ELSEVIER

On the Relation between NDVI, Fractional Vegetation Cover, and Leaf Area Index

Tobu N. Carlson” and David A. Riziley* J I

W e use a simple rudiative transfer model with vegetation, soil, and atmospheric components to illustrate hoW the normalized diflerence vegetation index @DVl), leaf area index (LAZ), and fractional vegetation cover are dependent. In particular, we suggest that LA1 and frac- tional vegetation cover may not be independent quan- titites, ut least when the fermer is dejïned without regard to the presence of hare patches between plante, and that the customary variation of LAI with NDVI can be er- plained as resulting from a variation in fractional vegeta- tion cover. The following points are made: i) Fractional vegetation cover and LA1 are not entirely independent yuantities, depending on how LA1 is defined. Gare must be taken in using LA1 and fractional vegetation cover in- dependently in a model because the forrner may partially take account of the latter; ii) A scaled NDVI taken be- tween the linzits of minimum (bare ,soil) and maximum fractional vegetation cover ia insenstive to atmosphetic correction for both clear und hazy conditions, at least f3r viewing angles less than aborct 20 degrees from nadir; iii) A ,simple relution between scaled NDVI and fractional vegetation cover, previously described in the literature, is ficrther c.on&-med by the .simulations; iv) The sensitice dependence of LA1 (~1 ND17 when the fonwr is below a value of about 24 may bc viewed as being dut to the variation in the bar{: .soil component. Ol%evier Sciencc lx., 1997

where amr and a.+ represent surface reflectances averaged over ranges of wavelengths in the visible (1-0.6 Pm, “red”) and near infrared, IR (2-0.8 Pm) regions of the spectrum, respectively. It is clear from its definition that the NDVI (like most other remotely sensed vegetation indices) is not an intrinsic physical quantity, although it is indeed correlated with certain physical properties of the vegetation canopy: leaf area index (LAI), fractional vegetation cover, vegetation condition, and biomass. As such, vegetation indices are highly useful measurements despite their limitations.

The NDVI lias been criticized because of the follow- ing perceived defects:

BACKGROUND

The normalized differente vegetation index (NDVI) is

defined as

Accordingly, various investigators have addressed these problems in light of indices that exhibit a better correla- tion with leaf area and less sensitivity to soil brightness changcs or to atmospheric attenuation thall does NDVI (Jasinski, 1996; L eprieur et al., 1996: Liu and Huete, 1995; Pinty and Verstraete, 1992).

’ Departnrent of Meteorology, The Pennsylvania State University, Universitv Park

That the relation between NDVI and L,AI undergoes a marked decrease in sensitivity above a loosely defined threshold is wel1 known from measurements. Carlson et al. (1990) stated that

NDVI incrcases almost linearly with increasing LA1

HEMOTE SENS. ENVIKON. 62241-252 (19973 OElswier Science Inc., 1997 655 Avw~~t~ of thr Atrwricas, New York, NY 10010

(1)

Differences between the “truc” NDVI, as would be measured at the surface, and that actually determined from space are sensitive to attenua- tion by the atmospheric and by aerosols. The sensitivity of NDVI to LA1 becomes in- creasingly weak with increasing LA1 beyond a threshold value, which is typically between 2 and 3. Variations in soil brightness may produce large variations in NDVI from one image to the next (Liu and Huete, 1995).

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Figuw 1. Schematic representa- tion of radiative flux coinponmts in the simple model. Solid lincs with arrows represmt the direct solar flux component at clrvaticm angle q,,, dashed lines represent diffuse flux coinponents over a bare soil fraction (right side) and a vegetated fraction (Fr; left sidIA). F111xrs are representecl by tlle s,ymbol F with subscripts denoting various flux coniponents either ahsorhed in the atmosphere, ah- sorbecl at the ground or within the vegetation cdnopy, scattercxd upwdrd or downward, or reflectrd upward. (Al1 ahsorbed fluxes arc> ~mderlined.) Thr fil~u to spacc measiirrtl bv a satellite radionie- trr is inclicdted by tlke dashed lirie lalde<1 F ,,,,, (sw Appen<l”).

