Download Radiative Transfer Theory at Optical and Microwave

Radiative Transfer Theory at
Optical and Microwave
wavelengths applied to
vegetation canopies: part 1
UoL MSc Remote Sensing
course tutors:
Dr Lewis
Dr Saich
[email protected]
[email protected]
Aim of this section
• Introduce RT approach as basis to
understanding optical and microwave
vegetation response
• enable use of models
• enable access to literature
Scope of this section
• Introduction to background theory
– RT theory
– Wave propagation and polarisation
– Useful tools for developing RT
• Building blocks of a canopy scattering
model
– canopy architecture
– scattering properties of leaves
– soil properties
Associated practical and reading
• Reading
– microwave leaf model
• Chuah, H.T., Lee, K.Y., and Lau, T.W., 1995, “Dielectric constants of
rubber and oil palm leaf samples at X-band”, IEEE Trans. Geoscience
and Remote Sensing, GE-33, 221-223.
– Optical leaf model
• Jacquemoud, S., and Baret, F., 1990, “PROSPECT: a model of leaf
optical properties spectra”, Remote Sensing of Environment, 34, 7591.
• Practicals investigating leaf scattering
– Optical OR microwave
Why build models?
• Assist data interpretation
• calculate RS signal as fn. of biophysical variables
• Study sensitivity
• to biophysical variables or system parameters
• Interpolation or Extrapolation
• fill the gaps / extend observations
• Inversion
• estimate biophysical parameters from RS
• aid experimental design
• plan experiments
Radiative Transfer Theory
• Approach optical and microwave case at
same time through RT
– ‘relatively’ simple & well-understood
– no other treatment in this way
– researchers tend to specialise in either field
• less understanding of other field / synergy
• Deal with other approaches in later lectures
Radiative Transfer Theory
• Applicability
– heuristic treatment
• consider energy balance across elemental volume
– assume:
• no correlation between fields
– addition of power not fields
• no diffraction/interference in RT
– can be in scattering
– develop common (simple) case here
Radiative Transfer Theory
• Case considered:
– horizontally infinite but vertically finite plane
parallel medium (air) embedded with
infinitessimal oriented scattering objects at low
density
– canopy lies over soil surface (lower boundary)
– assume horizontal homogeneity
• applicable to many cases of vegetation
Radiative Transfer Theory
• More accurate approach is to use Maxwell’s
equations
• difficult to formulate
• will return to for object scattering but not
propagation (RT)
Radiative Transfer Theory
• More accurate approach is to use Maxwell’s
equations
• difficult to formulate
• will return to for object scattering but not
propagation (RT)
Radiative Transfer Theory
• More accurate approach is to use
Maxwell’s equations
• difficult to formulate
• use object scattering but not propagation (RT)
• essentially wave equation for electric field
d E z  2
 k E z   0
dz
• k - wavenumber = 2p/l in air
 Ev  ikz
E  z    e
E 
 h
Plane wave
Radiative Transfer Theory
• Consider incident Electric-field Ei(r) of
magnitude Ei in direction k̂ to a position r:
i

 ik kˆr
E
i
i ik kˆr
v


E r   i e
Ee
E 
 h
• incident wave sets up internal currents in
scatterer that reradiate ‘scattered’ wave
• Remote sensing problem:
– describe field received at a sensor from an area
extensive ensemble average of scatterers
Scattering
• Define using scattering matrix:
ik0 r
ik0 r
 Svv
e
e
s
i

E 
SE 
r
r  S hv
Svh  i
 E
S hhv 
• elements polarised scattering amplitudes
– for discs:
k02V   1
 J1  x  
S 
 orientation,  2.0
pq
– for needles:
S pq
4p
d


 x 
k02V   1
 sin x  

n orientation,  

4p
 x 
• assume scattering in far field
Scattering
Bessel function
(complex)
permittivity of leaf
S pq
k02V   1
 J1  x  

d orientation,  2.0

4p
 x 
Wavenumber2 = 4p2/l2
Leaf volume
Scattering
Sinc
function
S pq
k V   1
 sin x  

n orientation,  

4p
 x 
2
0
Stokes Vector
• Can represent plane wave polarisation by
i
i
Ev , E h and phase term:
i

 ik kˆr
E
i
i ik kˆr
v


E r   i e
Ee
E 
 h
• h,v phase equal for linear polarised wave
Stokes Vector
• More convenient to use modified Stokes
vector:
2

Ev
 Iv  
 
2

I
 h
Eh
Fm     
*
U

  2 Re Ev Eh
 V  
   2 Im Ev Eh*











Stokes Vector
• Using this, relate scattered Stokes vector to
incident:
1
1
1
i
F  2 Fm  2 W  Fmi
r
r
r
m
 S *vv S vv
 *
 S hv S hv
W  *
 S hv S vv
 S *vv S
hv

