1. Radiative Transfer Theory at Optical and Microwave wavelengths applied to vegetation canopies: part 2
UoL MSc Remote Sensing
course tutors: Dr Lewis
Dr Saich

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
includes polarisation using modified Stokes Vector and Mueller Matrix

4. Medium 1: air
z = l cos q0=lm0 q0
Medium 2: canopy in air z
Pathlength l
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,W) at depth z in direction W wrt z:

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

7. Vector RT equation Source: I - modified Stokes vector
ke - a 4x4 extinction matrix
phase function replaced by a (4x4) phase matrix (averages of Mueller matrix)

8. 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).
Beer’s Law

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

10. Extinction Coefficient and Beers Law
Definition:
Qe() - extinction cross section for a particle (units of m2)
Nv - volume density (Np m-3)
can be defined for specific polarisation
subscript p for p-polarisation ().

11. Extinction Coefficient and Beers Law
Definition:
volume absorption and scattering coefficients

12. Optical Depth
Definition:

13. Single Scattering Albedo
Definition:
Effectively = reflectance + transmittance for optical
w(l)=rl(l)+tl(l)

14. Optical Extinction Coefficient for Oriented Leaves
Definition: extinction cross section:

15. Optical Extinction Coefficient for Oriented Leaves

16. Optical Extinction Coefficient for Oriented Leaves
range of G-functions small ( ) 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 50o and 60o
essentially invariant at 0.5 over different leaf angle distributions at 57.5o.

17. z = l cos q0=lm0 Medium 1: air q0 Medium 2: canopy in air z
Pathlength l
Medium 3:soil
Path of radiation

18. Optical Extinction Coefficient for Oriented Leaves

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

20. Extinction and Scattering in a Rayleigh Medium
Consider more general vector case
use example of Rayleigh medium
small spherical particles
can ‘modify’ terms for larger & non-spherical scatterers
e.g. discs, cylinders

21. Extinction and Scattering in a Rayleigh Medium
Scattering coefficient:
f - volume fraction of scatterers
es- is the dielectric constant of the sphere material
k is the wavenumber in air.

22. Extinction and Scattering in a Rayleigh Medium
Absorption coefficient:
f - volume fraction of scatterers
es- is the dielectric constant of the sphere material
k is the wavenumber in air.

25. Rayleigh Optical Thickness
For spherical scatterers:
extinction is scalar
no cross-polarisation terms

27. Rayleigh Phase Matrix

28. Solution to VRT Equation
Iterative method for low albedo
Do for Rayleigh here

29. Solution to VRT Equation

31. Solution to VRT Equation
Boundary Conditions:

32. Solution to VRT Equation
Rephrase as integral equations:
T(-x)=e-x is an attenuation (extinction) term due to Beer’s Law

33. Insert boundary conditions
VRT

34. VRT: upward terms

35. VRT: downward terms

36. Zero Order Solution
Set source terms in
to zero:

37. First Order Solution
Iterative method: Insert zero-O solution as source

38. First Order Solution
Result:
k1= ke(1/m0+1/ms); k2= ke(1/m0-1/ms)

39. First Order Solution
Upward Backscattered Intensity at z=0

40. First Order Solution direct intensity reflected at canopy base,
doubly attenuated through the canopy;

41. First Order Solution ‘double bounce’ term involving:
a ground interaction,
(downward) volumetric scattering by the canopy,
additional ground interaction.
Includes a double attenuation
Term is generally very small and is often ignored.

42. First Order Solution
downward volumetric scattering term, followed by a soil interaction,
including a double attenuation on the upward and downward paths

43. First Order Solution
Ground interaction followed by volumetric scattering by the canopy
in the upward direction, again including double attenuation

44. Pure volumetric scattering by the canopy.
First Order Solution
Pure volumetric scattering by the canopy.
‘path radiance’

45. First Order Solution Now have first order solution
Express as backscatter
plug in extinction coefficient & phase functions for Rayleigh
consider different scatterers
consider different soil scattering

46. Second+ Order Solution
2+ solutions similarly obtained
set the first-order solutions as the source terms.
Note whilst no cross-polarisation terms for spherical (Rayleigh) scatterers, they do occur in second+ order scattering,
i.e. cross-polarisation for spherical scatterers is result of multiple scattering.
2+ O used in RT2 model (Saich)

47. A Scalar Radiative Transfer Solution
Attempt similar first Order Scattering solution
N.B. - mean different to microwave field
in optical, consider total number of interactions
with leaves + soil
in microwave, consider only canopy interactions
Already have extinction coefficient:

48. SRT Phase function: ul - leaf area density;
m’ - cosine of the incident zenith angle
G - area scattering phase function.

49. SRT Area scattering phase function:
double projection, modulated by spectral terms

50. SRT Phrase in scalar form of VRT:
rsoil – soil directional reflectance factor

51. SRT
Insert Phase function definition:
so:

52. SRT
Note Joint Gap Probability:

53. SRT
Integrate to give intensity at z=0:
so:

54. Optical Extinction Coefficient
Insert into k3:

55. SRT
Since the LAI, L=ulH:

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

57. SRT: 1st O mechanisms Canopy only scattering Direct function of w
Function of gl, L, and viewing and illumination angles

58. 1st O SRT
Special case of spherical leaf angle:

59. 1st O SRT
Further, linear function between leaf reflectance and transmittance:

60. Multiple Scattering Albedo not always low (NIR)
cant use interative method
range of approximate solutions available

61. RT Modifications Hot Spot joint gap probabilty: Q
For far-field objects, treat incident & exitant gap probabilities independently
product of two Beer’s Law terms

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

63. 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.

64. 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

65. Summary SRT / VRT formulations Beer’s Law extinction
scattering (source function)
Beer’s Law
exponential attenuation
rate - extinction coefficient
LAI x G-function for optical
considered Rayleigh for microwave

66. Summary VRT 1st O solution 2 stream iterative approach
work 0th-O solution
plug into VRT eqn
work 1st-O solution
calculate for upward intensity at top of canopy
5 scattering mechanisms

67. Summary SRT 1st O solution Modification to SRT:
similar, but 1 x canopy or 1x soil solutions only
use area scattering phase function
simple solution for spherical leaf angle
2 scattering mechanisms
Modification to SRT:
hot spot at optical