# Microwave emission of vegetation: sensitivity to leaf characteristics

J.-P. Wigneron, Jean-Christophe Calvet, Yann H. Kerr, A. Chanzy, A. Lopes

► To cite this version:

J.-P. Wigneron, Jean-Christophe Calvet, Yann H. Kerr, A. Chanzy, A. Lopes. Microwave emission of vegetation: sensitivity to leaf characteristics. IEEE Transactions on Geoscience and Remote Sensing, 1993, 31 (3), pp.716-726. 10.1109/36.225537 meteo-05337483

HAL Id: meteo-05337483

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Distributed under a Creative Commons Attribution 4.0 International License
# Microwave Emission of Vegetation: Sensitivity to Leaf Characteristics

Jean-Pierre Wigneron, Jean-Christophe Calvet, Yann Kerr, Member, IEEE, André Chanzy, and Armand Lopes

***Abstract***—This paper presents an analysis of leaf-characteristics effects on the microwave emission of land surfaces. In order to simulate these effects, a radiative transfer model is presented. The medium consists of a vegetated layer containing randomly oriented leaves, modeled as elliptic-shaped scatterers, over the ground surface. Radiative transfer equations are solved with a discrete ordinate-eigenanalysis method. The calculation of the phase matrix of the elliptic scatterers is based on the generalized Rayleigh-Gans approximation which increases the frequency range of the modeling.

The sensitivity of brightness temperature and polarization ratio to leaf characteristics, volume fraction, gravimetric moisture, size, shape, and inclination distribution is investigated at L-, C-, and X-bands. The behavior of the simulated emission of a soybean canopy versus frequency and incidence angle is studied for different soil moisture levels. These computations are consistent with measurements performed with a multifrequency radiometer. Up to 10 GHz the microwave emission appears to contain significant information on underlying soil moisture.

## I. INTRODUCTION

Many parameters influence the microwave emission of land surfaces, the most important being soil and vegetation water content, vegetation structure and biomass, surface temperature, and soil roughness. Theoretical models can be most useful to understand the effect of these different parameters on the microwave remotely sensed data. The microwave thermal emission of vegetation-covered soil can be divided into three main components: the upward canopy emission, downward and soil reflected canopy emission, upward soil emission [1]–[3]. The vegetation canopy attenuates and scatters these radiations. Therefore, theoretical models which have been developed to simulate the thermal emission of vegetation canopies have taken into account volume absorption and scattering by the discrete scatterers of the vegetation: leaves, branches and stalks.

Volume scattering can be modeled through two types of approaches: the wave approach and the radiative transfer approach. Even though the radiative transfer approach does not take into account diffraction effects in the computation

of multiple scattering contributions and considers only far-field interactions, it is more appropriate for modeling strongly scattering media like vegetation canopies [4].

The vegetated canopy can also be modeled through two types of approaches: either as a continuous random medium [5], [6] (the so-called continuous approach) or as an homogeneous medium containing discrete scatterers (discrete approach). In the discrete approach every contribution to the microwave emission (from soil, leaves, branches) is precisely described. So this approach is useful to better understand and analyze the sensitivity of the canopy microwave thermal emission to soil and vegetation characteristics.

The numerous discrete models which have been developed describe more and more precisely the vegetation structure. In developing the discrete approach for passive microwave remote sensing, England [7] solved the radiative transfer equations for a half-space medium containing Rayleigh point scatterers. Tsang and Kong [8] solved the same equations for spherical scatterers using Mie scattering phase functions. Scattering from randomly oriented discs or circular cylinders with application to active remote sensing of vegetation was described by Karam and Fung [9]. Choe and Tsang [10] solved radiative transfer equations for a one-layer medium containing both circular-shaped discs (leaves) and cylinders (branches) and derived the thermal emission of crop canopies. Lang et al. [11] modeled passive emission of a soybean canopy using the Peake approach which relates the bistatic scattering coefficients of vegetation to its emissivity. Vegetation is modeled as a half-space medium containing lossy dielectric discs.

In this study the vegetation canopy is modeled as a one-layer medium containing randomly oriented leaves. The radiative transfer equations are solved using a discrete ordinate-eigenanalysis method [6], [12]. The leaf-scattering modeling used in this study is based on the model developed by Karam and Fung [13] for active microwaves. The modeling of leaf scattering is significantly improved with regards to other approaches since:

* The scattering amplitude of the leaves is computed with the generalized Rayleigh-Gans (GRG) approximation which is valid in the frequency region where leaf dimensions are of the order of the emitted wavelength.

* Scattering is accounted for in the extinction formulation since the optical theorem accounts only for absorption under low frequency approximations.

* Leaves are modeled as elliptic rather than circular-shaped scatterers.

The model presented in this paper is developed in order

Manuscript received April 17, 1992; revised August 24, 1992; rerevised November 25, 1992.

J.-P. Wigneron is with INRA, Bioclimatologie, BP91, 84143 Montfavet cedex, France.

J.-C. Calvet is with Meteo-France/CNRM, Toulouse, France.

Y. Kerr is with LERTS, CNES-CNRS, 31055 Toulouse cedex, France.

A. Chanzy is with INRA, Science du sol, BP91, 84143 Montfavet cedex, France.

A. Lopes is with CESR, UMR-CNRS BP4346, 31029 Toulouse cedex, France.

IEEE Log Number 9207390.
to study the effects of leaf parameters such as volume fraction, gravimetric moisture, dimensions, and distribution of orientation, on the microwave thermal emission of a vegetation canopy. Experimental data consist of multifrequency microwave radiometric measurements over a soybean field. Measurements were performed with the multifrequency PORTOS radiometer [14]. The most significant parameters influencing the thermal emission of a soybean field are investigated and discussed at L-, C-, and X-bands in order to better define the potential use of a sensor at such frequencies. The frequency and incidence angle dependence of brightness temperature are discussed and compared to actual radiometric measurements (PORTOS).

## II. MODEL FORMULATION

### A. Solving the Radiative Transfer Equations

A numerical quadrature method [6] is used in this model to solve the radiative transfer equations for a vegetation layer limited by two boundaries at the soil and atmosphere interfaces. This three-layer medium is noted as depicted in Fig. 1(a) and the subscripts 0, 1, and 2 will refer respectively to atmosphere, vegetation, and soil. Within the canopy layer, the radiative transfer equations for the intensity T (expressed as a brightness temperature) traveling in the direction $\theta$, $0 < \theta < \pi/2$, can be written:

$$ 
\begin{aligned}
\mu \cdot dT_v(\theta, z)/dz &= Ka \cdot T_1(z) - Ke_v(\theta) \cdot T_v(\theta, z) \\
&\quad + \int_{-1}^{1} ((v, v') \cdot T_v(\theta', z) + (v, h') \cdot T_h(\theta', z)) d\mu' \quad (1) \\
&\quad - \mu \cdot dT_v(\pi - \theta, z)/dz \\
&= Ka \cdot T_1(z) - Ke_v(\pi - \theta) \cdot T_v(\pi - \theta, z) \\
&\quad + \int_{-1}^{1} ((v(\pi - \theta), v') \cdot T_v(\theta', z) \\
&\quad + (v(\pi - \theta), h') \cdot T_h(\theta', z)) d\mu'
\end{aligned}
\quad (2)
$$

with $\theta = \cos^{-1}(\mu)$, $\theta' = \cos^{-1}(\mu')$; the subscripts v and h denote the vertical and horizontal polarizations (equations are written for the vertical polarization and are identical for the horizontal polarization); Ka, Ke are the absorption and extinction coefficients of the scatterers; (v, v'), (v, h') are scattering phase functions.

