## 2.3.1. Canopy Radiative Transfer[¶](#canopy-radiative-transfer "Permalink to this headline")
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Radiative transfer within vegetative canopies is calculated from the two-stream approximation of [Dickinson (1983)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#dickinson1983) and [Sellers (1985)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#sellers1985) as described by [Bonan (1996)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#bonan1996)

(2.3.1)[¶](#equation-3-1 "Permalink to this equation")\\\[-\\bar{\\mu }\\frac{dI\\, \\uparrow }{d\\left(L+S\\right)} +\\left\[1-\\left(1-\\beta \\right)\\omega \\right\]I\\, \\uparrow -\\omega \\beta I\\, \\downarrow =\\omega \\bar{\\mu }K\\beta \_{0} e^{-K\\left(L+S\\right)}\\\]

(2.3.2)[¶](#equation-3-2 "Permalink to this equation")\\\[\\bar{\\mu }\\frac{dI\\, \\downarrow }{d\\left(L+S\\right)} +\\left\[1-\\left(1-\\beta \\right)\\omega \\right\]I\\, \\downarrow -\\omega \\beta I\\, \\uparrow =\\omega \\bar{\\mu }K\\left(1-\\beta \_{0} \\right)e^{-K\\left(L+S\\right)}\\\]

where \\(I\\, \\uparrow\\) and \\(I\\, \\downarrow\\) are the upward and downward diffuse radiative fluxes per unit incident flux, \\(K={G\\left(\\mu \\right)\\mathord{\\left/ {\\vphantom {G\\left(\\mu \\right) \\mu }} \\right.} \\mu }\\) is the optical depth of direct beam per unit leaf and stem area, \\(\\mu\\) is the cosine of the zenith angle of the incident beam, \\(G\\left(\\mu \\right)\\) is the relative projected area of leaf and stem elements in the direction \\(\\cos ^{-1} \\mu\\), \\(\\bar{\\mu }\\) is the average inverse diffuse optical depth per unit leaf and stem area, \\(\\omega\\) is a scattering coefficient, \\(\\beta\\) and \\(\\beta \_{0}\\) are upscatter parameters for diffuse and direct beam radiation, respectively, \\(L\\) is the exposed leaf area index, and \\(S\\) is the exposed stem area index (section [2.2.1.4](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Ecosystem/CLM50_Tech_Note_Ecosystem.html#phenology-and-vegetation-burial-by-snow)). Given the direct beam albedo \\(\\alpha \_{g,\\, \\Lambda }^{\\mu }\\) and diffuse albedo \\(\\alpha \_{g,\\, \\Lambda }\\) of the ground (section [2.3.2](#ground-albedos)), these equations are solved to calculate the fluxes, per unit incident flux, absorbed by the vegetation, reflected by the vegetation, and transmitted through the vegetation for direct and diffuse radiation and for visible (\\(<\\) 0.7\\(\\mu {\\rm m}\\)) and near-infrared (\\(\\geq\\) 0.7\\(\\mu {\\rm m}\\)) wavebands. The absorbed radiation is partitioned to sunlit and shaded fractions of the canopy. The optical parameters \\(G\\left(\\mu \\right)\\), \\(\\bar{\\mu }\\), \\(\\omega\\), \\(\\beta\\), and \\(\\beta \_{0}\\) are calculated based on work in [Sellers (1985)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#sellers1985) as follows.

The relative projected area of leaves and stems in the direction \\(\\cos ^{-1} \\mu\\) is

(2.3.3)[¶](#equation-3-3 "Permalink to this equation")\\\[G\\left(\\mu \\right)=\\phi \_{1} +\\phi \_{2} \\mu\\\]

where \\(\\phi \_{1} ={\\rm 0.5}-0.633\\chi \_{L} -0.33\\chi \_{L}^{2}\\) and \\(\\phi \_{2} =0.877\\left(1-2\\phi \_{1} \\right)\\) for \\(-0.4\\le \\chi \_{L} \\le 0.6\\). \\(\\chi \_{L}\\) is the departure of leaf angles from a random distribution and equals +1 for horizontal leaves, 0 for random leaves, and –1 for vertical leaves.

