The derivation of a simplified reflectance model for the estimation of leaf area index

REMOTE SENSING OF ENVIRONMENT 25:53-69 (1988)

53

The Derivation of a Simplified Reflectance Model for the Estimation of Leaf Area Index

J. G. P. W. CLEVERS Agricultural University Wageningen, Department of Landsurveying and Remote Sensing, P.O. Box 339, 6700 AH Wageningen, The Netherlands

Information about crop reflectance obtained from the literature suggests that reflectance factors in the near-infrared are most suitable for estimating leaf area index (LAI). A problem arises if a multitemperal analysis is required. Soil moisture content is not constant during the season, and differences in soil moisture content greatly influence soil reflectance. A correction for soil background has to be made when ascertaining the relation between reflectance and crop characteristics. Since in the literature no satis|actory solution for correcting for soil background was found, an appropriate simplified reflectance model for estimating LAI is presented. First of all, an apparent plant cover is defined. Then, a corrected infrared reflectance is calcadated by subtracting the contribution of the soft from the measured reflectance. The assumption that there is a constant ratio between the reflectances of bare soil in different spectral bands for a given soil background, independent of soil moisture content, enables the corrected infrared reflectance to be calc~ated without knowing soil reflectances. Subsequently, this corrected infrared reflectance is used for estimating LAI according to the inverse of a special case of the Mitscherlich function, Simulations with the SAIL model confirmed the potential of this simplitied (semi-empirical) reflectance model for estimating LAI.

I. Introduction Remote sensing techniques provide information about agricultural crops quantitatively, instantaneously and, above all, nondestructively. During the past decades, knowledge about remote sensing techniques and their application to fields such as agriculture has improved considerably. Bunnik (1978) demonstrated the possibilities of applying remote sensing in agriculture, particularly with regard to its relation with crop characteristics such as plant cover and leaf area index (LAI). LAI is defined as the total one-sided green leaf area per unit soil area ("number of leaf layers"), and it is regarded as a very important plant characteristic because photosynthesis takes place in the green plant parts, One of the main results of the work done by Bunnik (1978) was the identification of five wavelengths based on optimum information about variation in ©Elsevier Science Publishing Co., Inc., 1988 52 Vanderbilt Ave., New York, NY 10017

relevant crop characteristics. These wavelengths were: one in the green at 550 nm, one in the red at 670 nm, one in the near-infrared at 870 nm, and (to a less extent) two in the mid-infrared--one at 1650 nm and the other at 2200 nm. In the literature there is a certain consensus that bands in the green, red, and nearinfrared regions are optimal if information about vegetation is to be obtained (e.g., Allen et al., 1970; Kondratyev and Pokrovsky, 1979; Wiegand et al., 1972). From the literature (e.g., Knipling, 1970) it is evident that, in the visible region, vegetation absorbs much radiation (for photosynthesis) and shows a relatively low reflectance. This is especially true in the red region, because of the large absorption of this radiation by the chlorophyll in the leaves. In the nearinfrared region the opposite occurs. The spectral reflectance in this region is high. In the visible and near-infrared region the reflectance and transmittance of a green 0034-4257/88/$3.50

54

leaf are approximately equal (e.g., Goudriaan, 1977; Knipling, 1970; Wiegand et al., 1972; Youkhana, 1983). The low transmittance of a green leaf in the visible region implies that in this region only the reflectance of the upper layer of leaves determines the contribution of the canopy to the total measured reflectance. In the near-infrared region the transmittance of a green leaf is in the order of 50%, and there is very little infrared absorptance by a green leaf (Gausman, 1974). In this situation leaves or canopy layers underneath the upper layer contribute significantly to the total measured reflectance. This contribution decreases with increasing depth in the canopy and is negligible from LAI 6-8 onwards (Chance and LeMaster, 1977). This multiple reflectance indicates that the infrared reflectance may be a suitable estimator of LAI. Soil reflectance influences the relation between scene reflectance and LAI. At low plant cover, soil reflectance contributes strongly to the composite canopy-soil reflectance in the different spectral bands. For a given soil type, soil moisture will be the main factor determining soil reflectance. Reflectance decreases with increasing moisture content of the soil, but the relative effect of soil moisture on the reflectance at distinct wavelengths is similar (Bowers and Hanks, 1965). Janse and Bunnik (1974) noticed that the decrease is almost independent of wavelengths between 400 and 1000 nm for a sandy soil. This means that the ratio of the reflectance in two spectral bands in this interval was nearly independent of soil moisture content. Results obtained by Condit (1970) and Stoner et al. (1980) confirm that the ratio of the re-