The radiative transfer equations are solved with the Gauss quadrature method and (1) and (2) become:

$$ 
\begin{aligned}
\mu_j \cdot dT_{v_j}(\theta, z)/dz &= Ka \cdot T_1(z) - Ke_{v_j}(\theta) \cdot T_{v_j}(\theta, z) \\
&\quad + \sum_{k=1}^{n} a_k \cdot (((v_j, v_k) \cdot T_{vk} + (v_j, v_{-k}) \cdot T_{v-k}) \\
&\quad + ((v_j, h_k) \cdot T_{hk} + (v_j, h_{-k}) \cdot T_{h-k}))
\end{aligned}
\quad (3)
$$

$$ 
\begin{aligned}
\mu_{-j} \cdot dT_{v-j}(\theta, z)/dz &= Ka \cdot T_1(z) - Ke_{v_j}(\theta) \cdot T_{v-j}(\theta, z) \\
&\quad + \sum_{k=1}^{n} a_k \cdot (((v_{-j}, v_k) \cdot T_{vk} + (v_{-j}, v_{-k}) \cdot T_{v-k}) \\
&\quad + ((v_{-j}, h_k) \cdot T_{hk} + (v_{-j}, h_{-k}) \cdot T_{h-k}))
\end{aligned}
\quad (4)
$$

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 1. (a) Geometrical configuration of a three-layer medium. (b) Geometrical configuration of scattering.</figcaption></figure>

where the $2n$ zeros $\mu_j$ and the $2n$ Christoffel weighting function $a_j$ obey the relations ($j = 1, 2, \dots, n$):

$$ 
\begin{aligned}
-1 < \mu_j < 1, \quad & \mu_j = -\mu_{-j} \\
& a_j = a_{-j}
\end{aligned}
$$

To simplify notations we can define:

$$ T_{a_{v_j}} = T_{v_j} + T_{v_{-j}} \quad (5) $$

$$ T_{s_{v_j}} = T_{v_j} - T_{v_{-j}} \quad (6) $$

In the absence of the source term $Ka \cdot T_1$, we obtain combining (3)–(6), the matricial linear system of homogeneous equations:

$$ MU \cdot dTs/dz = A \cdot Ta \quad (7) $$

$$ MU \cdot dTa/dz = W \cdot Ts \quad (8) $$

with the following notations:

$$ A = -Ke + F \cdot a + B \cdot a $$

$$ W = -Ke + F \cdot a - B \cdot a $$

$$ MU = \operatorname{diag}[\mu_1, \dots, \mu_n, \mu_1, \dots, \mu_n] $$

$$ 
\begin{aligned}
T_a &= [T_{v_1} + T_{v_{-1}}, \dots, T_{v_n} + T_{v_{-n}}, T_{h_1} \\
&\quad + T_{h_{-1}}, \dots, T_{h_n} + T_{h_{-n}}]^t
\end{aligned}
$$

$$ 
\begin{aligned}
T_s &= [T_{v_1} - T_{v_{-1}}, \dots, T_{v_n} - T_{v_{-n}}, T_{h_1} \\
&\quad - T_{h_{-1}}, \dots, T_{h_n} - T_{h_{-n}}]^t
\end{aligned}
$$

$$ a = \operatorname{diag}[a_1, \dots, a_n, a_1, \dots, a_n] $$

$$ Ke = \operatorname{diag}[Ke_{v_1}, \dots, Ke_{v_n}, Ke_{h_1}, \dots, Ke_{h_n}] \quad (9) $$
$$
\begin{align}
F &= \begin{bmatrix}
(v_1, v_1) \cdots (v_1, v_n)(v_1, h_1) \cdots (v_1, h_n) \\
\vdots \\
(v_n, v_1) \cdots (v_n, v_n)(v_n, h_1) \cdots (v_n, h_n) \\
(h_1, v_1) \cdots (h_1, v_n)(h_1, h_1) \cdots (h_1, h_n) \\
\vdots \\
(h_n, v_1) \cdots (h_n, v_n)(h_n, h_1) \cdots (h_n, h_n)
\end{bmatrix} \\
B &= \begin{bmatrix}
(v_1, v_{-1}) \cdots (v_1, v_{-n})(v_1, h_{-1}) \cdots (v_1, h_{-n}) \\
\vdots \\
(v_n, v_{-1}) \cdots (v_n, v_{-n})(v_n, h_{-1}) \cdots (v_n, h_{-n}) \\
(h_1, v_{-1}) \cdots (h_1, v_{-n})(h_1, h_{-1}) \cdots (h_1, h_{-n}) \\
\vdots \\
(h_n, v_{-1}) \cdots (h_n, v_{-n})(h_n, h_{-1}) \cdots (h_n, h_{-n})
\end{bmatrix} \\
&\tag{10}
\end{align}
$$

The homogeneous equations (7) and (8) are solved using
matricial diagonalization; the jth eigenvector $Ta_j$ and the jth
eigenvalue $\alpha_j$ are solutions of the 2 · n eigenvalue equation:

$$
(MU^{-1} \cdot W \cdot MU^{-1} \cdot A - \alpha_j^2) \cdot Ta_j = 0. \quad (11)
$$

The solutions **T**<sub>a</sub> and **T**<sub>s</sub> are written in terms of eigenvalues and eigenvectors as:

$$
\mathbf{T}\mathbf{a} = 2 \cdot [\mathbf{E} \cdot \mathbf{D}(z) \cdot \mathbf{x}_0 + \mathbf{E} \cdot \mathbf{U}(z+d)\mathbf{y}_0] \quad (12)
$$

$$
\mathbf{T}\mathbf{s} = 2 \cdot [\mathbf{Q} \cdot \mathbf{D}(z) \cdot \mathbf{x}_0 - \mathbf{Q} \cdot \mathbf{U}(z+d) \mathbf{y}_0] \quad (13)
$$

where

$$
\mathbf{Q} = \mathbf{M}\mathbf{U}^{-1} \cdot \mathbf{A} \cdot \mathbf{E} \cdot \alpha^{-1}
$$

$$
\alpha = \operatorname{diag}[\alpha_1, \dots, \alpha_j, \dots, \alpha_{2n}]
$$

$$
\mathbf{E} = [Ta_1, \dots, Ta_j, \dots, Ta_{2n}] \text{ is the } 2n \times 2n \text{ eigen matrix}
\mathbf{D}(z) = \operatorname{diag}[\exp(\alpha_1 z), \dots, \exp(\alpha_j z), \dots, \exp(\alpha_{2n} z)]
$$

$$
\begin{equation}
\begin{split}
U(z) = \operatorname{diag} [ & \exp(-\alpha_1 z), \dots, \exp(-\alpha_j z), \dots, \\
& \exp(-\alpha_{2n} z)].
\end{split}
\tag{14}
\end{equation}
$$

The vectors $\mathbf{x}_0$ and $\mathbf{y}_0$ are given by the boundary conditions.

To find simple particular solutions for the system of (1)
and (2), we assume that the temperature profile in the canopy
layer is linear:

$$
T_1(z) = T_1^0 + (z + d) \cdot \Gamma_1 \quad (15)
$$

where $T_1^0$ is the temperature of the canopy near the soil
boundary.