*
S vh S vh
*
S hh S hh
*
S hh S vh
*
S vh S hh
*
S vh S vv
*
S hh S hv
*
S hh S vv
*
S vh S hv
*
S vv S vh 

*
S hv S hh 

*
S hv S vh 
*
S vv S hhv 
 1 1 0 0


 1 1 0 0

0 0 1 1


0 0  i i 


N.B S2 so 1/l4 for discs etc
Stokes Vector
• Average Mueller matrix  over all
scatterers to obtain phase matrix for use in
RT
Building blocks
for a canopy model
• Require descriptions of:
– canopy architecture
– leaf scattering
– soil scattering
Canopy Architecture
• 1-D: Functions of depth from the top of the canopy (z).
Canopy Architecture
• 1-D: Functions of depth from the top of the canopy (z).
1.
2.
3.
Vertical leaf area density ul  z  (m2/m3)
OR
the vertical leaf number density function, Nv(z)
(number of particles per m3)
the leaf normal orientation distribution
function, (dimensionless).
leaf size distribution
• defined as:
– area to relate leaf area density to leaf number density, as well as
thickness.
– the dimensions or volume of prototype scattering objects such as discs,
spheres, cylinders or needles.
Canopy Architecture
• Leaf area / number density
– ul  z  (one-sided) m2 leaf per m3
– Nv(z) - number of ‘particles’ per m3
ul z   N v z Al
zH
LAI
L
 u z dz
l
z 0
Canopy Architecture
• Leaf Angle Distribution
z
Inclination to vertical
 p g  d 
2 
l
l
l
1
ql
l
Leaf normal vector
y
fl
azimuth
x
Leaf Angle Distribution
• Archetype Distributions:
 planophile  g l l   3 cos 2 l
 erectophile 
 3 2


g l l    sin l
 spherical
g l l   1

 plagiophile 
 extremophile
 2
 15  2
g l l     sin 2l
8
 15 
g l l     cos 2 2l
7
Leaf Angle Distribution
• Archetype Distributions:
3.0
2.5
g_l(theta_l)
2.0
1.5
1.0
0.5
0.0
0
10
20
30
40
50
60
70
leaf zenith angle / degrees
spherical
plagiophile
planophile
extremophile
erectophile
80
90
Leaf Angle Distribution
• Elliptical Distribution:
gl l  


m


1  
2
eccentricity of distribution
modal leaf angle

sin 2 l  m 
1
2
:
0   1
:
0  m 
p
2
Leaf Angle Distribution
• Elliptical Distribution:
2.0
1.8
1.6
g_l(theta_l)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
10
20
30
40
50
60
70
80
leaf zenith angle / degrees
erectophile
planophile
plagiophile
Elliptical leaf angle distributions:
=0.9; qm=0 (erectophile), p/2 (planophile), p/4 (plagiophile)
90
Leaf Dimension
• RT theory: infinitessimal scatterers
– without modifications (dealt with later)
• Scattering at microwave depends on leaf volume
for given number per unit area
– on leaf ‘thickness’ for given LAI
• In optical, leaf size affects canopy scattering in
retroreflection direction
– ‘roughness’ term: ratio of leaf linear dimension to
canopy height
also, leaf thickness effects on reflectance
/transmittance
Leaf Dimension
• RT theory: infinitessimal scatterers
– without modifications (dealt with later)
• Scattering at microwave depends on leaf volume
for given number per unit area
– on leaf ‘thickness’ for given LAI
• In optical, leaf size affects canopy scattering in
retroreflection direction
– ‘roughness’ term: ratio of leaf linear dimension to
canopy height
also, leaf thickness effects on reflectance
/transmittance
Canopy element and soil
spectral properties
• Scattering properties of leaves
– scattering affected by:
• Leaf surface properties and internal structure;
• leaf biochemistry;
• leaf size (essentially thickness, for a given LAI).
Scattering properties of leaves
• Leaf surface properties and internal
structure
optical
Specular
from surface
Dicotyledon leaf structure
Smooth (waxy) surface
- strong peak
hairs, spines
- more diffused
Scattering properties of leaves
• Leaf surface properties and internal
structure
optical
Diffused
from scattering at
internal air-cell
wall interfaces
Depends on total area
of cell wall interfaces
Dicotyledon leaf structure
Depends on refractive index:
varies: 1.5@400 nm
1.3@2500nm
Scattering properties of leaves
• Leaf surface properties and internal
structure
optical
More complex structure (or thickness):
- more scattering
- lower transmittance
- more diffuse
Dicotyledon leaf structure
Scattering properties of leaves
• Leaf surface properties and internal
structure
microwave
Thickness (higher volume)
- higher scattering
Dicotyledon leaf structure
Scattering properties of leaves
• Leaf biochemstry
Scattering properties of leaves
• Leaf biochemstry
Scattering properties of leaves
• Leaf biochemstry
Scattering properties of leaves
• Leaf biochemstry
Scattering properties of leaves
• Leaf biochemstry
– pigments: chlorophyll a and b, a-carotene, and
xanthophyll
• absorb in blue (& red for chlorophyll)
– absorbed radiation converted into:
• heat energy, flourescence or carbohydrates through
photosynthesis
Scattering properties of leaves
• Leaf biochemstry
– Leaf water is major consituent of leaf fresh weight,
• around 66% averaged over a large number of leaf types
– other constituents ‘dry matter’
• cellulose, lignin, protein, starch and minerals
– Absorptance constituents increases with concentration
• reducing leaf reflectance and transmittance at these
wavelengths.
Scattering properties of leaves
• Optical Models
– flowering plants: PROSPECT
Scattering properties of leaves
• Optical Models
– flowering plants: PROSPECT
Scattering properties of leaves
• Leaf water
Scattering properties of leaves
• Leaf water
 PROSPECT:
 leaf water content parameterised as equivalent water
thickness (EWT)
 approximates the water mass per unit leaf area.
 related to volumetric moisture content (VMC, Mv)
(proportionate volume of water in the leaf) by multiplying
EWT by the product of leaf thickness and water density.
Scattering properties of leaves
• Microwave:
– water content related to leaf permittivity, .
 n  1.7  3.2M v  6.5M v2


