Solar Radiation Transfer through Vegetation, part I • Beam Radiation transfer through Ideal canopies – Beer-Bouguer’ s Law – Poisson Probability Distribution – The G function, the Leaf-Sun Direction Cosine – Gap Distributions • Beam Radiation transfer through non-ideal canopies – Penumbra • Diffuse Radiation ESPM 129 Biometeorology 1 Primary Roles of Sunlight in Biomet • Photosynthesis • Stomatal conductance • Transpiration • Leaf Energy Balance – leaf temperature • RitiRespiration • VOC volatilization • Photosynthesis • Saturation vapor pressure at leaf surface • Stable Isotope ESPM 129 Biometeorology 2 Secondary Roles of Sunlight • Plant growth • Seedling regeneration • Vertical structure and crown shape of forest stands • Phenology • Leaf morphology • Uptake and emission of trace gases that participate in biogeochemical cycling and atmosppyheric chemistry. ESPM 129 Biometeorology 3 Light transmission through a Vegetation • Beam Radiation penetrating through gaps in the foliage in the direction of the sun • Diffuse radiation penetrating through gaps in the foliage in the direction of the sky hemisphere • Complementary radiation generating by the reflection and transmission of light by leaves Pt. Reyes National Seashore, Allomere Falls trail, August 2002 ESPM 129 Biometeorology 4 Compos i ti on of Sunli gh t in a Can opy From Ross, 1981 ESPM 129 Biometeorology 5 Radiative transfer througggh vegetation is a function of: • iidtditiincident radiation; • the optical properties of the leaves and stems; • the optical properties of the underlying ground surface or litter; • the architecture of the stand (which includes, leaf area index, leaf angle distribution and the spatial dispersion of leaves, e.g. random, clumped or regular). ESPM 129 Biometeorology 6 Sources of Heterogeneity in Sunlight • clumping and gapping of foliage, • gaps in canopy crowns due to treefall or cultivation practices, •sppgatial variations in leaf orientation angles, • penumbra • leaf flutter • clouds • topography • seasonal trends in plant phenology • seasonalddill and diurnal movemen tftht of the sun • directional and non-isotropic and wavelength dependent scattering of light. ESPM 129 Biometeorology 7 Gap Fraction, Probability of Beam Penetration, P0 Patterns of Sunflecks, Umbra and Penumbra ESPM 129 Biometeorology 8 Umbra ESPM 129 Biometeorology 9 Umbra and Penumbra Umbra Penumbra Penumbra ESPM 129 Biometeorology 10 Canopy Representation Probability of Beam Penetration or the Poisson Probability of Zero Contacts, P0 LG LG PkL0 exp( ) exp( ) exp( ) cossun sin sun spherical distribution, G=0.5 1.0 o 0.9 90 600 L: leaf area index 0.8 45o 30o G: direction cosine between sun 0.7 and leaf normal 0.6 : solar elevation angle (L) 0 0.5 P : Solar zenith Angle 0.4 k: extinction coefficient 0.3 0.2 0.1 0.0 0123456 leaf area index ESPM 129 Biometeorology 12 • Probability of Beam Penetration, P0, through n layers of leaves with an incremental amount of leaf area, a, relative to the total amount of area, A a a a P (1 ) (1 ) (1 ).... 0 A A A a P (1 )n 0 A ESPM 129 Biometeorology 13 Probability of Zero Contacts, P0 A = 30 by 36 = 1080 a = 16 x 21 = 336 a 336 P (1 )n (1 ) 0.688 0 A 1080 ESPM 129 Biometeorology 14 A = 30 by 36 = 1080 a = 16 x 21 = 336 a 336 P (1 )n (1 ) 0.688 0 A 1080 ESPM 129 Biometeorology 15 Con ceptual vi sualiz ati on of li gh t tr an sf er thr ough a r an doml y spaced m edi um Non-Random Distribution Random Distribution, Case 1 Random Distribution, Case 2 2 46 8 10 ESPM 129 Biometeorology 16 Beer’ s Law, v1. 0 a P (1 )n Limit n goes to infinity 0 A I(L) exp( Na ) A L=leaf area index =aN/A I(L) exp(L) I is the flux density of sunlight for a given wavelength ESPM 129 Biometeorology 17 Beer’ s Law, v2. 