A multiple-layer canopy scattering model to simulate shortwave radiation distribution within a homogeneous plant canopy

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1 WATER RESOURCES RESEARCH, VOL. 41, W08409, doi: /2005wr004016, 2005 A multiple-layer canopy scattering model to simulate shortwave radiation distribution within a homogeneous plant canopy Wenguang Zhao and Russell J. Qualls Department of Biological and Agricultural Engineering, University of Idaho, Moscow, Idaho, USA Received 3 February 2005; revised 22 April 2005; accepted 15 May 2005; published 12 August [1] A multiple-layer canopy scattering model to estimate shortwave radiation distribution within a wheat canopy was derived mathematically, which incorporated processes of radiation penetration through gaps between leaves, and radiation absorption, reflection, and transmission in leaf layers. The model is able to simulate the multiple scattering processes that occur among different canopy layers and to predict the vertical distributions of upward, downward, and reflected shortwave radiation within the canopy, as well as the magnitude of subcanopy radiation that penetrates to the soil surface. One of the primary advantages of this model, in contrast to other models, is that the multiple scattering processes are represented by a set of linear simultaneous equations that can be solved in a single pass through the equations, without iteration. This achieves computational economy while still accounting for the details of multiple scattering of radiation within the canopy. Stability analyses of the model showed that the canopy, with a leaf area index within a normal field range from 0 to 7, needed to be divided into about 50 or more layers in order to converge upon its final solution. Satisfactory agreement was obtained between model results and field measurements for downward shortwave radiation impinging on the soil surface below the canopy, and upward reflected radiation above the canopy, both for daily total values and for the 20-min averages throughout the diurnal cycle. Citation: Zhao, W., and R. J. Qualls (2005), A multiple-layer canopy scattering model to simulate shortwave radiation distribution within a homogeneous plant canopy, Water Resour. Res., 41, W08409, doi: /2005wr Introduction [2] Radiation energy is the driving force for land surface meteorological and micrometeorological processes, such as sensible and latent heat (evapotranspiration) transfer. The surface radiation balance is one of the major factors that determines the skin temperature of a crop canopy and the underlying soil substrate. The penetration of solar radiation into vegetation canopies plays a crucial role in surface energy balance. In cooperation with other physical and physiological controls on surface conductance, canopy radiative transfer also affects how the total energy input to the system is dissipated by radiation, sensible, latent, and soil heat fluxes, and changes in thermal storage in vegetation and soil substrate. Shortwave radiative transfer in terrestrial communities influences the way in which the land surface interacts with the atmosphere and is of consequent importance in models of atmospheric dynamics and boundary layer processes [Hanan, 2001]. [3] Remote sensing technology has tremendous potential for use in natural resource studies, agriculture (e.g., irrigation and yield prediction), and water and land use management because of the spatial information contained in remote sensing images and because of the ease and/or frequency of acquiring vast amounts of surface information. However, Copyright 2005 by the American Geophysical Union /05/2005WR W08409 the quantitative application of remotely sensed data is restricted by several problems. For example, the directional radiometric surface temperature measured from above a vegetated field represents neither the skin temperature of the crop nor the surface temperature of the soil substrate. This is because the temperatures of the soil surface and the various canopy elements viewed are not single-valued. For example, the skin temperature of a crop can exhibit a difference in excess of 10 C between the leaves at the bottom and those at the top of the canopy [Qualls and Yates, 2001]. When a remote sensing device views a vegetated surface from different view angles, different combinations of canopy and soil elements at different temperatures will be seen, producing different values of remotely sensed surface temperature. In order to deal with this view angle effect and develop methods to assimilate remote sensing into land surface energy exchange models, the vertical profile of skin temperature down through the canopy must be known. [4] The model presented here is the first component of a set of coupled models to simulate the vertical profile of canopy temperatures. This model will be extended to simulate longwave radiation absorption, scattering, and emission processes within the canopy layers, between the canopy layer and soil, and the upward longwave radiation above the canopy (W. Zhao and R. J. Qualls, manuscript in preparation, 2005). A multilayer turbulent flux model is also under development. The latter two models are coupled 1of16

2 W08409 ZHAO AND QUALLS: MULTILAYER CANOPY SCATTERING MODEL W08409 together by means of the canopy skin temperature profile. Canopy temperature profiles may be used in at least two ways to assist in assimilating spatially and temporally distributed remotely sensed surface temperatures to simulate land-atmosphere energy exchanges. One of these builds on the method of Crago [1998], which was further developed by Zibognon et al. [2002], which converts remotely sensed radiometric temperatures to aerodynamic surface temperatures by means of simulations of canopy temperature profiles. These spatially distributed aerodynamic surface temperatures may then be used in a soil-vegetationatmosphere transfer (SVAT) model to simulate sensible and latent heat fluxes. [5] The second method follows the approach of Bastiaanssen et al. [1998], which has been built upon by Allen et al. [2002] in their surface energy balance algorithms for land (SEBAL) model. In SEBAL, the surface-air temperature difference is modeled as a linear function of remotely sensed surface temperature. The coefficients of the linear relationship are determined from calculations at cold and hot reference pixels. SEBAL calculates scalar roughnesses (z oh ) as a function of normalized difference vegetation index. [6] Scalar roughness is an important quantity; Verhoef et al. [1997] note that all meteorological weather models currently include a parameterization of z oh. Numerous studies have shown that z oh is neither constant nor proportional to the momentum roughness length [e.g., Qualls and Brutsaert, 1996; Qualls and Hopson, 1998; Kustas et al., 1989; Sun and Mahrt, 1995; Verhoef et al., 1997; Matsushima and Kondo, 1997; Chen et al., 1997]. Consequently, we are developing methods to use the vertical profile of canopy skin temperatures, which will be produced by our coupled models, together with the vertical distribution of within-canopy turbulence intensity and stomatal conductance to produce z oh values for a given time of day and canopy architecture and density. These scalar roughnesses will be incorporated into SEBAL to account for diurnal variability. Furthermore, the ubiquity of z oh in SVATs will ensure widespread applicability of our results. This will advance our ability to assimilate remote sensing data into land-atmosphere exchange models and make use of the spatial information they contain. [7] Likewise, the remotely sensed albedo within a specified wavelength range for a vegetated field is neither a simple reflection from crop leaves nor a simple reflection from underlying soil substrate. Rather, it is a result of a series of complicated multiple reflection and transmission processes between the soil and different layers of crop leaves. Analysis of these multiple reflection and transmission processes allows a better retrieval of major canopy characteristics [Bacour et al., 2002]. [8] SVAT models are widely used in estimation of evapotranspiration, sensible heat flux, and photosynthesis from meteorological data. On the basis of the number of energy exchange layers that the model deals with, SVAT models can be divided into single-source models [Penman, 1948; Monteith, 1963, 1965], dual-source models [Shuttleworth and Wallace, 1985; Wang and Leuning, 1998], and multiple-source models [Choudhury and Monteith, 1988; Smith and Goltz, 1994; Flerchinger et al., 1998]. It has been demonstrated that single-source models are inadequate because of the error introduced by treating the canopy and the underlying soil substrate as a single source [Acs, 1994]. Dual-source models deal separately with transpiration from vegetation and the evaporation from soil substrate. However, treating the whole canopy, from the top to the bottom, as a single big leaf with uniform temperature and humidity characteristics has been shown inadequate, especially when the canopy skin temperature profile exhibits significant vertical variability [Qualls and Yates, 2001]. Multiple-source models are required in order to simulate the vertical profile of canopy skin temperatures. Multiple-source SVAT models require a multiple-layer radiation model to distribute radiation throughout the canopy, which is the focus of this paper. [9] Another application of our shortwave model relates to carbon sequestration. Photosynthesis and biomass accumulation rates are primary concerns for crop production and have attracted the interests of crop researchers. Leaf photosynthesis rates respond nonlinearly to absorbed photosynthetically active radiation (PAR) [Smith and Goltz, 1994]. The distribution of absorbed PAR within a crop canopy profile is extremely uneven. Therefore traditional big leaf photosynthesis models may create errors up to 44% when utilizing only the absorbed radiation by leaves [Myneni and Ganapol, 1992]. In order to model photosynthesis adequately, the vertical distribution of absorbed PAR within a canopy is required, and this is best provided by modeling. [10] In order to simulate the radiation distribution within a canopy profile, two categories of methods have been used up to now. The first category is the loop method represented by Norman [1979]. Norman s layer equations must be applied one layer at a time (except for canopies with horizontal leaves), through every layer of the canopy. This process is iterated until a stable result is obtained for each layer [Norman, 1979]. The other category contains the approximation methods. Among them, the two-stream approximation is the most well known [e.g., Dickinson, 1983; Sellers, 1985]. This method was originally developed for modeling radiative transfer through the atmosphere [e.g., Coakley and Chylek, 1975] and was adapted to model radiative transfer within vegetation canopies [Ross, 1981]. The upward and downward radiation is expressed as two differential equations for each layer and the coefficients used for the equations differ among researchers [Meador and Weaver, 1980]. Except under special conditions, the equations produce approximate results rather than an exact solution. In addition, two-stream approximation methods are reliable only for integrated quantities, like diffuse radiation, as opposed to angular quantities, like direct radiation [Meador and Weaver, 1980]. Even for diffuse radiation, depending on the coefficients used, the estimation error of the two-stream approximation can be as high as 25% [Thomas and Stamner, 2002]. Wang [2003] compared three different canopy radiation models commonly used in plant modeling and concluded that in the estimation of absorbed visible radiation, the two-stream approximation method is not superior to Beer s law if direct beam and diffuse radiation are considered separately with the appropriate extinction coefficients. [11] This paper presents a newly developed radiation model that treats a crop canopy as multiple layers of leaves. In dealing with the solar radiation penetration into the 2of16