J.G.P.w. CLEVERS

flectance in two spectral bands is independent of the soil moisture content for a given soil type. The main aim of this study was to investigate some index or model, derived from reflectance factors, for estimating LAI in a multitemporal analysis. Huete et al. (1984; 1985) indicated the complexity of correcting for different soil types with different physical properties. They suggested the use of "multiple soil lines" based upon soil type. Therefore, it was assumed that in the multitemporal analysis the soil type is given, and soil moisture content is the only varying property of the soil. In order to enable a multitemporal analysis, a correction for soil background should be made that is simple and requires few input variables. For instance, if the agronomist has to ascertain a leaf angle distribution (necessary for some existing models), he will probably prefer to collect the conventional field data as he has always done. 2. Literature on Estimating LAI 2.1. Indices for estimating LAI The green, red, and infrared reflectances may be used as variables for estimating LAI. Much recent research has been aimed at establishing combinations of the reflectance factors in different wavelength bands (vegetation indices), to minimize the undesirable disturbances of differences in soil background or atmospheric conditions. These disturbances are particularly prevalent when spatial and temporal analysis of reflectances is performed. The vegetation indices should also be sensitive to variations in LAI alter complete plant cover has been reached. This also means that the infrared reflec-

LAI ESTIMATION MODEL

tance should play a dominant role in such an index, The earliest investigations involved the infrared/red ratio [MSS(7/5)] by Rouse et al. (1973; 1974). Rouse and his colleagues found in their studies that this ratio was useful for estimating crop characteristics by eliminating seasonal sun angle differences and by minimizing the effect of atmospheric attenuation on radiances measured by earth-observation satellites (e.g., Landsat). The same authors also used the "vegetation index" [VI = M S S ( 7 - 5 ) / ( 7 + 5 ) ] for this purpose. In order to avoid negative values a transf o r m e d vegetation index [TVI = ~/VI + 0.5] was also used in practical applications. Wiegand et al. (1974)were the first to relate spectral observations to LAI. They concluded that Landsat MSS(7- 5) and MSS(5/7) in addition to MSS 7 and MSS 6 bands would be practical indicators of plant cover and density. Model simulations done by Bunnik (1978; 1981) show that these indices may be useful for estimating plant cover, but are only slightly sensitive for variations in LAI after complete plant cover has been reached. This is also confirmed by the resuits of, e.g., Asrar et al. (1984), Haffield et al. (1984), and Holben et al. (1980). In order to find an index independent of soft influence, Richardson and Wiegand (1977) introduced the perpendicular vegetation index (PVI). However, in order to apply the PVI, scatter plots of a sampie of data points containing soft and vegetation should be made in order to locate a "soft line." Often such a "soft line" cannot be determined. A similar approach for suppressing variations in soft background was developed by Kauth and Thomas (1976). They applied a heuristic

55

linear transformation in the four-dimensional data space provided by Landsat MSS measurements of agricultural terrain differing in softs. Application of the transformation to the four Landsat MSS bands produced a brightness index dominated by soft ("plane of softs") and a greenness index dominated by green vegetation. Gray and McCrary (1981a,b) applied the concept of a difference between an infrared and a visible spectral band using data from NOAA satellites. This index is comparable to the greenness index of Kauth and Thomas, insofar as greenness is a weighted difference between infrared and visible bands. The mathematical description of the relation of such an index to crop characteristics such as LAI differs from author to author, since the relations were mostly derived empirically ("soft line" or "plane of softs" are not physical). 2.2. Reflectance models The main aim of physical reflectance models suitable for agricultural crops is a better understanding of the complex interaction between solar radiation and plant canopies. Essentially, there are two classes of physical reflectance models: numerical and analytical models. Bunnik (1984) has reviewed several models. The model described by Idso and De Wit (1970), wherein radiative transfer is determined by scattering and absorption for discrete leaf layers, is an example of a numerical model. Goudriaan (1977) improved and extended this model by calculating a numerical solution for upward and downward diffuse fluxes within nine sectors of each hemisphere for each discrete layer.

