at
Optical and Microwave wavelengths applied to vegetation canopies: Part
1
P. Lewis & P. Saich, RSU, Dept. Geography, University College London,
26 Bedford Way, London WC1H 0AP, UK.
Email: plewis@geog.ucl.ac.uk,
psaich@geog.ucl.ac.uk
1. Introduction
1.1 Aim and Scope of these Notes
The purpose of these notes is to introduce concepts of the radiative
transfer approach to modelling scattering of electromagnetic radiation
by vegetation canopies and to review alternative approaches to modelling.
Whilst we concentrate on application to optical and microwave wavelengths
here, the theory is also appropriate to considerations in the thermal regime.
Since the underlying theory was first formulated for stellar atmospheres
(Chandrasekhar, 1960) the theory is of course also applicable to modelling
of atmospheric scattering.
Fung (1994) presents a range of motivations for the development of theoretical
models of scattering such as those presented here. These can be stated
as:
The notes are aimed at MSc students taking the University of London
MSc in Remote Sensing. We have primary educational aims here of enabling
students to gain an understanding of the theory and its applications and
permitting students access to the vast literature in this and related areas.
We aim the notes at non mathematical experts from a wide variety
of disciplines who wish to develop or (more typically) use models for remote
sensing applications, although some understanding of vectors, matrices,
dot products, basic calculus and complex numbers is required. For more
detail and options within the theory, readers are referred in particular
to the following references: Sobelev (1975), Ross (1981), Myneni et
al. (1989), Ulaby and Elachi (1990), and Fung (1994). A major motivation
for these notes is the lack of an introductory treatment of radiative transfer
in other texts that deals explicitly with both optical and microwave wavelengths.
Because of the tendency of researchers to specialise in modelling and applications
in either the optical or microwave domain, education in these areas
tends to lead to a lack of appreciation in both fields of the commonality
of aspects of available approaches. This leads in turn to a lack of exploitation
of synergy of information in these domains. Of course there are many more
approaches, often of more relevance, to modelling canopy scattering than
just radiative transfer. However, since the approach is a point of departure
for several of these, the authors believe this to be a convenient starting
point for the education of future modellers and applications scientists.
As a starting point, we assume students have at least a basic understanding of remote sensing and physical concepts in radiation such as polarised waves. A useful introduction to polarimetry is provided by Ulaby and Elachi (1990) (Chapter 1). Concepts in radiometry are usefully dealt with by Slater (1980) (Chapters 3-5).
1.2 Applicability of the Theory
The radiative transfer (RT) approach (otherwise known as transport theory)
is a heuristic treatment of multiple scattering of radiation which assumes
that there is no correlation between fields considered and so that the
addition of power terms, rather than the addition of fields, is appropriate
(Ulaby and Elachi, 1990). Although diffraction and interference effects
can be included in consideration of scattering from and absorption by single
particles, RT theory does not consider diffraction effects (ibid.,
p. 134). A more accurate, but difficult to formulate, approach is to start
with a consideration of basic differential equations such as Maxwell?s
equations (ibid.; chapter 1; Slater, 1980; p. 55).
We will develop here the radiative transfer equation for the case of
a plane parallel medium (of air) embedded with infinitessimal oriented
scattering objects at low density (leaves, stems etc., and an
underlying soil or other dense medium) ?suspended? in air (a ?turbid medium?).
We consider only absorption and scattering events (i.e. no emission),
We consider the canopy to be of horizontally infinite but vertically finite
extent filled with scattering elements defined continuously over the canopy
space (no explicit gaps which are correlated between any canopy layers).
We will further assume the canopy to be horizontally homogeneous (i.e.
scatterer density is constant over the horizontal extent of the canopy)
although this is not a strict requirement of the theory (see Myneni et
al., 1989; p.6). We will also deal only with a random (Poisson) distribution
of vegetation in detail in these notes. The reader is referred to Myneni
et al. (1989; p. 8) for consideration of other spatial distributions.