The average inverse diffuse optical depth per unit leaf and stem area is

(2.3.4)[¶](#equation-3-4 "Permalink to this equation")\\\[\\bar{\\mu }=\\int \_{0}^{1}\\frac{\\mu '}{G\\left(\\mu '\\right)} d\\mu '=\\frac{1}{\\phi \_{2} } \\left\[1-\\frac{\\phi \_{1} }{\\phi \_{2} } \\ln \\left(\\frac{\\phi \_{1} +\\phi \_{2} }{\\phi \_{1} } \\right)\\right\]\\\]

where \\(\\mu '\\) is the direction of the scattered flux.

The optical parameters \\(\\omega\\), \\(\\beta\\), and \\(\\beta \_{0}\\), which vary with wavelength (\\(\\Lambda\\) ), are weighted combinations of values for vegetation and snow, using the canopy snow-covered fraction \\(f\_{can,\\, sno}\\) (Chapter [2.7](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Hydrology/CLM50_Tech_Note_Hydrology.html#rst-hydrology)). The optical parameters are

(2.3.5)[¶](#equation-3-5 "Permalink to this equation")\\\[\\omega \_{\\Lambda } =\\omega \_{\\Lambda }^{veg} \\left(1-f\_{can,\\, sno} \\right)+\\omega \_{\\Lambda }^{sno} f\_{can,\\, sno}\\\]

(2.3.6)[¶](#equation-3-6 "Permalink to this equation")\\\[\\omega \_{\\Lambda } \\beta \_{\\Lambda } =\\omega \_{\\Lambda }^{veg} \\beta \_{\\Lambda }^{veg} \\left(1-f\_{can,\\, sno} \\right)+\\omega \_{\\Lambda }^{sno} \\beta \_{\\Lambda }^{sno} f\_{can,\\, sno}\\\]

(2.3.7)[¶](#equation-3-7 "Permalink to this equation")\\\[\\omega \_{\\Lambda } \\beta \_{0,\\, \\Lambda } =\\omega \_{\\Lambda }^{veg} \\beta \_{0,\\, \\Lambda }^{veg} \\left(1-f\_{can,\\, sno} \\right)+\\omega \_{\\Lambda }^{sno} \\beta \_{0,\\, \\Lambda }^{sno} f\_{can,\\, sno}\\\]

The snow and vegetation weights are applied to the products \\(\\omega \_{\\Lambda } \\beta \_{\\Lambda }\\) and \\(\\omega \_{\\Lambda } \\beta \_{0,\\, \\Lambda }\\) because these products are used in the two-stream equations. If there is no snow on the canopy, this reduces to

(2.3.8)[¶](#equation-3-8 "Permalink to this equation")\\\[\\omega \_{\\Lambda } =\\omega \_{\\Lambda }^{veg}\\\]

(2.3.9)[¶](#equation-3-9 "Permalink to this equation")\\\[\\omega \_{\\Lambda } \\beta \_{\\Lambda } =\\omega \_{\\Lambda }^{veg} \\beta \_{\\Lambda }^{veg}\\\]

(2.3.10)[¶](#equation-3-10 "Permalink to this equation")\\\[\\omega \_{\\Lambda } \\beta \_{0,\\, \\Lambda } =\\omega \_{\\Lambda }^{veg} \\beta \_{0,\\, \\Lambda }^{veg} .\\\]

For vegetation, \\(\\omega \_{\\Lambda }^{veg} =\\alpha \_{\\Lambda } +\\tau \_{\\Lambda }\\). \\(\\alpha \_{\\Lambda }\\) is a weighted combination of the leaf and stem reflectances (\\(\\alpha \_{\\Lambda }^{leaf},\\alpha \_{\\Lambda }^{stem}\\) )

(2.3.11)[¶](#equation-3-11 "Permalink to this equation")\\\[\\alpha \_{\\Lambda } =\\alpha \_{\\Lambda }^{leaf} w\_{leaf} +\\alpha \_{\\Lambda }^{stem} w\_{stem}\\\]

where \\(w\_{leaf} ={L\\mathord{\\left/ {\\vphantom {L \\left(L+S\\right)}} \\right.} \\left(L+S\\right)}\\) and \\(w\_{stem} ={S\\mathord{\\left/ {\\vphantom {S \\left(L+S\\right)}} \\right.} \\left(L+S\\right)}\\). \\(\\tau \_{\\Lambda }\\) is a weighted combination of the leaf and stem transmittances (\\(\\tau \_{\\Lambda }^{leaf}, \\tau \_{\\Lambda }^{stem}\\))