,56

One of the earliest analytical models was described by Allen and Richardson (1968). It is based on a theory of Kubelka and Munk (1931) which describes the transfer of isotropic diffuse flux in perfectly diffusing media. In the analytical model, upward and downward fluxes are expressed by differential equations. Allen et al. (1969) extended this model in order to include scattering of direct solar flux by using the Dunfley equations (Duntley, 1942). The first analytical model incorporating both illumination and observation geometry was developed by Suits (1972) and is an extension of the model developed by Allen and his colleagues. Suits' model also incorporates plant canopy structural (with a drastic simplification) and optical properties. When model simulations are carried out with varying view angle, Suits' simplifications appear to be too drastic (Verhoef and Bunnik, 1981). Therefore, Verhoef (1984) extended the Suits model further by including scattering and extinction functions for canopy layers containing fractions of oblique leaves (inclined leaves). He did not introduce the drastic simplification of canopy geometry to exclusively horizontal and vertical components as used by Suits, but he used a discretized set of frequencies at distinct leaf angles. This model is called the SAIL model (Scattering by Arbitrarily Inclined Leaves). Complicated physical reflectance models usually simulate reflectances for varying crop characteristics. They are difficult to invert and too complicated for practical applications. Although a simple function relating reflectances to LAI is sought, it is best sought through a physical basis, In this study, a new model will be introduced that results in a simple eorrection for soil background. A mathematical

J.G.P.W. CLEVERS

relation of the resulting index and LAI will be described. This will be verified by means of calculations with the SAIL model. 3. Simplified Reflectance Model for Vegetation The main requirements for a simplified reflectance model for vegetation are that it: 1. Permits estimation of LAI; 2. Relates reflectances and LAI by more or less physically defined parameters; 3. Corrects for soil background (i.e., soil moisture content) in a way that enables multitemporal analysis; 4. Is simple and results in a meaningful vegetation index. Conventionally, for a green canopy, plant cover is defined as the relative vertical projection of the canopy on the soil surface. In practice, however, the researcher will estimate the relative amount of plants visible to him. When one looks vertically downwards at a well-developed wheat crop, for instance, some shadows may be visible between the stems of the plants, and when looking from a height of a few meters or more above the top of the canopy, one is unable to decide whether there are leaves present in the shadows just above the soil. In a situation where no soil is actually visible to the researcher, he will judge that plant cover is complete. However, this same situation may be judged to be an incompletely covered soil if the conventional definition referring to the vertical projection of the canopy is strictly applied. If a remote sensing technique is used in a visible band while looking downwards from some distance and the direction of

IAI ESTIMATION MODEL

observation is unequal to the direction the sun is shining, the sensor will also be unable to detect whether soil in the shadows is obscured by leaves. For this reason plant cover is redefined by taking the sun-sensor geometry into account. A prerequisite for ascertaining bare soil according to the new definition is that the soil must contrast with the vegetation. When the sun is shining the soil should be directly illuminated by the sun [Fig. l(a)]. If the soil is not directly sunlit, it will be impossible to say whether it is obscured by green vegetation or by shadows. Both have a low reflectance value in the visible region of the electromagnetic spectrum (much lower than sunlit soil), with little contrast. Further, the soil must also be in the line of sight of the sensor to detect it [Fig. l(b)]. The fraction of soil that satisfies both conditions (i.e., soil that is sunlit as well as directly detectable by the sensor) will be classified as the fraction of soil that is not covered [Fig. l(e)]. The complementary fraction will now be defined as plant cover ("apparent plant cover"). It can be described as the fractional horizontal area of ground obscured and shaded by vegetation at the particular combination of sun and view angles existing at the instant of the observation. This definition is valid for a uniform crop as well as for a nonuniform crop (including row crops). In the special situation of the sensor looking vertically downwards (as is often the case in remote sensing), this definition of plant cover is equivalent to the relative vertical projection of green vegetation, the relative area of the shadows (shaded soil) included, In order to ascertain whether there is a useful relation between infrared reflectance of the composite canopy-soil scene