3 W08409 ZHAO AND QUALLS: MULTILAYER CANOPY SCATTERING MODEL W08409 canopy, scattering of the direct solar beam and diffuse sky radiation are treated separately and include multiple reflection and transmission processes among different layers of the canopy. Not only is the model able to predict both downward and upward solar radiation and their components, PAR and NIR (near-infrared radiation), at any height within the canopy and above the soil surface, but also it is able to calculate the PAR and NIR absorbed by each layer of the canopy. [12] One of the primary advantages of the new canopy scattering model presented here is that it results in a set of linear simultaneous equations that can be solved in a single pass through the equations, without iteration. A further advantage is their adaptability for parallel solution of the tridiagonal system [Press, 1996]. This achieves tremendous computational economy, with respect to the Norman loop method, while still accounting for the details of multiple scattering of radiation within the canopy. The solution obtained is exact, as opposed to the single-scattering method [Thomas and Stamner, 2002] and the two-stream approximation method. In addition, the computational economy of linear equations, as compared with the differential equations used in the two-stream approximation, makes this method appropriate for regional application, e.g., calculation of the radiation distribution for each representative canopy type in a region. [13] The model results were validated with two independent sets of data measured within a wheat field: The modeled radiation reflected upward from the top of the canopy was compared with measured reflected radiation; the modeled downward, subcanopy radiation was compared with the measured data from a tube solarimeter placed below the canopy, just above the soil surface. 2. Radiative Transfer Model Description [14] The shortwave radiation inside a crop canopy is composed of two parts, directional radiation and hemispherical radiation. Directional radiation comes directly from the Sun and penetrates through the canopy until it is intercepted by any vegetation or the soil surface. Upon interception, directional radiation is scattered or absorbed and ceases to be directional. Hemispherical radiation is either the diffuse radiation from the sky or the reflected or transmitted radiation whose source is either hemispherical radiation within the canopy or directional radiation that has been intercepted by vegetation or the soil surface. The directional radiation has a single valued direction at any point in time, which may be defined by a local solar zenith angle and azimuth. The hemispherical radiation is assumed to emanate from all points on a hemisphere toward the center of the sphere (i.e., toward an elemental area of interest) or vice versa, and is separated into two components, upward moving and downward moving hemispherical radiation. [15] As a result of the fact that leaves scatter a portion of any radiation that they intercept, a series of multiple reflections and transmissions (i.e., radiation scattered by passing through a leaf without being absorbed) occurs as radiation propagates through a canopy. We refer to the combined reflection and transmission upon interception simply as scattering. Radiation within plant canopies experiences an infinite series of scattering between canopy elements. These multiple scatterings cause the canopy to absorb a higher fraction of incident radiation than if only a single interception of radiation occurred. Furthermore, in certain wavelength bands of radiation, most notably NIR, these multiple scatterings are significant in altering the distribution of energy within the canopy [Ross, 1975]. For NIR, on the order of 80% of the energy involved in each leaf interception is scattered [Campbell and Norman, 1998]. [16] We have derived a simple analytical model that explicitly includes the infinite series of interactions that occurs when radiation is repeatedly scattered within a canopy. The series is convergent and can be represented analytically as a mathematical expression with only two terms. [17] In the following, we present the multiple scattering model. For clarity, a standard model for penetration of directional radiation from the literature is briefly explained, followed by its application to the interception of hemispherical radiation. Then the single scattering of directional radiation and the single scattering of hemispherical radiation are described, and finally we present the multiple scattering radiation model Interception of Radiation by a Canopy Layer Interception of Directional Radiation [18] Assume that there is a canopy layer i with a leaf area index (LAI) L i in the layer. When the directional radiation beam reaches the canopy layer, a portion of it is intercepted by the canopy and the remaining portion penetrates through the gaps of the canopy. The interception function, I(y) i, defined as the fraction of radiation beam that is intercepted by the canopy layer, is a function of the zenith angle y of the directional radiation beam and can be expressed [Campbell and Norman, 1998] as 3of16 IðyÞ i ¼ 1 exp KðyÞ i L i ; ð1þ where K(y) i is the directional extinction coefficient. It is a function of the source direction of the directional radiation beam and the leaf distribution character of the canopy. K(y) i is often represented in the literature as K(y) i = G(y)/ cos(i) where G(y) is the so-called G-function, which describes the mean leaf projection per unit leaf area in the direction of the solar radiation beam [Ross, 1975; Dickinson, 1983; Sellers, 1985]. [19] A broad range of vegetation species leaves can be represented by an ellipsoidal distribution [Campbell, 1986; Campbell and van Evert, 1994]. The directional extinction coefficient for an ellipsoidal leaf distribution can be expressed [Campbell and Norman, 1998] as KðyÞ i ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þ tan 2 y x þ 1:774ðx þ 1:182Þ 0:733 ; ð2þ where x is a constant related to canopy architecture. A few idealized cases which can be accommodated by the ellipsoidal distribution are the spherical distribution, x = 1; the vertical distribution, x = 0; and the horizontal distribution, x approaches infinity. Values of x have been determined for many crops. For example, wheat has x = 0.96, which is very close to the spherical case [Campbell and van Evert, 1994].

