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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 kˆr E i i ik kˆr 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 kˆr E i i ik kˆr 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 zH 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 2l 8 15 g l l cos 2 2l 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 Imr 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