports the predlchon that Ama~ for each leaf should be proportional to the hme-averaged or integrated PAR that that leafexpenences Addl- honal studies, Including DeJong and Doyle (1985), Hlrose and Werger (1987), Hlrose et al (1989), Klttel et al (1990). and Schlmel et al (1991), reported that leaf nitrogen, a strong corre- late of A .... parallels Integrated PAR These stud- les extend the generahty of the relahonshlp, because they involve a multlspecles prame (Schlmel et al . 1991) and single-species swards In which the leaf-age gradient places the youngest leaves at either the top of the canopy (Hlrose and Werger, 1987) or the bottom (Hlrose et al, 1989) Over a broad range of species and ecosystems, and with few exceptions (e g . Leverenz and Jarvls, 1980), the general pattern matches the predlchon from theory-Am~x and leaf nitrogen [N]- should scale with the hme-lntegral of the absorbed local PAR

These arguments and Eq (22) suggest that the profile of leaf nitrogen, [N]. and V, ..... down through the canopy should follow some hme average of the PAR flux. most hkely the radmtlon- weighted, time-mean F However. whde this para- digm describes the relative distribution of [N], it does not say anything about the absolute values of [N] and Vm~. within the canopy, which are more hkely to be constrained by the overall avallabdlty of nutrients within the plants environment For optimal efficiency, canopies can be expected to allocate [N] such that for a specified hght regime (F n) operating over a period T on the order of several days to a few weeks

U1 A dt + 3o[N]

ls maxlmlzed for all L U1 and U2 are cost-benefit weighting factors

which would be largely determined by the local avallablhty of mtrogen In a nitrogen-rich environment. where there is httle cost in malntalnmg high values of Vm... UI should be much greater than U2 and so maximizing photosynthesis [first term m Eq (23)] would have a higher priority than maximizing efficiency [second term in Eq (23)] Conversely, m nutrient-poor environments. we can expect Uz to be greater than U~ Whatever the condlhons, the value of Vm~. arrwed at from (23) can be assocmted with a reference value of PAR flux, ~', which hght-saturates the leaves, that

IS, F : F when w, = w, Under most eondlhons, when UI>>Ue, this reference value of P would be close to. if not equal to. the radiation-weighted hme-mean value of F. F. for smoothly varying fluxes for slmphclty, we shall assume that ~'--F- from now on

INTEGRATING THE COUPLED LEAF STOMATAL-PHOTOSYNTHESIS MODEL OVER THE CANOPY

From the analysis and observahons discussed m the previous seehon, we propose that the profiles of leaf mtrogen, [N], V, ...... and hence V., within the plant canopy are dlstnbuted accordmg to the radlahon-welghted hme-mean profile of PAR

V,,, = V,,.,f(L) (24a) where

Vm0 = maximum, that is. "'top" leaf, value of Vm m the canopy (/tmol m -2 s -l)

= product of Vm..0 and a temperature mhlbmon funehon (see Table 2),

f (L) = time-averaged vanahon of PAR flux w~th LAI [ eg , e -~ Eq (7)]

Also, for most condlhons where nutnents are not m drashcally short supply, that is. UI>>U2 m Eq (23). the value of V,,., wdl be determined by the radmhon-welghted hme-mean flux of PAR, F0, at the top of the canopy, that ~s.

V,,,o = function ot (w, = w,) (24b) when F0 = F0

To obtain V,,., as defined in (24b), F0 is inserted mto (12) and the derived value of w, used m place of w, m (11) to calculate V,,, (in this case V,,,,,)

The overbar on f(L) in (24a) denotes "radmhon-welghted time-mean value" Using the example of Eq (7) to describe f(L). we then have

f (L) = e- ~'. (25a) F = Fof(L), (25b)

= [G(/z) //.t](1 - wt,) 1/2 (25c) Note that Eq (25) also ~mphes a hme-mean zemth angle fi for the flux F

The mstantaneous hght-hmlted value of photosynthesis, we, is gwen by Eq (12) A full treatment of canopy photosynthesis would take mto

Canopy Reflectance, Photosynthesis, and Transpzratwn ] 99

account the entire range of leaf angles and orien- tations at every level on the canopy in addition to carrying out the top-to-bottom integration (PAR attenuation as described by Eq (7)), in other words, the F n term would be subjected to full azimuth and zenith integrations However, as discussed in Sellers (1985), the adoption of a single mean leaf projection G(/~) in place of n gives very similar results numerically The expression (F n) in (12) can therefore be replaced by a simpler function

l -n = C(u)

= FG(o-~)f(L) (26)

We can now construct a canopy-scale model of photosynthesis Equation (24) is combined with Eq (11) to give an expression for the variation of Wc with canopy depth

wc = aof(L), ao = Vmo C, + K~(1 + 02 / Ko)

where Vmo = mean value of Vm for the ensemble

of leaves at the top of the canopy (btmol m -2 s -1)

Equation (26) is combined with (12) to give an equivalent expression for We

we = bo f(Z),

(27b) The f (L) terms in (27a) and (27b) are different

i In (27a), the profile of Vm within the canopy is given by the product of Vm at the top of the canopy, Vmo, and the radiation-weighted t~me-mean profile of PAR flux d o w n t h r o u g h the canopy, j~L) [V~ is related to F0, the reference or t~me-mean (overbar) PAR flux as gaven by (24a)]

u In (27b), We varies with f(L), the instantaneous (no overbar) attenuation function for PAR down through a canopy, and F0, both of which vary with solar angle and cloud conditions