(2.3.12)[¶](#equation-3-12 "Permalink to this equation")\\\[\\tau \_{\\Lambda } =\\tau \_{\\Lambda }^{leaf} w\_{leaf} +\\tau \_{\\Lambda }^{stem} w\_{stem} .\\\]

The upscatter for diffuse radiation is

(2.3.13)[¶](#equation-3-13 "Permalink to this equation")\\\[\\omega \_{\\Lambda }^{veg} \\beta \_{\\Lambda }^{veg} =\\frac{1}{2} \\left\[\\alpha \_{\\Lambda } +\\tau \_{\\Lambda } +\\left(\\alpha \_{\\Lambda } -\\tau \_{\\Lambda } \\right)\\cos ^{2} \\bar{\\theta }\\right\]\\\]

where \\(\\bar{\\theta }\\) is the mean leaf inclination angle relative to the horizontal plane (i.e., the angle between leaf normal and local vertical) ([Sellers (1985)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#sellers1985)). Here, \\(\\cos \\bar{\\theta }\\) is approximated by

(2.3.14)[¶](#equation-3-14 "Permalink to this equation")\\\[\\cos \\bar{\\theta }=\\frac{1+\\chi \_{L} }{2}\\\]

Using this approximation, for vertical leaves (\\(\\chi \_{L} =-1\\), \\(\\bar{\\theta }=90^{{\\rm o}}\\) ), \\(\\omega \_{\\Lambda }^{veg} \\beta \_{\\Lambda }^{veg} =0.5\\left(\\alpha \_{\\Lambda } +\\tau \_{\\Lambda } \\right)\\), and for horizontal leaves (\\(\\chi \_{L} =1\\), \\(\\bar{\\theta }=0^{{\\rm o}}\\) ), \\(\\omega \_{\\Lambda }^{veg} \\beta \_{\\Lambda }^{veg} =\\alpha \_{\\Lambda }\\), which agree with both [Dickinson (1983)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#dickinson1983) and [Sellers (1985)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#sellers1985). For random (spherically distributed) leaves (\\(\\chi \_{L} =0\\), \\(\\bar{\\theta }=60^{{\\rm o}}\\) ), the approximation yields \\(\\omega \_{\\Lambda }^{veg} \\beta \_{\\Lambda }^{veg} ={5\\mathord{\\left/ {\\vphantom {5 8}} \\right.} 8} \\alpha \_{\\Lambda } +{3\\mathord{\\left/ {\\vphantom {3 8}} \\right.} 8} \\tau \_{\\Lambda }\\) whereas the approximate solution of [Dickinson (1983)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#dickinson1983) is \\(\\omega \_{\\Lambda }^{veg} \\beta \_{\\Lambda }^{veg} ={2\\mathord{\\left/ {\\vphantom {2 3}} \\right.} 3} \\alpha \_{\\Lambda } +{1\\mathord{\\left/ {\\vphantom {1 3}} \\right.} 3} \\tau \_{\\Lambda }\\). This discrepancy arises from the fact that a spherical leaf angle distribution has a true mean leaf inclination \\(\\bar{\\theta }\\approx 57\\) [(Campbell and Norman 1998)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#campbellnorman1998) in equation [(2.3.13)](#equation-3-13), while \\(\\bar{\\theta }=60\\) in equation [(2.3.14)](#equation-3-14). The upscatter for direct beam radiation is