With these assumptions we obtain a particular solution:

$$
Ta_0 = 2[T_1^0 + \Gamma_1 \cdot (z + d)] \quad (16)
$$

$$
Ts_0 = -2\Gamma_1[\mu/Ka] \quad (17)
$$

The general solution of the system of radiative transfer equa-
tions (1), (2) can be written as the sum of the homogeneous
solutions and a particular solution as

$$
\mathbf{T}\mathbf{a} = 2[\mathbf{T}_1^0 + \Gamma_1(z+d) + \mathbf{E} \cdot \mathbf{D}(z) \cdot \mathbf{x}_0 + \mathbf{E} \cdot \mathbf{U}(z+d) \cdot \mathbf{y}_0] \quad (18)
$$

$$
\mathbf{T}\mathbf{s} = 2[-\Gamma_1\mu/\mathbf{K}\mathbf{a} + \mathbf{Q} \cdot \mathbf{D}(z) \cdot \mathbf{x}_0 - \mathbf{Q} \cdot \mathbf{U}(z+d) \cdot \mathbf{y}_0] \quad (19)
$$

with

$$
\begin{align}
T_1^0 &= [T_1^0, \dots, T_1^0]^t \\
\Gamma_1 &= [\Gamma_1, \dots, \Gamma_1]^t \\
\Gamma_1 \mu / K \mathbf{a} &= [\Gamma_1 \mu_1 / K a_{v1}, \dots, \Gamma_1 \mu_n / K a_{vn}, \nonumber \\
&\qquad \Gamma_1 \mu_1 / K a_{h1}, \dots, \Gamma_1 \mu_n / K a_{hn}]^t. \tag{20}
\end{align}
$$

The boundary conditions at the interfaces soil–canopy ($z = -d$) and canopy–atmosphere ($z = 0$) with respective reflectivity $r_{12}$ and $r_{10}$, are given by

$$
\begin{align*}
\text{*at } z = 0: \quad T_p(\pi - \theta_1) \\
&= (1 - r_{10p}) \cdot \text{T sky}(\theta_0) + r_{10p} \cdot T_p(\theta_1)
\end{align*}
$$

$$
\begin{align*}
\text{*at } z = -d: \quad T_p(\theta_1) &= r_{12p} \cdot T_p(\pi - \theta_1) + (1 - r_{12p}) \cdot T_2 \\
&= r_{12p} \cdot T_p(\pi - \theta_1) + (1 - r_{12p}) \cdot T_2
\end{align*}
$$

$T$ sky ($\theta_0$) is the downwelling atmospheric emission at the
incidence angle $\theta_0$ hence

$$
\begin{align}
T_{-jp} &= (1 - r_{10p}) \cdot \text{T sky}(\theta_0) + r_{10p} \cdot T_{jp} \\
T_{jp} &= r_{12p} \cdot T_{-jp} + (1 - r_{12p}) \cdot T_2 \tag{22}
\end{align}
$$

where $\theta_1$ and $\theta_2$ are related by Snell's law; $p$ denote vertical or horizontal polarization $v$ or $h$. Letting $p_{ijp} = 1 + r_{ijp}$ and $v_{ijp} = 1 - r_{ijp}$, the boundary conditions become for vector variables $\mathbf{T}\mathbf{a}$ and $\mathbf{T}\mathbf{s}$:

$$
\begin{equation}
\text{*at } z = 0 \quad \mathbf{V}_{10} \cdot \mathbf{T}\mathbf{a} - \mathbf{P}_{10} \cdot \mathbf{T}\mathbf{s} = 2 \cdot \mathbf{V}_{10} \cdot \mathbf{T}\mathbf{sky} \tag{23}
\end{equation}
$$

$$
\begin{equation}
\begin{split}
\text{*at } z = -d \quad \mathbf{V}_{12} \cdot \mathbf{T}\mathbf{a} + \mathbf{P}_{12} \cdot \mathbf{T}\mathbf{s} = 2 \cdot \mathbf{V}_{12} \cdot \mathbf{T}_{2} & \\
\mathbf{T}\mathbf{sky} = [\mathbf{T}\mathbf{sky}_{1v}, \dots, \mathbf{T}\mathbf{sky}_{nv}, \mathbf{T}\mathbf{sky}_{1h}, \dots, \mathbf{T}\mathbf{sky}_{nh}]^{\mathrm{t}} &
\end{split}
\tag{24}
\end{equation}
$$

where

$$
\begin{align*}
\mathbf{V}_{12} &= \operatorname{diag}[v_{v12}(\theta = \theta_1), \dots, v_{v12}(\theta = \theta_n), \\
&\qquad v_{h12}(\theta = \theta_1), \dots, v_{h12}(\theta = \theta_n)]
\end{align*}
$$

$$
\begin{align*}
\mathbf{V}_{10} &= \operatorname{diag} [v_{v10}(\theta = \theta_1), \dots, v_{v10}(\theta = \theta_n), \\
&\qquad v_{h10}(\theta = \theta_1), \dots, v_{h10}(\theta = \theta_n)]
\end{align*}
$$

$$
\mathbf{P}_{12} = 2 \cdot \mathbf{I} - \mathbf{V}_{12}
$$

$$
\mathbf{P}_{10} = 2 \cdot \mathbf{I} - \mathbf{V}_{10}
$$

$$
\mathbf{T} \text{ sky} = [\mathbf{T} \text{ sky}_{1v}, \dots, \mathbf{T} \text{ sky}_{nv}, \mathbf{T} \text{ sky}_{1h}, \dots, \mathbf{T} \text{ sky}_{nh}]^t \quad (25)
$$

Substituting (23) and (24) into the radiative transfer equations
(18) and (19) we obtain

$$
\begin{align}
(V_{10}E - P_{10}Q)\mathbf{x}_0 + (V_{10}E + P_{10}Q)U(d)\mathbf{y}_0 \nonumber \\
&= V_{10}(T \text{ sky} - T_1^0) - V_{10}\Gamma_1 d - P_{10}\Gamma_1 \mu / K \mathbf{a} \tag{26}
\end{align}
$$

$$
\begin{align}
(V_{12}E + P_{12}Q)D(-d)\mathbf{x}_0 + (V_{12}E - P_{12}Q)\mathbf{y}_0 \nonumber \\
&= V_{12}(T_2 - T_1^0) + P_{12}\Gamma_1\mu/K\mathbf{a} \tag{27}
\end{align}
$$

Solving (26) and (27) gives $\mathbf{x}_0$ and $\mathbf{y}_0$.

The brightness temperature $T_{Bp}$ at the top of the vegetation
canopy is given by

$$
\text{at } z = 0: \quad T_{Bp}(\theta_0) = v_{10p} \cdot T_p(\theta_1) + r_{10p} \cdot T \text{ sky}(\theta_0)
$$

hence

$$
T_{Bjp} = v_{10p}(Ta_{jp} + T_{s_{jp}})/2 + r_{10p} \cdot T \text{ sky}_j \quad (28)
$$
The matricial formulation is calculated as follows

$$T_B = (1/2) \cdot V_{10}(T\mathbf{a}(z=0) + T\mathbf{s}(z=0)) + R_{10} \cdot T\mathbf{s}ky$$

with

$$T_B = [T_{Bv1}, \dots, T_{Bvn}, T_{Bh1}, \dots, T_{Bhn}]^t$$

hence

$$T_B = V_{10} \cdot ((E + Q)\mathbf{x}_0 + (E - Q)U(d)\mathbf{y}_0) + V_{10}(T_1^0 + \Gamma_1 d - \Gamma_1 \mu / K\mathbf{a}) + R_{10m} \cdot T\mathbf{s}ky. \quad (29)$$

Finally we substitute $\mathbf{x}_0$ and $\mathbf{y}_0$ in this last equation to calculate $T_B$. Each value of the vector $T_B$ correspond to incidence angle $\theta_j = \cos^{-1} \mu_j$.