In reahty, leaves within a canopy are subject to variations in leaf surface relative humidity (h~),

leaf surface CO2 concentration (Cs), boundary layer conductance (gb), leaf temperature (Ts) and a profile of temperature (Ta), water vapor concentration (Ha), and carbon dioxide concentration (Ca) in the canopy air space The resulting resistance network for this "real" situation is shown in Figure 5a Complete integration of Eq (27) over a canopy IS fairly complex, requiring a numerically solved multdayer model of the type described in Sellers and Lockwood (1981) or Goudrlaan (1977) In most cases, however, it can be assumed that variations in leaf temperature (T~) and the canopy air space profile gradients of Ta, Ha, and Ca are small so that the resistance network can be simplified to that shown in Figure 5b, where Ts, Ta, ea, and Ca are assumed to be lnvarlant with depth Sensltlwty studies by Sellers and Lockwood (1981) indicated that, for the case of modeled transpiration from dry canopies, the difference between results produced by the Integration schemes represented by Figures 5a and 5b is relatively small

Following Figure 5b, the solution of the equation set for the entire canopy is now a relatively simple matter of numerical integration, more or less following the procedure used in Figure 4 Typical values of the forcing variables and leaf parameters listed in Table 2 were extracted from Collatz et al (1991) to construct a model copy It will be remembered that Vm IS given by the product of a leaf physiological property, Vm~x, and a temperature function (see Table 2) In this case, the value of Vma~0 was taken to be equal to the value of Vmax In Collatz et al (1991) and the profile of Vm~x was given by Eq (24) For this worked example ~, G(~) = 0 5 This corresponds to a canopy of spherically distributed leaves exposed to a t ime-mean flux of F0 with direction of ~ = 0 5 (solar zenith angle = 60 °) In Table 2, the quantum efficiency coefficient e is assumed to be constant with canopy depth, following the observations of Ehleringer and Bjorkman (1977) A maximum leaf area index LT of 8 was chosen for the study which represents a near maximum for normal broad-leaf conditions and provides a se- vere test of the integration schemes over the depth-varying PAR regime In this and all subsequent calculations, the vegetation is assumed to be free of soil moisture stress

Figure 6 shows the light response curves for leaves at different levels in the model canopy note In Figures 6c and 6d how the leaves all

2 0 0 Sellers et al

(a) F

I I

Leaf Canopy ] Leaf Boundary Interiors Stomates I Layers

I 1' ~ --. ~VVV"

C ' I I

'VVVVVV' • 'VV~VV~ -.., , 1/gs I 1/gb I

'VVVVVV~ = 'VVVVVV~, / , I I I

C,, H, C s, H s, T s I I

(b ) Leaf Interiors

C V H I I M

Canopy Leaf Boundary Stom ales Layers

I I I

~NVVVV' Z ~ T I

1/gs ~ 1/gb I I

I r w I I

"VVVVV~ANV' = I

Cs, H s, T s

(@) Canopy Leaf Boundary Leaf Stom ates I Layer

Interiors (Bulked) 1 (Bulked) I

C , , H , = ~ , ~ Z I

1/gc I 1 /g B I

Cs, Hs, Ts

Canopy Air Space

./

\ /

C a, H a, T a

Canopy Air Space

I C a, H a, T a

Canopy Air Space

C a, H a, T a

Ftgure 5 Resistance network corresponding to different canopy integration schemes a) Full scheme with developed profiles of T., H., C,,. and T, b) Restricted numermal scheme with T,,, Ho, C,,, and T, mvanant with canop~ depth c) Bulk (semlanalytlcal) scheme with T.. H~, C~, and T, mvanant with depth and bulk values of C,. H,, and C, In a) and b). F has direction/2, where /2 ~s not necessardv ~ In c),/2 =~

saturate at one value of F0, thus sahsfylng Eq (22) [The curves for the uppermost leaves in Figures 6e and 6f (O, fl = 0 8) were used to fit Eqs (5) and derive the parameters a l , bx, a2, b2, and c2 hsted in Table 1 and used m Fig 2]

Figures 7a-f show some results from slmula- hons based on numerically integrating (27) over the c a n o p y - 8 0 layers were used Note how the

behavior of the complete canopy follows that of the top leaves when h, ~ 1 (compare Figs 6 and 7) Figure 8 shows how various biophysical states and rates can vary with depth in the model canopy the profiles of C,, h,, g~, and A are faMy well developed In Figure 9, the effect of PAR intensity and incidence angle is shown, except for the extreme and physmally implausible cause of high

~7 0

E

E <I:

60 60

50

40

30

20

10

(a)

0 I 0 500

" , I , 1 = I J I

1 O0 200 300 4 0 0

F (Win 2)

50

10

4O

30

E

20

0 0 1 O0 200 300 400

F (Wm z)

(b)

Canopy Reflectance, Photosynthests, and Transpwatum 201

L

500

b

,y E

"5 E <

6Oso f (e) 40

3O

2 0 -

10

0 ~ - . . . . . . . . J . . . . . . . . . . . . . . I . . . . . . . = ....... 1 ....... i . . . . . . . T °'L~J'3 ....... 0 1 O0 200 300 400 500

F o (Wm 2)

60

50

4O

30

E

20

10

0 0

(d)

I ~ ~ . . . . . , . . . . . ~ . . . . . . ~-. ~ . . . . .

1 O0 200 300 400 500

F o (Wm 2)

Fzgure 6 Dependence of leaf photosynthesis and leaf conductance on PAR as calculated using the coupled stomatal-photosynthesls model (Fig 4) In all cases, relatwe humidity m the canopy air space was held at umty, h ,~ l , and leaf respiration Rd was set to zero The numbers m boxes refer to leaf position m terms of cumulative leaf area index a,b) A and g, versus the mean value of incident, PAR, F,, for leaves at different depths m the canopy, L = 0 refers to "top" leaves 0, ]/-- 1 0 c,d) Same as a) and b) except that all leaf responses are plotted against F0, note how all leaves saturate at roughly the same value of F0 O, fl = 1 0

PAR fluxes at glancing angles (1 e , to the left of the numerical solut]on max]ma), the canopy process rates show little variation with p