4 W08409 ZHAO AND QUALLS: MULTILAYER CANOPY SCATTERING MODEL W08409 plane XOY. The differential area da at point P, defined by the differential angles da and dq, can be expressed as da ¼ cos q dq da: ð4þ The angle d between the normal direction of the differential area da and the solar beam can be expressed as cos d ¼ sin q cos y þ cos q sin y cos a: ð5þ [22] The radiation flux received on the differential area da can therefore be expressed as ds ¼ S 0 cos d da: ð6þ Figure 1. Directional radiation on a spherical leaf distribution model Interception of Hemispherical Radiation [20] Hemispherical radiation comes from all directions, and the canopy layer intercepts radiation from each individual direction of the hemisphere according to the interception function for that direction. Assuming that the hemispherical radiation is isotropic, the interception coefficient of a canopy layer i can be obtained by integrating equation (1) over the whole hemisphere as Z p 2 I i ¼ exp KðyÞ i L i sin y cos ydy: ð3þ Equation (3) can be integrated analytically for simple leaf distribution models and numerically for complicated leaf distribution models Single Scattering of Directional Radiation Derivation of the Function for Single Reflection of Directional Radiation [21] For an arbitrary geographic position on the Earth, we denote the radiation flux density on a plane perpendicular to the direction of sunlight as S 0 and the local solar zenith angle as y. Both vary with time of day and day of year. Consider a unit spherical leaf distribution model, that is, a spherical leaf distribution model with a radius of 1. The radius is taken to be 1 for simplicity only. We could assign an arbitrary length or radius, r, since it eventually cancels itself out of the equations. We define a set of Cartesian axes whose origin is located at the center of the leaf sphere, whose z-axis corresponds with the local vertical or zenith direction and whose x-axis is horizontal and points toward the azimuth of the Sun. The rays of the Sun lie in or parallel to the plane XOZ. The y-axis is perpendicular to the plane XOZ (see Figure 1). The position of any point P on the leaf distribution sphere can be expressed as P(a, q), where a is the azimuthal angle between the plane POZ and the plane XOZ, and q is the vertical angle between the line OP and the 4of16 Using equations (4) and (5) to eliminate cosd and da, equation (6) can be rewritten as ds ¼ S 0 ðsin q cos y þ cos q sin y cos aþcos q dq da: ð7þ [23] It is assumed that the reflection of a leaf surface is isotropic and the reflection coefficient of the leaf surface is independent of the direction of incident radiation and is given by b L. We denote the function, f u (q), representing the upward fraction of reflected radiation, i.e., the ratio of upward reflected radiation to the total reflected radiation. Therefore, under the above assumptions, f u (q) is a function of a single variable q, the complement of the inclination of the leaf element with respect to the local horizontal, and is independent of the direction of incident radiation. Therefore the total upward reflected radiation flux from da can be expressed as dr us ¼ b L f u ðþds: q Substituting for ds in equation (8) from equation (7), one obtains dr us ¼ b L f u ðþs q 0 ðsin q cos y þ cos q sin y cos aþcos q dq da: ð9þ [24] By integrating equation (9) over the sunlit hemisphere of the unit leaf distribution sphere, we can obtain the total upward reflected radiation flux. The sunlit hemisphere on the leaf distribution sphere is the region for which d p/2 or sin q cos y + cos q sin y cos a 0. Therefore ZZ R us ¼ b L f u ðþs q 0 ðsin q cos y þ cos q sin y cos aþcos q dq da: d p 2 ð8þ ð10þ [25] Equation (10) is difficult to integrate analytically because d is a function of a, q, and y, and a and q are the real variables to be integrated. By slightly changing its form, equation (10) can be expressed identically as follows: R us ¼ 1 2 b LS 0 Z 2p a¼0 Z p 2 q¼ p 2 f u ðþcos q q dq da: j sin q cos y þ cos q sin y cos aj ð11þ

5 W08409 ZHAO AND QUALLS: MULTILAYER CANOPY SCATTERING MODEL W08409 [26] The upward reflected radiation flux R us is part of the directional radiation intercepted by a unit leaf sphere. The directional radiation intercepted by the unit leaf sphere is equivalent to the interception by an area of p perpendicular to the radiation beam, which is equivalent to the interception by a horizontal area of p/cosy. We denote the upward reflected radiation flux density as R u ; that is, R u corresponds to R us per unit area. Therefore resulting from the scattering of directional radiation from a single leaf layer: R u þ T u S ¼ 0:5ðb L þ t L Þþ0:3334ðb L t L Þcos y: ð18þ Similarly, the downward fraction of scattered radiation that originates from the scattering of directional radiation from a single leaf layer can be expressed as R u ¼ 1 p R us cos y: ð12þ R d þ T d S ¼ 0:5ðb L þ t L Þþ0:3334ðt L b L Þcos y: ð19þ [27] If the incident direct radiation flux density on the horizontal surface is denoted as S, then S ¼ S 0 cos y: Combining equations (11), (12) and (13), one obtains R u b L S ¼ 1 Z 2p Z p 2 j sin q cos y þ cos q sin y cos aj 2p a¼0 q¼ p 2 f u ðþcos q q dq da: ð13þ ð14þ 2.3. Single Scattering of Hemispherical Radiation [32] Assuming that the reflection coefficient of a leaf surface is the same on both sides, the upward and downward fractions of the radiation that originates from the scattering of hemispherical radiation from a single, spherically distributed leaf layer can be calculated from equations (20) and (21) following Norman and Jarvis [1975]: R u þ T u D ¼ 2 3 b L þ 1 3 t L ð20þ Numerical Integration of the Single Scattering of Directional Radiation [28] As inferred from the analysis in the appendix of Norman and Jarvis [1975], the upward fraction of reflected radiation f u (q) can be expressed as f u ðþ¼ q 1 h 2 1 þ cos p i 2 q : ð15þ [29] The function f u (q) in equation (14) was replaced by equation (15), and a series of numerical integrations was performed for 101 different directional radiation zenith angles, y, ranging from 0 to p/2 in increments of p/200. At each y value, q and a were integrated from (p/2) to p/2 and 0 to 2p, respectively, in increments of p/1000. The results of the numerical integration of equation (14) with equation (15) incorporated can be very accurately approximated by R u ¼ 0:5 þ 0:3334 cos y b L S ð16þ with an R-squared value of [30] Similarly, we obtained the upward fraction of transmitted directional radiation as T u ¼ 0:5 0:3334 cos y; t L S ð17þ where T u is the upward transmitted radiation flux density through the spherically distributed leaves in the layer, and t L is the transmission coefficient of the leaves. [31] Equations (16) and (17) can be added together to obtain the total upward fraction of scattered radiation, including both the reflected and transmitted components, 5of16 R d þ T d D ¼ 1 3 b L þ 2 3 t L; where D is hemispherical radiation. ð21þ 2.4. Multiple Scattering of Hemispherical Radiation [33] As noted earlier, once directional radiation is intercepted and scattered by any leaf or soil substrate, it ceases to be directional and becomes hemispherical radiation. Therefore directional radiation can only be scattered once, after which it becomes hemispherical radiation. Therefore we treat directional radiation independently from hemispherical radiation in our model until the first interception, which occurs at different depths for different rays of direction radiation. Following the interception, the scattered radiation energy is collected into the hemispherical radiation component of the model. [34] Consider the shortwave radiation exchange processes that occur between adjacent vegetation layers i and i + 1. Each layer has a penetration function t i, defined as the fraction of hemispherical shortwave radiation that penetrates through layer i without being intercepted by any leaf, relative to the total hemispherical shortwave radiation that enters layer i. We define absorptivity a i as the fraction of absorbed to total intercepted shortwave radiation within layer i. We also define backward scattering, r i, and forward scattering, (1 r i ) functions for each layer. Here r i represents the fraction of scattered radiation that is moving in an aggregate direction (i.e., upward versus downward) that is opposite to the direction it was propagating prior to being scattered. Conversely, (1 r i ) represents the fraction of scattered radiation that is moving in the same aggregate direction as it was prior to being scattered. Both of these functions may include both reflected and transmitted components. For example, the backward scattering func-