tapdd+ . . . . . . . . . . . . . . . . . . . . . . . . .

tqd vegddim Fd IIIIIIIIIII / I I I 11111 I A

can l)e largely explaincd by a variation in fiac- tional vegetation cover; that NDVI decreases much less rapidly with in-

creasing glol~l LA1 wllen the fractional vegetation cover reachcs 100%; that ding the NDVI between values for hare soil and fol- 100% vegetation cover factors out most of thc atmospheric correction; LlIIC 1

fiirtlwr ~W&IK~ to confirm the existente of a sini- plr relation kween scaled NDVI and fractional qetation cover.

THE MODEL

The radiative transfer model, described in the Appendix,

is a very simple two-stram (upwarcl and downward) rep resentation of fluxes through a one-layer atmosphere over ;i surface consisting of bare soil and a single layer of veg- etation. The latter is represented by an unbroken vegetu- tion canopy (represented 1)~ a local LAI) occupying a fraction, Fr, of the total domain whosc global LA1 is equal to Fr times the local LAI.

Figure 1 illustrates the various flux streams that un- dergo absorption, scattering, and reflection as they move through the atmosphere and the vegetation lqer. Scatter-

ing and reflection are either upward or downward. Re- flectances and their derivative product, NDVI, are deter- mixed at thv surfacr and at the top of the atmospherp.

NDVI is derived from reflectante values that are calculated separately in two wavelength hands in the visi- He (0.5-0.7 pin) and near infrared (0.X1.9 Pm) regions of tlw spectn1111. The radiation scheme is as follows. A

heain of direct solar radiation at solwr elrvation angle p,, is incident at the top of the atmosphere. Some of that inci-

dent flux in the downward direct heam is ahsorbed by the

atmosphere (F,,),), a part is scattered upward as diffuse flux (FI,,), and another component is scattrrecl <lownward as

cliffusc flux(F,~,), which, along with the unattenuated di- rect flux, is incident on the ground anti the top of thr

vegetation canopy. The comhinrd diffke and direct flux

incident at tlie ground, whose all~do is a,, is divided be- Ween ii con~l~onent absorhed at thr gromid in the hare soil area (F,) and a component reflected upward (F,,.).

An identical incident flux is absorkd hy the vegeta- tion canopy (F,,,,,) and by the ground underneath the veg- etation c~mopv (F,,.). The remaining fll~x, tltat not al)- sorhecl in thcOvegetation a1ic1 at thr gronnd, is refiected upward (F,,). Upwarci flux streams l-eflected from the at- mosphere. the hare soil, and the vrgetation canopy (and its lmtkrlving soil) combiw to fornr thr‘ total upwad flux

(F,). The total upward flux is further attenuated by ah-

sorption, which rrmoves im amount of flux (F,,,,,). Mo ac- comlt is taken of backwartl scattering of this upward ra- diation stream.

The flux reaching the satellite is determined indi- rectly by subtracting al1 absorbed atmosphere and sur- face components from the incident flux at the top of the atmosphere. The apparent reflectante is calculated by di- viding the upward flux at the top of the atmosphere (F,,,,,) by the exoatmospheric solar flux [Eq. (AlO)]. Surface re- flectance is determined as a ratio of the sum of reflected fluxes at the surface divided by the incident surface flux [ Eq. (AS)]. Satellite angle is considered only insofar as it affects the path lengt11 of the reflected flux component F, toward the satellite. NDVI is calculated, by Eq. (1).

Model initialization requires specifying the reflec- tances for bare soil and leaves, the fractional vegetation cover, time of day, satellite viewing angle, local LAI, lati- tude, longitude, horizontal visibility (from which aerosol optica1 depth is calculated), and a few other variables of much less importance, such as surface pressure and ozone concentration (see Table Al).

