Kubelka–Munk equations for multispecies canopies

H. Sinoquet et al. Agricultural and Forest Meteorology 101 2000 251–263 253 multiple species canopies. Multilayer models, how- ever, need many more input parameters i.e. LAI of each component in each layer than of Wallace 1997 i.e. total LAI and height of each compo- nent. The first objective of this paper was to propose a simple light partitioning model for horizontally homo- geneous canopies, where parameters could be explic- itly related to canopy structure and optical properties of the leaves and the soil surface. Such a model would be useful as a sub-model in growth simulation mod- els. For this purpose, we derived the KM equations Kubelka and Munk, 1931 for the case of a mixture of N vegetation components, and we simplified the solu- tion in order to obtain simple equations for light par- titioning. As a test, we compared the simplified KM equations KMS to the detailed model of Sinoquet et al. 1990 on contrasting canopy structures. The second objective was to assess the requirements in structure parameters for an accurate estimation of light partitioning. For this, the detailed and simplified models were run on mono and multilayer canopies, and compared with the model of Wallace 1997.

2. Kubelka–Munk equations for multispecies canopies

2.1. Kubelka–Munk equations in a multispecies monolayer canopy Equations were derived according to Bonhomme and Varlet-Grancher 1977. The differences with the present model are the presence of N components in the vegetation layer and taking into account only two hemispherical fluxes, i.e. downwards and upwards. The latter means that the model could only apply to overcast sky conditions. However model simula- tions have shown that daily interception is correctly estimated by the interception computed in overcast conditions, both in monocrops Varlet-Grancher and Bonhomme, 1979 and mixtures Sinoquet et al., 1990. The model is therefore aimed at estimating light partitioning at a daily scale, as required by growth analysis methods and most simulation models. KM equations are based on the balance of transmit- ted T and ascendant A radiation of an infinitesimal dimensionless vegetation layer dz dT = − T N X i=1 K i L i dz + A N X i=1 σ i 2 K i L i dz + T N X i=1 σ i 2 K i L i dz 5a dA = − A N X i=1 K i L i dz + A N X i=1 σ i 2 K i L i dz + T N X i=1 σ i 2 K i L i dz 5b where σ i is the scattering coefficient of leaves of species i, i.e. the sum of leaf reflectance and transmit- tance e.g. Goudriaan, 1977. The first term of right member of Eq. 5 accounts for radiation interception while the two last terms describe scattering of radia- tion interception within the small layer dz. Eq. 5 as- sumes that radiation is equally scattered upwards and downwards. Extinction coefficients K i can explicitly be related to foliage inclination θ i of species i, by using values computed by Bonhomme and Varlet-Grancher 1977 see Table 3, p. 572 in the case of a standard overcast sky SOC, Moon and Spencer, 1942. Bon- homme and Varlet-Grancher’s extinction coefficients were related to leaf inclination as follows K = 0.988 cos θ 2 2.4 r 2 = 0.997 6 Maximum deviation between tabulated values and those computed from Eq. 6 was 2.8. Kubelka and Munk 1931 gave the solution to Eq. 5 in the case of a turbid medium including only one compo- nent see Bonhomme and Varlet-Grancher, 1977, for details. In the case of a mixture of N components, similar considerations can be used and lead to T z = C 1 α exp K ′ z + C 2 β exp −K ′ z 7a A z = C 1 β exp K ′ z + C 2 α exp −K ′ z 7b where K ′ = v u u t N X i=1 K i L i N X i=1 K i L i 1 − σ i 8 254 H. Sinoquet et al. Agricultural and Forest Meteorology 101 2000 251–263 α = N X i=1 K i · L i − K ′ and β = N X i=1 K i · L i + K ′ 9 Constants C 1 and C 2 have to be derived from the boundary conditions, i.e.At the top of the canopy