(2.3.15)[¶](#equation-3-15 "Permalink to this equation")\\\[\\omega \_{\\Lambda }^{veg} \\beta \_{0,\\, \\Lambda }^{veg} =\\frac{1+\\bar{\\mu }K}{\\bar{\\mu }K} a\_{s} \\left(\\mu \\right)\_{\\Lambda }\\\]

where the single scattering albedo is

(2.3.16)[¶](#equation-3-16 "Permalink to this equation")\\\[\\begin{split}\\begin{array}{rcl} {a\_{s} \\left(\\mu \\right)\_{\\Lambda } } & {=} & {\\frac{\\omega \_{\\Lambda }^{veg} }{2} \\int \_{0}^{1}\\frac{\\mu 'G\\left(\\mu \\right)}{\\mu G\\left(\\mu '\\right)+\\mu 'G\\left(\\mu \\right)} d\\mu '} \\\\ {} & {=} & {\\frac{\\omega \_{\\Lambda }^{veg} }{2} \\frac{G\\left(\\mu \\right)}{\\max (\\mu \\phi \_{2} +G\\left(\\mu \\right),1e-6)} \\left\[1-\\frac{\\mu \\phi \_{1} }{\\max (\\mu \\phi \_{2} +G\\left(\\mu \\right),1e-6)} \\ln \\left(\\frac{\\mu \\phi \_{1} +\\max (\\mu \\phi \_{2} +G\\left(\\mu \\right),1e-6)}{\\mu \\phi \_{1} } \\right)\\right\].} \\end{array}\\end{split}\\\]

Note here the restriction on \\(\\mu \\phi \_{2} +G\\left(\\mu \\right)\\). We have seen cases where small values can cause unrealistic single scattering albedo associated with the log calculation, thereby eventually causing a negative soil albedo.

The upward diffuse fluxes per unit incident direct beam and diffuse flux (i.e., the surface albedos) are

(2.3.17)[¶](#equation-3-17 "Permalink to this equation")\\\[I\\, \\uparrow \_{\\Lambda }^{\\mu } =\\frac{h\_{1} }{\\sigma } +h\_{2} +h\_{3}\\\]

(2.3.18)[¶](#equation-3-18 "Permalink to this equation")\\\[I\\, \\uparrow \_{\\Lambda } =h\_{7} +h\_{8} .\\\]

The downward diffuse fluxes per unit incident direct beam and diffuse radiation, respectively, are

(2.3.19)[¶](#equation-3-19 "Permalink to this equation")\\\[I\\, \\downarrow \_{\\Lambda }^{\\mu } =\\frac{h\_{4} }{\\sigma } e^{-K\\left(L+S\\right)} +h\_{5} s\_{1} +\\frac{h\_{6} }{s\_{1} }\\\]

(2.3.20)[¶](#equation-3-20 "Permalink to this equation")\\\[I\\, \\downarrow \_{\\Lambda } =h\_{9} s\_{1} +\\frac{h\_{10} }{s\_{1} } .\\\]

With reference to [Figure 2.4.1](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/Radiative_Fluxes/CLM50_Tech_Note_Radiative_Fluxes.html#figure-radiation-schematic), the direct beam flux transmitted through the canopy, per unit incident flux, is \\(e^{-K\\left(L+S\\right)}\\), and the direct beam and diffuse fluxes absorbed by the vegetation, per unit incident flux, are

(2.3.21)[¶](#equation-3-21 "Permalink to this equation")\\\[\\vec{I}\_{\\Lambda }^{\\mu } =1-I\\, \\uparrow \_{\\Lambda }^{\\mu } -\\left(1-\\alpha \_{g,\\, \\Lambda } \\right)I\\, \\downarrow \_{\\Lambda }^{\\mu } -\\left(1-\\alpha \_{g,\\, \\Lambda }^{\\mu } \\right)e^{-K\\left(L+S\\right)}\\\]

(2.3.22)[¶](#equation-3-22 "Permalink to this equation")\\\[\\vec{I}\_{\\Lambda } =1-I\\, \\uparrow \_{\\Lambda } -\\left(1-\\alpha \_{g,\\, \\Lambda } \\right)I\\, \\downarrow \_{\\Lambda } .\\\]

These fluxes are partitioned to the sunlit and shaded canopy using an analytical solution to the two-stream approximation for sunlit and shaded leaves [(Dai et al. 2004)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#daietal2004), as described by [Bonan et al. (2011)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#bonanetal2011). The absorption of direct beam radiation by sunlit leaves is

