There are various shapes and sizes of leaves in rangelands and farming fields, such as blade-type and disk-type lossy dielectric leaves. Those natural and agricultural leaves usually have curved shapes and/or irregular surfaces. The reduction of radar cross section (RCS 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].

where S_c is the scattering matrix element for a curved lossy dielectric sheet, S_fp is the scattering matrix element for a flat plate, F(γ) is the Fresnel integral with an argument γ, and k₀ 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_factor, to the scattering matrices for leaves and stems to compensate for the RCS reduction because of the leaf-curvature effect in the RTM.

where [S]_m is the modified scattering matrix of a leaf or a branch.

The curve factor may also cover the RCS reduction that would be caused by the irregularity deviating from the flatness of a leaf.

The RTM does not include the higher-order multiple scattering as shown in Fig. 1, whereas the actual microwave backscatter will include the higher-order multiple scattering inside the vegetation layer and between the vegetation layer and the underlying ground surface. A full-wave analysis by the moment method for a relatively sparse vegetation field revealed that the effect of multiple scattering is negligible for co-polarization, whereas the cross-polarized backscattering coefficients are significantly influenced by the effect of multiple scattering [8]. For example, the difference between a full-wave analysis that includes all orders of multiple scattering and a theoretical model that includes only single scattering is about 9 dB for 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].

The effect of multiple scattering can be accounted by multiplying a factor, the so-called “multiple factor” M_factor, to the transformation matrix elements for the higher-order multiple scattering from scatters in a vegetation layer.

where [T]_km is the modified k^th transformation matrix, and the multiple factor may be obtained from an extensive database of experimental measurements.

For the backscattering from an underlying soil surface, the microwave radiates from an antenna of a radar far from the rough surface. Therefore, the surface roughness should be considered for the whole antenna footprint that provides a largescale roughness. However, for the reflection from the underlying soil surface in the scattering mechanisms 2a, 2b, and 3 in Fig. 1, the scattering particles are positioned near the surface. Therefore, the area of reflection might be small, which provides a small roughness. The retrieval of the RMS height depends on the length of the surface profile, and the measured RMS height rapidly decreases with a decrease in the profile length of the surface [13, 14]. Therefore, the use of large-scale roughness parameters for mechanism 4 and small-scale roughness parameters for mechanisms 2a, 2b, and 3 can be recommended.

The effect of small roughness would be implemented with the decrease in the RMS height for the computation of the microwave reflection, such as h_RMS–s = roughness factor (R_factor)×h_RMS, where h_RMS–s is the small-roughness RMS height for reflection, and h_RMS 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.

where [R]_m is the modified reflectivity matrix, and

with X=2(kh_RMS–scosθ)², where Γ_sp is the reflection coefficient of a rough soil surface, Γ_fp is the Fresnel reflection coefficient of a flat plane, the subscript p is the polarization, θ is the incidence angle, and I₀[···] is the modified first-kind Bessel function.

The Hongik polarimetric scatterometer (HPS) was used to acquire the full-polarimetric backscattering coefficients of vegetation fields to closely examine the RTM. The backscattering coefficients of a cornfield in Suwon, Korea, were acquired at the C-band and five different incidence angles (20°, 30°, 40°, 50°, and 60°) from May 29 to July 11, 2013, covering a whole growing season. The HPS is a vector-network-analyzer-based full-polarimetric L-, C-, and X-band scatterometer, and it is calibrated by the differential Mueller matrix calibration technique (DMMCT) [15] with a single calibration target, namely, a trihedral corner reflector. The backscattering coefficients of a bean field in Suwon, Korea, were also collected by the COSMO-SkyMed SAR at the X-band at 40° for hh-polarization during the bean growth cycle from July 22 to September 24, 2010.

We also collected 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.

By minimizing the root-mean-square error (RMSE) between the RTM and the scatterometer measurements of a cornfield at the C-band, we can obtain optimum curve, multiple, and roughness factors. It was found that the optimized curve factors for considering the effect of leaf curvatures are about 0.7 for leaves and 1.0 for stems at the C-band for scattering from cornfields. The curve factor depends on frequency (or wavelength) [6], and the curve factor becomes about 0.5 for leaves and 0.7 for branches (stems) at the X-band for scattering from soybean fields.

The co-polarized backscattering coefficients that were measured from vegetation fields agree well with the RTM, and consequently, the multiple factor is about 1.0 at 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.

The RMSE technique leads us to the rough factor of about 0.7. The effect of the rough factor is negligible for very sparse and very dense vegetation canopies, because the interaction between the vegetation canopy and the underlying surface in these cases is negligible.

The RTM is modified with the induced parameters from the effects of the leaf curvature, multiple scattering, and small roughness for the reflection, and its accuracy is examined by comparisons between the modified RTM and the scatterometer measurements. Fig. 2 shows the comparison among the scatterometer measurements on May 29, the traditional RTM (“Old Model”), and the new RTM modified with the “curve factor,” “multiple factor,” and “roughness factor” as explained in the previous section, and with 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.

Fig. 3 shows the comparison among the scatterometer measurements on July 11, 2013, the traditional RTM, and the modified RTM for the multi-polarimetric data. The full-grown cornfield on July 11 was a dense vegetation field with a plant height of 250 cm and an LAI of 2.62. Fig. 3 shows that the traditional RTM provided about 2–3 dB higher co-polarized backscattering coefficient than the radar measurements without correction for the effect of leaf curvature, whereas the modified model and the measurements agree quite well for the co-polarized backscattering coefficients. The improvement of the cross-polarized backscattering coefficients with the modification for the higher-order multiple scattering is very large as shown in Fig. 3 and Table 1, especially for higher incidence angles, because of (1) the vertical structures of the corn-plant stems and (2) the effective depth increase at higher incidence angles. The discrepancy between the measurements and the modified RTM for 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.

The modified RTM is also compared with the COSMO-SkyMed data at the X-band, which were acquired from a bean field in Suwon, Korea, at 40° for 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.

There are no changes in the multiple and rough factors: i.e., a multiple factor of 1.0 for hh-polarization and a rough factor of 0.7 for the small-roughness effect.