## B. Phase Matrix of the Leaves

In this model the leaves of the vegetation canopy are modeled as circular or elliptic scatterers. The calculation of the phase matrix of the leaves is based on the results of Karam and Fung [13] who used the generalized Rayleigh-Gans approximation to derive the scattering amplitude tensor of elliptic discs. Whereas Rayleigh and Rayleigh-Gans approximation are valid for very small or tenuous scatterers, the generalized Rayleigh-Gans approximation is valid for scatterers with at least one of their dimensions small compared to the wavelength (in this study the leaf thickness varies around 0.2 mn). Karam and Fung [13] used a first-order radiative transfer solution to deduce the backscattering coefficient of a vegetated layer from the scattering matrix of discs. The passive microwave model we describe here uses the same formulation for the phase matrix and a numerical solution.

The phase matrix, the scattering and absorption coefficients, are deduced from the scattering amplitude tensor. The tensor is first calculated in the local frame of the leaf and then in a reference frame. The description of the incident and scattered radiations in the local and reference frames is given in Appendix A. The pq component of the scattering amplitude tensor in the reference frame $F_{pq}(\mathbf{s}, \mathbf{i})$ is deduced from the expression of the scattering amplitude in the local frame $F'(\mathbf{s}, \mathbf{i})$ by writing the dot product:

$$F_{pq}(\mathbf{s}, \mathbf{i}) = \mathbf{p} \cdot F'(\mathbf{s}, \mathbf{i}) \cdot \mathbf{q} \quad (30)$$

where $\mathbf{p}$ and $\mathbf{q}$ are the polarization vectors of the scattered field which are deduced from the equations (A1-A5) ($p, q = v$ or $h$ for respectively vertical or horizontal polarization). The thorough formulation of the scattering amplitude tensor in the local and reference frame for elliptic scatterers is described in detail by Karam and Fung [13] and is given in Appendix B for easy reference.

For randomly oriented ellipsoids the phase matrix is the average over the probability distribution of leaf orientation of the Stokes matrix [14, p. 138] (we assume that the third and fourth Stokes parameters are equal to zero).

$$(p, q') = n_0 \cdot \langle |F_{pq}(\mathbf{s}, \mathbf{i})|^2 \rangle_{\alpha, \beta, \gamma} \quad (31)$$

where $(p, q')$ denotes the pq component of the phase matrix $P$; $\langle \rangle_{\alpha, \beta, \gamma}$ denotes the average over the Euler angles $\alpha, \beta, \gamma$

which are used to describe the leaf orientation; $n_0$ denotes the number of leaves per unit volume. Hence

$$(p, q') = n_0 \int_0^{2\pi} d\alpha \int_0^{2\pi} d\beta \int_0^{2\pi} d\gamma |F_{pq}(\mathbf{s}, \mathbf{i})|^2 \cdot p_d(\alpha, \beta, \gamma) \quad (32)$$

where $p_d(\alpha, \beta, \gamma)$ is the probability density function of the leaf orientation.

The scattering coefficient can be deduced from the phase matrix components [14], [15]

$$K_{sv}(\mathbf{i}) = \int_{0}^{\pi} \sin \theta_s d\theta_s \int_{0}^{2\pi} d\phi_s [(v, v') + (h, v')] \quad (33)$$

$$K_{sh}(\mathbf{i}) = \int_{0}^{\pi} \sin \theta_s d\theta_s \int_{0}^{2\pi} d\phi_s [(v, h') + (h, h')]. \quad (34)$$

The ohmic extinction coefficient is obtained via the optical theorem.

$$K_{av}(\mathbf{i}) = [4\pi/k] \cdot \mathrm{Im}[(F_{vv}(\mathbf{i}, \mathbf{i}))_{\alpha, \beta, \gamma}] \quad (35)$$

$$K_{ah}(\mathbf{i}) = [4\pi/k] \cdot \mathrm{Im}[(F_{hh}(\mathbf{i}, \mathbf{i}))_{\alpha, \beta, \gamma}] \quad (36)$$

where $\mathrm{Im}(\mathbf{i})$ denotes the imaginary part operator and $k$ is the wave number ($k = 2\pi/\lambda$).

The total extinction coefficient is taken as the sum of the scattering and the ohmic extinction coefficient as

$$K_{ev} = K_{sv} + K_{av} \quad (37)$$

$$K_{eh} = K_{sh} + K_{ah}. \quad (38)$$

The radiative transfer equations (1) and (2) are solved using these formulations for $K_{ap}$ and $K_{ep}$ and the phase matrix $P$.

## III. MATERIAL AND METHOD

### A. Experiment Description

Microwave radiometric measurements over a soybean field were carried out with the new radiometer PORTOS during the summer 1991 in Avignon [14]. PORTOS is a six-frequency (1.4, 5.05, 10.65, 23.8, 36.5, and 90 GHz) dual polarized radiometer designed by CNES (Centre National d'Etudes Spatiales, France) and Matra Marconi Space. The radiometer 3 dB beamwidth is 12.5° and allows measurements at various incidence angles from 0° to 60° when mounted on a 20 m crane boom. The theoretical absolute accuracy, achieved using a sole external calibration during the experiment, is +/- 0.5 K with an integration time of 100 or 200 ms depending on the channel. Regular external calibrations showed that because of problems in the thermal control of the radiometer, this absolute accuracy was only about +/- 3 K during the whole experiment.

The experiment was conducted during a 2.5-month period from sowing (Day of Year 205) to the ripeness state of the soybean field (DoY 273). Microwave measurements were made daily between 1 and 4 pm. The soybean canopy characteristics were measured early in the morning (between 8 and 11 a.m.) and throughout its growth (weekly). The measurements
include: dry and wet biomass; gravimetric water content; size, surface and volume of leaves, stems and fruits; probability distribution of leaves and stems orientations (in terms of three Euler angles).

Measurements of soil moisture content, soil, and vegetation temperatures were made concurrently with the microwave measurements. Radiosoundings from Meteo France performed at 11 a.m. provided us with the atmosphere characterization which is used to calculate the downwelling atmospheric emission ($T_{Bsky}$). The impact of this term on the simulated data is very small in the range of frequencies 1–10 GHz.

## B. Parameters of the Discrete Model

The parameters of the discrete model which are used to compare the modeled brightness temperature of the soybean canopy with PORTOS data were measured during the experiment and are detailed in this section. The vegetated canopy was fully developed from DoY 245 to DoY 253. We will assume that these parameters do not change during this 9-day period.

The dielectric behavior of the soil is calculated using a semiempirical mixing approach [16]. To account for soil roughness we use a modified Fresnel reflection formulation [17] ($h_r = 0.1$ for a rather smooth soil). The downwelling atmospheric radiation $T_{Bsky}$ is calculated in clear-sky conditions [15, pp. 269–288]. The relative permittivity $\epsilon_{r,v}$ of the scatterers is estimated based on dual-dispersion theory [18].