As presented in the Appendix, the radiative transfer formulation does not constitute a rigorous treatment of the radiation physics; nor is the treatment particularly rtew. The purpose of the radiation model is to provide a reasonable estimate of the major radiation components in the atmosphere, in a vegetation layer, and at a bare soil surfwce. Our confidence in the model’s efficacy is based on its long-term satisfactory performance as a solar radiation component within a SVAT model (Gillies et al., 1997). Direct (unpublished) comparisons of simulated and measured solar fluxes have been made, and these comparisons show agreement within about 5%. More im- portant, however, the SVAT model lias been widely em- ployed in conjunction with remote and in situ measurr- ments to investigate land-surface processes, particularly the role of soil moisture in modifying the surface-ener&? budget (Gillies et al.. 1997), the determinatiou of train- spiration fluxes over plants, and the intake of carbon in plaut canopies (Olioso et al., 1996). Further validation by comparison with anothcr model is presented in the ncxt section.

RESULTS

Initial Conditions

Radiative transfer simulations were made for the latitude and longitude of State College, PA (approximately 40” N ad ï6” W), for a July day over a range of times from IIOOII until 2:00 P.M. and satellite zenith angles from 0 to 20 degrees from nadir. Because results were similar for all satellite and sun angles investigated, al1 illustrations refer to one time and one viewing angle-I:OO P.M. locd

time 20 degrees from nadir.

7’&le Z. Albedos of Bare Soil and Leaves (%)

Visible

x 5

Near IK

11 50

Fractional vegetation cover (Fr) was varied from 0.0 to 1.0, the local LA1 remaining fixed. Global LAI, equal to Fr times the local LAI, is not directly used in the cal- culations, but this parameter is referred to in the illustra- tions. For example, a local LA1 of 3 and a value of Fr

equal to 50% corresponds to a global LA1 of 1.5. At Fr=lOO%, global and local LA1 are identical. Values of the local LA1 for an Fr value of less than 1.0 were fixed at a value of 3.0 in most simulations, but a few simula- tions were made by using a local LAI of 2.0 and of 4.0. Additional simulations were made at Fr= 1, by varying LA1 in increments from the local LA1 for the partial cover case to LAI=lO. Al1 calculations refer to clear sky conditions.

Albedos pertaining to the bare soil surface ((lg, in- cluding the albedo beneath the canopy) and to the leaves (ar) were fixed for al1 simulations; they are identical for diffuse and direct flux. Table 1 refers to these fixed re- flectances, thereby furing NDVI for bare soil and at the limiting values in the asymptotic reg.ime (infinite LAI). The purpose in choosing this particular combination of albedos is that they yield the range of NDVI typically measured by satellite over a mixture of bare soil and veg- etation (Cillies and Carlson, 1995). (The values are not meant to be representative of any particular soil or vegc- tation types.)

To simulate normal and hazy conditions, two differing visibility values were used: 15 and 5 km. They were COII-

verted into aerosol optical depth, as indicated in the Al’- pendix. A summary of parameters used to perform the simulations and selected values of reflectante obtained thereof are presented in the Appendix in Table Al.

Comparisons with MODTRAN

MODTRAN (Kneizys et al., 1996), a more recent version of LOWTRAN (Kneizys, 198X), is considered a standard 1node1 for making atmospheric correction to satellite ra- diance mrasurements. As such, it is normally rmployetl to correct al1 om satellite and aircraft images for atmo- spheric attenuation for both the solar and the thermal IR part of the spectrum (Gillirs et al., 1997). MODTRAN does not account for vegetation, but it does requircx a temperature and humidity sounding and an cstimate of horizontal visibility.

A comparison between NDVI simulated with the simple model described in the Appendix and MOD- TRAN was made as follows. Tables of surf&, and appar- ent (at sensor) reflectances were generated by using both the simple model ad MODTRAN ancl were subst~-

NBVI, Fractional Vegetation Cowr, crnd LA1 245

Figure 2. NDVI converted into at- surface values for atmospheric atten- uation by using MODTRAN for four atmospheric temperature and mois- ture soundings made over Pennsylva- nia on differing days (see key) and the corrected NDVI based on the ra- diative transfer model simulations (Nsim) described in the text (solid

0.8

0.6

. .

0.1 --

0 I I . . . I . . I. I . I. I. I. I . I.. I . 1. . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

a Apparent NDVI

0.8 ,

0.7 -.

0.6 -*

$0.5 -.