Considering only low density canopies (1% or less by volume) of small scatterers
aids the applicability of the no field correlation assumption (scatterers
are assumed to be in the ?far field? of one another). This assumption means
that the theory presented is not directly applicable for dense media such
as snow or sea ice (see Fung, 1994; p. 373 on how to approach this problem).
Assuming the canopy to be in air allows for power absorption by the surrounding
medium to be ignored (Ulaby and Elachi, 1990; p. 136). Assuming horizontal
homogeneity allows us to deal with radiation transport in only one dimension
(a ?1-D solution?), although the theory can be applied to 3-D scattering
problems. Consideration of a medium containing oriented scatterers (developed
for optical wavelengths by Ross, 1981) is appropriate for vegetation canopies
and provides a point of departure from consideration of atmospheric scattering.
See Ulaby and Elachi (1990) pp. 185-186 for a further discussion of the
deficiencies of the radiative transfer approach. Later in the course, we
will review alternative approaches to modelling which overcome some of
these issues.
We will develop both scalar and vector forms of the RT equation. In the microwave domain, waves are not unpolarised, and we consider polarisation by using the vector form. At optical wavelengths, the scalar form is generally used, although the vector form is also used as the basis for many atmospheric examples and e.g. for considering aspects of specular effects from soils and leaves. The vector form provides four coupled intergo-differential equations, whereas a single equation is used in the scalar form.
1.3 Fundamentals of Wave Propagation and Polarization
The starting point for theoretical models for the scattering and propagation
of electromagnetic fields is Maxwell?s Equations (Fung, 1994; chapter 1).
The equations define electric and magnetic fields and magnetic flux density
and electric displacement over time and space, and are coupled to the law
of conservation of charge relating charge and current densities associated
with free charges at the location under consideration. Scattering and propagation
of electromagnetic (EM) waves within a medium is controlled by the permittivity, 
,
permeability, 
, and conductivity, 
,
of the medium.
An important point about the use of Maxwell's equations (and wave theory
more generally) is that it is typically encountered in one of two ways:-
(i) as a way of determining the total scattered field from an ensemble
of scattering elements (which is effectively the "exact" solution) and
(ii) as a way of determining the scattering properties of individual discrete
objects (in order to embed these into some other solution to the scattering
problem). The first of these is the strategy one would use in the design
of (e.g.) antenna, where the propagation and interactions of electric fields
is critical.
In a source free medium, Maxwell?s equations can be combined to give
the wave equation. For a wave travelling along the z direction,
if the electric and magnetic fields are assumed not to be functions of
x and y (Ulaby and Elachi, 1990; p. 3):
(1.1)
k is the wavenumber in the medium, which is equal to 2p/l
in air (denoted k0) or more generally 
,
where l is the wavelength of the radiation in
air. E is described by vertically and horizontally polarised components,
denoted by subscripts v and h respectively. The exponential
term can be considered as describing the phase of the wave.
We consider first scattering by an incident electric field Ei(r)
of magnitude Ei (a plane wave) propagating in
a direction defined by a unit vector 
to
a position r (Ulaby and Elachi, 1990; chapter 1):
(1.2)
We can represent the incident and scattered field relationship through
a scattering matrix 
for
a scattered plane wave (an approximation to the scattered spherical wave
over the small aperture of the receiving antenna) (Ulaby and Elachi, 1990;
p.21):
(1.3)
The terms in the matrix are polarised scattering amplitudes, which will
in general depend on the orientation of the scatterers (see below) and
the incident and scattered directions of propagation.
As noted above, in remote sensing of vegetation, however, we often exploit
the simpler radiative transfer theory, which deals only with the propagation
of energy (intensities). However, we still encounter the solution to Maxwell's
equations at microwave wavelengths when we wish to know how individual
particles scatter energy, since ultimately this is used to build up the
total scattering from an ensemble of scatterers.