(2.3.23)[¶](#equation-3-23 "Permalink to this equation")\\\[\\vec{I}\_{sun,\\Lambda }^{\\mu } =\\left(1-\\omega \_{\\Lambda } \\right)\\left\[1-s\_{2} +\\frac{1}{\\bar{\\mu }} \\left(a\_{1} +a\_{2} \\right)\\right\]\\\]

and for shaded leaves is

(2.3.24)[¶](#equation-3-24 "Permalink to this equation")\\\[\\vec{I}\_{sha,\\Lambda }^{\\mu } =\\vec{I}\_{\\Lambda }^{\\mu } -\\vec{I}\_{sun,\\Lambda }^{\\mu }\\\]

with

(2.3.25)[¶](#equation-3-25 "Permalink to this equation")\\\[a\_{1} =\\frac{h\_{1} }{\\sigma } \\left\[\\frac{1-s\_{2}^{2} }{2K} \\right\]+h\_{2} \\left\[\\frac{1-s\_{2} s\_{1} }{K+h} \\right\]+h\_{3} \\left\[\\frac{1-{s\_{2} \\mathord{\\left/ {\\vphantom {s\_{2} s\_{1} }} \\right.} s\_{1} } }{K-h} \\right\]\\\]

(2.3.26)[¶](#equation-3-26 "Permalink to this equation")\\\[a\_{2} =\\frac{h\_{4} }{\\sigma } \\left\[\\frac{1-s\_{2}^{2} }{2K} \\right\]+h\_{5} \\left\[\\frac{1-s\_{2} s\_{1} }{K+h} \\right\]+h\_{6} \\left\[\\frac{1-{s\_{2} \\mathord{\\left/ {\\vphantom {s\_{2} s\_{1} }} \\right.} s\_{1} } }{K-h} \\right\].\\\]

For diffuse radiation, the absorbed radiation for sunlit leaves is

(2.3.27)[¶](#equation-3-27 "Permalink to this equation")\\\[\\vec{I}\_{sun,\\Lambda }^{} =\\left\[\\frac{1-\\omega \_{\\Lambda } }{\\bar{\\mu }} \\right\]\\left(a\_{1} +a\_{2} \\right)\\\]

and for shaded leaves is

(2.3.28)[¶](#equation-3-28 "Permalink to this equation")\\\[\\vec{I}\_{sha,\\Lambda }^{} =\\vec{I}\_{\\Lambda }^{} -\\vec{I}\_{sun,\\Lambda }^{}\\\]

with

(2.3.29)[¶](#equation-3-29 "Permalink to this equation")\\\[a\_{1} =h\_{7} \\left\[\\frac{1-s\_{2} s\_{1} }{K+h} \\right\]+h\_{8} \\left\[\\frac{1-{s\_{2} \\mathord{\\left/ {\\vphantom {s\_{2} s\_{1} }} \\right.} s\_{1} } }{K-h} \\right\]\\\]

(2.3.30)[¶](#equation-3-30 "Permalink to this equation")\\\[a\_{2} =h\_{9} \\left\[\\frac{1-s\_{2} s\_{1} }{K+h} \\right\]+h\_{10} \\left\[\\frac{1-{s\_{2} \\mathord{\\left/ {\\vphantom {s\_{2} s\_{1} }} \\right.} s\_{1} } }{K-h} \\right\].\\\]

The parameters \\(h\_{1}\\) –\\(h\_{10}\\), \\(\\sigma\\), \\(h\\), \\(s\_{1}\\), and \\(s\_{2}\\) are from [Sellers (1985)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#sellers1985) \[note the error in \\(h\_{4}\\) in [Sellers (1985)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#sellers1985)\]:

(2.3.31)[¶](#equation-3-31 "Permalink to this equation")\\\[b=1-\\omega \_{\\Lambda } +\\omega \_{\\Lambda } \\beta \_{\\Lambda }\\\]

(2.3.32)[¶](#equation-3-32 "Permalink to this equation")\\\[c=\\omega \_{\\Lambda } \\beta \_{\\Lambda }\\\]

(2.3.33)[¶](#equation-3-33 "Permalink to this equation")\\\[d=\\omega \_{\\Lambda } \\bar{\\mu }K\\beta \_{0,\\, \\Lambda }\\\]

(2.3.34)[¶](#equation-3-34 "Permalink to this equation")\\\[f=\\omega \_{\\Lambda } \\bar{\\mu }K\\left(1-\\beta \_{0,\\, \\Lambda } \\right)\\\]