Figure 10 explores the effects of varying g~ within the canopy Three depth dependences of gb are used gb is assumed to be lnvar]ant with depth (gb = 0 04 m s- t), g~ vanes linearly with depth,

and g~ varies exponentially with depth, more or less as in the Simple Biosphere Model (S]B) of Sellers et al (1986) In all three cases, the integral of gb over the depth of the canopy is the same, 0 32 m s- 1 It appears that variations m the profile form of g~ have some influence on the estimate of canopy-integrated conductance g~ and Ec (In

202 Sellers et al

60

50

(e)

40 0

30 E '6 E

20

10

o

7

E

o~

L: -J l , ] J 1 ~ I 100 200 300 400

60

50

40

30

20

10

0 0 500 0

F (Wm 2)

60 60

50

6- 40

e3 30

E -6 E

20

10

(g)

(f)

1 O0 200 300 400 500 F ( W m 2)

o

E

50

40

~....~...'i~.....l. ,4~. I ~ " I r ~ r . . ~ r O0 1 O0 200 300 400 500 0 0

F,, (Win 2)

30

20

10

(h)

1 O0 200 300 400 500 F o (Win 2)

Fzgure 6 (~ontmued) e,t) Same as a) and b) except 0, fl = 0 8 The top leaf", L = 0, responses were used to fit E q (5) and derlve values of aL, bl, az, bz, and c2, see Table 1 g,h) Same as e) and d) except 0, ]~ = 0 8

these figures and all subsequent equations, the use of a capital subscript refers to a bulk or canopy-integrated quantity)

The solutions shown in Figures 7-10 for the numerical integration scheme of Figure 5b were used to evaluate the accuracy of a simpler bulk integration scheme suitable for operational apph- cations

Figure 5c shows the bulk integration scheme for the canopy Essentially, bulk or integral values of C,, g,, and g~ (C,, g~, g~) are specified which in turn imphes bulk values of C,, h,, A, and E The

canopy transfers of COz and H20 are thus treated as bulk integrated fluxes as m SIB It is further assumed that p---/7, so that f(L) is replaced by f(L) in Eq (27b) The PAR and Vm~, extinction terms are now identical [both are defined by f(L)] so the canopy-depth portion of the integral can be separated from the physiological portion of the equation set Ecluatlon (27) can then be substituted into Ec 1 (14a) and solved to give

6 0

5 0

40 0

~, 30 E "5 E

20

10

0 0

1 5

- - 1 0 L r-

E E J

05

0 0

6 0

5 0

E " 4 0

,5

30 E oV

2O

10

(a) Numerncal

~ . - - . - 1 O0 ./.>.,- o 75

. ~ J f ~ . s ' - - - - 0 50

/ S / • ## s ~ -o°'°

/ j / ~ * * o . O O o o ~ ° ' ~

°s °

, I J I , I , I , 1 O0 200 300 400 500

F o (Wm -2)

_ (c) Numerical ,...~----nO 50

0 75

/ 7 .-'~'~ ,~/ /:.+ oo , , /~/ ~,, '~" . . . . . //..,,.,,," //C......::;7 .,?.;--/

/ . /

I , [ , I , I , 1 O0 200 300 400 500

F o (Win 2)

`5

o

E 0 E v o <

60

50

40

30

20

10

0 0

60

50

40

30

20

10

0 0

(b) Numerical

(d )

f . ~ . . . . 1 0(3

/ . / . ~ 0 7E

. / " ~ .....,- . . . . . . 0 5(3

. . . . . . . . . . . . . . . . . . . .

, 1 , I I I , I , 1 O0 200 300 400 500

F o ( W m 2)

Numerical 1 001

. ~ - ~ 0 7. = _ . ~ _ _ . 0 50

/ ' ~ / J / . . . . . . . . . . . 0 25 ~ . . o.---° . . . . . . . .o-

/ ' . i - ......... ..................... ¢ . - . . .... ....

, I , [ i I , I , 1 O0 200 300 400 500

F o (Wm -2)

(e) Numerical

t r-

E E ~ . ~ . ~ . . . . . . . 1 O0

. ~ . ~ . ~ ' - ~ 0 75

~ . ~ j ~ ' ~ _ _ ~ _ _ ~ - - 0 5(3 • ....,....."

................... 0 25 , I , I , I , I ,

1 O0 200 300 400 500 F o ( W m - 2 )

15

10

05

- ( f ) Numerical

. . . . . 050 . . . ~ . . . " $ I 0 75

j l 1 1

~ ' ..._. 1 O0 ~ p . f ~ _ . . . ........

#z';: ;7- / . /

0 0 , I , I , I , I , 0 0 1 O0 200 300 400 500

F o (Wm -2)

Figure 7 Canopy (Lr = 8 0) photosynthesns Ac, conductance gc, and transpiration Ec plotted agaanst PAR (/t = 0 5) for chfferent values of canopy mr space relabve hum](hty (h~), ho lS marked on each curve Numer=cal method a) A~, b) g~, c) Ec for 0,/~ = 1, d) A~, e) g~, f) Ec for 0,/~ = 0 8

0

E "5 E

60

50

4 0

30

20

10

0 0

(g) Bulk / - - - - - 1 O0

,SI :: . . . . . . . . . . . . . . . . . . . .

I , I ~ I , I ,

1 O0 200 300 400 500 F o (Wm 2)

60

50

40 f 0

30 E

20

(h) Bulk

10

.... 1 O0 7"

7 . 7 " 075

7 " J ~ - - - o 50

........................... 0 2!

i I ~ I J I J I ,

1 O0 200 300 400 500 F o (Wm 2)

15

~ 1 0

E E

0 5

_ ( i ) 0 75 Bulk / / ~ - - - - - 0 50

./~" ..-- --100 .i.i /..:" i.I"

, . , S , " ' " ........ o ~

¢ ' ; , / /"

I t I L I J I J 1 O0 200 300 400 500

F o (Wm 2)

~T 0

E

E L, <

60

50

40

30

20

10

0 0

(I) Bulk

I 1 1 ~ I 7~

/ ~ ~ 05o

111 . . . . . . . . . . . . . . . . . - 0 25

' i \ . - - . . . . . . .