In our context, Maxwell's equations effectively boil down to an equation
to be solved, that defines the electric field scattered (
)
by a particle of given size and shape, when there is some electric field
incident on it:
(1.4)
is a unit polarization
vector, and 
is the internal
field of the scatterer. e is the relative
dielectric constant of the scatterer 
where 
and 
are
the real and imaginary parts of the term, defined as functions of the medium
permittivity, , and conductivity, and
the permittivity of free space (air), 
. 
is
the angular frequency of the wave 
for
wavelength l. As we are ultimately interested
only in electric fields of either horizontal or vertical polarisation,
the dot products allows us to consider only the component of the electric
field in these directions. 
represents
waves propagating from points across the scatterer over the volume V
under consideration from location 
to 
(c.f.
equation 1.2).
The meaning of the whole equation is therefore, that this component
of the scattered electric field (in a particular polarisation) is given
by an integration over the volume of an individual scatterer. The integration
depends upon the polarised strength of the field internal to the scatterer
(
). The other term in the integrand
represents waves propagating from lots of points across the scatterer -
the integration therefore adds all of these up.
In order to solve this equation, we therefore need to do two things:
(i) specify the internal field, and (ii) perform the integration. Neither
of these is easy (analytically) and it is at this stage that we usually
encounter simplifications. Typical of these are:
For example, in the latter of these, we assume that the far-field is:
(1.5)
is
a particular direction of propagation. Under this approximation, the scattered
wave looks like a spherical wave located at the object origin. The integral
solution takes the form
(1.6)In this case, the integration is much simpler. Notice now that the scattered
field has an approximate k02 dependence (i.e.
1/l2) though this is in the
long wavelength limit - there are other important terms in the integration.
Note also that the complex permittivity e determines
the strength of the scattered field. This term depends on the frequency
of the field and (effectively) on the moisture content of the object. We
return to this dependence later.
We can show that (e.g.) for thin discs, the polarised scattering amplitudes
Spq (seen in equation 1.4) can be closely related to
the Rayleigh approximation (see later), but with an additional factor related
to the shape and size of the scatterer (compared with the incident wavelength)
which we can view as a correction to the Rayleigh approximation for when
the wavelength is no longer much larger than the scatterer size.
for discs:
(1.7)
(1.8)
Here, V is the volume of the object, and both functions 
and 
and
the final scattering amplitudes are functions of the orientation of the
scatterer with respect to the incident wave, the incident wavelength, and
the permittivity of the object. They also take slightly different forms
for different combinations of incident and scattered field polarisations.
1.4 Modified Stokes Vector and Mueller Matrix Representations
The polarisation state of a plane wave can be represented by horizontally
and vertically polarised magnitudes
and
and a phase factor term as
in equation 1. This latter term can also be described by a phase term d,
being equal for horizontal and vertical components for a linear polarised
wave (Ulaby and Elachi, 1990; p.4). It is found to be generally more convenient
for completely polarised waves to represent the electric field by a modified
Stokes vector (Ulaby and Elachi; pp.11-13). We define the modified Stokes
vector Fm:
(1.7)
where Re() and Im() represent the real and imaginary parts of the bracketed
terms respectively and the electric field components are understood to
include the magnitude and phase terms given in equations 1.1 and 1.2. Superscript
* represents a complex conjugate.
Using this representation of the electric field, a scattered (spherical)
wave Stokes vector 
can be
related to an incident wave vector 
through
the use of a Mueller matrix 
(Ulaby
and Elachi, 1990; p. 25).
(1.8)
where:
(1.9)
(1.9)
The terms in 
are the
complex scattering terms defined in equation 1.3. Note that when the Mueller
matrix is averaged over all scatterer orientations and sizes, it becomes
the phase matrix that we shall use in radiative transfer.