z=0, T 0 = 1 10a At the ground level z=1, A 1 = ρ s T 1 10b where ρ s is the soil reflectance. Rewriting Eq. 10 from expressions of T and A given in Eq. 7 leads to a system of two equations where C 1 and C 2 are the unknowns. Solving the system results in C 1 = −exp −K ′ α − βρ s 1 11a C 2 = exp K ′ β − αρ s 1 11b where 1 = β 2 exp K ′ − αβρ s exp K ′ − exp −K ′ −α 2 exp −K ′ 12 Transmitted and reflected fluxes i.e. Tz and Az, respectively may finally be written from Eq. 7 where parameters are given by Eqs. 8, 9, 11a, 11b and 12. In particular, canopy transmittance T c and reflectance ρ c are T c = β + α β − α β 2 exp K ′ − αβρ s exp K ′ − exp −K ′ −α 2 exp −K ′ 13 ρ c = β 2 ρ s exp −K ′ − αβ exp −K ′ − exp K ′ −α 2 ρ s exp K ′ β 2 exp K ′ − αβρ s exp K ′ − exp −K ′ −α 2 exp −K ′ 14 In case of large LAI, the coefficient K ′ tends towards 0 and the canopy reflectance becomes that of an infinite canopy ρ ∞ ρ ∞ = α β 15 Canopy transmittance T c and reflectance ρ c can thus be rewritten from Eqs. 13 and 14 in terms of ρ ∞ , ρ s and K ′ i.e. by dividing both numerator and de- nominator of Eq. 14 by β 2 expK ′ T c = 1 + ρ ∞ 1 − ρ ∞ exp −K ′ 1 − ρ ∞ ρ s + ρ ∞ ρ s − ρ ∞ exp −2K ′ 16 ρ c = ρ ∞ 1 − ρ ∞ ρ s + ρ s − ρ ∞ exp −2K ′ 1 − ρ ∞ ρ s + ρ ∞ ρ s − ρ ∞ exp −2K ′ 17 Eq. 17 makes clear that canopy reflectance ρ c ranges from ρ s to ρ ∞ according to canopy development. Light absorption efficiency of the whole canopy is then e.g. Varlet Grancher et al., 1989 ε = 1 − ρ c − T c 1 − ρ s 18 It can be seen from Eq. 5 that light is absorbed by component i in proportion of K i L i 1 – σ i , hence the light absorption efficiency of component i is ε i = K i L i 1 − σ i P N j =1 K j L j 1 − σ j ε 19 2.2. Simplification of Kubelka–Munk equations KMS in a multispecies monolayer canopy For dense enough canopies, neglecting exp−K ′ with regard to expK ′ is an approximation which allows computations to be simplified. Eq. 11 then become C 1 = 0 and C 2 = 1 β 20 Downward and upward fluxes may then be written from Eq. 7 as T z = exp −K ′ z 21a A z = α β = ρ ∞ 21b If the canopy includes only one component i.e. N=1, Eq. 21 are the very same as the equations of Goudri- aan 1977. In particular, the apparent extinction coef- ficient K ′ = K √ 1 − σ see Goudriaan, 1977 p. 26 can be retrieved from Eqs. 21a and 8. Eq. 21 is thus an extension of Goudriaan’s equations to the case of a mixture of N components. When applied to reflected radiation, the simplification leads to the reflectance of an infinite canopy ρ ∞ . H. Sinoquet et al. Agricultural and Forest Meteorology 101 2000 251–263 255 2.3. Simplified Kubelka–Munk equations KMS in a multispecies multilayer canopy The multilayer canopy is described in terms of LAI L ik and mean leaf inclination θ ik of each vegeta- tion component i in each layer k k=1, . . . , M. Light transmission is computed from the top to the bottom of the canopy by successive applications of Eq. 21a. Light transmitted at the bottom of layer k is thus T k = k Y l=1 exp −K ′ l 22 Upward fluxes at the layer boundaries are then re- cursively computed from the bottom to the top of the canopy by successive applications of Eq. 17. At ground level i.e. the bottom of layer M, A M = ρ s T M 23 For other layers k, A k = T k ρ ∞k 1 − ρ ∞k A k+1 T k+1 + A k+1 T k+1 − ρ ∞k exp −2K ′ k 1 − ρ ∞k A k+1 T k+1 + ρ ∞k A k+1 T k+1 − ρ ∞k exp −2K ′ k 24 Eq. 24 is similar to Eq. 17 where incident radia- tion is T k and soil reflectance ρ s is replaced by the ratio A k +1 T k +1 , i.e. equivalent to reflectance at the bottom of layer k+1. Notice that the reflectance of the multilayer canopy can be computed from Eq. 24 with k=0. Light absorption by vegetation layers and light partitioning between vegetation components are then computed from fluxes T k and A k by using relation- ships similar to Eqs. 18 and 19.

3. Multispecies canopies for model application