E0.4-. ar t 0 0.3 -. 0

0.2 -.

0.1 -.

+ Nsim 8 02Jul-89 -iF 12Jul-89 -s 02-Aug-90 * Ol-!%p-90

rectaneles) versus the uncorrected n-l : :::::::::: ;:; : I, - .

(apparent) NDVI for va!ues used in the simulations: (a) 15-km visibility;

0 0.1

(b) 5-km visibility. b

0.2 0.3 Apparti NDVI

0.5 0.6 0.7 0.8

quently used to calculate corresponding values of surface and apparent NDVI. In this manner, a set of values of corrected and uncorrected or apparent NDVI were made for both models over a range of values of fractional vegetation cover and LAI. Four different sets of initial temperature and moisture soundings were used for the MODTRAN calculations.

Figure 2 shows the results of the comparison of the corrected (at-surface) NDVI and the apparent (at-satel- lite) NDVI for normal and hazy conditions (visibility=I5 km and 5 km, respectively). The relation is nearly linear for both models, and the results are very similar, al- though agreement was not quite as good for the hazy conditions. As in Pinty and Verstraete (1992), the cor- rected NDVI is approximately 0.15-0.2 greater than the apparent (measured) NDVI.

NDVI as a Function of LAI Local LA1 was set at a fixed value of 3, and the fractional vegetation cover was varied from 0 to 1. In the asymp-

totic regime (Fr= l), LA1 was increased incrementdlly from 3 to 10. Figure 3a (15-km visibility) shows that the apparent NDVI increases from 0.54 to 0.61 and the cor- rected NDVI from 0.72 to 0.75 as LA1 increases from the 100% vegetation cover threshold to LAI=lO in the asymptotic regime. Figure 3b is similar but shows a lower NDVI in the asymptotic regime for this hazy case (visibility=5 km).

Additional simulations for the 15-km visibility case were made for a local (threshold) LA1 of 2 and of 4 (Fig. 4). These simulations show an increase of apparent NDVI from 0.47 to 0.60 and the corrected NDVI from 0.57 to 0.76 in the asymptotic regime (LA1 greater than 2). For a threshold LA1 of 4, the increase in NDVI in the asymptotic regime (LA1 between 4 and 10) was from 0.57 to 0.60 for the apparent NDVI and from 0.74 to 0.76 for the corrected NDVI. Little change occurred in any- of the simulations for a LA1 greater than about 6.

Because the most likely values of local LA1 for vege- tated surface just reaching 100% cover are between 2

0.8 ,

Ncorr(Fr=l)

0.6

0 2 4 6 8 10 a LAI

0.8 , 1

0.6

0 2 4 6

b LAI

;md 4, we can infer froin Figures 3 and 4 that the appar- ent NDVI at the asyrnptotic threshold is likely to be be- tween 0.05 and 0.10 helow the values pertaining to au infinitely lnrgc LAI. It seems reasonable to suppose, therefore, that the NDVT for areas in which thc vegeta- tiou cover is just reaching 100% wil1 be adequately rep- rrscnt(~d by vahles slightly less than the maximum tin~~l in the image over densr vegetation.

Staling the NDVI

Thr advantage of a scaled NDVI in minimizing uncer- tainty in the initial conditions in models that yield soil water content and surfidce energy fluxes by inversion of a WAT model bas been pointed out by Gillies ït al. (1997). An added benefit in staling NDVI can now be sren. Scaled NDVI (N”) is defined as

N” _ NDVI-NDK NDVI,- NDVI,,’

(2)

8 10

Ezg~re 3. Uncorrected (apparent; Napp) NIWI ad NDVI corrected to surftic<, val- WS (Ncorr) versus glohl LAI; [global LA1 is cqual to the fractiorlal wgrtatio~~ cover (Fr) times a fixed value of local LA1 of 3 up to 100% vegetation cover, dmv~~ whicll locd ad globd LA1 are ecpd]: (a) fix 15-km visibility; (17) for R 5-km visibil- ih-. Horizontal dotted lines denote thr vulue of NDVI at the threshold to thr as- ymptotic rrgirnca in which fractional vegc- tation cu\vr is jllst rexhing 100%~ (Fr= 1).

where NDVZ,, and NDVZ, correspond to the valurs of‘ NDVI fi)r bare soil (LAI=O) and a surface with ;t frac- tional vegetation cover of lOO%, respectively.