, I L i , I ~ I L _ 1 O0 200 300 400 500

F,, (Win ")

6

"T u)

E

60

50

40

30

20

10

0 0

_ (k) Bulk

.......... 100'

1 . i o I ~ 0 75~

i ~ . . , , ' '~ ' . ' ' ' '~ . . . . . . . . . 05C

~ - ? ; .......... o~ i I , I i I i I i

1 O0 200 300 400 500 F o (Wm 2)

E E

15

10

0 5

0 0

( I ) Bulk

0 75 _..1...t'jCS-- o ~

/ " - I . . - . :~C ....... o~ , ~ s . f . . . - ......

/

I i ~ ~ L , 1 t 1 O0 200 300 400 500

F o (Wm 2)

Figure 7 (continued) Bulk method g) A~, h) g,, 1) E~ for 0, 17 = 1, j) A~, k) g<, 1) E, tor 0, 17 = 0 8

Canopy Reflectance, Photosynthesis, and Transpwatzon 2 0 5

0 / 2

3

4

5

6

7

8 a 0 0 0 0 0 0

I" \ \

\ \.

\ \ \

\

I

I I

I I !

~ F

. . . . . E l

E

. . . . . g S

D B A

(a)

I I i I I 100 0 (Wm 2) 300 0 400

25 0 h s (~) 75 0 100 12 5 C I (Pa) 37 5 50 0 5 E ( m m hr l) 1 5 2

5 0 0 gs(m s 1 " 1 0 3 ) 1 5 0 0 200 12 5 A (mol m 2 s 1"10 6) 37 5 50

. . .1

O

1

2

3

4

5

0 0 0 0 0 0

; / 2 " / . . . . . 7' ! 1 / ~ E

I I \ , A \

\ \, '\

\ \

t

\ I t

100 O F (Wm 2i 300 0 400 25 0 h s (~o) 75 0 100 12 5 Cf (Pa) 37 5 50

0 5 E(mm hr 1) 1 5 2 5 0 0 gs(m s 1-103) 1 5 0 0 200 12 5 A(mol rn 2 s-1-10 6} 37 5 50

Figure 8 Profiles of PAR flux, F, leaf surface relatwe huml&ty h,, leaf internal CO2 concentration C,, leaf transpira- taon E, leaf conductance g~, leaf asslmdahon A 0, f l = 1 0, h , ffi 0 5, it ffi 0 5, Rd ffi 0 all other parameters as m Table 2 a) F0ffi50 W m -2, b) F 0 = 4 0 0 W m -2

The use of the bulk values C,, h,, and Cs allows us to integrate (28) over the depth of the canopy to obtain w~ simply by integrating the f (L) term

= [!a0 + b0) - 4(ao + bo) 2 - 40aobo- We

L 20

x f~Tf(L ) dL (29)

In the simple case of a homogeneous canopy, we can substitute Eq (25) into (29) to yield

w~ = I(a° + b°) - ~/(a° + b°)z - 4Oa°b°

Note that the second term in parenthesis can be wntten as

In Eq (30), the first term in parenthesis is simply the photosynthetic rate of the "top" leaves in the canopy, that is, those with the highest photosynthetic capacity, Vmaxo, exposed to the highest time-mean PAR flux, Fo, and subject to C,, /is, and Cs The second term, defined as H in (31), acts as a simple scaling-up factor to relate the "top" leaf performance to canopy performance

The canopy equivalent of w~, w~, is given by rearranging Eq (12) to give

Ws "~- l LrVm dL Jo 2

= V;~I~?(L ) dL (32)

w~ and w~ can then be inserted into (14b) to determine A_~, the canopy gross assimilation rate

The canopy respiration rate can be given by

"Rj f(C Ro = dL = R~o ) dL (33) 0

Equations (30), (32), and (33) can be combined to gave the net canopy assimilation rate AN As the integral off(L) occurs in all three equations, AN can be wntten as

AN = A.o" ~ (34)

A~o = f( ao,bo, Vmo, Rao), where FI is given by Eq. (31)

A.0 is effectively the "single leaf" solution to Eqs (11)-(15) where constants appropriate to the

2 0 6 Sellers et al

60

50

¢ o 40 r

'E 30

# <~ 20

10

0 O0

(a) Bulk

Numerical

~ 4 0 0

200

I O0 , , , . , . , , .--

50

[ I I I I I I I I I 02 0 4 0 6 0 8 10

6O (b)

50

40 ¢ o

30 E

20

I0

0 O0

Bulk

- - Numerical

~ 4 0 0

. . . . . . . :'~ 200 " - - - I O0 ~--- 50 I I I I I I I I I )

0 2 0 4 0 6 0 8 10 #

E E

15

10

05-

0 O0

_(c) ~ 4 O O

"- " ~ " ~ . _ 200

~ : ......................................... I O0

Bulk Numerical

1 I [ I 02 04

....................................... 50

[ ; I I 1 0 6 0 8 10

Fzgure 9 Dependence of canopy photosynthesis A,, con- duetanee g,, and transpiration E,, on the zenith angle p and intensltv of the above canopy PAR flux, F0 Sofid hnes denote the numerical solution, dotted lines the bulk soluhon (The bulk soluhon shows no varlatmn with/ t ) Num- bers on lmes refer to F0 Note that the combmatmn of low/.t values and high F0 values (i e , the areas on the figures to the left of the numerical soluhon maxima) are physically mplauslble (0,/~ = 1 0, h° = 0 5, L; = 8 O, Ra = 0 0) a) a, , b) g,, c) E,

"top" leaves are used, H ~s the canopy-integral term whmh is dependent not only on FPAR but also on how the PAR is absorbed through the canopy, that ~s, canopy architecture, as specified by