6 W08409 ZHAO AND QUALLS: MULTILAYER CANOPY SCATTERING MODEL W08409 Figure 2. Diagram of multiple scattering of shortwave radiation between vegetation layers. tion corresponding to radiation that is initially propagating downward will include radiation reflected upward off of leaves, and radiation that is transmitted through an inclined leaf and exits the underside of the leaf in an upward direction. This upward transmitted component only occurs from inclined leaves. [35] It is assumed that the reflection coefficient of leaves, b L, and the transmission coefficient of leaves, t L, are uniform throughout the canopy, taking the same value in each layer. According to the definition above, the backward scattering functions for directional and hemispherical shortwave radiation from the canopy layer i can be obtained from equations (18) and (20), respectively, as rðyþ i ¼ 0:5 þ 0:3334 b L t L cos y b L þ t L r i ¼ 2 3 b L b L þ t L þ 1 3 t L b L þ t L ð22þ : ð23þ [36] We label the downward directional shortwave radiation flux density from vegetation layer i + 1 to layer i as S i, the original downward hemispherical shortwave radiation flux density from vegetation layer i + 1 to layer i, before taking the multiple scatterings between layers i and i + 1 into account, as SWd0 i+1, and the original hemispherical upward shortwave radiation flux density from vegetation layer i to layer i + 1, before taking multiple scatterings between layers i and i + 1 into account, as SWu0 i (see Figure 2). The total downward hemispherical shortwave radiation flux density from vegetation layer i + 1 to layer i, resulting from all the interactions between layers i and i + 1, is the sum of all the downward components shown in Figure 2. This infinite series may be written as a recursion formula that converges to SWd0 iþ1 SWd iþ1 ¼ 1 r i r iþ1 ð1 a i Þð1 t i Þð1 a iþ1 Þð1 t iþ1 Þ r iþ1 ð1 a iþ1 Þð1 t iþ1 ÞSWu0 i þ 1 r i r iþ1 ð1 a i Þð1 t i Þð1 a iþ1 Þð1 t iþ1 Þ ð24þ [37] Similarly, the total upward hemispherical shortwave radiation flux density from vegetation layer i to layer i +1, after taking the multiple scatterings between layers i and i + 1 into account, converges to SWu0 i SWu i ¼ 1 r i r iþ1 ð1 a i Þð1 t i Þð1 a iþ1 Þð1 t iþ1 Þ r i ð1 a i Þð1 t i ÞSWd0 iþ1 þ 1 r i r iþ1 ð1 a i Þð1 t i Þð1 a iþ1 Þð1 t iþ1 Þ ð25þ Therefore the original downward hemispherical shortwave radiation flux density from vegetation layer i to layer i 1, before taking the multiple scatterings between layers i and i 1 into account, is SWd0 i ¼ t i SWd iþ1 þ ð1 t i Þð1 a i Þð1 r i ÞSWd iþ1 þ 1 tðyþ i ð 1 ai Þ 1 rðyþ i Si : ð26þ [38] The right-hand side of equation (26) includes three terms. The first term is the downward hemispherical shortwave radiation flux density from layer i + 1 penetrating through the gaps of canopy layer i without being intercepted within layer i. The second term is the downward hemispherical shortwave radiation flux density passing through the canopy layer i by means of leaf scattering. The third 6of16

7 W08409 ZHAO AND QUALLS: MULTILAYER CANOPY SCATTERING MODEL W08409 term is the hemispherical flux density resulting from the directional shortwave radiation that is intercepted in layer i and scattered downward by means of leaf reflection and transmission. Because of scattering by vegetation, this shortwave radiation flux changes from directional radiation into downward hemispherical radiation. [39] It is important to note that although SWd0 i, which emanates down out of layer i and enters layer i 1, does not include multiple scatterings between layer i and i 1, it is composed of the total downward radiation flux density coming from layer i + 1, including the multiple scatterings between layers i + 1 and i by virtue of the definition of the input variable SWd i+1. It also includes multiple scatterings between all other pairs of layers higher in the canopy. Substituting equation (24) into (26), we can eliminate the multiple scattering variable, SWd i+1, and retain only the original directional and hemispherical radiation flux density variables: t i þ ð1 t i Þð1 a i Þð1 r i Þ SWd0 i ¼ 1 r i r iþ1 ð1 a i Þð1 t i Þð1 a iþ1 Þð1 t iþ1 Þ SWd0 iþ1 þ r iþ1½t i þ ð1 t i Þð1 a i Þð1 rþšð1 a iþ1 Þð1 t iþ1 Þ 1 r i r iþ1 ð1 a i Þð1 t i Þð1 a iþ1 Þð1 t iþ1 Þ SWu0 i þ 1 tðyþ i ð 1 ai Þ 1 rðyþ i Si : ð27þ Equations (27) and (29) can be rearranged to yield ½1 r i r iþ1 ð1 a i Þð1 t i Þð1 a iþ1 Þð1 t iþ1 ÞŠSWd0 i r iþ1 ½t i þ ð1 t i Þð1 a i Þð1 r i ÞŠð1 a iþ1 Þð1 t iþ1 ÞSWu0 i ½t i þ ð1 t i Þð1 a i Þð1 r i ÞŠSWd0 iþ1 ¼ ½1 r i r iþ1 ð1 a i Þð1 t i Þð1 a iþ1 Þð1 t iþ1 ÞŠ 1 tðyþ i ð1 a i Þ 1 rðyþ i Si ð30þ ½t iþ1 þ ð1 t iþ1 Þð1 a iþ1 Þð1 r iþ1 ÞŠSWu0 i r i ½t iþ1 þ ð1 t iþ1 Þð1 a iþ1 Þð1 r iþ1 ÞŠð1 a i Þð1 t i Þ SWd0 iþ1 þ ½1 r i r iþ1 ð1 a i Þð1 t i Þð1 a iþ1 Þð1 t iþ1 ÞŠ SWu0 iþ1 ¼ ½1 r i r iþ1 ð1 a i Þð1 t i Þð1 a iþ1 Þð1 t iþ1 ÞŠ rðyþ iþ1 1 tðyþ iþ1 ð 1 aiþ1 ÞS iþ1 ð31þ [41] We divide the vegetation canopy above the soil surface into m layers and designate the layer immediately above the soil surface as layer 1 and the topmost layer of the canopy as layer m. The soil layer is denoted as layer 0. We use subscripts to indicate layers. The boundary conditions at the soil surface can be expressed as Similarly, the original upward hemispherical shortwave radiation flux density from vegetation layer i + 1 to layer i + 2, before taking the multiple scatterings between layers i + 1 and i + 2 into account, is t 0 ¼ 0 SWu0 0 ¼ alb 0 S 0 ð32þ ð33þ SWu0 iþ1 ¼ t iþ1 SWu i þ ð1 t iþ1 Þð1 a iþ1 Þð1 r iþ1 ÞSWu i þ rðyþ iþ1 1 tðyþ iþ1 ð 1 aiþ1 ÞS iþ1 ð28þ [40] The right-hand side of equation (28) includes three terms also. The first term is the upward hemispherical radiation flux density from layer i penetrating through the gaps of canopy layer i + 1 without being intercepted. The second term is the upward hemispherical radiation flux density scattered upward through the canopy layer i + 1 by means of leaf reflection and transmission. The third term is the flux density resulting from the directional shortwave radiation that is intercepted in layer i + 1 and scattered upward by means of reflection and transmission. Becauser of the scattering by leaves, part of the intercepted directional radiation becomes upward hemispherical shortwave radiation. As before, it is important to note that SWu0 i+1 includes all multiple scatterings among lower layers and only excludes those multiple scatterings between layers i + 1 and i + 2. Substituting equation (25) into (28), we can eliminate the multiple scattering variable and only retain the original hemispherical and directional radiation flux density variables as we did previously: t iþ1 þ ð1 t iþ1 Þð1 a iþ1 Þð1 r iþ1 Þ SWu0 iþ1 ¼ 1 r i r iþ1 ð1 a i Þð1 t i Þð1 a iþ1 Þð1 t iþ1 Þ SWu0 i þ r i½t iþ1 þ ð1 t iþ1 Þð1 a iþ1 Þð1 rþšð1 a i Þð1 t i Þ 1 r i r iþ1 ð1 a i Þð1 t i Þð1 a iþ1 Þð1 t iþ1 Þ SWd0 iþ1 þ rðyþ iþ1 1 tðyþ iþ1 ð 1 aiþ1 ÞS iþ1 : ð29þ a 0 ¼ 1 alb 0 r 0 ¼ 1; ð34þ ð35þ where S 0 is the directional radiation flux density penetrating to the soil surface and alb 0 is the albedo of the soil surface. [42] Above the topmost layer m of the canopy, an imaginary layer m + 1 is introduced to represent a thin atmospheric layer above the canopy. This atmospheric layer is transparent so that both directional and hemispherical shortwave radiation penetrate downward or upward through the layer without being intercepted. Therefore the boundary conditions for the topmost imaginary transparent layer are t mþ1 ¼ 1 r mþ1 ¼ 0 SWd mþ1 ¼ SWd0 mþ1 ¼ SW sky ; ð36þ ð37þ ð38þ where SW sky is the diffuse radiation flux density from the sky. [43] Equations (30) and (31), together with their boundary conditions (33) and (38), can be rewritten in matrix form as A SW ¼ C ð39þ 7of16