2. Building Blocks for a Canopy Reflectance Model
In this section, we develop the ?building blocks? that we will require to describe scattering from a vegetation canopy using RT theory. This involves (i) a description of canopy architecture; (ii) a description of the scattering properties of vegetation as a function of wavelength (and polarisation); (iii) a description of scattering by an underlying (e.g. soil) surface.
2.1 Description of Canopy Architecture
For the present, we consider for simplicity a canopy composed of only
leaf elements. In canopies comprising e.g. a mixture of leaf and
branch elements, the following can be generalised by averaging effects
over proportionate distributions of these. Under the assumptions given
above, we can describe the canopy using only two terms (Myneni et al.,
1989; p. 8):
(m2/m3)
OR
the vertical leaf number density function, Nv(z) (number of particles per m3)
(dimensionless).All terms can be defined as a function of depth from the top of the
canopy (z).
2.1.1 Leaf Area/Number Density
, the leaf area density function,
describes the vertical profile of one-sided leaf area density (m2
of leaf area per m3 of volume). This term is typically more
convenient for optical vegetation applications, where we are typically
concerned with projection of leaf areas as the scatterers are large compared
to the wavelength under consideration. For a constant leaf area, Al,
it is related to Nv(z) by:
(2.1)
A number density is a more convenient description for dealing with extinction
and scattering by individual ?particles?, as in atmospheric radiative transfer
or when dealing with vegetation at microwave wavelengths. The integral
of over the vertical extent
of the canopy (H) is known as the leaf area index (LAI), L
(unitless) - one sided leaf area (m2) per unit of ground area
(m2):
(2.2)
Figure 2.1 Turbid Medium Representation of Canopy
There are many models of (or
Nv(z)) we could apply. Many canopies tend to have
a higher lead area density towards the top of the canopy (Myneni et
al., 1989). This can either be modelled as a continuous function or
by considering scattering between canopy layers. The simplest form of canopy
scattering model is obtained by assuming that density is constant over
canopy height (
).
Figure 2.2 Leaf Geometry
showing normal vector and spherical and Cartesian representations
2.1.2 Leaf Angle Distribution
The leaf normal orientation distribution function (?leaf angle distribution?)
is defined as a function of the leaf normal vector 
.
It is defined so that its integral over the upper hemisphere (
sr) is unity:
(2.3)
Myneni et al. (1989; pp. 13-23) describe a range of methods for
the measurement of leaf angle distribution. Many models of leaf angle distribution
can be applied. It is most typical to assume that the leaf normal azimuthal
distribution is independent of the leaf zenith angle distribution (inclination
to the local vertical), i.e., 
, 
.
Most often, the azimuthal distribution is assumed to be uniform (
).
This allows for a simpler description of the inclination function of 
as 
, 
.
Strebl et al. (1985) and Otterman (1990) suggest however that this
may often not be a valid assumption, citing significant diurnal variations
in the azimuthal dependance for cotton and soybean crops (Kimes and Kirchner,
1982), as well as intrinsic azimuthal variations seen in tree species such
as balsam fir and other factors such as wind, stress effects and heliotropism
(Ross, 1981). Goel and Strebel (1984) suggest that for some canopies (e.g.
soybean) the assumption of independent azimuthal and inclination angles
may be valid, although for others (e.g. fir tree needles) it is
not.
Typically, we tend to use either simple parameterised distributions
or ?archetype? distributions (figure 2.3, after Ross, 1981):
);
);
;
);
Figure 2.3 Archetype Leaf Angle Distributions
Several of these have simple mathematical forms and mathematically convenient
solutions for average projection and (scalar) scattering functions.
A typical parameterised model distribution is the elliptical function
(Nilson and Kuusk, 1989), which provides a convenient and flexible representation:
where
is a scaling factor (see
equation 2.2). This distribution is defined through the two parameters:
¾
eccentricity of distribution : 
¾
modal leaf angle : 
The distribution can be visualised as the equivalent of ?pasting? the
leaves over an ellipse of eccentricity (zero
eccentricity is a spherical leaf angle distribution ¾
;
and eccentricity of 1 is a ?needle? ¾
an erectophile distribution for
and planophile for 
), inclined
so that most common angle is described by .