The combined canopy model is then com- pleted with an integral form of the conductance equahon (17)

- mANh + bLT, (35) g ' - C----~ sp

where AN, h~, and C~ are bulk canopy values In

Eq (35), it Is assumed that all leaves have the same "leakage" conductance b when AN = 0

The system of equations corresponchng to Fig- ure 5c now consists of (27), (29), (31), (32), and (33) [summarized m (34)] and (35) These can be solved m exactly the same way as for a single leaf (see Fig 4)

The assumptions and simplifications involved m going from the system shown in Figure 5b to that m Figure 5c are nontnvlal , / t is set equal to

and the variables C,, hs, C,, g~, and gb are all replaced by bulk canopy values It is therefore

Canopy Reflectance, Photosynthesm, and Transpwatmn 207

60

6- o 40

~ 30 E -6 E ~ 20 <

(a)

50

10

0 0

gb -Constant - - - - - - gb-Dnear

~ gb-Exponentml . . . . . . . gB-Bulk

7"

[ I [ I I I I [ I 1 O0 200 300 400 500

Fo(Wm -2)

(c) g b - C o n s t a n t ~ . . . . . . . . . . . . . .

- - - - - - gb-Dnear /......-- ~ gb-Exponentml o, ," _

. . . . . . gB Bu lk n t l ~ o j -

= . /

J o g

I I I I 0 I I I t 0 1 O0 200 300 400 500

F o (Wm -2)

6

I

60

50

40

30

20

10

0 0

(b) gb-C°nstant

- - - - - - gb-Dnear ~ . ~ gb-EXponentml

. . . . gs-Bulk

f . . ° . . . . . . . . . . . . . . . . .

1 I I I I I 1 I I 1 O0 200 300 400 500

Fo(Wm -2)

Ftgure 10 Effec ts of t h e va r i a t ion of g~ w i th c a n o p y d e p t h on va lues o f A~, gc, a n d Ec as a f u n c h o n o f PAR (O, f l= l O, ha=O5,1~=O5, LT=80, Rd=O0) a) Ac, b) g~, c) Ec Numemcal soluttons ( ) gb = 0 04 m s - 1 (cons tan t ) , ( - - - - ) g ~ = 0 06 - 0 04L / Lr ( h n e a r dec rea se ) m s - 1, ( - - - - ) gb -- 0 0 9 2 5 e - 2L / LL ( e x p o n e n h a l d e c r e a s e ) rn s - 1 T h e in t eg ra l of gb ove r t h e d e p t h of t h e c a n o p y is 0 32 m s-1 ( = g~) m all t h r e e cases Bulk method ( ) g~ = 0 3 2 m s -1

interesting to explore the differences between the solutions gaven by the two methods

F]gures 7g-1 show the bulk method estimates of A~, gc and Ec for the optimal case ~ =/g, the soluhons are always within a few percent of the numerical scheme solut]ons, see Figures 7a-f

Figures 9 and 10 compare the values of A~, gc, and Ec as given by the two methods over a range of PAR vectors and for a range of gh profiles Except for the extreme and physmally implausible case of low values of/~ combined w]th h]gh values

of F0, the bulk method ymlds fluxes that are close to those of the numerical scheme

Figure 11 compares the values of gc hs, and C, as given by the two mtegrahon schemes Weighted estimates ofgc, h~, and C, for the numer- Ical scheme are given by

I ~LTg, dL, (36a) g~ = L, Jo

1 ILT h hs=L--~3 ° , dL, (36b)

6 0 70

5 0

4O

c/ l

E 3O

2 0

10

O 0

3 0

2O

r l

cY

1 0

(a) Numer ica l Bulk

] 4 0 0

. . . . . . / loo ] ~ I ~ J ~ "/

2 4 6 8

Leaf Area Index

(c) N u m e r i c a l Bu lk

0 ~ I I I i t I 0 2 4 6

Leaf Area Index

5O

100 2 0 O 4 0 0

(b) i Numerical ol.

2 6O >.

"5 E

T

50 t r

g, 40

L9

208 Sellers et al

3 0 * 1 i I J I 0 2 4 6 8

Leaf Area Index

Ftgure 11 Comparison ot bulk canopy conductance values. g,, leaf surface specific humidity h~, and internal CO2 concentration C,, as given by the numerical scheme, Fig- ure 5b, and the bulk scheme, Figure 5c Estimates of and (7, for the numerical scheme are given by Eq (36) a) g~ comparison numbers denote PAR fluxes in W m -2, b) h~ comparisons symbols same as in a), e) C, comparisons, symbols same as in a) (0. fl = 1 0, h. = 0 5, p = 0 5)

4 0 0 2 0 0 100 50

A~LTJ0

The bulk model yields integrated estimates of these state variables that are close to those calculated with the numerical scheme

RELATING CANOPY BIOPHYSICAL PROPERTIES TO SPECTRAL REFi,ECTANCES

Equations (34) and (35) appear to be effective canopy-integral forms of the leaf or chloroplast

level formulations of Farquhar et al (1980) and Collatz et al (1991) The nonlinear effects in- duced by varying solar angles and the profile gradients of governing state variables (C,, hs, g~,, etc ) within the canopy do not prevent the bulk integral method from provichng useful estimates of the fluxes of CO2 and H20 under normal conditions

The inclusion of these more sophisticated physlologacal models into the analysis leads to different canopy biophysical-spectral reflectance relationships than those described by Sellers (1987)

We can rewrite the canopy-integral equations

Canopy Reflectance, Photosyntheszs, and Transp:ratmn 209

for the rate limiting processes governing photosynthesis as follows

wc --- Vmo Yl B1, (37a) w~ = Fo II Bz, (37b) ws = Vm0 YI B3, (37c)

Ro = Vmo FI B4, (37d)

A~ = A.o Fl B~ = f(wc,w~,w~) - Ro, (37e)

g~ -~ A.o Fl B6, (37f)

where C , -F* ]