8 W08409 ZHAO AND QUALLS: MULTILAYER CANOPY SCATTERING MODEL W08409 where 2 3 SWu0 0 SWd0 1 SWu0 1.. SWd0 SW ¼ i SWu0 i. 6 SWd0 m 7 4 SWu0 m 5 SWd0 mþ1 ð40þ (25). Finally, the net shortwave radiation absorbed by each vegetation layer can be obtained by calculating the difference between incoming and outgoing directional and hemispherical shortwave radiation. [46] As the shortwave radiation is composed of PAR and NIR wave bands, and they exhibit different canopy characteristic in terms of leaf absorption and scattering [Ross, 1975], the multiple scattering of shortwave radiation will be modeled independently in PAR and NIR wave bands. Theoretically, the user may divide the radiation into as many wave bands as desired and run the model separately for each in order to incorporate wavelength-dependent 2 alb 0 S 0 ½1 r 0 r 1 ð1 a 0 Þð1 t 0 Þð1 a 1 Þð1 t 1 ÞŠrðyÞ 1 1 tðyþ 1 ð 1 a1 ÞS 1 ½1 r 1 r 2 ð1 a 1 Þð1 t 1 Þð1 a 2 Þð1 t 2 ÞŠ 1 tðyþ 1 ð 1 a1 Þ 1 rðyþ 1 S1.. 1 r C ¼ ½ i 1 r i ð1 a i 1 Þð1 t i 1 Þð1 a i Þð1 t i ÞŠrðyÞ i 1 tðyþ i ð 1 ai ÞS i ½1 r i r iþ1 ð1 a i Þð1 t i Þð1 a iþ1 Þð1 t iþ1 ÞŠ 1 tðyþ i ð 1 ai Þ 1 rðyþ i Si.. 6 ½1 r m 1 r m ð1 a m 1 Þð1 t m 1 Þð1 a m Þð1 t m ÞŠrðyÞ m 1 tðyþ m ð 1 am ÞS m 4 ½1 r m r mþ1 ð1 a m Þð1 t m Þð1 a mþ1 Þð1 t mþ1 ÞŠ 1 tðyþ m ð 1 am Þ 1 rðyþ m SW sky Sm ð41þ are both one-column matrixes with (2m + 2) rows. [44] A is a (2m +2)by(2m + 2) matrix, all of whose elements are equal to 0 except those on the main diagonal or the immediate neighbors of the main diagonal elements. The nonzero elements can be expressed as a 1;1 ¼ 1 a 2i;2i 1 ¼ t ½ i þ ð1 t i Þð1 a i Þð1 r i ÞŠ ði ¼ 1; 2; 3;...mÞ a 2i;2i ¼ r i 1 ½t i þ ð1 t i Þð1 a i Þð1 r i ÞŠð1 a i 1 Þð1 t i 1 Þ ði ¼ 1; 2; 3;...mÞ a 2i;2iþ1 ¼ ½1 r i 1 r i ð1 a i 1 Þð1 t i 1 Þð1 a i Þð1 t i ÞŠ ði ¼ 1; 2; 3;...mÞ a 2iþ1;2i ¼ ½1 r i r iþ1 ð1 a i Þð1 t i Þð1 a iþ1 Þð1 t iþ1 ÞŠ ði ¼ 1; 2; 3;...mÞ a 2iþ1;2iþ1 ¼ r iþ1 ½t i þ ð1 t i Þð1 a i Þð1 r i ÞŠð1 a iþ1 Þ ð1 t iþ1 Þ ði ¼ 1; 2; 3;...mÞ a 2iþ1;2iþ2 ¼ t ½ i þ ð1 t i Þð1 a i Þð1 r i ÞŠ ði ¼ 1; 2; 3;...mÞ a 2mþ2;2mþ2 ¼ 1 ð42þ [45] Equation (39) is composed of (2m + 2) single equations with (2m + 2) unknowns. It can be solved by elimination techniques, such as the Gauss-Jordan method. After the original downward and upward hemispherical shortwave radiation flux densities are calculated for each layer, the total hemispherical downward and upward shortwave radiation flux densities, which take the multiple scatterings between each pair of adjacent vegetation layers into account, can be calculated from equations (24) and variation in absorption, reflection, and transmission characteristics of the vegetation. In practice, dividing the model into too many wave bands may cause unnecessary computational burden. [47] The significant feature of this new method is that it accounts for multiple scattering of radiation between all layers of the canopy, yet may be solved in a single pass through the equations without iteration. This is an advantage over other methods currently in existence such as the loop method [Norman, 1979], which must be iterated to account for multiple canopy interactions. Thus our approach achieves computational economy. [48] Because the matrix A only has nonzero elements on the main diagonal and its immediate neighbors, the computational time required to solve equation (39) can be decreased further by using the so-called parallel solution of tri-diagonal systems described by Press et al. [1996]. This method reduces the problem of solving a (2m + 2)by(2m + 2) matrix equation of this kind to that of solving an (m +1) by (m + 1) or smaller matrix equation. The ability to take advantage of parallel computation makes this multiple-layer radiation model superior to other existing models, such as the loop method and the two-stream approximation, especially for regional or large applications involving large numbers of grid cells. This is especially important since our goal is to assimilate spatially distributed remote sensing data into the coupled radiation and turbulent flux model. 3. Materials and Data Acquisition 3.1. Experiment and Data Measurement [49] The field trials were conducted during the two consecutive years, 2001 and In 2001, the experiment was performed from 30 May to 16 August on farmland northeast of Moscow, Idaho. The experiment site was 8of16

