Radiative Transfer Theory at Optical and Microwave wavelengths applied to vegetation canopies: part...

Radiative Transfer Theory at Optical and Microwave wavelengths applied to

vegetation canopies: part 1

UoL MSc Remote Sensing

course tutors: Dr Lewis plewis@geog.ucl.ac.uk

Dr Saich psaich@geog.ucl.ac.uk

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, 75-91.

• 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

• k - wavenumber = 2/ in air

02 zEkdz

zEd ikz

h

v eE

EzE

Plane wave

Radiative Transfer Theory

• Consider incident Electric-field Ei(r) of magnitude Ei in direction to a position r:

• 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

k̂

rkikirkik

ih

ivi eEe

E

ErE

ˆˆ

Scattering

• Define using scattering matrix:

• elements polarised scattering amplitudes– for discs:

– for needles:

• assume scattering in far field

i

hhvhv

vhvvrik

irik

s ESS

SS

r

eES

r

eE

00

x

xJnorientatio

VkS dpq

120 0.2,

4

1

x

xnorientatio

VkS npq

sin,

4

120

Scattering

x

xJnorientatio

VkS dpq

120 0.2,

4

1

Bessel function

(complex) permittivity of leaf

Leaf volumeWavenumber2 = 42/2

Scattering

x

xnorientatio

VkS npq

sin,

4

120

Sinc function

Stokes Vector

• Can represent plane wave polarisation by , and phase term:

• h,v phase equal for linear polarised wave

ihEi

vE

rkikirkik

ih

ivi eEe

E

ErE

ˆˆ

Stokes Vector

• More convenient to use modified Stokes vector:

*

*

2

2

Im2

Re2

hv

hv

h

v

h

v

m

EE

EE

E

E

V

U

I

I

F

Stokes Vector

• Using this, relate scattered Stokes vector to incident:

im

im

rm FW

rF

rF 1

22

11

hhvvvhvvhhhvhhvvv

vhhvvvhhvhhhvvhv

hhhvhvhhhhhhhvhv

vhvvvvvhvhvhvvvv

SSSSSSSS

SSSSSSSS

SSSSSSSS

SSSSSSSS

W

****

****

****

****

ii00

1100

0011

0011

N.B S2 so 1/4 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

Soil

H

zCanopy

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. Vertical leaf area density (m2/m3)

OR

the vertical leaf number density function, Nv(z) (number of particles per m3)

2. the leaf normal orientation distribution function, (dimensionless).

3. 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.

zul

Canopy Architecture

• Leaf area / number density– (one-sided) m2 leaf per m3

– Nv(z) - number of ‘particles’ per m3

zul

lvl AzNzu

dzzuLHz

z

l

0

LAI

l

x

z

y

ql

fl

Inclination to vertical

azimuth

Leaf normal vector

Canopy Architecture• Leaf Angle Distribution

12

lll dg

• Archetype Distributions:planophile

erectophile

spherical

plagiophile

extremophile

Leaf Angle Distribution

lllg 2cos3

lllg 2sin2

3

1llg

lllg 2sin8

15 2

lllg 2cos7

15 2

• Archetype Distributions:

Leaf Angle Distribution

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 10 20 30 40 50 60 70 80 90

g_

l(t

he

ta

_l)

leaf zenith angle / degrees

spherical planophile erectophileplagiophile extremophile

• Elliptical Distribution:

Leaf Angle Distribution

2122 sin1 ml

llg

eccentricity of distribution : 10mmodal leaf angle : 2

0m

• Elliptical Distribution:Leaf Angle Distribution

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 10 20 30 40 50 60 70 80 90

g_

l(t

he

ta

_l)

leaf zenith angle / degrees

erectophile planophile plagiophile

Elliptical leaf angle distributions:=0.9; qm=0 (erectophile), /2 (planophile), /4 (plagiophile)

• 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

Leaf Dimension

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

Dicotyledon leaf structure

opticalSpecular

from surface

Smooth (waxy) surface- strong peak

hairs, spines- more diffused

Scattering properties of leaves

• Leaf surface properties and internal structure

Dicotyledon leaf structure

opticalDiffused

from scattering at internal air-cell wall interfaces

Depends on refractive index:varies: 1.5@400 nm

1.3@2500nmDepends on total areaof cell wall interfaces

Scattering properties of leaves

• Leaf surface properties and internal structure

Dicotyledon leaf structure

optical

More complex structure (or thickness):- more scattering- lower transmittance- more diffuse

Scattering properties of leaves

• Leaf surface properties and internal structure

Dicotyledon leaf structure

microwave

Thickness (higher volume)- higher scattering

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, -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, .

25.62.37.1 vvn MM

166.082.0 vvf MMvf

2

2

5.591

4.31

v

vb M

Mvf

18.01

559.2

18

181

759.4

fi

vff

if

ivfM bfnv

Volume fractions

Offset factor

Scattering properties of leaves

• Microwave:– water content related to leaf permittivity, .

18.01

559.2

18

181

759.4

fi

vff

if

ivfM bfnv

Frequency / GHz

iconic conductivity of free water

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:

ir IRI12

Soil roughness effects• Smooth surface:

12*

1212*

12

12*

1212*

12

2

12

2

12

12

ReIm00

ImRe00

000

000

hvhv

hvhv

h

v

rrrr

rrrr

r

r

R

2211

221112

2112

211212

coscos

coscos

coscos

coscos

nn

nnr

nn

nnr

h

v

1122 sinsin nn

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: area/number, angle, size– leaf scattering: spectral & structural– soil scattering: roughness, type, water