Leaves are either considered as elliptic-shaped, $a_1$ and $a_2$ are the half-dimensions of the elliptic leaf and $a_3$ is its half-thickness ($a_1 > a_2 \gg a_3$); or as disc-shaped scatterers, $a_1 = a_2 = 2.8$ cm, $a_3 = 0.12$ mm with equivalent area $S = 24.65 \text{ cm}^2$. Only leaf inclination distribution is considered and its probability density $p_β$ is described by a simple beta representation [19] with following parameters: $μ = 1.5$ and $ν = 2.3$. The leaf volume fraction $f_v$ is 0.14%. Other measured parameters used in the computations are as follows:

<table><tr><td>d = 0.7 m</td><td>height of canopy</td></tr><tr><td>ρ<sub>s</sub> = 0.2 g/cm<sup>3</sup></td><td>bulk density of the dry leaves</td></tr><tr><td>M<sub>g</sub> = 0.8 (kg/kg)</td><td>leaf water content</td></tr><tr><td>m<sub>v</sub> = 0.2 (m<sup>3</sup>/m<sup>3</sup>)</td><td>volumetric soil moisture of the top 3-cm layer</td></tr><tr><td>T = 300 K</td><td>soil and vegetation temperature</td></tr></table>

# IV. RESULTS AND DISCUSSIONS

## A. Simulation of Leaf Effects

To better understand the sensitivity of the microwave signal to the leaf effects, a simulation of the microwave thermal emission of the soybean canopy is done at 1.4 GHz (L-band), 5 GHz (C-band), and 10.6 GHz (X-band). The theoretical behavior of the brightness temperature $T_{Bp}$ (polarization $p = V$ or $H$) and of the polarization ratio $R = (T_{Bv} - T_{Bh})/(T_{Bv} + T_{Bh})$ at the incidence angle $θ = 50°$ as a function of the leaf characteristics, size, shape, density inside the canopy, water content, and orientation is studied in this section. Soil and vegetation characteristics are described in the previous section. The soil moisture $m_v$, which has a large influence

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2. The brightness temperature at H-polarization (a) and the polarization ratio (b) vs. the leaf volume fraction at 1.4, 5.0, and 10.6 GHz.</figcaption></figure>

on the simulated data especially at 1.4 and 5.0 GHz, is kept constant at the arbitrary value: $m_v = 0.2$. So the results of this study are valid for rather wet soils. This analysis complements the study of Ferrazzoli et al. [20] carried out at 10 GHz and at different incidence angles with a modeling based on the Matrix Doubling Algorithm.

Figs. 2–6 show the effects of the leaves characteristics on the modeled soybean canopy emission at three frequencies: 1.4, 5.0, and 10.6 GHz. The parameters considered in this study are: volumetric fraction $f_v$, gravimetric water content $M_g$ (%), equivalent radius $a_1 (= a_2)$, half thickness $a_3$, ratio of the main axes of the elliptic leaf $a_1/a_2$. The range of the leaf parameters has been chosen to allow the description of various types of vegetated canopies: crop canopies, forest canopies with brushwood.

The simulation results indicate that:

1) at 1.4 GHz the thermal emission of the soybean canopy is mainly sensitive to leaf volume fraction $f_v$. The sensitivity to the gravimetric water content $M_g$ is lower but remains significant. Dimensions and shape of the leaves appear to have a very small effect on the emission.

2) at 5 GHz the thermal emission is roughly equally sensitive to $f_v$, $M_g$ and to $a_3$ and $a_1$. The effect of the shape of the leaf remains very small.
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 3. The brightness temperature at V-polarization (a) and the polarization ratio (b) vs. the leaf gravimetric water content at 1.4, 5.0, and 10.6 GHz.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 4. The brightness temperature (V-polarization, $\theta = 50^{\circ}$) vs. the leaf half-thickness at 1.4, 5.0, and 10.6 GHz.</figcaption></figure>

3) at 10.6 GHz the influence of $f_v$ is very small (limited to the range $0 < f_v < 10^{-3}$) while the microwave signal becomes very sensitive to $M_g$ and to the leaf thickness $a_3$. The signal is also significantly sensitive to the shape and to the surface of the leaf.

These results can be interpreted in terms of scattering and absorption inside the vegetated canopy. $f_v$ is a key driver

<figure><img src="image_9.png" /><figcaption>Fig. 5. The brightness temperature (V-polarization, $\theta = 50^{\circ}$) vs. the leaf equivalent radius at 1.4, 5.0, and 10.6 GHz.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 6. The brightness temperature (V-polarization, $\theta = 50^{\circ}$) vs. the ratio of the leaf main axis: $a_1/a_2$ at 1.4, 5.0, and 10.6 GHz.</figcaption></figure>

for absorption inside the canopy while it has no influence on the single scattering albedo $K_s/K_e$ which parameterizes scattering inside the canopy. So $f_v$ has dominant effects on the emission at low frequencies. As the frequency increases, the relative influence of scattering in comparison with absorption increases and the influence of $f_v$ on the microwave signal decreases. Moreover, at high frequencies (> 8–10 GHz), only a top layer of the canopy contributes to the microwave emission; the biomass under this top layer has almost no influence on the microwave signal. The shape and the dimensions of the leaves have increasing effects with the frequency on the microwave emission, as scattering also increases with the frequency. $M_g$ has both high effects on absorption and scattering and thus has a very significant effect on the microwave emission at L-, C-, and X-bands.

To illustrate the sensitivity of the brightness temperature $T_B$ to the distribution of leaf inclination, we will assume a uniform probability distribution function of the form:

$$p_{\beta}(\beta) = \begin{cases} 1/(\beta_2 - \beta_1) & \beta_1 < \beta < \beta_2 \\ 0 & \text{otherwise} \end{cases}$$

We choose $\beta_2 - \beta_1 = 20^{\circ}$ as angle interval, and $PB$ is defined as the mean angle of the leaf inclination $PB =$
<figure><img src="image_11.png" /><figcaption>Fig. 7. The brightness temperature vs. the incident angle for a mean leaf inclination angle PB = 10° (planophile canopy), 45°, 80° (erectophile canopy): (a) at 1.4 GHz (b) at 10.6 GHz.</figcaption></figure>

($\beta_2 + \beta_1$)/2. Figs. 7(a-b) show the behavior of $T_B$ versus the incidence angle $\theta$, with PB = 10° (planophile canopy), PB = 45°, PB = 80° (erectophile canopy) at L- and X-bands. At 1.4 GHz and for small incidence angles the attenuation of the microwave soil emission by leaves decreases as the canopy becomes more erectophile: the vegetation becomes more transparent to the soil emission. So the contribution of the soil emission to the total canopy emission increases while the contribution of the vegetation layer decreases. This explains the observed decrease in the brightness temperature especially for H-polarization while the polarization ratio R increases (Fig. 7(a)). At 10.6 GHz leaf-scattering effects are very important and depend strongly on the probability distribution of leaf inclination. We observe that both $T_B$, especially at V-polarization, and R are significantly sensitive to the leaf inclination distribution (Fig. 7(b)) [20], [21].

## B. Comparison With Radiometric Measurements

This section shows a comparison between simulated emissivity $e_p$ ($p$ = v or p = h) and the measurements performed with the multifrequency radiometer PORTOS. An estimate of the emissivity is calculated by dividing the measured brightness temperature by a surface temperature inferred from

thermal infrared measurements. Soil moisture $m_v$ which is used to simulate emissivity is a fitted parameter: it is calculated in order to obtain the best visual fit between simulated and measured emissivity. The soil moisture $m_v$ is the only fitted parameter and the vegetation characteristics are assumed not to change from DoY 245 to DoY 253.