(2.3.35)[¶](#equation-3-35 "Permalink to this equation")\\\[h=\\frac{\\sqrt{b^{2} -c^{2} } }{\\bar{\\mu }}\\\]

(2.3.36)[¶](#equation-3-36 "Permalink to this equation")\\\[\\sigma =\\left(\\bar{\\mu }K\\right)^{2} +c^{2} -b^{2}\\\]

(2.3.37)[¶](#equation-3-37 "Permalink to this equation")\\\[u\_{1} =b-{c\\mathord{\\left/ {\\vphantom {c \\alpha \_{g,\\, \\Lambda }^{\\mu } }} \\right.} \\alpha \_{g,\\, \\Lambda }^{\\mu } } {\\rm \\; or\\; }u\_{1} =b-{c\\mathord{\\left/ {\\vphantom {c \\alpha \_{g,\\, \\Lambda } }} \\right.} \\alpha \_{g,\\, \\Lambda } }\\\]

(2.3.38)[¶](#equation-3-38 "Permalink to this equation")\\\[u\_{2} =b-c\\alpha \_{g,\\, \\Lambda }^{\\mu } {\\rm \\; or\\; }u\_{2} =b-c\\alpha \_{g,\\, \\Lambda }\\\]

(2.3.39)[¶](#equation-3-39 "Permalink to this equation")\\\[u\_{3} =f+c\\alpha \_{g,\\, \\Lambda }^{\\mu } {\\rm \\; or\\; }u\_{3} =f+c\\alpha \_{g,\\, \\Lambda }\\\]

(2.3.40)[¶](#equation-3-40 "Permalink to this equation")\\\[s\_{1} =\\exp \\left\\{-\\min \\left\[h\\left(L+S\\right),40\\right\]\\right\\}\\\]

(2.3.41)[¶](#equation-3-41 "Permalink to this equation")\\\[s\_{2} =\\exp \\left\\{-\\min \\left\[K\\left(L+S\\right),40\\right\]\\right\\}\\\]

(2.3.42)[¶](#equation-3-42 "Permalink to this equation")\\\[p\_{1} =b+\\bar{\\mu }h\\\]

(2.3.43)[¶](#equation-3-43 "Permalink to this equation")\\\[p\_{2} =b-\\bar{\\mu }h\\\]

(2.3.44)[¶](#equation-3-44 "Permalink to this equation")\\\[p\_{3} =b+\\bar{\\mu }K\\\]

(2.3.45)[¶](#equation-3-45 "Permalink to this equation")\\\[p\_{4} =b-\\bar{\\mu }K\\\]

(2.3.46)[¶](#equation-3-46 "Permalink to this equation")\\\[d\_{1} =\\frac{p\_{1} \\left(u\_{1} -\\bar{\\mu }h\\right)}{s\_{1} } -p\_{2} \\left(u\_{1} +\\bar{\\mu }h\\right)s\_{1}\\\]

(2.3.47)[¶](#equation-3-47 "Permalink to this equation")\\\[d\_{2} =\\frac{u\_{2} +\\bar{\\mu }h}{s\_{1} } -\\left(u\_{2} -\\bar{\\mu }h\\right)s\_{1}\\\]

(2.3.48)[¶](#equation-3-48 "Permalink to this equation")\\\[h\_{1} =-dp\_{4} -cf\\\]

(2.3.49)[¶](#equation-3-49 "Permalink to this equation")\\\[h\_{2} =\\frac{1}{d\_{1} } \\left\[\\left(d-\\frac{h\_{1} }{\\sigma } p\_{3} \\right)\\frac{\\left(u\_{1} -\\bar{\\mu }h\\right)}{s\_{1} } -p\_{2} \\left(d-c-\\frac{h\_{1} }{\\sigma } \\left(u\_{1} +\\bar{\\mu }K\\right)\\right)s\_{2} \\right\]\\\]

(2.3.50)[¶](#equation-3-50 "Permalink to this equation")\\\[h\_{3} =\\frac{-1}{d\_{1} } \\left\[\\left(d-\\frac{h\_{1} }{\\sigma } p\_{3} \\right)\\left(u\_{1} +\\bar{\\mu }h\\right)s\_{1} -p\_{1} \\left(d-c-\\frac{h\_{1} }{\\sigma } \\left(u\_{1} +\\bar{\\mu }K\\right)\\right)s\_{2} \\right\]\\\]