Note that the sensitivity of the function to increases
with increasing ¾
for 
, the function is independent
of , for 
,
we essentially have all leaves inclined at a single angle ().
Because of this sensitivity, eccentricity is often used in a near-linearised
sense as 
.
Figure 2.4 Elliptical leaf angle distributions:
e =0.9; q m=0 (erectophile), p /2 (planophile), p /4 (plagiophile)
Other typical parameterised models include the trigonometric model (Bunnik,
1978) and Goudriaan?s model (Goudriaan, 1977). Note that although we may
generally call a particular parameterised distribution by its closest archetype
name (e.g. figure 2.4) these distributions are not generally exactly
equivalent.
2.1.3 Leaf Dimensions
Without modification, RT theory assumes the scatterers to be of infinitesimal
dimensions (point scatterers), thus no definition of leaf size is formally
required. However, at microwave wavelengths, scattering by objects depends
significantly on the dimensions of the leaves, so we must further define
leaf size. Simple analytical equations for scattering in the microwave
domain are only available for simple geometric primitives (e.g.
cylinders, needles, discs) (Fung, 1994; pp 451-473), so leaf dimensions
need to be generalised into equivalents for these forms. As apects of leaf
dimension relate mainly to scattering from individual objects, this is
dealt with below in section 2.2.
At optical wavelengths, scatterer dimensions can also significantly
impact canopy reflectance in a certain viewing/illumination configuration
(the ?hot spot? effect, see below) which can be accounted for by modification
of RT theory. In this case, it is not absolute leaf dimension that is important,
but rather leaf size (typically a disc radius) relative to canopy height
(Nilson and Kuusk, 1989). At optical wavelengths, we may therefore include
a relative leaf size parameter in the formulation. Leaf thickness can also
be of importance at optical wavelengths, although this is most typically
modelled as part of the leaf reflectance and transmittance function. Leaf
thickness, often defined as a leaf complexity term (Jacquemoud and Baret
1990) essentially affects the ratio of diffuse reflectance to transmittance
of the leaf, a thick/complex leaf having reduced transmittance.
We can generally include a distribution of (absolute, but generalised) element sizes, although this is rarely done for optical models..
2.2 Canopy element and soil spectral properties
In building a canopy scattering model, we consider leaves (and branches)
to be the main canopy ?primitives? and, in these notes, apply the radiative
transfer equation to describing scattering from an ensemble of primitives.
This section therefore describes the spectral and polarisation behaviour
of vegetation. Since a canopy is typically (lower) bounded by a soil medium,
this section also deals with spectral and structural effects of soil scattering.
2.2.1 Scattering properties of leaves
Leaf scattering properties are largely determined by: Leaf surface properties and internal structure; leaf biochemistry; leaf size (essentially thickness, for a given LAI).
Scattering at optical wavelengths occurs within a leaf occurs
at air-cell interfaces within the leaf - at refractive index differences
which have an overall effect of diffusing the incoming radiation. The refractive
index is seen to vary slowly with wavelength, decreasing essentially linearly
from around 1.5 at 400 nm to around 1.3 at 2500 nm (Jacquemoud et al.,
1996). The amount of scattering is seen to depend on the total area of
cell wall-air interfaces (Myneni et al., 1989; p.33). This can be
considered to related to the complexity of the internal structure of a
leaf (Jacquemoud and Baret, 1990): the more complex the structure (or the
thicker the leaf) the more scattering per unit area, the more diffuse the
radiation field (in general), and the lower the transmittance.
In addition, scattering occurs at the surface of a leaf. For optically smooth waxy leaves, a strong specular peak is noted in reflectance. As the leaf surface ?roughness? (due to hairs, spines etc.) increases, the specular peak becomes broader (more diffuse) (see e.g. graphs in Myneni et al., 1989; p.37).