0 Change in Light flux density, I, through a path x is a function of its current intensity, I, and its extinction coefficient, k, and leaf area density, a. dI k a I dx Note the negative sign, indicating I decreases with attenuation ESPM 129 Biometeorology 18 Beer’s Law, cont. dI k a dx Re-arrange derivatives I dI I() x ln( ) Integr at e both si des II(0) kadx kax I()x exp(ln( )) exp(kax ) Simplify by taking exp of both sides I(0) Ix() exp(kax ) I0 ESPM 129 Biometeorology 19 Beer’s Law, cont. I(L) I 0 exp(k L) Leaf area density,a, times ppg,,qathlength, x, equals Leaf area index, L ESPM 129 Biometeorology 20 How Much Leaf Area interceppgts 99% of incident light? IL() ln( ) ln(001. ) kL I0 If k = 1.00, then L = 4.6 ESPM 129 Biometeorology 21 The extinction coefficient, k, equals the fraction of hemi-surface leaf area (A) that is projected onto the horizontal (Ah), from a particular zenith angle. A k h A ESPM 129 Biometeorology 22 A Light interception by Inclined leaves k h A k goes to zero as A goes to zero k= Ah/A; it equals 1 h As Ah and A equal One another k goes to infinity A As Ah goes to A infinity h ESPM 129 Biometeorology 23 Define K in terms of Sun and Leaf Inclination Angles AGGcos horiz leaf sun Aleafsin sun sin sun cos sun ,angle between leaf normal leaf leaf sun And solar zenith leaf sun sun, solar elevation angle A leaf sun k, extinction coefficient A beam leaf leaf sun G, G-function or mean direction cosine sun A horizontal leaf leaf cosleaf sun ESPM 129 Biometeorology 24 1P1. Pro jec t the Area of a LfLeaf, Aleaf, onto the Area Normal to the Solar Beam, Abeam Abeam cosleaf sun Aleaf leaf leaf sun 2. Project the Area of The Sun’s Beam (Abeam) onto the horizontal (Ahoriz). A leaf sun A beam A leaf beam sin cos leaf sun sun sun sun Ahoriz A horizontal leaf leaf ESPM 129 Biometeorology 25 leaf leaf sun Aleaf sun Abeam leaf leaf sun sun Ahorizontal ESPM 129 Biometeorology 26 leaf leaf Algebraic Manipulation AAbeam leafcos leaf sun Abeam A horizsin sun AAcos sin leaf leaf leaf sun horiz sun leaf sun A leaf sun Ahoriz cosleaf sun cos leaf sun A beam leaf A sin cos leaf sun leaf sun sun sun A horizontal leaf leaf ESPM 129 Biometeorology 27 Voila’ AGGcos horiz leaf sun Alfleaf sinsun sin sun cos sun ESPM 129 Biometeorology 28 G function for Plant Canopies 2 /2 Gggnndd( , ) '( ) "( )|cos( )| z z ll lll 0 0 cos cos(nnl ) cos suncos leaf sin sunsin leaf cos( sun leaf ) Weighted by elevation angle Dist Weight by Azimuth Angle Probability Distr /2 gd'( ) 1 2 z ll gd"( ) 1 0 z ll 0 sun's azimuth leaf azimuth ESPM 129 Biometeorology 29 General Functions for Different Leaf Orientations Leaf Angle Distribution G, direction cosine K, extinction coefficient HiHorizonta l cos() 1 Vertical 2/π sin() 2 tan(/π) Conical cos() cos(l)cos() Spherical or random 0.5 1/(2 cos()) Heliotropic 1 1/ cos() Ellipsoidal * ** A cos GG horiz Aleafsin sun sin sun cos sun ESPM 129 Biometeorology 30 G functions for different leaf inclination angle distributions heliotropic horizontal spherical vertical 1.0 0.9 0.8 0.7 0.6 G 0.5 0.4 0.3 0.2 0.1 0.0 0 20406080 zenith angle All G functions Converge on G=0. 5 at zenith angle equal 1 Radian! ESPM 129 Biometeorology 31 Extinction coefficients with different solar zenith angles 10 k spherical k horizontal k vertical 1 k 0.1 0.01 0 20406080 cos() K converges to 1 when zenith angle equals 1 Radian, 180/pi ESPM 129 Biometeorology 32 Family of G-Functions Computed Y. Ryu ESPM 129 Biometeorology 33 G-Function for Oak Savanna Data of Ryu, Sonnentag, Vargas, 2008 ESPM 129 Biometeorology 34 Sun Angles and the probability of beam penetration, P0 spherical distribution, G0.5 G=0.5 1.0 90o LG LG 0.9 P exp( ) exp( ) 600 0 cossunsin sun 0.8 45o 30o 0.7 0.6 ) LL ( 0 0.5 P 0.4 0.3 0.2 0.1 000.0 0123456 leaf area indexESPM 129 Biometeorology 35 Probabili ty o f beam pen etr ati on wi th cl um ped an d r an doml y di stri buted f oli age spherical distribution, G=0.5, =45o 1.0 0.9 080.8 L G clumped distribution 0.7 P0 = exp (- ) random distribution sin 0.6 (L) 0 0.5 P 0.4 0.3 , clumping coef 0.2 0.1 000.0 0123456 leaf area indexESPM 129 Biometeorology 36 ESPM 129 Biometeorology 37 Clumping Factors Class Class names Min Max Mean 1 Tree Cover, broadleaf, evergreen 0.59 0.68 0.63 2 Tree Cover, broadleaf, deciduous, closed 0.59 0.79 0.69 3 Tree Cover, broadleaf, deciduous, open 0.62 0.78 0.70 4 Tree Cover, needleleaf, evergreen 0550.55 0680.68 0620.62 5 Tree Cover, needleleaf, deciduous 0.60 0.77 0.68 6 Tree Cover, mixed leaf type 0.58 0.79 0.69 7 Tree Cover, regularly flooded, fresh water 0.61 0.69 0.65 8 Tree Cover, regularly flooded, saline water 0650.65 0790.79 0720.72 9 Mosaic: Tree Cover / Other natural vegetation 0.64 0.82 0.72 10 Tree Cover, burnt 0.65 0.86 0.75 11 Shrub Cover, closed-open, evergreen 0.62 0.80 0.71 12 Shrub Cover , closed -open, deciduous 0620.62 0800.80 0710.71 13 Herbaceous Cover, closed-open 0.64 0.83 0.74 14 Sparse herbaceous or sparse shrub cover 0.67 0.84 0.75 15 Reg.