### I. Introduction

*θ*

*,*

_{i}*φ*

*;*

_{i}*θ*

*,*

_{s}*φ*

*;*

_{s}*θ*

*,*

_{p}*φ*

*) as well as the size of the particles relative to the wavelength [1].*

_{p}*vv*-,

*hh*-, and

*vh*- (

*hv*-) polarization combinations, and the effect of the surface roughness on the reflection coefficient of an underlying soil surface. Then, the RTM is modified with the optimized input parameters in relation with the leaf curvature, the higher-order multiple scattering, and the small roughness of underlying surface. Finally, the accuracy of the modified RTM is verified by comparing it with experimental data sets.

### II. Examination of RTM

(1) the direct backscatter from the vegetation layer,

(4) the direct backscatter from the underlying soil surface with the attenuation through the vegetation layer, and

(2)–(3) the interactions between the vegetation layer and the underlying soil surface: i.e.,

(2a) incidence – reflection from the ground – forward scattering from the vegetation layer – backscatter,

(2b) incidence – forward scattering from the vegetation layer – reflection from the ground – backscatter, and

(3) incidence – reflection from the ground –backscattering from the vegetation layer – reflection from the ground – backscatter.

*p*or

*q*denotes

*v*- or

*h*-polarization,

*m*or

*n*is 1 or 2, and

*vv*-,

*vh*-,

*hv*-, and

*hh*-polarizations correspond to 11, 12, 21, and 22 elements of the 4×4 transformation matrix

##### (3)

*k*= 1, 2, 3, 4, and 5 of the transformation matrix

_{kl,ij}can be obtained using the averaged scattering matrices 〈

*ij*

*elements of the matrices. The Stokes scattering operator matrix*

^{th}### 1. Effect of Leaf Curvature

*σ*) of a curved dielectric sheet can be analytically computed accurately by multiplying a curve factor to the scattering matrix of a flat dielectric disk, using a Fresnel integral with the argument of leaf length

*a*and the radius of curvature

*ρ*for a given frequency [6].

*S*

*is the scattering matrix element for a curved lossy dielectric sheet,*

_{c}*S*

*is the scattering matrix element for a flat plate,*

_{fp}*F*(

*γ*) is the Fresnel integral with an argument

*γ*, and

*k*

_{0}is the wavenumber. For example, the reductions of RCS at normal incidence are about 0.98, 0.75, and 0.4 in magnitude for L-, C-, and X-bands, respectively, when both the curvature radius and leaf length are 6 cm. The RCS reductions are 0.95, 0.75, and 0.32 for the curvature radii of 12, 6, and 3 cm, respectively, with a fixed leaf length of 6 cm at the C-band [6]. Therefore, we need to multiply a multiplicative factor, i.e., the “curve factor”

*C*

*, to the scattering matrices for leaves and stems to compensate for the RCS reduction because of the leaf-curvature effect in the RTM.*

_{factor}*S*]

*is the modified scattering matrix of a leaf or a branch.*

_{m}### 2. Effect of Multiple Scattering

*hv*-polarization and less than 0.5 dB for

*hh*-polarization for a sample vegetation canopy with only ten 2λ-length stems on a 2λ-diameter underlying surface [8]. The fully phase-coherent computation [11] for the backscattering coefficients of grasslands also showed that the higher-order scattering terms should be added to fit the radar measurements for the cross-polarized data [12].

*M*

*, to the transformation matrix elements for the higher-order multiple scattering from scatters in a vegetation layer.*

_{factor}*T*]

*is the modified*

_{km}*k*

*transformation matrix, and the multiple factor may be obtained from an extensive database of experimental measurements.*

^{th}### 3. Effect of Surface Roughness

*h*

_{RMS}_{–}

*= roughness factor (*

_{s}*R*

*)×*

_{factor}*h*

*, where*

_{RMS}*h*

_{RMS}_{–}

*is the small-roughness RMS height for reflection, and*

_{s}*h*

*is the RMS height for backscattering from the soil surface. For the reflection from a rough surface, the following form of the reflection coefficient is used for the reflectivity matrix.*

_{RMS}*R*]

*is the modified reflectivity matrix, and*

_{m}*X*=2(

*kh*

_{RMS}_{–}

_{s}*cosθ*)

^{2}, where

*Γ*

*is the reflection coefficient of a rough soil surface,*

_{sp}*Γ*

*is the Fresnel reflection coefficient of a flat plane, the subscript*

_{fp}*p*is the polarization,

*θ*is the incidence angle, and

*I*

_{0}[···] is the modified first-kind Bessel function.

### III. Modification and Verification of RTM

### 1. Experimental Data Sets

*hh*-polarization during the bean growth cycle from July 22 to September 24, 2010.

*in situ*measured ground-truth data for all the input parameters of the RTM on the same days when the radar data were collected. The parameters for vegetation fields were obtained by sampling. The surface roughness parameters such as the RMS height were obtained from surface profiles that were measured with a laser profilometer and a pin profilometer. The leaf-area index (LAI) values were acquired using AccuPAR LP-80 of Decagon Devices Inc., and the soil moisture contents were measured using EC-5 of the same company.

### 2. Determination of Parameters

*vv*- and

*hh-*polarizations as determined by the RMSE technique while considering the effect of multiple scattering, which means that the first-order multiple scattering is dominant for the co-polarized backscattering coefficients. However, the RMSE technique provides about 50 for the cross-polarized scattering from the corn fields.

### 3. Verification

*in situ*field-measured 21 input parameters, for multi-polarized backscattering coefficients of the cornfield. The early-stage cornfield on May 29, 2013 was a sparse vegetation field with a plant height of 30 cm and an LAI of 0.48. The backscattering from the underlying soil surface would be dominant in this case. Therefore, the improvement of the modified RTM is minimal for co-polarized backscattering coefficients. However, the cross-polarized backscattering coefficient of the modified RTM is remarkably improved, such as 2.2 dB at 50° and 4.4 dB at 60° as shown in Fig. 2.

*vh*-polarization at low incidence angles, especially 20° and 30°, may be from the fact that the RTM inherently does not include the shape and location of the full-grown leaves.

*hh*-polarization during the bean growth cycle from July to October 2010. The ground-truth data were also collected

*in situ*during the period for about two dozens of input parameters of the RTM. The LAIs of the bean fields were 0.73, 2.42, 3.21, 3.36, and 4.54 for 22, 37, 53, 70, and 86 days after planting, respectively. Fig. 4 shows the comparison between the COSMO-SkyMed data and the modified RTM for the bean field in a growing season. In this comparison, we used a curve factor of 0.5 for leaves and a curve factor of 0.7 for branches and stems, which are lower than the factors at the C-band, because the RCS reduction increases as frequency increases.

*hh*-polarization and a rough factor of 0.7 for the small-roughness effect.

### IV. Concluding Remarks

*in situ*measured ground-truth data for typical agricultural canopies. The modified RTM was multiplied by two multiplicative factors: (1) one is the curvature factor for the curvature effect of leaves and branches (or stems) for all polarizations and (2) the other is the multiple-scattering factor for the cross-polarized backscattering because the cross-polarized backscattering coefficient is very sensitive to the higher-order multiple scattering. In addition to the aforementioned two multiplicative parameters, the small-roughness parameter for microwave reflection from the underlying soil surface was also introduced.