9 W08409 ZHAO AND QUALLS: MULTILAYER CANOPY SCATTERING MODEL W08409 situated in a wheat field at N, W and 806 m above mean sea level (msl). The wheat field extended approximately 600 m from north to south and 500 m from east to west and was planted with an east-west row orientation. The distance between rows was 18 cm. The instrumentation was set up 120 m from the southern edge and 150 m from the western edge of the wheat field, providing at least 100 m of flat, homogeneous fetch in each direction from the instrumentation. The average vegetation height of the wheat during the experiment period progressed from about 35 cm, at the beginning of the experiment on 30 May, to a full development height of about 85 cm, on 4 July. [50] Two Eppley precision spectral pyranometers (model PSP) were set up at 2.2 m height above the ground, one of them facing up to measure global radiation flux density and the other facing down to measure reflected radiation flux density. (Manufacturer names and model numbers are provided for the convenience of the reader and do not imply affiliation with or endorsement by the authors.) A Vaisala analog barometer (model PTB101B) was used to measure atmospheric pressure at the experiment site. A Campbell Scientific CR10X data logger was used to record the data from the pyranometers and the barometer. The sensors were scanned once every 5 s and averaged every 20 min. A Li-Cor LAI-2000 plant canopy analyzer was used at six representative points around the radiation measurement area to measure leaf area index (LAI) above the soil surface. Total LAI was measured twice a week, and the mean value of the six measurements was fitted to a polynomial as a function of day of year (DOY). The LAI-2000 measures total LAI above the height of the sensor. Therefore we were able to determine the vertical profile of LAI by taking measurements with the sensor positioned in 10-cm increments from the ground surface to just below the top of the canopy. Profile measurements were carried out once per week. The results (not shown here) confirmed that the canopy was vertically uniform. [51] In 2002, the experiment was conducted from 29 May to 11 August in another wheat field immediately south of the 2001 experiment site ( N, W and 806 m above msl). The wheat field extended approximately 450 m from north to south and 350 m from east to west and was planted with a north-south row orientation. The distance between wheat rows was 18 cm. The instrumentation was set up 200 m from the southern edge and 150 m from the eastern edge of the wheat field, providing at least 150 m of flat, homogeneous fetch in every direction from the instrumentation. The average vegetation height of the wheat during the experiment period progressed from about 35 cm, at the beginning of the experiment on 29 May, to a full development height of about 85 cm, on 7 July. [52] The two Eppley pyranometers were set up at 2.2 m height to measure global and reflected radiation flux densities, as in Two tube solarimeters (model TSL, Delta-T Devices Ltd.) were used to measure solar radiation attenuation within the canopy from 8 June to 11 August. Both of the TSLs were mounted with an east-west orientation to avoid problems found by Mungai et al. [1997] to be associated with north-south oriented sensors. One was mounted above the top of the canopy as a reference; the other was placed below the canopy, immediately above the soil surface, to measure the radiation flux density penetrating to the bottom of the canopy, hereinafter referred to as subcanopy radiation. Since the wheat rows were oriented from north to south, the TSL ran perpendicular to the wheat row direction. The effective measurement length of the tube solarimeter was 85 cm; therefore measurements represented the average radiation flux density across nearly five rows. In order to limit the influence of spatial variability on our radiation measurements, the mounting location for the subcanopy TSL was selected to satisfy the following criteria: (1) homogeneous wheat canopy above and around the subcanopy TSL; and (2) representative of the growing conditions within the entire radiation measurement area. The measurements from the TSL below the canopy were not used directly. Instead, subcanopy radiation flux density was calculated as (TSL b /TSL a ) Q, where TSL a and TSL b are the TSL radiation flux densities measured above and below the canopy, respectively, and Q = S + D is the global radiation flux density measured by one of the Eppley pyranometers. [53] A Vaisala analog barometer (model PTB101B) was used to measure atmospheric pressure at the experiment site. A Campbell Scientific CR10X data logger was used to record the data from the barometer, the pyranometers, and the tube solarimeters. The sensors were scanned once every 5 s and averaged every 20 min. As in the 2001 experiment, LAI was measured twice weekly with a Li-Cor LAI-2000 plant canopy analyzer at four representative points around the radiation measurement area. LAI profile measurements were carried out as described for our 2001 experiments Estimation of Diffuse Radiation [54] In order to partition the measured global radiation into its diffuse and direct components, we estimated the diffuse radiation via atmospheric transmittance. Under clear-sky conditions, direct radiation S and diffuse radiation D, on a horizontal surface, can be estimated [Campbell and Norman, 1998] as S ¼ S po t m atm cos y ð43þ D ¼ 0:3 1 t m atm Spo cos y; ð44þ respectively, where S po is the extraterrestrial flux density of shortwave radiation normal to the solar beam, taken as 1360 W m 2, t atm is the atmospheric transmittance, and m is the optical air mass number. The optical air mass number m can be estimated from the solar zenith angle y and atmospheric pressure P a at the observation site as [Campbell and Norman, 1998] m ¼ P a P 0 cos y ; ð45þ where P 0 is the standard atmospheric pressure at sea level (=101.3 kpa). [55] Equations (43), (44), and (45) can be solved for the atmospheric transmittance: ln t atm ¼ P 0 cos y P a S þ D ln 0:7S po cos y 3 : ð46þ 7 9of16

10 W08409 ZHAO AND QUALLS: MULTILAYER CANOPY SCATTERING MODEL W08409 The atmospheric transmittance was estimated using the 20-min averages of global radiation and atmospheric pressure measurement data from cloud-free days (14 days in 2001 and 10 days in 2002) during midday periods for which the solar zenith angle y was smaller than 60. This produced an average t atm value of [56] The diffuse radiation flux density was estimated from equation (44) assuming that the atmospheric transmittance remains constant for all sky conditions. The direct radiation was then obtained by subtracting estimated diffuse radiation from measured global radiation. On overcast days, occasionally measured global radiation was smaller than the calculated diffuse radiation. Whenever this occurred, all of the measured global radiation was assumed to be diffuse radiation. [57] The simplicity of equation (44) together with our use of a constant t atm in it quite likely introduces error into our estimation of the diffuse radiation flux density. However, because the direct radiation is calculated by the difference between global and diffuse radiation, any estimation error in the diffuse component will be compensated by the direct component. Therefore the final modeling error is caused only by the different penetration and reflection characteristics between direct and diffuse radiation. When the Sun is in the lower middle altitude, between 45 and 65 degrees zenith angle, for example, the penetration and reflection characteristics of direct and diffuse radiation are very similar and the model is insensitive to errors in estimating the diffuse radiation. Only when the Sun is near the zenith, or near the horizon, are the penetration and reflection characteristics of direct and diffuse radiation different from one another. At these times, the error in estimating diffuse radiation may influence the model results. However, as the diffuse radiation itself is usually a small fraction in global radiation, the model error caused by incorrectly estimating diffuse radiation is minimal on clear days Other Parameters Used in the Model [58] Values for several model parameters were taken from published literature. In the following, all values were taken from Campbell and Norman [1998] except as noted otherwise: Shortwave radiation from the sky was taken to be composed of 50% PAR and 50% NIR; leaf absorptivity was assumed to be 80% in the PAR wave band and 20% in NIR wave band; reflection and transmission coefficients were taken to be 0.6 and 0.4, respectively, for both PAR and NIR wave bands [Ross, 1975]; and the reflectivity (albedo) of the soil was taken to be 0.11, based on data for dark soils. 4. Results and Discussion [59] The distribution of radiation flux density throughout the wheat canopy was modeled by solving the scattering equations given in matrix form by equation (39), using the measured 20-min average global radiation data, the solar zenith angle calculated from local time, day of year, the geographical location of the experiment site [Campbell and Norman, 1998], and the canopy input parameters. Our radiation data set includes downward global radiation above the canopy, reflected shortwave radiation coming off the canopy, and shortwave downward subcanopy radiation. Therefore our model validations focus on the modeled 10 of 16 subcanopy radiation and the total scattered upward shortwave radiation exiting out the top of the canopy Modeling Downward Subcanopy Radiation [60] The downward shortwave radiation, or global radiation, entering the canopy from above is composed of two components: direct radiation from the Sun and diffuse radiation from the whole hemisphere. In order to determine the minimum number of layers that the radiation scattering model requires to produce stable results, two extreme cases were investigated: In case 1, global radiation is composed entirely of diffuse radiation. This case represents overcast or cloudy conditions when direct radiation is completely blocked; in case 2, global radiation is composed entirely of direct radiation. This artificial condition represents a limiting case, which is most nearly achieved at local solar noon on clear days when the atmospheric transmittance is large. Under these conditions, the diffuse radiation occupies only a small fraction in the global radiation. Multiple model simulations were run which included permutations on LAI, solar zenith angle, and number of canopy layers. LAI was varied from 0 to 7 in increments of 0.05; for direct radiation, solar zenith angles of 0, 30, and 60 were used; and the canopy was divided into 1, 2, 3, 5, 10, 20, 50, 100, and 200 layers. [61] Figure 3 shows that for direct radiation, when LAI is small (i.e., less than 1), the number of the model layers is unimportant; the model produces approximately the same results regardless of the number of canopy layers. However, the influence of the number of modeling layers increases with increasing total LAI. As additional model layers are used, the modeled subcanopy radiation decreases. However, the rate of decrease diminishes with successively larger numbers of canopy layers. When the model includes 50 or more layers, the results converge so that for 50, 100, and 200 layers the results are indistinguishable, as shown in Figures 3a, 3b, and 3c. [62] The influence of the number of model layers is more significant with diffuse radiation than with direct radiation (Figure 3d). Even for LAI as low as 1, there is a large difference between 1 layer and, say, 50 layers. However, the modeling results stabilized again for 50 or more layers. In order to achieve more accuracy in this study, we chose to incorporate 100 model layers for both direct and diffuse radiation. However, to balance accuracy and computational economy, we believe 50 layers should be reasonable when the model is used in SVAT models for remote sensing regional computations. [63] Another obvious fact illustrated by Figure 3 is that for a given LAI, the subcanopy radiation decreases with the increasing solar zenith angle. [64] A value of LAI is required to run the model for any particular day. LAI was measured on 19 days throughout the 2002 growing season. Second-order polynomials were fitted to the LAI data as a function of day of year as LAI ¼ a 2 DoY 2 þ a 1 DoY þ a 0 ð47þ using the time variable, day of year (DOY) with 1 January as day 1, in order to interpolate between measurement days. One polynomial was fitted to the data prior to day 193, and a second was fitted to data from day 193 onward. The