The main pigments are chlorophyll a and b, a-carotene,
and xanthophyll (Myneni et al., 1989; p. 31), which absorb in the
blue part of the spectrum (around 445 nm). Chlorophyll also absorbs radiation
in the red, around 645 nm. Radiation absorbed by pigments may be converted
into heat energy, flourescence or converted into carbohydrates through
photosynthesis. Leaf water is a major consituent of leaf fresh weight,
representing around 66% averaged over a large number of leaf types (Jacquemoud
et al., 1996). Due to its importance, water is considered separately
below. The remainder of leaf weight is ?dry matter? mainly composed of
cellulose, lignin, protein, starch and minerals. Absorptance by these constituents
increases with increasing concentration, thereby reducing leaf reflectance
and transmittance at these wavelengths.
Various models of the dependence of leaf reflectance, transmittance
(and their sum, single scattering albedo) exist, though they typically
model only a hemispherical integral of reflectance/transmittance. Nilson
and Kuusk (1989) model scattering from the leaf surface by a Fresnel term,
simulating roughness effects with a facet model. A common model for flowering
plants is the PROSPECT model of Jacquemoud and Baret (1990) which models
leaf chlorophyll and water absoption effects in a leaf idealised to N elemental
?layers? using a ?plate model? (figure 2.8). In this approach, each ?layer?
(the solution can be generalised to non-integer numbers of layers) is characterised
by a discontinuity refractive index and an absorption coefficient defined
by the leaf biochemical composition. Jacquemoud et al. (1996) further
developed this model to include other leaf constituents such as lignin
and cellulose, and demonstrate how the model can be used to estimate biochemical
composition from measured leaf spectra. A similar model, developed to simulate
scattering by needle leaves is LIBERTY (Dawson, et al., 1997).
Figure 2.6 Main regions and factors of leaf absorption
(a) Chlorophyll |
(b) Cellulose + lignin |
(c) Protein |
(d) Water |
(a) chlorophyll; (b) cellulose + lignin; (c) protein; (d) water
(from PROSPECT-redux model: Jacquemoud et al., 1996)
 |
 |
At wavelengths used in active microwave sensing, water content, expressed
as VMC or Gravimetric Moisture Content (GMC) can be related to leaf permittivity,
e. The VMC relationship is given by (Fung,
1994; pp. 524-525) through the following:
The nondispersive residual component of the dielectric constant en:
(2.4)
(2.5)
(2.6)
(2.4)
where 
is the iconic conductivity
of free water (in Siemens/m), given by Fung (1994;p. 524) as 1.0, but as
1.27 (for X-band (10 GHz) rubber and oil palm leaf modelling) by Chuah
et al. (1995), and f is the frequency in GHz. Chuah et
al. (1995) demonstrate a good correspondence between the model and
their measurements, as further validation of the original work by El-Rayes
and Ulaby (1987). The general trend of the model is that dielectric constant
increases with increasing water content. Chuah et al. (1995) provide
useful reference to a range of other measurement and modelling studies
on leaf dielectric constant.
At microwave wavelengths, we can note a dependence on leaf dimension of the scattering behaviour of leaves. There is a direct dependence of object volume on scattering from simple geometric primitive leaf prototypes such as discs, spheres and needles (see section on Rayleigh). Thus, even if we characterise the canopy in terms of normalised leaf area (LAI) rather than the more usual number density, we still see an increase in scattering from a leaf ?primitive? as a linear function of leaf thickness (volume divided by area) for a given LAI.
For all but the most optically thick canopies, for which the canopy
medium may be considered semi-infinite, we require a definition of scattering
by a lower boundary surface. Although this may be, for example, snow covered,
we consider the more general case of a lower bounding soil surface here.
The main soil properties governing soil scattering are similar for both
the optical and microwave case, namely: soil moisture content, soil type/texture,
and soil surface roughness.