vf f  M v 0.82M v  0.166
vfb 
31.4M 
2
v
1  59.5M 
Offset
factor
Volume
fractions
2
v







75
18 
55


 M v    n  vf f 4.9 
i

vf
2
.
9



b
f
f
f




1 i
1 i
18



0.18 


Scattering properties of leaves
• Microwave:
– water content related to leaf permittivity, .
Frequency / GHz
iconic conductivity of free water







75
18 
55


 M v    n  vf f 4.9 
i
 vfb 2.9 


f
f
f




1 i
1 i
18



0.18 


Scattering properties of leaves
• leaf dimensions
– optical
• increase leaf area for constant number of leaves
- increase LAI
• increase leaf thickness - decrease transmittance
(increase reflectance)
– microwave
• leaf volume dependence of scattering
– volume for constant leaf number
– thickness for constant leaf area
Scattering properties of soils
• Optical and microwave affected by:
– soil moisture content
– soil type/texture
– soil surface roughness.
soil moisture content
• Optical
– effect essentially proportional across all wavelengths
• enhanced in water absorption bands
soil moisture content
• Microwave
– increases soil dielectric constant
• effect varies with wavelength
• generally increases volume scattering
– and decreases penetration depth
soil texture/type
• Optical
– relatively little variation in spectral properties
– Price (1985):
• PCA on large soil database
• 99.6% of variation in 4 PCs
– Stoner & Baumgardner (1982) defined 5 main soil types:
•
•
•
•
•
organic dominated
minimally altered
iron affected
organic dominated
iron dominated
• Microwave - affects dielectric constant
Soil roughness effects
• Simple models:
– as only a boundary condition, can sometimes use simple
models
• e.g. Lambertian
• e.g. trigonometric (Walthall et al., 1985)
Soil roughness effects
• Smooth surface:
– Fresnel specular reflectance/transmittance
– can be important at microwave
• due to double bounce in forest
– can be important at optical for viewing in close to
specular direction
– Using Stokes vector:
I  R12 I
r
i
Soil roughness effects
• Smooth surface:
 rv12 2

0
R12  
 0

 0
rv12
rh12


0
rh12
2

Imr
0
0
0
0
0
Re rv12r *h12
0
*
r
v12 h12
n2 cos 1  n1 cos 2
n2 cos 1  n1 cos 2
n1 cos 1  n2 cos 2
n1 cos 1  n2 cos 2



 Im rv12r *h12

Re rv12r *h12








n2 sin 2  n1 sin 1
Soil roughness effects
• Low roughness:
– use low magnitude distribution of facets
• apply specular scattering over distribution
– general effect:
• increases angular width of specular peak
Soil roughness effects
• Rough roughness:
– optical surface scattering
• clods, rough ploughing
– use Geometric Optics model (Cierniewski)
– projections/shadowing from protrusions
Soil roughness effects
• Rough roughness:
– optical surface scattering
• Note backscatter reflectance peak (‘hotspot’)
• minimal shadowing
• backscatter peak width increases with increasing roughness
Soil roughness effects
• Rough roughness:
– volumetric scattering
• consider scattering from ‘body’ of soil
– particulate medium
– use RT theory (Hapke - optical)
– modified for surface effects (at different scales of roughness)
Summary
• Introduction
–
–
–
–
Examined rationale for modelling
discussion of RT theory
Scattering from leaves
Stokes vector/Mueller matrix
• Canopy model building blocks
– canopy architecture:
– leaf scattering:
– soil scattering:
area/number, angle, size
spectral & structural
roughness, type, water