1) Incidence angle behavior: Figs. 8(a-b) show a comparison between calculated and measured emissivity versus the incidence angle at 1.4 and 5.05 GHz.

From the day 245 to the day 253 the soil moisture continuously decreased from $m_{vm} = 0.29$ to $0.13$ ($m^3/m^3$); the subscript “m” denotes measured values. The fitted values $m_v$ agree well with this regular soil drying up (Figs. 8(a-b)). They are lower at 5.05 GHz than at 1.4 GHz. This could be explained by the shorter penetration depth at C-band which makes the signal depend more on the dry surface crust. Agreement between measured and computed values is good though some discrepancies between the measured and best-fit soil-moisture are visible. This is especially so at C-band and could be caused by bias in the measurements of vegetation characteristics. In fact it was not possible to perform a complete vegetation characterization (a.m.) concurrently with the microwave measurements (p.m.). Vegetation moisture changes with the hour of the day and is lower around 2-4 p.m. when the radiometric measurements were made than in the morning. Moreover vegetation moisture decreased slowly between DoY 245 and DoY 253. Other parameters such as the leaf-orientation distribution and the leaf-thickness also depend strongly on the hour of the day and their accurate estimation is very difficult.

2) Frequency behavior: Figs. 9(a-b) show a comparison between the calculated and measured emissivity as a function of the frequency for three days: on day 245 $m_v = 0.3$ ($m_{vm} = 0.29$), on day 247 $m_v = 0.15$ ($m_{vm} = 0.25$), on day 253 $m_v = 0.05$ ($m_{vm} = 0.13$). The effective soil layer depth contributing to the microwave emission depends on the frequency $f$. In this study, the moisture content $m_v$ of these different effective layers is assumed not to depend on $f$. This approximation is only valid when the moisture profile in the soil is uniform. Before discussing about this comparison we will describe the simulated behavior of emissivity for these three different soil moistures.

It can be seen in Figs. 9(a-b) that the simulated emissivity becomes independent of the soil contribution when the frequency goes beyond 8-10 GHz. In this frequency range the soil emission is entirely screened by the vegetated layer. At lower frequencies the trend of $e_p$ is strongly dependent upon the soil moisture. When the soil is dry ($m_v = 0.02$) $e_p$ continuously decreases with the frequency whereas at higher soil moisture levels $e_p$ increases up to 4-6 GHz and then decreases. From 1 to 4-6 GHz, for rather wet soils the main contribution to the total emission is the direct and indirect vegetation emission which increases with the frequency (the absorption and thus the emission increase with the frequency inside the vegetated layer while scattering is very weak). Whereas for very dry soils the main contribution comes from the soil and is more and more attenuated by vegetation as the frequency increases. So the difference $e_C - e_L$ (the indices L and C denote the L-
<figure><img src="image_12.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(b)</figcaption></figure>

Fig. 8. A comparison between calculated and measured brightness temperature vs. the incidence angle over soybean field at 1.4 GHz (a) and 5.0 GHz (b).

<figure><img src="image_14.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(b)</figcaption></figure>

Fig. 9. A comparison between calculated and measured brightness temperature vs. frequency over soybean field at $\theta = 0°$ (a) and at $\theta = 50°$, H-polarization (b).

and C-bands) increases with the soil moisture content and seems to be less dependent on the vegetation effects than $e_C$ or $e_L$. This property can be particularly useful for the interpretation of remotely sensed data when little information about the vegetation biomass or moisture is available. For frequencies higher than 4-6 GHz, scattering effects inside the canopy becomes significant. The consequence is a decrease in emissivity whatever the soil moisture.

When we compare these simulated data with the PORTOS measurements, discrepancies appear to be higher at 10.65 GHz than at lower frequencies. At 10.65 GHz errors in the estimation of leaves characteristics (i.e., thickness, moisture content, distribution of orientation) have strong effects on the result (Figs. 3-7). Moreover, as the vegetation coverage is not quite continuous the radiometer also measures direct soil emission through holes in the soybean canopy. This effect is not taken into account by the model which assumes the media is homogeneous and so completely attenuates soil emission at 10.65 GHz.

In this section all the vegetation parameters were set before the comparison. A more accurate estimation of leaves characteristics seems to be needed to compute the emissivity at C- and X-bands when using discrete modeling. Nevertheless,

results show good agreement between measured and calculated emissivity.

V. CONCLUSIONS

A radiative transfer model is used in this paper to study the leaf effects on the microwave thermal emission of a mature soybean canopy at L-, C-, and X-bands. The levels and trends of brightness temperature and of polarization ratio versus frequency are analyzed for different soil moisture conditions. The main results of this sensitivity analysis are:

* the trend of the brightness temperature versus frequency is largely conditioned by soil moisture. Future investigations are required to verify whether this trend is a significant indicator of soil moisture under vegetation cover when little information about the vegetated layer is available.

* the leaf gravimetric moisture content appears to be a key factor of the vegetation emissivity especially at C- and X-bands.

* for a well-developed vegetated canopy the effects of the leaf volume fraction are limited to the lower frequencies (L- and C-band).

* the leaf size effects increase with frequency.
* the effects of the distribution of leaf inclination are significant at high frequencies (X-band: V-polarization), weaker at low frequencies (L- and C-bands: H-polarization).

Results of the measurements with the radiometer PORTOS are consistent with the computations from the model. It is seen that the estimation of the structure and of the dielectric characteristics of the soybean canopy must be performed concurrently with the radiometric measurements to have the precision required in discrete modeling validation. The next step will be to study stems, branches, or stalks whose contributions are effective at low frequencies and to compare the levels and trends of the microwave emission for different types of vegetated layers: from short crop canopies to forest canopies.

## VI. APPENDIX A

The orientation of the local frame $R'(\mathbf{x}', \mathbf{y}', \mathbf{z}')$ is that of the principal axes of a random leaf. $R'$ is related to the reference frame $R(\mathbf{x}, \mathbf{y}, \mathbf{z})$ using the Eulerian angles $\alpha, \beta, \gamma$ through [9]:

$$ \mathbf{x}' = (\cos \gamma \cdot \cos \beta \cdot \cos \alpha - \sin \gamma \sin \alpha) \cdot \mathbf{x} + (\cos \gamma \cdot \cos \beta \cdot \sin \alpha + \sin \gamma \cdot \cos \alpha) \cdot \mathbf{y} - \cos \gamma \cdot \sin \beta \cdot \mathbf{z} $$

$$ \mathbf{y}' = -(\sin \gamma \cdot \cos \beta \cdot \cos \alpha + \cos \gamma \cdot \sin \alpha) \cdot \mathbf{x} + (\cos \gamma \cdot \cos \alpha - \sin \gamma \cdot \sin \alpha \cdot \cos \beta) \cdot \mathbf{y} + \sin \gamma \cdot \sin \beta \cdot \mathbf{z} $$

$$ \mathbf{z}' = \sin \beta \cdot \cos \alpha \cdot \mathbf{x} + \sin \beta \cdot \sin \alpha \cdot \mathbf{y} + \cos \beta \cdot \mathbf{z} \quad (A1) $$