2.3.1 Soil moisture
At optical wavelengths, the general effect of increasing soil (near
surface) moisture is to decrease soil reflectance. The effect is similar
in proportion across the optical spectrum, being enhanced only in the water
absorption bands.
At microwave wavelengths, the main effect of increasing soil moisture
is to increase the soil dielectric constant. This effect, however, varies
with wavelength. As such, increasing soil moisture generally increases
the volumetric scattering from the soil. Longer wavelength microwaves penetrate
further into the soil and are therefore sensitive to soil moisture at lower
depths. Decreasing the incidence angle, however, increases the path length
in the soil and so decreases the penetration depth. Additionally, increasing
the soil moisture increases the extinction coefficient and also decreases
penetration depth.
Figure 2.9 Response of soil reflectance to soil moistute
2.3.2 Soil type/texture
The soil type or texture essentially controls the ?spectral? behaviour
of soils at optical wavelengths. Perhaps surprisingly, there is generally
very little variation in soil spectra: Price (1985) performed a principle
components analysis on more than 500 soils from different regions and found
that only four spectral basis functions were required to describe 99.6%
of the variation in the soil spectra. Stoner and Baumgardner (1982) classified
around 500 soil samples from the USA and Brazil and found only five mineral
soil spectral curve types (organic dominated, minimally altered, iron affected,
organic dominated and iron dominated). These and other results suggest
that only a simple parameterisation of soil spectral reflectance (single
scattering albedo) is required for canopy reflectance models at optical
wavelengths. The basis functions of Price (1985) are well-suited to this
purpose.
2.3.3 Soil roughness effects
For a smooth (relative to the wavelength of the radiation) soil surface,
lower boundary scattering is simply a Fresnel specular term. This can be
important in treating multiple interactions in the microwave case, particularly
in forest canopies where the specular scattered field can interact with
vertical tree trunks to produce an enhanced backscatter effect. For smooth
surface at optical wavelengths, the effect will not generally be significant,
except very close to the viewing geometries in the specular direction.
Using the Stokes vector, we can relate the incident Ii
and reflected Ir Stokes vectors via:
(2.4)
(2.5)
and:
(2.6)
where 
is the incidence
angle, 
is the refraction
angle, n1 and n2 are the refractive
indices of medium 1 (air: n1=1) and medium 2 (soil) respectively
(Ulaby and Elachi, 1990). The incidence and refraction angles are related
by Snell?s law:
(2.7)
When we are not concerned with polarisation effects (e.g., typically
at optical wavelengths), we can simply use the Fresnel power reflectance
(Shepard et al., 1993):
In many canopy models, a very simple soil scattering model is all that
is required. Typical examples at optical wavelengths are either to simply
assume a Lambertian (perfectly diffuse) lower boundary, or to use a simple
empirical model such as the modified Walthall model used by Nilson and
Kuusk (1989), parameterised by fitting it to soils with a range of roughnesses.
For low (but not zero) roughness, specular effects can be incorporated
at by using a distribution of facet orientations and performing an integral
over the distribution (Shepard et al., 1993). At optical wavelengths,
increasing RMS roughness increases the angular width of the specular reflectance
peak. The same effect is generally true at microwave wavelengths.
For rougher surfaces, such as where there are surface clods or rough
ploughing effects, we can use geometric optics models at optical wavelengths
for surface scattering. Here, the main phenomenon is protrusion projection
and shadow-casting effects. There are few models of soil reflectance which
deal explicitly with anisotropic roughness effects. One study which does
is the numerical model of Cooper and Smith (1985) which calculates scattering
from an arbitrary geometry surface defined using rectangular or triangular
facets over an explicitly-defined height field. More recent numerical models
such as that of Lewis (1999) can also simulate scattering from such surfaces
in investigating soil or topographic effects (Burgess et al., 19XX). Various
simpler optical surface scattering models exist, such as the ?clod? macrosctructure
model of Cierniewski (1987) that models roughness (projection and shadowing)
effects for a distribution of spheres or other simple forms on a planar
surface. In this model, roughness is parameterised as the ratio of the
projected area of a sphere to the square of the sphere spacing and the
cosine of the slope angle.