(2.3.51)[¶](#equation-3-51 "Permalink to this equation")\\\[h\_{4} =-fp\_{3} -cd\\\]

(2.3.52)[¶](#equation-3-52 "Permalink to this equation")\\\[h\_{5} =\\frac{-1}{d\_{2} } \\left\[\\left(\\frac{h\_{4} \\left(u\_{2} +\\bar{\\mu }h\\right)}{\\sigma s\_{1} } \\right)+\\left(u\_{3} -\\frac{h\_{4} }{\\sigma } \\left(u\_{2} -\\bar{\\mu }K\\right)\\right)s\_{2} \\right\]\\\]

(2.3.53)[¶](#equation-3-53 "Permalink to this equation")\\\[h\_{6} =\\frac{1}{d\_{2} } \\left\[\\frac{h\_{4} }{\\sigma } \\left(u\_{2} -\\bar{\\mu }h\\right)s\_{1} +\\left(u\_{3} -\\frac{h\_{4} }{\\sigma } \\left(u\_{2} -\\bar{\\mu }K\\right)\\right)s\_{2} \\right\]\\\]

(2.3.54)[¶](#equation-3-54 "Permalink to this equation")\\\[h\_{7} =\\frac{c\\left(u\_{1} -\\bar{\\mu }h\\right)}{d\_{1} s\_{1} }\\\]

(2.3.55)[¶](#equation-3-55 "Permalink to this equation")\\\[h\_{8} =\\frac{-c\\left(u\_{1} +\\bar{\\mu }h\\right)s\_{1} }{d\_{1} }\\\]

(2.3.56)[¶](#equation-3-56 "Permalink to this equation")\\\[h\_{9} =\\frac{u\_{2} +\\bar{\\mu }h}{d\_{2} s\_{1} }\\\]

(2.3.57)[¶](#equation-3-57 "Permalink to this equation")\\\[h\_{10} =\\frac{-s\_{1} \\left(u\_{2} -\\bar{\\mu }h\\right)}{d\_{2} } .\\\]

Plant functional type optical properties ([Table 2.3.1](#table-plant-functional-type-optical-properties)) for trees and shrubs are from [Dorman and Sellers (1989)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#dormansellers1989). Leaf and stem optical properties (VIS and NIR reflectance and transmittance) were derived for grasslands and crops from full optical range spectra of measured optical properties ([Asner et al. 1998](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#asneretal1998)). Optical properties for intercepted snow ([Table 2.3.2](#table-intercepted-snow-optical-properties)) are from [Sellers et al. (1986)](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#sellersetal1986).

Table 2.3.1 Plant functional type optical properties[¶](#id7 "Permalink to this table")          
| Plant Functional Type
 | \\(\\chi \_{L}\\)

 | \\(\\alpha \_{vis}^{leaf}\\)

 | \\(\\alpha \_{nir}^{leaf}\\)

 | \\(\\alpha \_{vis}^{stem}\\)

 | \\(\\alpha \_{nir}^{stem}\\)

 | \\(\\tau \_{vis}^{leaf}\\)

 | \\(\\tau \_{nir}^{leaf}\\)

 | \\(\\tau \_{vis}^{stem}\\)

 | \\(\\tau \_{nir}^{stem}\\)

