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

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Presentation on theme: "Radiative Transfer Theory at Optical and Microwave wavelengths applied to vegetation canopies: part 2 UoL MSc Remote Sensing course tutors: Dr Lewis plewis@geog.ucl.ac.uk."— Presentation transcript:

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.

23

24

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

26

27 Rayleigh Phase Matrix

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

29 Solution to VRT Equation

30

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


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