Using entirely different approaches and data sources from this paper, Clwudhury et al. (1994) and Gillies and (hrkon (1995) independently obtained UJ~ identical square root rrlation lwtween N” and Fr, which is stated:

Fi4P. (3)

E’igure 5 shows N*” versus Fr for both the cwrected and uncorrected NDVI vahws for the I5-km visibility and S-km visibility cases. Figure 6 is identical in farm with Figure 5a but fol- the other two threshold values (2 and 4) of LAl, both corrected and uncorrected. In al1 cases, the curves conform closely to Eq. (3), with the maximum deviation in Fr hing generally less than 0.1. An impor- tant implication Henk is that atmospheric correction ot’ the scaled NDVI is unnecessary for drttwnining fiac- tional vcgetarian cm’er and LAI, b~causr Ai” [anti thew

I,AI 247

Figurc, 4. Same as Figure 3a (lrj-km visibility) hut for dif’fering LA1 thrrsh«lds of’ 2 anti 4, c’or- rwted and uncorrected radiances.

fiere Eq. (3)] ‘. 1 p IS a ) roximately the same for both corrected and uncorrected NDVI. That the atmospheric correction effectively cancels in making the staling from NDVI to N” is a consequente of the linearity between corrected and uncorrected NDVI, as indicated in Figure 2.

CONCLUSIONS

We show with the aid of a simple radiative transfer model that the charactcristic behavior of NDVI as a function of LA1 can be simulated by changing only fractional vegeta- tion cover when it is less than 100%. NDVI is sensitive to changes in the fractional vegetation cover until a full cover is reached, beyond which a further increase in LA1 results in an additional smal1 and asymptotic increase in NDVI. The change in regimes from one that is affect& primarily by changes in fractional vegetation cover to an asymptotic one can be simulated by varying fractional vegetation cover while keeping LA1 fîxed at a value of about 3 in thc vegetated fraction.

The importante of this finding, which by itself is not very startling, is that the identification of the NDVI threshold between a full and a partial vegetation cover allows one to scale NDVI between hare soil and 100% vegetation cover, for which there is a simple square root relation with fructional vegetation cover [Eq. (3)]. More- over, our calclllations suggest that this relation holds equally wel1 for NDVI corrected or uncorrected for at- mospheric attenuation. The latter is of practica1 signifi- cance in view of the seeming importante of fractional vegetation covc’r. which may be more caasily obtainable from satellite tntasurenwnts than is LAI, ad the diffi-

culty in accurately correcting for atmospheric attenua- tion. Staling nut only minimizes the importante of choosing the initial conditions when using land surface model to obtain the surface energy fluxes and soil water content (Gillies et al., 1997), hut also mav eliminate the need to accurately correct the satellite radiances for at- mospheric attenuation. Furthermore, we suggest that fractional vegetation cover, not LAI, is the key variable in determining surface enerm fluxes over partial vegeta- tion cover. Indeed, varying both fractional vegetation cover and LA1 in a land surface model may be somewhat redundant when the vegetation cover is less than 100%.

Identification of this full-cover vah~e of NDVI (re- f&rcld to here as NDVI,) may not be straightforward un- lcss the image contains a full range of vegetation cover. In this case, our calculations suggest that the likely value of NDVI wil1 be about 0.05 below the largest values of NDVI. In any case, a reasonable estimate of NDVI, based on a qualitative inspection of a histogram wil1 likely leave an uncertainty of ahout +0.05 in choosing NDVI,. For a range of NDVI from 0.0 to 0.6, an error of 0.05 would correspond to an error of less than 0.1 in Fr. This stil1 leakes some uncertainty in choosing the bare soil valse of NDVI, however.

