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

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

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

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

5. 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:

6. 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

7. 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

8. Extinction Coefficient and Beers Law No source version of SRT eqn

9. 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

10. Optical Extinction Coefficient for Oriented Leaves

11. 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.

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

13. 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:

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

15. 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

16. SRT

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

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

19. 1st O SRT Special case of spherical leaf angle:

20. Multiple Scattering LAI 1 Scattering order Contributions to reflectance and transmittance

21. Multiple Scattering LAI 5 Scattering order Contributions to reflectance and transmittance

22. Multiple Scattering LAI 8 Scattering order Contributions to reflectance and transmittance

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

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

25. 2 nd Order scattering: i0i0 i 0 p i 0 p(1-p)

26. single scattering albedo of canopy

27. 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

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

29. Canopy with shoots as fundamental scattering objects: Clumping

30. Canopy with shoots as fundamental scattering objects:

31. p2p2 p canopy Smolander & Stenberg RSE 2005 p shoot =0.47 (scots pine) p 2 <p canopy Shoot-scale clumping reduces apparent LAI

32. 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

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

34. 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.

35. 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

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

37. 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