11 W08409 ZHAO AND QUALLS: MULTILAYER CANOPY SCATTERING MODEL W08409 Figure 3. Modeled subcanopy radiation for wheat as a function of leaf area index (LAI) and number of modeling layers. Figures 3a 3c represent different solar zenith angles, and Figure 3d represents diffuse radiation. measured LAI and their fitted polynomials are displayed in Figure 4. For each model simulation, LAI was uniformly distributed among the 100 model layers. This distribution was justified by our in situ measurements of LAI profiles (W. Zhao and R. J. Qualls, manuscript in preparation, 2005); however, the model can easily accommodate different vertical distributions of LAI. This can be accomplished by dividing the canopy into layers of fixed increments in LAI rather than by fixed increments of thickness of the layer. [65] Figure 4 also shows a comparison of modeled daily total downward subcanopy shortwave radiation and the measurements from the tube solarimeter below the canopy. The time series runs from 8 June (DOY 159), the first day the tube solarimeters were deployed in the field, until 11 August (DOY 223), the day prior to harvest. The large values of subcanopy radiation occurring at the beginning of the measurement period no doubt result because LAI was small at that time. The rapid decline in subcanopy radiation coincides with a rapid increase in LAI. During the first 15 days after the solarimeters were deployed, the measured subcanopy radiation was higher, by 10% or less, than the modeled values. This might have been caused by the disturbance to the canopy during the installation of the solarimeters. It is more likely the result of the fact that the canopy exhibited a discontinuous row structure while LAI was small. That is, the wheat plants had not yet grown sufficiently to close the gap between adjacent rows. This deviates from the implicit model assumption that leaf elements are distributed randomly through each horizontal layer. As a result, radiation penetrates unobstructed to the soil surface between the rows, concentrating a large 11 of 16 Figure 4. Modeled and measured daily total downward subcanopy shortwave radiation and LAI for 2002.

12 W08409 ZHAO AND QUALLS: MULTILAYER CANOPY SCATTERING MODEL W08409 Figure 5. Diurnal cycle of modeled and measured downward subcanopy shortwave radiation. Figures 5a 5c represent clear days at different stages of foliage development; Figure 5d represents an intermittently cloudy day with nearly full foliage development. Rg is global radiation; Rsc is subcanopy radiation; Psc is subcanopy radiation as a percentage of global radiation. amount of radiation on portions of the TSL lying between the rows. This results in a larger measurement of subcanopy radiation than if the leaves in each layer were distributed across the row gaps. During the first 15 days, as the wheat grew and filled the row gaps, the discrepancy between measured and modeled values decreased until it was less than a few percent. Figure 4b presents a scatterplot of modeled versus measured subcanopy radiation superimposed on a line with 1:1 slope. All of the measured values of subcanopy radiation in excess of 5 MJ m 2 d 1, where the errors are largest, were collected during the first 13 days, prior to row gap closure. Even including the data with the larger errors, the root-mean-square error (RMSE) between the modeled and measured values is only MJ m 2 d 1. Consequently, the model results agree well with the measurements. [66] The four panels in Figure 5 demonstrate the diurnal cycles of the above-canopy measured global radiation, measured and modeled subcanopy radiation, and the percentage of subcanopy radiation relative to global radiation for three clear days (Figures 5a, 5b, and 5c) and a day with intermittent clouds (Figure 5d). The three clear-day figures represent a sequence of growth stages: Figure 5a, 13 June 2002 (DOY 164), represents the early, vigorous growth stage of the wheat with an LAI of 3.12; Figure 5b, 2 July (DOY 183), represents the canopy at a point in time near peak LAI; and Figure 5c, 2 August (DOY 213), represents the drying stage of the canopy, shortly before harvest. On 2 August, two thirds of the wheat leaves were yellowed, wilted, and curled. The LAI measured during this period had decreased from its midseason peak. The cloudy day, Figure 5d, was 30 June 2002 (DOY 181), which was close to the time of maximum LAI. Figure 5a corresponds to the first clear day observed during the 2002 experiment and was within the first week of the experiment. We can see that the measured subcanopy radiation was higher than the modeled values from around 1000h until 1330h. As noted regarding Figure 4, this may have been caused by the row structure of the wheat canopy during the early growing stage, when the gaps between rows were not filled in. More directional radiation reached the soil surface through the gaps between rows around solar noon, when the solar azimuth was aligned with the row direction. [67] Figure 5 shows that the model simulates the subcanopy radiation quite well both for clear days and cloudy days. The RMSE for Figures 5a, 5b, 5c, and 5d are 20.6, 7.59, 8.84, and 3.17 W m 2, respectively. In general, the diurnal cycle is accurately modeled. [68] Around noontime on clear days, the measured subcanopy radiation is slightly larger than the model results. This is most prevalent early in the measurement period (Figure 5a), although is still observable in Figures 5b and 5c as well. This is the result of the alignment of the solar azimuth with the north-south row orientation of the wheat, which occurs around the time of solar noon. This alignment produces an effect where direct radiation penetrates to the bottom of the canopy through the narrow gaps between rows more strongly than predicted by the model. The model does not capture this effect because it is based on a randomly organized spherical leaf distribution. We intend to incorporate this type of inhomogeneity into a later version of the model. 12 of 16

13 W08409 ZHAO AND QUALLS: MULTILAYER CANOPY SCATTERING MODEL W08409 Figure 6. Model results of reflected radiation from the wheat canopy as a function of LAI and number of modeling layers. Figures 6a 6c represent different solar zenith angles, and Figure 6d represents diffuse radiation. [69] Figure 5d shows results from a day with intermittent clouds. The deviation between modeled and measured subcanopy radiation does not appear on this day owing to the intermittency of the direct radiation component; diffuse solar radiation does not exhibit the row effect. [70] Prior to peak foliage development, the gaps between rows are larger. Therefore the row effect around solar noon lasts for a longer duration as exhibited in Figure 5a, in contrast to Figure 5b. As the canopy senesced, the leaves shriveled and curled slightly, so that the LAI was reduced as indicated in Figure 5c. However, the leaves still occupied much of the space between the rows, so the duration of the row effect was no more significant than when the canopy s foliage was at its peak LAI. This evidence supports our earlier conclusion that gaps between rows cause the daily aggregated measurements of subcanopy radiation flux density to exceed the modeled values. [71] Though the model validation of downward shortwave radiation is performed only at the bottom of the canopy, we believe the model is also valid within the canopy for the following three reasons: (1) Given the observed vertically uniform LAI distribution, theoretically there should be no sudden deviation in the middle of the canopy. (2) When shortwave radiation propagates downward through the canopy, modeling errors produced in any layer of the canopy accumulate. Therefore the maximum modeling error is observed at the bottom of the canopy, and comparison of model results with measurements at the bottom of the canopy provides an upper bound for errors in higher layers within the canopy. (3) The validation is performed for the whole growing season, during which time LAI changes from about 2.3 to about 4.8 (Figure 4). The validation results for smaller LAI can be viewed, in some way, as the within canopy validation for larger LAI situations Modeling Reflected Radiation [72] In this section we discuss the sensitivity analysis and validation of the model output corresponding to reflected radiation. Although our model includes numerous reflections and transmissions of radiation within individual leaf layers, we restrict our discussion in this section to the traditional concept of reflected radiation, that is, to the radiation which exits the top of the canopy in an upward direction and can be measured with a downward facing pyranometer. A comparison of modeled and measured albedo, or ratio of reflected to incoming global radiation, is also discussed. [73] The convergence tests of the model for albedo were done similarly to those for the downward subcanopy radiation presented in the preceding section. The model was run for canopies with different total LAI values ranging from 0 to 7 in increments of For each assigned total LAI value, the sensitivity of the model to the number of canopy layers was tested. These included tests for 1, 2, 5, 10, 20, 50, 100, 200, and 500 layers. Furthermore, permutations of each of the above test cases were run for four different incoming radiation scenarios: 100% direct radiation with zenith angles of 0, 30, and 60 ; and 100% diffuse radiation. [74] Sensitivity analysis results are shown in Figure 6. When LAI is 0, the albedo of the soil-vegetation system is equal to the soil albedo at the experiment site (11%). The albedo of the soil-vegetation system increases with increas- 13 of 16