Considering an incident radiation over a leaf, represented by its incident vector $\mathbf{i}$, characterized by the inclination and azimuth angles ($\theta_i, \phi_i$) in the reference frame, and ($\theta'_i, \phi'_i$) in the local frame (Fig. 1(b)):

$$ \begin{aligned} \mathbf{i} &= \sin \theta_i \cdot (\mathbf{x} \cdot \cos \phi_i + \mathbf{y} \cdot \sin \phi_i) - \mathbf{z} \cdot \cos \theta_i \\ \mathbf{i} &= \sin \theta'_i \cdot (\mathbf{x}' \cdot \cos \phi'_i + \mathbf{y}' \cdot \sin \phi'_i) - \mathbf{z}' \cdot \cos \theta'_i \quad (A2) \end{aligned} $$

where

$$ \cos \theta'_{i} = -\mathbf{i} \cdot \mathbf{z}' = \cos \beta \cdot \cos \theta_{i} - \sin \beta $$

$$ \cdot \cos(\alpha - \phi_i) \cdot \sin \theta_i = -t_i $$

$$ \sin \theta'_{i} = (\mathbf{i} \cdot \mathbf{x}) / \cos \phi'_{i} = (1 - \cos^2 \theta'_{i})^{1/2} $$

$$ \cos \phi'_{i} = (\mathbf{i} \cdot \mathbf{x}') / \sin \theta'_{i} = [\sin \theta_i(\cos \gamma \cdot \cos \beta) $$

$$ \cdot \cos(\phi_i - \alpha) + \sin \gamma \cdot \sin(\theta_i - \alpha)] $$

$$ + \cos \theta_i \cdot \cos \gamma \cdot \sin \beta] / (1 - t_i^2)^{1/2} $$

$$ \sin \phi'_{i} = (\mathbf{i} \cdot \mathbf{y}') / \sin \theta'_{i} = [\sin \theta_i(-\sin \gamma \cdot \cos \beta) $$

$$ \cdot \cos(\phi_i - \alpha) + \cos \gamma \cdot \sin(\phi_i - \alpha)] $$

$$ - \cos \theta_i \cdot \sin \gamma \cdot \sin \beta] / (1 - t_i^2)^{1/2} \quad (A3) $$

Similar expressions for the scattered wave (along vector $\mathbf{s}$) can be written. The inclination and azimuth angles ($\theta_s, \phi_s$) and ($\theta'_s, \phi'_s$) respectively in the reference and local frame can be obtained by replacing $\theta_i$ and $\phi_i$ respectively by $\pi - \theta_s$ and $\phi_s$ in (A3).

$$ \mathbf{s} = \sin \theta_s \cdot (\mathbf{x} \cdot \cos \phi_s + \mathbf{y} \cdot \sin \phi_s) + \mathbf{z} \cdot \cos \theta_s $$

$$ \mathbf{s} = \sin \theta'_s \cdot (\mathbf{x}' \cdot \cos \phi'_s + \mathbf{y}' \cdot \sin \phi'_s) + \mathbf{z}' \cdot \cos \theta'_s \quad (A4) $$

with

$$ \cos \theta'_{s} = \mathbf{s} \cdot \mathbf{z}' = \cos \beta \cdot \cos \theta_{s} + \sin \beta \cdot \cos(\alpha - \phi_{s}) \\ \cdot \sin \theta_{s} = +t_{s} $$

$$ \sin \theta'_{s} = (\mathbf{s} \cdot \mathbf{x}) / \cos \phi'_{s} = (1 - \cos^2 \theta'_{s})^{1/2} $$

$$ \cos \phi'_{s} = (\mathbf{s} \cdot \mathbf{x}') / \sin \theta'_{s} = [\sin \theta_s(\cos \gamma \cdot \cos \beta) \\ \cdot \cos(\phi_s - \alpha) + \sin \gamma \cdot \sin(\phi_s - \alpha)] \\ - \cos \theta_s \cdot \cos \gamma \cdot \sin \beta] / (1 - t_s^2)^{1/2} $$

$$ \sin \phi'_{s} = (\mathbf{s} \cdot \mathbf{y}') / \sin \theta'_{s} = [\sin \theta_s(-\sin \gamma \cdot \cos \beta) $$

$$ \cdot \cos(\phi_s - \alpha) + \cos \gamma \cdot \sin(\phi_s - \alpha)] $$

$$ + \cos \theta_s \cdot \sin \gamma \cdot \sin \beta] / (1 - t_s^2)^{1/2} \quad (A5) $$

## VII. APPENDIX B

The scattering amplitude tensor for elliptic discs can be written in the local frame as

$$ F'_{vv}(\mathbf{s}, \mathbf{i}) = [k^2 v_0 (\epsilon_r - 1) / 4\pi] [a_{33} \sin \theta'_i \sin \theta'_s - \cos \theta'_i \cos \theta'_s \\ \cdot (a_{11} \cos \phi'_i \cos \phi'_s + a_{22} \sin \phi'_i \sin \phi'_s)] \mu(\mathbf{s}, \mathbf{i}) $$

$$ F'_{hv}(\mathbf{s}, \mathbf{i}) = [k^2 v_0 (\epsilon_r - 1) / 4\pi] \cos \theta'_i \\ \cdot [a_{11} \cos \phi'_i \sin \phi'_s - a_{22} \sin \phi'_i \cos \phi'_s] \mu(\mathbf{s}, \mathbf{i}) $$

$$ F'_{vh}(\mathbf{s}, \mathbf{i}) = [-k^2 v_0 (\epsilon_r - 1) / 4\pi] \cos \theta'_s \\ \cdot [a_{11} \sin \phi'_i \cos \phi'_s - a_{22} \cos \phi'_i \sin \phi'_s] \mu(\mathbf{s}, \mathbf{i}) $$

$$ F'_{hh}(\mathbf{s}, \mathbf{i}) = [k^2 v_0 (\epsilon_r - 1) / 4\pi] [a_{11} \sin \phi'_i \sin \phi'_s \\ + a_{22} \cos \phi'_i \cos \phi'_s] \mu(\mathbf{s}, \mathbf{i}) \quad (B1) $$

where $k$ denotes the wave number ($k = 2\pi/\lambda$), $\lambda$ denotes the wavelength; $v_0$ the leaf volume $v_0 = (4\pi/3)a_1a_2a_3(a_1 > a_2 \gg a_3)$; $\epsilon_r$ denotes the dielectric constant of the leaf; $\mu(\mathbf{s}, \mathbf{i})$ is the Debye interference function.

$a_{ii}$ and $g_i$ ($i = 1, 2, 3$) are the polarizability tensor elements and the demagnetizing factors for an elliptic-shaped leaf:

$$ a_{ii} = 1 / [(\epsilon_r - 1)g_i + 1] \quad (B2) $$

$$ g_1 = (a_3/a_1)(1 - e^2)^{1/2} (K(e, \pi/2) - E(e, \pi/2)) / e^2 $$

$$ g_2 = (a_3/a_1)(-1 - e^2)K(e, \pi/2) + E(e, \pi/2)) / (e^2(1 - e^2)^{1/2}) $$

$$ g_3 = 1 - (a_3/a_1)E(e, \pi/2) / (1 - e^2)^{1/2} \quad (B3) $$

with

$$ e = (1 - (a_2/a_1)^2)^{1/2}. \quad (B4) $$

$K(e, \pi/2)$ and $E(e, \pi/2)$ are elliptic integrals of the first and second kind.

$$ K(e, \pi/2) = \int_{0}^{\pi/2} d\psi / (1 - e^2 \sin^2 \psi)^{1/2} $$

$$ E(e, \pi/2) = \int_{0}^{\pi/2} d\psi \cdot (1 - e^2 \sin^2 \psi)^{1/2} \quad (B5) $$