 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| NET Temperate

 | 0.01

 | 0.07

 | 0.35

 | 0.16

 | 0.39

 | 0.05

 | 0.10

 | 0.001

 | 0.001

 |
| NET Boreal

 | 0.01

 | 0.07

 | 0.35

 | 0.16

 | 0.39

 | 0.05

 | 0.10

 | 0.001

 | 0.001

 |
| NDT Boreal

 | 0.01

 | 0.07

 | 0.35

 | 0.16

 | 0.39

 | 0.05

 | 0.10

 | 0.001

 | 0.001

 |
| BET Tropical

 | 0.10

 | 0.10

 | 0.45

 | 0.16

 | 0.39

 | 0.05

 | 0.25

 | 0.001

 | 0.001

 |
| BET temperate

 | 0.10

 | 0.10

 | 0.45

 | 0.16

 | 0.39

 | 0.05

 | 0.25

 | 0.001

 | 0.001

 |
| BDT tropical

 | 0.01

 | 0.10

 | 0.45

 | 0.16

 | 0.39

 | 0.05

 | 0.25

 | 0.001

 | 0.001

 |
| BDT temperate

 | 0.25

 | 0.10

 | 0.45

 | 0.16

 | 0.39

 | 0.05

 | 0.25

 | 0.001

 | 0.001

 |
| BDT boreal

 | 0.25

 | 0.10

 | 0.45

 | 0.16

 | 0.39

 | 0.05

 | 0.25

 | 0.001

 | 0.001

 |
| BES temperate

 | 0.01

 | 0.07

 | 0.35

 | 0.16

 | 0.39

 | 0.05

 | 0.10

 | 0.001

 | 0.001

 |
| BDS temperate

 | 0.25

 | 0.10

 | 0.45

 | 0.16

 | 0.39

 | 0.05

 | 0.25

 | 0.001

 | 0.001

 |
| BDS boreal

 | 0.25

 | 0.10

 | 0.45

 | 0.16

 | 0.39

 | 0.05

 | 0.25

 | 0.001

 | 0.001

 |
| C3 arctic grass

 | \-0.30

 | 0.11

 | 0.35

 | 0.31

 | 0.53

 | 0.05

 | 0.34

 | 0.120

 | 0.250

 |
| C3 grass

 | \-0.30

 | 0.11

 | 0.35

 | 0.31

 | 0.53

 | 0.05

 | 0.34

 | 0.120

 | 0.250

 |
| C4 grass

 | \-0.30

 | 0.11

 | 0.35

 | 0.31

 | 0.53

 | 0.05

 | 0.34

 | 0.120

 | 0.250

 |
| C3 Crop

 | \-0.30

 | 0.11

 | 0.35

 | 0.31

 | 0.53

 | 0.05

 | 0.34

 | 0.120

 | 0.250

 |
| Temp Corn

 | \-0.50

 | 0.11

 | 0.35

 | 0.31

 | 0.53

 | 0.05

 | 0.34

 | 0.120

 | 0.250

 |
| Spring Wheat

 | \-0.50

 | 0.11

 | 0.35

 | 0.31

 | 0.53

 | 0.05

 | 0.34

 | 0.120

 | 0.250

 |
| Temp Soybean

 | \-0.50

 | 0.11

 | 0.35

 | 0.31

 | 0.53

 | 0.05

 | 0.34

 | 0.120

 | 0.250

 |
| Cotton

 | \-0.50

 | 0.11

 | 0.35

 | 0.31

 | 0.53

 | 0.05

 | 0.34

 | 0.120

 | 0.250

 |
| Rice

 | \-0.50

 | 0.11

 | 0.35

 | 0.31

 | 0.53

 | 0.05

 | 0.34

 | 0.120

 | 0.250

 |
| Sugarcane

 | \-0.50

 | 0.11

 | 0.35

 | 0.31

 | 0.53

 | 0.05

 | 0.34

 | 0.120

 | 0.250

 |
| Tropical Corn

 | \-0.50

 | 0.11

 | 0.35

 | 0.31

 | 0.53

 | 0.05

 | 0.34

 | 0.120

 | 0.250

 |
| Tropical Soybean

 | \-0.50

 | 0.11

 | 0.35

 | 0.31

 | 0.53

 | 0.05

 | 0.34

 | 0.120

 | 0.250

 |
| Miscanthus

 | \-0.50

 | 0.11

 | 0.35

 | 0.31

 | 0.53

 | 0.05

 | 0.34

 | 0.120

 | 0.250

 |
| Switchgrass

 | \-0.50

 | 0.11

 | 0.35

 | 0.31

 | 0.53

 | 0.05

 | 0.34

 | 0.120

 | 0.250

 |

Table 2.3.2 Intercepted snow optical properties[¶](#id8 "Permalink to this table")   
| Parameter
 | vis

 | nir

 |
| --- | --- | --- |
| \\(\\omega ^{sno}\\)

 | 0.8

 | 0.4

 |
| \\(\\beta ^{sno}\\)

 | 0.5

 | 0.5

 |
| \\(\\beta \_{0}^{sno}\\)

 | 0.5

 | 0.5

 |