APPENDIX

The following mathematica1 development refers to Fig- tire 1, which depicts streams of radiant fluxes above a partial vegetation cover. Al1 fluxes mov~ either upward or downward with respect to a horizontal surface. The fluxes are distributed as follows. A pencil beam of sun-

1

0.9

0.8

0.7

0.6

r 2 0.5

0.4

0.3

0.2

0.1

0 0

1

0.9

0.8

0.7

0.6

T! 4 0.5

0.4

0.3

0.2

0.1

0.1 0.2 0.8 0.9 0.3 0.4 0.5 0.6 0.7 Fractional Vegetation Cover

1

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

b Fractional Vegetation Cover

light (solar constant S0 adjusted for solar distance) at so- lar elevation angle p0 passes through the atmosphere. The direct flux reaching the surface just above the bare soil and vegetation canopy (Fd) is

Fd= S,,sinqq,T,hT,. (Al.1)

The symbol T represents a broad-band transmittance, defined in general terms as

T=ë’“‘, (A1.2)

where r is a normalized optica1 depth for that band and m is a path length. Tab and T, refer to absorption by the direct beam (water vapor, carbon dioxide, ozone, and aerosols) and forward scattering (by air and aerosols), re- spectively.

Unlike Beer’s Law, which it resembles [Eq. (A1.2)],

0.8 0.9 1

Figure 5. Scaled NDVI squared (N”‘) versus fractional vegetation cover based on the apparent [uncorrected, N” (a); solid circles] and corrected radiances [N” (c); open rectangles] for a threshold LA1 of 3. The dashed line indicates the 1:l correspondence; (a) 15-km visibility; (b) 5-km visibility.

the transmittances refer to a broad-band radiant flux, ei- ther diffuse or direct and in the visible or near IR. By definition, the remainder of the incident flux transmitting through a medium (1-T) is split between an absorptance and a reflectante, R. The latter is subdivided into a for- ward-scattering component and a backward-scattering component, which refer to the fraction of the incident beam scattered upward or downward.

The diffuse flux reaching the surface (F,,f) is

Fdf=s,sinyl”[T,b(l-T,)(l-T,~~)l, L42) where Tbs refers to the backscattering component of the transmittance. Some of the flw incident on the bare soil surface is reflected back and forth between the atmo- sphere and soil. If absorption by these multiple reflec- tions and other second-order effects are ignored, the to-

-_

M(local)=P (Uncomcted)

LAl(lmal)=2 (comcied)

lAl(local)=4 (Uncorrected)

LAl(local)=4 (Corrected)

NDVI, Fractional Veptation Cooer, n~td LA1 249

0 0.2 0.4 0.6 0.6 1 Fl

Figure 6. Same as Figure 5a (15-km visibility) but for simulations with threshold values of LA1 equal to 2 and 4, for both apparent (uncorrected) N” and corrected radiances. (The 1:l line is omitted.)

tal direct plus diffuse flux absorbed at the surface in the bare soil portion (F,) is

F,=y_+(l-u,), (A3.1)

where uR is the bare soil albedo (identical for direct and diffuse flux) and X is the correction for the internal re&ctions:

X=a&L I - ~~Ld)~abdsinv)o* (A3.2)

TbSa, TSpd, and Tak1 respectively refer to transmittance components for backward scattering of diffuse flux by air and aerosols, forward scattering of diffuse flux, and the absorption of diffuse flux. (Mathematical definitions for the transmittances and the method of calculating them are presented later.) The value of X tends to be very smak-about 1 or 2% of the incident flux at the surface.

The flux reflected at the surface over the bare soil (F,,) is approximately

F,,=(F<,+F&,. (A3.3)

Both direct (FJ and diffuse (Fdf) flux components are also incident at the top of the vegetation canopy. In ac- cord with Taconet et al. (1986), the direct flux penetrat- ing the plant canopy and absorbed at the ground beneath the vegetation (F,i) is

Fgcd=F<l[(I-~O(I-ag)/(I-~~~gal)l, (A4) where ug and af refer to the albedo of the ground surface and the vegetation, respectively.