The Debye interference function can be written as

$$ \mu(\mathbf{s}, \mathbf{i}) = 2J_1(Q_e)/Q_e \quad (B6) $$
with

$$Q_e = [(q_{xa} \cos \beta + q_{za} \sin \beta)^2 (a_1^2 \cos^2 \gamma + a_2^2 \sin^2 \gamma) \\ + q_{ya}(q_{xa} \cos \beta + q_{za} \sin \beta)(a_1^2 - a_2^2) \sin^2 \gamma \\ + q_{ya}^2 (a_1^2 \sin^2 \gamma + a_2^2 \cos^2 \gamma)]^{1/2} \quad (B7)$$

$$\begin{aligned}
q_{xa} &= k[\sin \theta_i \cos (\phi_i - \alpha) - \sin \theta_s \cos (\phi_s - \alpha)] \\
q_{ya} &= k[\sin \theta_i \sin (\phi_i - \alpha) - \sin \theta_s \sin (\phi_s - \alpha)] \\
q_{za} &= k[\cos \theta_i - \cos \theta_s]
\end{aligned} \quad (B8)$$

and $J_1(\mathbf{i})$ is the first order cylindrical Bessel function.

The elements of the scattering amplitude tensor in the reference frame $F_{pq}(\mathbf{s}, \mathbf{i})$ are related to the same tensor in local frame $F'_{pq}(\mathbf{s}, \mathbf{i})$ as follows

$$\begin{aligned}
F_{vv}(\mathbf{s}, \mathbf{i}) &= [t_{vs}(F'_{vv}(\mathbf{s}, \mathbf{i})t_{vi} - F'_{vh}(\mathbf{s}, \mathbf{i})t_{hi}) \\
&\quad - t_{hs}(F'_{hv}(\mathbf{s}, \mathbf{i})t_{vi} - F'_{hh}(\mathbf{s}, \mathbf{i})t_{hi})]/D(\mathbf{s}, \mathbf{i}) \\
F_{hv}(\mathbf{s}, \mathbf{i}) &= [t_{hs}(F'_{vv}(\mathbf{s}, \mathbf{i})t_{vi} - F'_{vh}(\mathbf{s}, \mathbf{i})t_{hi}) \\
&\quad + t_{vs}(F'_{hv}(\mathbf{s}, \mathbf{i})t_{vi} - F'_{hh}(\mathbf{s}, \mathbf{i})t_{hi})]/D(\mathbf{s}, \mathbf{i}) \\
F_{vh}(\mathbf{s}, \mathbf{i}) &= [t_{vs}(F'_{vv}(\mathbf{s}, \mathbf{i})t_{hi} + F'_{vh}(\mathbf{s}, \mathbf{i})t_{vi}) \\
&\quad - t_{hs}(F'_{hv}(\mathbf{s}, \mathbf{i})t_{hi} - F'_{hh}(\mathbf{s}, \mathbf{i})t_{vi})]/D(\mathbf{s}, \mathbf{i}) \\
F_{hh}(\mathbf{s}, \mathbf{i}) &= [t_{hs}(F'_{vv}(\mathbf{s}, \mathbf{i})t_{hi} + F'_{vh}(\mathbf{s}, \mathbf{i})t_{vi}) \\
&\quad + t_{vs}(F'_{hv}(\mathbf{s}, \mathbf{i})t_{hi} + F'_{hh}(\mathbf{s}, \mathbf{i})t_{vi})]/D(\mathbf{s}, \mathbf{i}) \quad (B9)
\end{aligned}$$

$$D(\mathbf{s}, \mathbf{i}) = [(t_{vs}^2 + t_{hs}^2)(t_{vi}^2 + t_{hi}^2)]^{1/2} \quad (B10)$$

where

$$\begin{aligned}
t_i &= \sin \beta \cdot \sin \theta_i \cdot \cos (\alpha - \phi_i) - \cos \beta \cdot \cos \theta_i \\
t_{vi} &= -[\sin \beta \cdot \cos \theta_i \cdot \cos (\alpha - \phi_i) + \cos \beta \cdot \sin \theta_i] \\
t_{hi} &= \sin \beta \cdot \sin (\alpha - \phi_i)
\end{aligned} \quad (B11)$$

and $t_h$, $t_{vs}$, $t_{hs}$ can be obtained from (B11) by replacing $\theta_i$ and $\phi_i$ with respectively $\pi - \theta_s$ and $\phi_s$.

ACKNOWLEDGMENT

The authors would like to thank B. Seguin and L. Bruckler for their encouragement and support. They also wish to thank the anonymous reviewers for their comments and suggestions.

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Jean-Pierre Wigneron was born in Aix en Provence, France. He received the engineering degree from Sup’Aero “Ecole Nationale Superieure de l’Aeronautique et de l’Espace,” Toulouse, France, in signal processing and radar.

In 1990 he joined the INRA, Institut National de Recherches Agronomiques, Station de Bioclimatologie, Avignon, France. He received the Ph.D. degree from the Université Paul Sabatier, Toulouse, France, in 1993. His research interests include passive microwave remote sensing applied to vegetation and microwave modeling.

Jean Christophe Calvet received the engineering degree from the "Institut National Agronomique Paris-Grignon," Paris, France, in agronomy, an engineering degree from the "Ecole Nationale de la Météorologie, Toulouse, France, in meteorology.

In 1990 he joined the Centre National de Recherches Météorologiques, Toulouse, France, where from September 1990 to March 1991 he was on leave of absence to work at the Laboratoire d’Etudes et de Recherches en Télédétection Spatiale, Toulouse, France, and from April 1991 to July 1992 at the Instituto Nacional de Pesquisas Espaciais, Sao José dos Campos, Brazil. His applications interests are in the use of remote sensing over land surfaces for meteorology.
**Yann Kerr** (M'88) was born in Paris, France. He received the engineering degree from the "Ecole Nationale Supérieure de l'Aéronautique et de l'Espace," Toulouse, France, in radar and telecommunications, and the M.Sc. degree from Glasgow University, Glasgow, Scotland, in electronics and electrical engineering.

From 1980 to 1985 he was employed by the Centre National d'Etudes Spatiales, Toulouse, France. In 1985 he joined the Laboratoire d'Etudes et de Recherches en Télédétection Spatiale as a Research Scientist, where from May 1987 to December 1988 he was on leave of absence to work at the Jet Propulsion Laboratory, California Institute of Technology, Pasadena. His fields of interest are in the theory and techniques for microwave and thermal infrared remote sensing of the Earth, with emphasis on hydrology and vegetation monitoring.

**André Chanzy** received the B.S. and Ph.D. degrees in physics of the environment from the "Institut National Agronomique," Paris-Grignon, France in 1985 and 1991, respectively.

Since 1986, he has been working at the INRA, station de Science du sol (Avignon, France). His main field of interest is the application of remote sensing to hydrological fluxes.

**Armand Lopes** received the Ingenieur degree from the Ecole Nationale Supérieure de l'Aéronautique et de l'Espace, and the Ph.D. degree from the Université Paul Sabatier, Toulouse, France, in 1983.

He is presently Assistant Professor at the Université Paul Sabatier and a Researcher at the Centre d'Etude Spatiale des Rayonnements, Toulouse. His research activities are in the domain of active microwaves remote sensing and are concerned with physics and image processing: microwave extinction and scattering by vegetation, SAR image calibration, edge detection, speckle filtering, and texture analysis.