57

and LAI for green vegetation on a given soil type, the former should be corrected for fractional soil reflectance, because soil background may influence infrared reflectance independently of the LAI. The infrared reflectance is then calculated for the situation of the soil background in line of sight being completely black or nonreflective. This corrected infrared reflectance value is then used to estimate LAI. Let us consider the simple situation of a surface, partly covered with green vegetation and partly bare (Fig. 2). This is not restricted to row crops, but it includes the situation with row crops. The fraction of the surface obscured and shaded by vegetation is called plant cover B. If the reflectance of the soil is called rs and the reflectance of the vegetation (complete cover) r~, then the total measured reflectance r will equal

r = r v . B + r s . ( 1 - B ).

(1)

A green band will be denoted by the subscript g and a red band by the subscript r. Then Eq. (1) is written as

rg=rv.g.B+r~,g.(1-B ),

(2)

r,=r~,~.B+r,,~.(1-B)

(3)

for the green and red bands, respectively. The reflectance of plants (complete cover) in a green and red band (rv, g and rv, r) may be regarded as being independent of the number of leaf layers, because leaf transmittance in the green and red are assumed to be negligible. Hence Eqs. (2) and (3) describe the linear relation of the reflectance in a visible band to plant cover, if the soil reflectance can be con-

58

J . G . P . w . CLEVERS

sun

/

/"~ '

~

i

/

/

/'~

/

/

sensor

\

\

\

,

\

,

\ ~lJ

\

[

\

/\ /

/ \

\ \

\

/

i

\

/\ 1"~1

/ \1

F I G U R E 1. Schematic presentation to illustrate aspects of the new definition of plant cover (this is not restricted to row crops): A) soil illuminated by the sun; B) soil in the line of sight of a sensor; C) sunlit soil in the line ot sight of this sensor.

LAI ESTIMATION M O D E L

59

\1/ /1\ r

/

?

F I G U R E 2. Schematic presentation o| a simplified reflectance model for a composite vegetation and soil scene, r is the composite scene reflectance; rs is the reflectance of the sunlit soil; rv is the reflectance of the plants and shadows combined (complete cover).

sidered constant (in case of a constant soft moisture content), For an infrared band the subscript ir will be used, and Eq. (1) is then written as

r~r=rv,,r.B+r~,ir'(1-B).

(4)

In this equation rv,ir depends on the number of leaf layers (or LAI), and so it may not be regarded as a constant. It is composed of a single scattering component and a multiple scattering component. This means that the reflectance measured in an infrared band (fir) is not a linear func-

tion of plant cover and continues to increase even after complete plant cover. For estimation of LAI the difference r' between the composite reflectance r and the soil component of the scene could be used. This difference (corrected reflectance) is [according to Eq. (1)] defined as

r'=r-r~.(1-B)=rv.B.

(5)

The corrected reflectance is the reflectance one would have obtained with a black background. In order to obtain the corrected inflared reflectance, Eq. (5) first has to be

60

J, G. P. W. CLEVERS

apphed to the infrared band: ri;=rir-rs,ir'(1-B

).

(6)

B can be ascertained by means of Eq. (2) or (3). However, the reflectance of bare soil and of vegetation (complete cover) in a green or red band should be known for applying Eq. (2) or (3). Mthough it is quite often possible to ascertain a good estimate for the reflectance of vegetation at complete cover, estimating the reflectance of bare soil poses greater difficulties. The reflectance of a soil may change very rapidly, according to soil moisture content. Mso, very large local differences in soil moisture content may occur since the whole ground area does not dry uniformly (soil beneath plants remains wet longer). At low plant cover this may cause large inaccuracy if neither the soil moisture content nor the actual reflectance of the soft are known. To obtain an accurate estimate of LAI, one either has to know or to measure the reflectance of the bare soil, or one has to derive a relation that is less dependent on differences in soil rnoishtre content, For many soil types (with some moisture content) there is a monotonic increase in reflectance with increasing wavelength (e.g., Condit, 1970). However, the ratio of the reflectance in two spectral bands is essentially independent of the soil moisture content: r~.,g/r~,r = C l

(7)

r~.ir/rs, ~ = C2.