14 W08409 ZHAO AND QUALLS: MULTILAYER CANOPY SCATTERING MODEL W08409 Figure 7. Modeled and measured daily total reflected radiation above a wheat field. (a) Year (b) Year (c) Modeled against measured radiation. ing LAI, because the reflectivity of a leaf is higher than that of the soil surface, for both direct and diffuse radiation. When LAI is small, for example, LAI less than 0.5, the number of modeling layers does not influence the model results significantly. However, for larger LAI values, the modeled albedo of the soil-vegetation system depends on the number of model layers. In general, the albedo decreases as the number of model layers is increased. The only exception to this trend is the single-layer model, which eliminates all scattering processes between vegetation layers, and which may produce a lower albedo at higher LAI in the single-layer model than in the two-layer and fivelayer models. However, as the number of model layers increases, there is a lower limit to the decrease of the modeled albedo. When the number of layers exceeds 100, the modeled albedo essentially converges to an equilibrium state, and does not change with the addition of more layers. As with subcanopy radiation, using 100 canopy layers in the model balances accuracy and computational economy. [75] In contrast to downward subcanopy radiation, the number of model layers has a more significant effect on the albedo corresponding to direct radiation than on the albedo of diffuse radiation. For example, when LAI = 3, the difference between the albedos of zero-degree zenith angle direct radiation for 1 layer and 100 layers is about 7% (Figure 6a). However, the corresponding difference in the albedos of diffuse radiation is only about 3% (Figure 6d). Among the direct radiation cases, the influence of the number of canopy layers decreases with increasing solar zenith angle. Considering only the stabilized model results for albedo, i.e., the results for 100 layers, at the same LAI, the modeled albedo for direct radiation increases with increasing solar zenith angle and the modeled albedo for diffuse radiation is very close to the albedo for direct radiation at a 60 solar zenith angle. [76] The time series of reflected radiation of the soilvegetation system was modeled with 100 canopy layers by solving the matrix equation (39) for both the 2001 and 2002 growing seasons. The model input values were measured 20-min average global radiation data, total LAI interpolated in time from equation (47), and divided equally into 100 sublayers, and the solar zenith angle calculated from local time, season, and geographical location of the experiment site [Campbell and Norman, 1998]. [77] Figure 7 shows a comparison of modeled daily total reflected radiation above the soil-vegetation system and the measurements from the downward facing Pyranometers in year 2001 (Figure 7a) and year 2002 (Figure 7b). In the year 2001, the measurements of reflected radiation were collected from 30 May (DOY 150) to 15 August (DOY 227), except for DOY 155 when the system was down. In the year 2002, the measurements of reflected radiation were collected from 30 May (DOY 150) to 11 August (DOY 223). Both years include the main development stage and the seasonal variation of LAI of wheat after the pesticides were sprayed. Figure 7c provides a comparison of measured and modeled reflected radiation by means of a scatterplot. The overall RMSE between measured and modeled values is about 0.2 MJ m 2, which is satisfactory. [78] Figure 8 compares the diurnal cycles of both measured and modeled reflected radiation and the corresponding albedos for three clear days and an intermittently cloudy day in The four days presented in Figure 8 are the same as those displayed in Figure 5 which include a range of LAI values from different wheat growth stages. [79] From the Figure 8 we can see that the model simulates the upward reflected radiation above the canopy quite well both on clear and cloudy days. The RMS errors in Figures 8a, 8b, and 8c are about 5.5 W m 2, and in Figure 8d it is 3.7 W m 2. The modeled values depict the general shape of the measured diurnal cycles: For albedo, both exhibit concave upward shapes; both reach minimum albedos around solar noon and increase toward early morning and late afternoon; both albedos exhibit a slight downturn near sunrise and sunset on all days with the exception of the measured data near sunrise in Figure 8c. These are consistent with previous studies [e.g., Ross, 1975; Sellers, 1985]. The concave upward shape of the diurnal cycle of albedo is more evident on clear days than on cloudy days, although it is present in both. [80] The concave upward albedos may be partially attributed to the fact that early and late in the day, the solar zenith angle is large because the Sun is low on the horizon. This influences the albedo in two ways. First, when the Sun is low on the horizon, more of the direct radiation is intercepted in the upper canopy layers. Consequently, more 14 of 16

15 W08409 ZHAO AND QUALLS: MULTILAYER CANOPY SCATTERING MODEL W08409 Figure 8. Diurnal cycle of modeled and measured reflected shortwave radiation above a wheat field for the same days and conditions as in Figure 5. Rg is global radiation; Ru is upward reflected radiation; Alb is albedo. 15 of 16 scattering occurs near the top of the canopy, and more radiation ends up being reflected out of the canopy. This phenomenon is captured both by the measurements and by the model, as evidenced by the concave upward shape of the albedo. Second, when the Sun is low on the horizon, the direct radiation impinges upon a canopy at very oblique angles. Under these circumstances, the reflectance of the leaves is usually increased. Since the leaves are assumed to be spherically distributed, there are leaves oriented obliquely to the Sun s rays for any zenith angle and this enhanced leaf reflectance occurs at all zenith angles. However, when the Sun is high above the horizon, the additional reflected radiation propagates downward into the canopy. When the Sun is low on the horizon, the additional reflected radiation is reflected off of horizontal or near-horizontal leaves and is immediately scattered out of the canopy, where it is measured by the pyranometers. In the model, we assume reflectances and transmittances for individual leaves are constant regardless of the angle of incidence. Consequently, the model fails to capture this latter increased reflectance. As a result, the measured and modeled albedos tend to diverge from one another early and late in the day, especially on clear days. However, this divergence is insignificant because it occurs at times of day when the global radiation is quite small. Therefore the model error with respect to reflected radiation is quite small as well. This can be seen from the close match of the asterisks and the solid line (Figure 8). [81] On the intermittently cloudy day 181 from 0900h to 1100h PST, when the reflected radiation itself is low, it seems that a deviation appeared between the modeled and measured albedo. The maximum difference between modeled and measured albedos during this period is 1.8%, which corresponds to an absolute simulation error of reflected radiation of only 6 W m 2. That might have been caused by measurement error. Generally speaking, the model provides very good predictions of reflected radiation with absolute simulation errors of less than 10 W m Conclusions [82] A multiple-scattering shortwave radiation model was derived mathematically that incorporated processes of radiation penetration through gaps between leaves, and radiation absorption, reflection, and transmission in leaf layers. The model is able to simulate the shortwave radiation distribution within the canopy, as well as the upward reflected radiation and downward subcanopy radiation. Good agreement was obtained between model results and measurements for downward subcanopy radiation and reflected radiation above the canopy, for both daily total values and diurnal cycles of the 20-min averages. The model can easily compute the radiation distribution of distinct wavelength bands such as PAR and NIR, so that they may be separated out for photosynthesis and biomass accumulation models. [83] The ability to assimilate remotely sensed surface temperatures into SVAT models, and thereby simulate spatially distributed energy fluxes across the landscape, including evapotranspiration, would harness the tremendous spatial information content of remote sensing. However, the vertical variation of canopy temperature profiles causes remotely sensed surface temperatures to exhibit a significant view angle effect, and induces variability in scalar roughness lengths for heat transfer. This hindrance may be overcome through the ability to model canopy skin temper-