Volumetric scattering from soils is generally treated through modifications
of radiative transfer theory at optical wavelengths. Hapke (1981) and Hapke
and Welles (1981) develop a set of models for scattering from particulate
soil-like (e.g. lunar) surfaces using such an approach. The model
is modified in Hapke (1984) to deal with multiple scales of surface roughness
and in Hapke (1986) to deal more explicitly with the soil extinction coefficient
and the so-called ?hot spot? effect. This latter term can be important
for rough soil surfaces, as it can be considered to arise from a decrease
in shadow hiding in and around the retro-reflection (?backcsatter?) direction.
This results in a peak in reflectance in this region, which increases in
angular width with increasing roughness. The half width is shown by Hapke
(1986) to be equal to the ratio of average particle radius to extinction
length at unit slant path optical depth: effectively a normalised roughness
measure. This effect is clearly seen in geometric optics approaches that
explicitly consider shadowing.
There are various approaches to modelling surface ?roughness? scattering and volumetric scattering terms at microwave wavelengths. As with optical wavelengths, the rougher the soil surface, the more diffuse the surface scattering. Examples of approaches are the Kirchhoff approximation and the small perturbation model (see chapter 4 of Ulaby and Elachi (1990) and chapter 2 of Fung (1994)).
References
Chandrasekhar, S., 1960, "Radiative Transfer", Dover, New York, USA.
Fröhlich, C. and Shaw, G.E.,1980, "New determination of Rayleigh scattering in the terrestrial atmosphere", Applied Optics, 19(11), 1773-1775.
Fung, A.K., 1994, "Microwave Scattering and Emission Models and Their Applications", Artech House, Norwood MA, USA.
Goel and Streble, 1984
Goudriaan, 1977
Myneni, R.B., Ross, J., and Asrar, G., 1989, "A Review of the Theory of Photon Transport in Leaf Canopies", Agriculture and Forest Meteorology, 45, 1-153.
Nilson and Kuusk, 1989
Otterman, 1990
Ross, J., 1981, "The Radiation Regime and The Architecture of Plant Stands", Dr. W. Junk Publ., The Netherlands.
Slater, P.N., 1980, "Remote Sensing: Optics and Optical Systems", Addison-Wesley, Reading, MA, USA.
Sobolev
Strebl et al 1985
Ulaby, F.T. and Elachi, C. (eds), 1990, "Radar Polarimetry for Geoscience
Applications", Artech House, Norwood MA, USA.
Jacquemoud, S., Ustin, S.L., Verdebout, J., Schmuck, G., Andreoli, G., and Hosgood, B., 1996, "Estimating leaf biochemsitry using the PROSPECT leaf optical properties model", Remote Sensing of Environment, 56, 194-202.
Jacquemoud, S., and Baret, F., 1990, "PROSPECT: a model of leaf optical properties spectra", Remote Sensing of Environment, 34, 75-91.
El-Rayes, M.A., and Ulaby, F.T., 1987, "Microwave dielectric spectrum of vegetation - Part 1: Experimental observations", IEEE Trans. Geoscience and Remote Sensing, GE-25, 541-549.
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
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Appendix 1 - Some Notes on Maths
There are many examples in developing RT theory where we need to integrate
projections, fluxes etc. over a range of directions. We typically
use vector notation for this where we integrate (?sum?) over infinitessimal
solid angles 
around direction 
.
To solve the integral, we replace the vector 
by
a spherical coordinate representation, 
.
For example, the leaf projection function (?G-function?) in direction 
for
a spherical leaf angle distribution (equation 2.X) where 
is:
Figure A.1.1 Spherical coordinate geometry