(8)

and

If we assmne that we are able to determine the constants C~ and C2, merely

by measuring the required reflectance values at the same soil moisture content, thenEqs.(2),(3),(6),(7),and(8)offerus five equations with five unknown variables: the corrected infrared reflectance (ri'r), plant cover (B), and the soil reflectance in the three bands (r~,g, r~,r, and r,,~r). After solving these equations for the corrected infrared reflectance, we obtain

rir' = r, -

C 2. ( rg, r,,, - r,. r,: g) C~.r, - r,, '

(9)

For the case of bare soil, rg, rr, and r~r equal r~,g, re, r, and G,ir, respectively, and Eq. (9) results in r(r = 0. In the situation of complete plant cover, rg and rr equal r~,g and r,.... respectively, and Eq. (9) results in ri'~ = rir; in other words, no correction for soil background is applied if no sunlit soil is in the line of sight. Since C l and C 2 may vary for different soil types, this procedure is not based on one "soil line." Huete et al. (1984) found individual soil lines for different soil types and suggested the use of multiple soil background lines, which coincides with the approach in this paper. In order to deduce the relation between plant cover and LAI, the process of extinction of radiation in a canopy should be considered. If a canopy has a certain extinction coefficient per leaf layer as well as a certain LAI (abbreviated as L in the formulae), the product of both factors equals the mean extinction of the canopy. The mean extinction consists of two components: 1. Extinction in the direction of the sensor, indicated by K . L , where K is the extinction coefficient per leaf layer in the sensor direction (so K varies with view direction).

LAI ESTIMATION MODEL

61

2. Extinction in the direction of the sun, indicated by k . L , where k is the extinction coefficient per leaf layer in the direction of the sun (k varies with position of the sun). Consider the process of extinction in a very small part (or element) of the canopy. In the visible spectral bands, extinction in an element occurs w h e n a leaf is hit by radiation. The probability of hitting i elements a m o n g n independent elements has a binomial distribution (an element is hit or not). If the number n of independ e n t elements per unit area increases to infinity while the probability of hitting a specific element decreases to zero, the binomial distribution can be approxim a t e d by a Poisson distribution. The Poisson distribution states that

e-x.)ki P(x = i)=

i!

'

i =0,1,2,3,.... (10)

T h e random variable x is the number of i n d e p e n d e n t elements of the canopy in which extinction occurs; • is the mean or expected n u m b e r of elements in which extinction occurs. The probability that no element is hit (i = 0) equals:

P(x = 0) = e -x.

(11)

T h e probability of soil being visible in the direction of the sensor equals e - K L [which follows from Eq. (11) with K . L as the mean n u m b e r of elements in which extinction occurs in the sensor direction]. T h e probability of soil being illuminated by the sun equals e -k'L. If one assumes both events to be independent, the probability of sensing sunlit soil is the product of e -K'L and e -k'L, which equals e -(K+k)'L.

The complementary probability is equal to the apparent plant cover (new definition as introduced at the beginning of this section). This means that plant cover may be described as B = 1 - e-(K ÷ k)-L.

(12)

Inserting Eq. (12) into (5) gives r ' = r~.(1 - e -(K÷k)'L)

(13)

T h e relation between LAI and infrared reflectance (cf. Bunnik, 1978) greatly resembles a "Mitscherlich curve" [y = A b-exp( - k .t)]. In the special situation of such a curve running through the origin (A = b), the curve defined by y = A.[1 - e x p ( - k.t)] has only two parameters (Mitscherlich, 1923). For describing the relation between the corrected infrared reflectance and LAI (which runs through the origin) an empirical equation similar to Eq. (13) could be used: r~'r = roo,ir- (1 - e - ~ L),

(14)

the parameter ro~.ir being the asymptotically limiting value for the infrared reflectance and a a complex combination of extinction and scattering coefficients. As long as there is no data base of r~,~r and a values for different crops, these should be estimated empirically from a training set. Finally the LAI is solved from Eq. (14): L= -1/a.ln(1-

r~'Jr~.ir ).