The light attenuation factor (canopy transittance) for

direct solar flux (of) is calculated by using the formula

(if=e -KI/SII$0~ (A5) where IC is assigned a value for isotropic leaf orientation of 0.4. For the diffuse flux component (Fgd), the diffuse solar angle (po) direction is assigned a value of 54 de- grees (path length 1.7). Calculation of the diffuse flux ab- sorbed at the ground beneath the canopy (Fgc$ is identi- cal with that of Fg,d in Eq. (A4) but with F,f replacing Fd and with a canopy transmittance factor (of) for diffuse light (sinv0=1/l.7).

The direct flux absorbed by the plant canopy (FJ is

F,,=F~(l-a~)a~l+u,(l-a~)/(l-a~ugu~). (A6.1)

The diffuse flux absorbed by the canopy (F& is similarly calculated but with incident diffuse flux (FdJ replacing the incident direct flux (F,!) on top of the canopy and the diffuse path length in erf.

The total flux absorbed at the ground under the can- opy (F,) is therefore the sum of F,d and F,, and the total flux absorbed in the canopy (FC,,,) is the sum of the diffuse and direct flux components absorbed in the can- opy (F‘d+FJ. The total flux absorbed in and below the plant canopy (F,) is the sum of Fo,,, and F,,..

Weighted for the fraction of vegetation cover (Fr), the absorbed flux at the surface over both vegetated and bare soil fractions (F,,J is

F,,,,=FrF,+(l-Fr)F,. (A6.2)

The total reflected flux from the canopy (F,,) is not calculated directly. Instead, it is assumed to be the dif-

f;m~ic.(> I)rtwcwr tho iiicident flur at thct top 01’ tlrcs cxii-

Ol>\ Lln(l tllilt ill)SO~l>f’tl ill ml IK~lO\~~ tIlc c’alrop’:

1;,.,=l;,,+F,,,-F,. (i\Í.l)

Thca total rrfiected flilx over the canopy d Ixwt~ soil weiglitetl for vrgetatioii c’ovcar (F,) is

F, = FrFs, + ( 1 - Fr)F,, (A7.2)

Thr reflectivity of that surf& (H,,,) is

(As)

Th upwding dar flux at the top of the atmo-

sphere (F,,,,,) is calculated indirectly. It is assumed to be the differente betweeii the incoining solar flux ancl al1 thc flux components absorbed in the atmosphere ad at thc surfk.

F,,,,,=Sl,sinyl,,-F,,~,-F,,~,,-F,,,~,. (AY)

Here, F,,,, = [S,,sin(o,,( 1 -T,,J] is the al)sorbed flux hom th tlownwelling direct beam, F,, ,, 1 is the absorbed flux fronl the upvelling diffuse radiation streani reflected by tlie caiiopy ad bare soil surfaces (F,), and F,<,,, is the total absorbrd flux at the surface.

Tbr reflectivity at the top of thc atlnosphcm~ (H,,,,,) is thrn

H,,,,,=- !L-

S,,sinq,\’ (A 10)

wllich is to lw compared with that at thr ground, as el- pressed by Eq. (A8). Equation (AlO) is the at-sensor (al’- purent) reflectivity measured by satellite, whereas Eq. (AX) is cousidered to be “corrected” reflectitity that is independent of the intrrvening atmosphere.

Trwnsrnittances are calculated separately for scatter- ing ad absorption. For Hayleigh scattering by air mok- cules. the norinalized optica1 depth rII is expressrd ils

7,{ =O.OOXX?.““, (All.l)

where s(i), a function of wavelength A (in micrometers), is qua1 to (-4.15+O.eA)P,/1013, where P, is the surf& pressurt’ in millibars. Path kmgth, 1)~ [sec Eq. (A1.2)] is coriiputecl from the ecpition

llr~~=sin~,,+O.lS(y7,,+3.8H)-““‘. (All.2)

Then the transmittance for Hayleigh scattering (Tl<) is

(A11.3)

where po is taken in degrees, AA. is the radiation band width in micrometers pertaining to the limits of the inte- gal, and S,,, is the extraterrestrial solar flux as a function of wavelength. S,, represents the integrated flux in the band. Th<> integral thereforc represents a weighted mean transrnittance over the wavelength band, 1, =O.5-0.7 pul11

(visible fxd) ad A=O.ï-0.9 pn (near IR band).

