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Radiative Transfer Theory at Optical wavelengths applied to vegetation canopies: part 2 UoL MSc Remote Sensing Dr Lewis plewis@geog.ucl.ac.uk

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Radiative Transfer equation Used extensively for (optical) vegetation since 1960s (Ross, 1981) Used for microwave vegetation since 1980s

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Radiative Transfer equation Consider energy balance across elemental volume Generally use scalar form (SRT) in optical Generally use vector form (VRT) for microwave

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z Pathlength l z = l cos l Medium 1: air Medium 2: canopy in air Medium 3:soil Path of radiation

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Scalar Radiative Transfer Equation 1-D scalar radiative transfer (SRT) equation –for a plane parallel medium (air) embedded with a low density of small scatterers –change in specific Intensity (Radiance) I(z, ) at depth z in direction wrt z:

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Scalar RT Equation Source Function: - cosine of the direction vector ( with the local normal –accounts for path length through the canopy e - volume extinction coefficient P() is the volume scattering phase function

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Extinction Coefficient and Beers Law Volume extinction coefficient: –total interaction cross section –extinction loss –number of interactions per unit length a measure of attenuation of radiation in a canopy (or other medium). Beers Law

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Extinction Coefficient and Beers Law No source version of SRT eqn

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Optical Extinction Coefficient for Oriented Leaves Volume extinction coefficient: u l : leaf area density –Area of leaves per unit volume G l : (Ross) projection function

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Optical Extinction Coefficient for Oriented Leaves

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range of G-functions small (0.3-0.8) and smoother than leaf inclination distributions; planophile canopies, G-function is high (>0.5) for low zenith and low (<0.5) for high zenith; converse true for erectophile canopies; G-function always close to 0.5 between 50 o and 60 o essentially invariant at 0.5 over different leaf angle distributions at 57.5 o.

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Optical Extinction Coefficient for Oriented Leaves so, radiation at bottom of canopy for spherical: for horizontal:

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A Scalar Radiative Transfer Solution Attempt similar first Order Scattering solution –in optical, consider total number of interactions with leaves + soil Already have extinction coefficient:

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SRT Phase function: u l - leaf area density; - cosine of the incident zenith angle - area scattering phase function.

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SRT Area scattering phase function: double projection, modulated by spectral terms l : leaf single scattering albedo –Probability of radiation being scattered rather than absorbed at leaf level –Function of wavelength

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SRT

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SRT: 1st O mechanisms through canopy, reflected from soil & back through canopy

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SRT: 1st O mechanisms Canopy only scattering Direct function of Function of g l, L, and viewing and illumination angles

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1st O SRT Special case of spherical leaf angle:

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Multiple Scattering LAI 1 Scattering order Contributions to reflectance and transmittance

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Multiple Scattering LAI 5 Scattering order Contributions to reflectance and transmittance

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Multiple Scattering LAI 8 Scattering order Contributions to reflectance and transmittance

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Multiple Scattering –range of approximate solutions available –Recent advances using concept of recollision probability, p Huang et al. 2007

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Q0Q0 s i0i0 i 0 =1-Q 0 p s 1 =i 0 (1 – p) p : recollision probability : single scattering albedo of leaf

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2 nd Order scattering: i0i0 i 0 p i 0 p(1-p)

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single scattering albedo of canopy

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Average number of photon interactions: The degree of multiple scattering p : recollision probability Absorptance Knyazikhin et al. (1998): p is eigenvalue of RT equation Depends on structure only

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For canopy: Smolander & Stenberg RSE 2005 p max =0.88, k=0.7, b=0.75 Spherical leaf angle distribution

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Canopy with shoots as fundamental scattering objects: Clumping

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Canopy with shoots as fundamental scattering objects:

31 p2p2 p canopy Smolander & Stenberg RSE 2005 p shoot =0.47 (scots pine) p 2

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Other RT Modifications Hot Spot –joint gap probabilty: Q –For far-field objects, treat incident & exitant gap probabilities independently –product of two Beers Law terms

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RT Modifications Consider retro-reflection direction: –assuming independent: –But should be:

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RT Modifications Consider retro-reflection direction: –But should be: –as have already travelled path –so need to apply corrections for Q in RT e.g.

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RT Modifications As result of finite object size, hot spot has angular width –depends on roughness leaf size / canopy height (Kuusk) similar for soils Also consider shadowing/shadow hiding

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Summary SRT formulation –extinction –scattering (source function) Beers Law –exponential attenuation –rate - extinction coefficient LAI x G-function for optical

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Summary SRT 1st O solution –use area scattering phase function –simple solution for spherical leaf angle –2 scattering mechanisms Multiple scattering –Recollison probability Modification to SRT: –hot spot at optical

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