B~ = C, +/~[1 + O2 / Ko] '

B2 = G(_~)(10)v'e [ C ' - F * ]

B3=0 5, B4 = 0 015, B s = l 0,

B6 = mhsp us

In Eq (37), the canopy biophysical variables (left-hand side) are calculated as the product of three parameters or forcing variables (right-hand side) These parameters/forcing variables can be described as follows

First Variable: Plant Physiology or Radiation Rate Limit Variable (Vmo, Fo, A.o)

Vmo or F0 are the rate limiting factors governing the canopy response at saturating PAR fluxes and less than saturating PAR fluxes respectively Ano, appeanng in (37e) and (37f), lS a direct function of these two variables Previous sections have discussed the functional dependence of V~o on Fo n The time history of Fo'n can be obtmned from satelhte chmatologlcal studies, for example, Frouln and Gautier (1990) demonstrated how Geostatlonary Operational Environmental Satel- hte (GOES) data could be used to compute the diurnally varying PAR flux over a Kansas grassland area (the FIFE site) with a surface resolution of around 1 km 2 Such data could be composlted to provide a global PAR chmatology from which seasonally-varying, time-mean fields of Fo" n could

be produced The dependence of A~ and g~ on F0 is obvious

a Ac 0 g~ = YIBs, 1-IB6, O Fo' O Fo

=0,

R, for F0< Vmo-" (38a)

Bz

for F0 > Vm0 B1 (38b) Bz

Equation (38) indicates that for nonsaturatmg PAR fluxes, the unstressed canopy photosynthetic rate and conductance respond almost linearly with changes in PAR with a slope that is directly proportional to Fl and FPAR The effects of environmental stress or forcing are contained within the B5 and B6 terms Equation (38) holds for 0, /~--" 1, lower values of 0 and ]~ will give a gradual transition from (38a) to (38b) with increasing Fo

Equation (38) and the supporting analysis indicate that all the leaves in the canopy saturate at the same value of F0 = Vmo(B~/B2) This is because Vm IS scaled according to the time-mean profile of PAR within the canopy

Second Variable: Canopy PAR Use Parameter (H) This parameter corresponds to FPAR Chvlded by the PAR extinction parameter k It is the scaling parameter that relates canopy performance to the performance of the "top" leaves Typically, k varies between 0 4 and 1, so that a continuous fully developed canopy (FPAR --- 1) will perform at between one and three times the rate defined by the ensemble of "top" leaves

FPAR is the vegetation parameter most readily amenable to remote sensing [see Asrar et al (1984), Tucker et al (1981), Sellers (1985, 1987), Hall et al (1990), and papers m Asrar (1990)], From this analysis, It appears that the value of FPAR associated with the radiation- weighted time-mean PAR flux vector F0 [see Eq (25)] combined with the corresponding value of k, is the most useful canopy parameter for biophysical calculations This implies that multlangle data should be acquired over vegetation canopies at solar angles corresponding to ~ to obtain estimates of FPAR and

If we neglect second-order feedback effects (see next section), all of the canopy biophysical rates are linear m I7 [see Eqs (37) and (38)]

210 Sellers et al

This represents a distinct improvement over the analysis of Sellers (1985) as now the contribution of canopy density and morphology (H) IS cleanly separated from those of leaf physiology and radiation flux (V.,o, Fo) Sellers (1987) showed how under ideal conditions the simple ratio vegetation index, SR, was linearly related to FPAR provided that Eq (3) holds for the vegetation / type viewing sensor combination and the soil background is fmrly dark The chain of relatmnships

g~,A~ oc H oc FPAR ~ SR (39a)

or, simply,

A, =f.(SR), (39b)

g, =f~(SR). (39c) where f~ and f, are near-linear {unctions when /t ~ ~, indicates that SR or other SVI images may be used to calculate fields of A,, g~, and E, using a simple linear transform of the image combined with environmental forcings (F0, H., T,,, C,,, soil moisture stress, etc ) and some knowledge of the leaf physiology (V..,, m, etc) In grasslands or areas with drought-deciduous vegetation, It is reasonable to assume that chronic soil moisture stress will be expressed as a decrease in H so that direct

. J

o 0

0 , I , I O 0 0 2 0 8

(a) Uniform

Z X

, I , I 0 4 0 6

Cover Frac t ion , "Y

8 (c) Intermedmte

6

,-J

2

0 I , O 0 0 2 0 4 0 6 0 8

C o v e r Frac t ion ,

1 0

o 0

_J

>

( b ) Clumped

0 , I ~ I L ~ L , O 0 0 2 0 4 0 6 0 8 1 0

Cover F rachon , "Y

F:gure 12 Local area index versus coyer relahonshlps for increasing area-averaged LAI, (L) Arrow denotes increasing (L) The heavy hnes m each figure denote area- averaged leaf area mdex values, (L) of 0 1, 0 5, 1 0, 2 0, 4 0, and 8 0, Increasing m the dlrechon of the arrows Values of (L) are eqmvalent to integrals under the heavy lines or the product of Cover Frachon and Local LAI a) Umform (plane-parallel) canopy, b) clumped canopy constant local LAI, Lm=--8, cover fraction ),, vanes wtth (L) , c) mterme&ate ease, Lm~ = 8y These canopy configura- hons are used in Figure 13

Canopy Reflectance, Photosynthesis, and Transptratmn 211

knowledge of the soil moisture content might not be essential

In nature, variations in area-averaged leaf area index can occur in two ways First, a horizontally uniform canopy may vary ]n thickness (vertical dimension) This was the case studied by Sellers (1987), which led to the result that FPAR would be hnearly related to SR (see F]g 12a) A second way is for the canopy to vary in area-averaged cover fraction This is often the case for coniferous trees or desert shrubs where the leaf area of an lndlwdual tree remains relatively constant, but plant abundance varies depending on ecolog]cal conditions (see Fig 12b) Hall et al (1990) used this second scenario to explore the utlhty of spectral second derivatives and SR data to determine FPAR, once again, these radices were more or less linear with FPAR A simple analysis is presented below which demonstrates that while the relation- sh]p between area-averaged leaf area index and FPAR (and therefore spectral vegetation indices, SVI) will vary depending on the horizontal and vertical distribution of the vegetation, the relationship between FPAR, SVI and A~, g~ remains mvarlant