(15)

This is the inverse of the special case of the Mitscherlich function. Equation (15) is not defined at infinite LAI (ri~ = r~,i~ ), but holds for all other LAI values.

62

.l. (;. P. W. CLEVERS

4. Comparing the Model with the SAIL Model In this section calculations with the more complicated SAIL model (Verhoef, 1984) were used as a data set for verifying the model derivations presented earlier. The SAIL model simulates reflectances as a function of plant variables and measurement conditions (cf. Sec. 2.2). The following variables for the SAIL model have been used: - - T h r e e simulated soil types: dry soil (green reflectance = 20.0%, red reflectance = 22.0%, infrared reflectance = 24.2%); wet soil (green reflectance = 10.0%, red reflectance = 11.0%, infrared reflectance = 12.1%); "black" soil (green, red, and infrared reflectance = 0%). --Spherical leaf angle distribution. --Direct sunlight only (solar zenith angle: 45°). --Direction of observation was assumed to be vertically downwards, --Reflectance and transmittance of a single leaf were assumed to be equal: green reflectance = 8%, red reflectance = 4%, and infrared reflectance = 45%. Model calculations were carried out using the following LAI values: 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 6.0, 7.0, and 8.0. The green, red, and infrared reflectance factors were calculated according to the SAIL model for each of the above situations. At a solar zenith angle of 45 °, there will be shaded soil and vegetation as well as sunlit soil and vegetation in the field of view. The SAIL model was also able to calculate the complement of the

sunlit soil detectable by the sensor (this equals plant cover with the new definition introduced in this paper). The results obtained with the SAIL model clearly show that the relation between plant cover, according to the new definition, and green or red reflectance was nearly peliectly linear for a dry soil (Fig. 3). Similar results were obtained for a wet soil (Clevers, 1986). These results support the validity of Eqs. (2) and (3) if the new definition of plant cover is used. Clevers (1986) showed that Eqs. (2) and (3) are not valid with the conventional definition of plant cover. The infrared reflectance is corrected for soil background and then used to estimate LAI. This latter step may be investigated by using the simulations with the SAIL model for a black background. Then the infrared reflectance does not require correction and the vahdity of Eq. (15) may be checked. The results, shown in Fig. 4, support the validity of this equation for describing the relation between "corrected" infrared reflectance and LAI at constant leaf angle distribution. Results presented by Clevers (1986) show that distinct leaf angle distributions cause quite distinct asymptotic values for the infrared reflectance, calculated from the SAIL model (cf. also Bmmik, 1978). A changing leaf angle distribution during the growing season of a crop may disturb the relation between corrected infrared reflectance and LAI. However, Clevers also showed with real field data that leaf angle distribution of cereals could be considered constant during the vegetative and reproductive stages, respectively. A correction can be made for differences in soil moisture content by subtracting the contribution of the soil in the line of sight of the sensor from the mea-

L A I E S ~ M A T I O N MODEL

PLANT COVER (Z) I00

SAIL MODEL

~

SPHERICAL SUNLIGHT

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SOIL 80

60

40

20

0 0

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I

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5

I

10

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I

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PLANT COVER (7.)

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I

20 25 GREEN REFL. (%)

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SPHERICAL SUNL I GHT DRY S O I L

80

60

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0

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0

5

10

15

20 RED REFL.

i

25 (7.)

FIGURE 3. Plant cover (new definition) as a function of green and red reflectance, respectively, for a spherical leaf angle distribution. CV is the coefficient of variation (x) calculated points SAIL model; ( - - ) simplified reflectance model.

30

50

(%)

FIGURE 4. LAI as a function of the infrared reflectance for a black soil with a spherical leaf angle distribution. CV (coefficient of variation) = 0.034, a = 0.456; r~,ir = 42.0. (x) calculated points SAIL model; ( - - ) simplified reflectance model.