‘I‘railsiriitt~ur~(, l;)r iltIIIOS~>II(‘ïi~~ ii(~rosoIs (TI,) is cd~ikkl~tl iii a siiriilar Iii;iiin<7.

‘1’,,=~ J “? 1 s,,A3 S,,,[f,‘~il)““Jf/~, i

u\l2.1> i ‘1

dierc, t,, is thr riormalizcd opticd tlepth íi)]. WJ-osols (dwt). In accord with Paltridgr and l’lwtt, we writca tlrcl Augstron~ turbidity law:

s=/O?‘. (‘412.2)

where ,8 is the Angstrom turbidity coefficient and Ibr ex- ponent a depends on the type of aerosol: a valer of I .O was chosen for CL, which is typicd fi)r natura1 continental aerosols. Coefficient p is related to the aërosol 0ptic;il dept11 at 05 Pm through Eq. (12.2):

~=t,,(X=o.FjlUI/1)/(~.~5~ “), (A12.3)

where 7,1(Eb=0.5 pin) is deterinined from the horízoi~tal visibility (1’: km) iis

t,,(i,=o.s~,,,)=:3.~lAi. ( A 12.4)

Thus aerosol transmittance is varied as ;I function of the aerosol optica1 depth at O.Ej pirl as indica~ed by tlic hori- zontal visibility. We divide 5,) into two componeiits, oiit~

fol- scattering’ (t,) and ene fol- absoi-ption (7,). wlicrc t,=O.‘ii-ir,, and r,=0.25sl,, wliich yield thr tr~iiisinittailct,s for scaterring antl absorpfion by- (lust, rrspr~ctivrl\~ TI,> a11<1 Tl>,,. Then the scattering transmittancr (7:) is wlw- lated as the product of scattcring tntnslrlitt;nlces.

‘I:= ï’,{ï’,>\. ~\AlXI)

Absorption of radiation by wwter vapor, carbon dioxide, and oxygen is neglecM, bccausc these effects are as- siiiriïd to he small-betwfwi O.Fj and 0.9 pin, altholq$ a weak water vapor band cbxists nar 0.74 pm a11t1 hm, is a very weak oxygen band at 0.68 Pm. A smal1 c’orrec- tion is made for ozoiiï (norrn;ilized clepth 0.:3 CJ~I) in thr Chappuis (vis&) band by using a method outlined 1)). Paltridge ad Platt (1976): ser Lacis ad H~SPJI ( 1 Yï4).

Ths correction wil1 not be tliscllssed cxcept to indicak

that T,,, is computrd only for the visible band and its n11- ~iierical ~duc~ is vcry nearly ~yi~il to I.O.

The only significant absorber therefi)re i u cither thr visible or tht, nc’ar IK band is acrosol. Thus,

‘1:,,>= K,ï’,,,. (AIX2)

111 calculating thc diffuse transmittance functions in Eq. (A3.2), we calculate Z’,,.,, and ?‘,J,<~ in a marmer identical with tliat for ï: anti Yy,,,,, vxcept tliat ;i dar anglc of Fi4 degrees is used for thtx quivalent solar angle of’ diffuse isotropic solar radiation.

For backscattered flux (Ft,,), which does not reach the surhce, thr transmittance (Y,,,) is calculated by as- sluning that half of thrh K&igh scattering by air rnolr- cults is directed backward (toward the top of the ;itnlo- sphclrï) and half downwarcl. For clllst, \v(’ ;LssIllII(’ tIlat

the fraction backscattered is only 0.2, leaving 0.8 of the scattered flux directed toward the ground. Therefore, the hulk hackscattering transmittance for dust und air is

(A14.1)

where the eosine function accounts for the highly aniso- tropie nature of’ aerosol scattering in which increasing amounts of scatteting occur in the upward direction as the solar elevation angle decreases from solar noon (20% of the scattered flux directed upward) to near the hori- zon (FjO% direct& downward).

Values of parameters used to perform the simula- tions from which the illustrations in this article were oh- tained are given in Tahle 2.

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