Up to now, we have considered a plane parallel canopy of leaf area mdex LT Let us now consider a landscape made up of ldenhcal homogeneous vegetation units, rather hke box hedges, each with a local leaf area index of Lmax and covering a total fraction 7 of the landscape (or instrument field of view) Thus,

(FPAR) = y(1 - e -~max) (40a)

The area-averaged leaf area index is given by

(LT) = 7L . . . . (40b)

where the ( ) symbols denote "area average" Figure 13 shows the relationship between

FPAR and (L~) for different degrees of "clump- lng" (values of 7) Clearly, as 7 decreases, (L~) must increase to maintain the same value of FPAR

However, an inspection of Eq (37) and the supporting analysis shows that A~ and g~ are dependent on 1-I that is, on FPAR/k, and not on the spatial distribution of the vegetation density, provided that one ignores variations in the forcing/ feedback terms B1-B6 If we look at a vegetated region of area S, contmnlng clumps of vegetation of varDng size and density, but all having the same

. J

V < . d "O

<

(,) Uniform , . ' / / I

- - - - - Intermediate ,-• t / / / ........ Clumped ,•••• / / /

• - / ' • / / / • J° /¢

," o• •" i i I • o• f

• " " s 4

,•o" SI ,.** s t

,•." $1 so" •

~ I t ss oo° #,

0 1 , " s S " .?--" . - " • .- " - ~ o 5

0 O0 0 5 1 0

FPAR

14 (b) Uniform

12 --- - - - Intermediate / ¢ ~ ........ Clumped j / ~ 4

10 .~,

8 . . " 2 t r

o .

E 6 y " 1

4

2

01 0 I I i I I I ~ I i 0 0 20 40 60 80 0

FPAR

F=gure 13 Vartahon of a) area-averaged total leaf area Index (Lr ) with FPAR, and b) simple ratio vegetahon index wRh FPAR for different levels of "clumpmess" parameters from Table 1 In a) and b), the thin hnes en&ng m numbers refer to values of area-averaged leaf area index, (Lv) ( ) Spatially uniform canopy, y = 1 LT varms from 0 to 8, ( ) c lumped canopy, y = 0 to 1, Lm~ = 8, ( - - - ) mtermethate case, Lm~ = 8y Refer also to Fig 12

baseline physiology (Vm~x0) and leaf geometric/spectral properties (k, tow) overlyang a soil background of umform reflectance, we can write

- - f l s (A~,g~) = - | A~,g~ ds (41a)

SJ0

o 4O

30 E -6 E

v <o 20

m l l i l I SVI ds, (41b)

where SVI, the spectral vegetation index, is SR or a spectral second derivative index (see Hall et al , 1990)

Now for a hnear function, the area-Integral and spatial average (multlpled by the area) opera- tors are equivalent

< , , d, (42)

Since the functions relating A,, g, and I1, SVI are linear or nearly so, we can rewrite (41) as

- - C 1 (A~,g~)=-| A~,g~ dsoc <I-I) = (SVI} (43) SJo

Equahon (43) represents a powerful and

Figure 14 Variation of canopy photosynthesis A, and conductance g, with FPAR as calculated using the coupled stomatal-photosynthesls model (bulk scheme) shown m Figure 5c Different spatial distributions of (L} are used corresponding to those shown in Figure 12 Dots denote values of (L} = 0 1, 0 5, 1 0. 2 0, 4 0, 8 0 ( ) Spatmllv umform canopy, 7 = 1, L~ ~anes from 0 to 8 ( ) c lumped canopv, 7 = 0 to 1. Lm.. = 8, ( - - - ) Intermediate case, L,,,.,, = 87 a b) A, and g,, h .= 1, c,d) A, and g,, h . = 0 5 (0, f l = 0 8, E)= 400 W m --~,/~=0 5)

6O (a) ~ Uniform

------ Intermedmte 50 Clumped

10

0 O0

60

..J

, / j *

A I L I ~ I i I ,

02 0 4 0 6 0 8 10 FPAR

60

50

4O

b

30 E

20

10

0 O0

(b) ~ U n i f o r m

------ Intermedmte Clumped

~ ~ 1 i t , t , t L

02 0 4 0 6 0 8 10 FPAR

o

E -6 E

50

40

30

20

10

0 O0 02 0 4 0 6 08 10

FPAR

( C ) Uniform - - - Intermediate

Clumped

212 Sellers et al

60,

b

E o~

50

40

30

20

10

0 O0

(d) , Uniform - - - Intermediate

Clumped

02 0 4 0 6 08 1 0 FPAR

Canopy Reflectance, Photosynthesm, and Transp=ratzon 213

countenntu i t ive result" The mean value of SVI as measured over an area should vary linearly w]th the areal integral a n d / o r mean values of 11 A~, and g~, to first order In areas with very h]gh local values of L~, higher-order effects (feedbacks be tween the vegetation and its immediate environmental through G, hs, etc ) may act to vary the coefficients BI-B6 and cause some local distor- tion of this relationship However, ]t is interesting that the inclusion of the ecophyslologlcal opt]- mahty assumption of Eq (21) into the equation set maintains the hnear l ty of the SVI versus A~, g~ relationship over a wide range of spatial scales and over he terogeneous vegetation dens]ty dlstrlbut]ons (see Fig 14) This is a very &fferent result from that obtained by Sellers (1985), who assumed lnVarlant leaf physiology within the canopy, that is, Vm~x = Vm~0 for all L, and therefore der ived a range of SVI versus A~, g~ relationships depending on the value of y (see Fig 15)