MODEL

lO

CV

20

30

CORR.

- O. 045

O. 621

SUNLIGHT DRY SOIL METHOD O roo,ir- 41.3

SPHER ICAL

SAIL

40

50

INFRARED REFL.

(%) FIGURE 5. Method 0 (soil reflectance known) and Method 1 (constant ratio of soil reflectance in two wavelengths) for correcting for differences in soil moisture content in estimating LAI. Spherical leaf angle distribution. 90% Plant cover occazrs at LAI = 2.0. CV is the coefficient of variation.

0

40

INFRARED REFL.

0

20

4:

41

10

B

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I

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LAI

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LAI

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CV

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-

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1

20

40

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FIGURE 5.

CORR. INFRARED REFL. (Z)

,

(Continued)

oo

2

4

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CV 6

~ -

-

20

30

,

40

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CORR. INFRARED REFL. (Z)

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8

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SPHERICAL SUNLIGHT DRY SOIL METHOD I r = , i r - 41.3

SAIL MODEL

z

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E

66

J . G . P . W, CLEVERS LAI

I0

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MODEL

SPHERI CAL SUNL I GHT WET SOIL

8

METHOD I ro=,ir- 41.5 - O. 528 cv - O. 013

8

4

2

,

O~

10

20

FIGURE 5.

LAI I0

SAIL

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30 40 5'0 CORR. INFRARED REFL. (Z) (Continued)

MODEL

SPHERI CAL SUNLIGHT

8

6

j

,

.S// ///

O0

10

20

30 CORR.

410 INFRARED

'

510 REFL.

(%)

FIGURE 6. Influence of soil background on the regression of LAI on corrected infrared reflectance: ( - - ) dry soil; 6--) wet soil; ( . . . . ) black soil.

LAI ESTIMATION MODEL

sured infrared reflectance [Eq. (6)]. If soft reflectance is known, Eq. (6) may be combined with, e.g., Eq. (3) in order to ascertain this corrected infrared reflectance. This method will be called Method 0 (indicating that it cannot be applied without knowing soil reflectances explicitly). In practice, however, soil reflectances often are not known. Then Eq. (9) can be applied, taking into account the constant ratios of soil reflectance between spectral bands. This method will be called Method 1. Results for both methods are given in Fig. 5. Both methods gave essentially the same results, which supports the validity of Eq. (9) for correcting the infrared reflectance for soil background, Because the only correction made is for soft in the line of sight of the sensor and not for the soil underneath vegetation, some influence of soil background will still remain. This is illustrated in Fig. 6. Even with such a large range in soil reflectances, differences between curves were not very large. In reality, fluctuations in soil moisture content underneath vegetation will be less than those on bare

67

tional definition of plant cover, the relation is not linearl 2. It was shown to be possible to get around the problem of an unknown soil mois~re content (and so an unknown soil reflectance) in estimating LAI. Under the assumption that there was a constant ratio between the reflectance factors of bare soil in different spectral bands, independent of soil moisture content, a combination of green, red, and infrared reflectances was applied for correcting the infrared reflectance for soil background [Eq. (9)]. This assumption holds for many soil types. 3. For a green vegetation, the inverse of a special case of the Mitscherlich function, namely, the one running through the origin [Eq. (15)], described the regression of LAI on the infrared reflectance corrected for background. In this semi-empirical model, two parameters (r~,ir and a) of a physical nature have to be estimated empirically.

SOil.

A more extensive verification of the new model by means of calculations with the SAIL model for several leaf angle distributions and also for skylight only are presented by Clevers (1986). In a future paper a verification with practical field data will be given. 5. Conclusions 1. If plant cover is redefined as in Sec. 3, then the reflectance in a spectral band in the visible region of the electromagnetic spectrum decreases linearly with increasing plant cover [Eqs. (2) and (3)]. With the conven-

The author wishes to acknowledge the help and advice given by all colleagues, in particular Wouter Verhoef, who provided the data set for the SAIL model. References

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