Thi rd Variable: Environmenta l Forcing or Feedback Term (B,)

For a given set of environmental conditions, the variables Ba, B4, and B5 can be effectively consid- e red as constants over the area of mtegration, S The variables B1, B2, and B6 are the result of interactions be tween the vegetation biophysical process rates and the environmental forclngs, the linkages being through C,, hs, and Cs These feedbacks are functionally the same as those for a single leaf (see Collatz et a l , 1991)

SUMMARY

The leaf photosynthetic model of Farquhar et al (1980) and the leaf conductance model of Collatz et al (1991) can be analytically integrated over the depth of a vegetation canopy provided some slmphfylng assumptions are made In practice, the bulk analytical canopy model )rtelds values of net canopy assimilation rate A~, canopy conductance g~, and canopy transpiration Ec that are close to those provided by an exact (numerical) integration of the leaf models for normal environmental conditions

In defining the properties of the model canopy, the arguments of ecophyslological optimality, as invoked by Farquhar (1989) to describe the

60

50

~, 40

30 E 0

E v

&o 20

10

0 O0

(a) Uniform

- - - - - - Intermediate ........ Clumped . ~

j " 9 ° , I , I , I , I ,

0 2 0 4 0 6 0 8 FPAR

60

0

50

4O

(b) Unlform

- - - - - - Intermedmte . . . . . . . Clumped

7 30 . /

*~ 20

I0 •

O 0 0 2 0 4 0 6 0 8 1 0

FPAR

Figure 15 Variation of canopy photosynthesis Ac and conductance gc with FPAR, as calculated with empirical leaf models m Eq (5), followmg methods descnbed m Sellers (1985) All symbols and conditions are the same as m Fig- ures 14a and 14b a,b) A~ and g~, f(E) = 1 (Canopy, sod properties from Table 1, F0 = 400 W m-2,/~ = 0 5) Note that these values of A~ and g~ are higher than those shown in Figures 14a and 14b This is because all the leaves m the canopy have the same biophysical properties as the "top leaf", see Figures 1, 6c, and 6g Additionally, A~ and g~ are shown to vary markedly with vegetatmn heteroge- neity usmg the methods of Sellers (1985) whereas, m the new formulation, they do not, compare wlth Figures 14a and 14b

profile of chloroplast Rubisco content within a leaf, were used to define the profile of leafRub]sco content, Vm~, within the canopy so that the re-

214 Sellers et al

suiting profile of Wmax follows the time-mean profile of PAR through the canopy

As a result, the contributions of leaf physiology, canopy density and geometry, and environmental forcing are separable m the integrated bulk canopy model, this was not the case for the "simpler" models used by Sellers (1985, 1987) This finding, Eq (37), may be summarized by

1 Canop~ hyslolog~

rate radiation use • , a r l a b l e | r a t e | [paramete d

~ hmlt At,g( = [Vma,,o, Fo] [I-[]

rEn'• lronment,d] forcing /

feedback ]

[BI, B~] (3z)

This revised canopy model has the following interesting and useful properties

i. Near-linear relationship between Ac, gc and incident PAR flux, F0, for values of Fo less than saturation. All leaves within the canopy saturate at the same value of Fo due to the variation of Vm,~ with depth.

ii. Linear relationship between Ac, g~ and the canopy PAR use parameter If, where II is equal to the radiation-weighted time-mean of canopy FPAR divided by the extinction coefficient for PAR, It should be possible to estimate YI by taking

multlangle reflectance data over a target area for solar angles that correspond to the oplamal lame- mean PAR flux (/t = p) as defined by Eq (26) iii. Linear relationships between A~, go II, and

SVI simplify area-averaged biophysical calculations. The SVI can be configured to ymld a hnear

relalaonshlp with FPAR and hence 1I [see Eq (2)] This is true whether one considers a canopy varying in depth (Sellers, 1985, 1987) or areal cover fraction (Hall et al, 1990) or a heteroge- neous combmalaon of both Gwen such a condI- tmn (SVIo:FPAR, l-I) and an area contamlng vegetation of uniform physiology, leaf geometry, and spectral properlaes overlying a uniform background, the mean SVI for an area can be used to directly calculate the area integrals of the canopy photosynthelac rate A, and conductance g~

-~ A~.~ ds =f~((SVI)). F,((SVI)). (44) o

where the functions f,, fg are the same (near- linear) functions relating canopy assimilation and conductance, respectively, to the SVI as those derived for a small-scale homogeneous vegetation cover 0 e , a sample in the area) The angle brack- ets denote "area average"

The slmphclty of these relationships should permit straightforward transformation of satellite imagery - t ime series ofF0 from GOES data super- posed on fields of SVI - to calculate regional fields of the (sod-moisture stress-free) canopy photosynthelac rates A,, conductances g,, and transpiration rates Ec The llnearlty of the SVI-II relationship should permit the use of coarse spalaal resolution satellite imagery for this application

Funding support for th~s work was suppled by an Earth Observing System/Interdmczphnary Science (EOS-IDS) grant from NASA and a NASA grant for FIFE-related studies (NAG- 5-892) The support of many people at NASA HQ and NASA / GSFC ts gratefully acknowledged At Carnegu~ Inst~tutzon, Cyml Gmvet provuted technwal assmtance Dawd Sch~mel and Fred Huemmrwh are thanked for interesting d~scussugns Much of the analyms was conducted at the Unwers~ty of Maryland under the aegm of the Center for Ocean-Land-Atmosphere Interactions (COLA) and the Department of Meteorology The support of these mst~tuttons and the help of Mark Hemer, who prepared the graphs, and Marlene Schhcht~g, who typed the paper, ks gratefully acknowledged

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Canopy Reflectance, Photosynthesis, and Transpiration. III