The radiative transfer equation is used in active remote sensing to calculate the backscattering coefficient for a theoretical model of the earth's topography, such as vegetation, soil, snow media, and other earth terrains. In this study, a second-order radiative transfer equation is incorporated into a theoretical snow medium model.
Background of the Research
In order to calculate the current and EM fields of scatterers from a given ground terrain, an appropriate scattering mechanism must be determined. Unfortunately, the use of Rayleigh scattering theory does not take into account the coherent effect of the wave interaction between the scatterers.
Problem Statement
In this research, two types of hybridized CEM methods are applied to validate the accuracy of the backscattering coefficient result of snow medium. The backscattering coefficient results of these two CEM techniques will be compared and analyzed as each CEM technique has a different way of numerical approach and they can affect the accuracy of the result.
Objective of the research
Another aspect of the research is the type of computerized electromagnetic technique used to substantiate the accuracy of the data. To validate an improved radiative transfer (RT) model by comparing backscatter coefficient results with ground truth satellite data.
Outline of the Dissertation
These results are compared with each other and further compared with theoretical Mie results to study the accuracy of the CEM techniques and the suitability of the diffuser shapes for the snow environment. In the sixth chapter, the comparison of the theoretical data with the ground truth measurement is presented to validate the accuracy of the theoretical results for different shapes of ice diffusers of the snow environment.
RT Equation
The scattering by neighboring particles is taken into account and the intensities of the multiple scattering are taken into account. The phase correction factor was further developed for non-spherical scatterers such as ellipsoidal and disc-shaped scatterers to calculate the backscatter coefficient of the vegetation medium.
Second-Order RT Equation for Snow Medium
Surface Scattering
As explained in Equation 2.18, there are two scattering contributions representing the upper and lower surfaces. Where 𝜎𝑝𝑞𝑜1 and 𝜎𝑝𝑞𝑜2 are the bistatic scattering coefficient of the upper and lower surfaces, which is based on the rough surface model of the integral equation method (IEM) (Ewe, Chuah, and Fung, 1998).
Surface-Volume Scattering
𝑃𝑢𝑞 and 𝜎𝑝𝑢𝑠2 are the phase matrix of the ice scatterers of the snow medium and the scattering from the lower surface, respectively.
Volume Scattering
In the (up, down, down) mechanism, the incident wave is scattered toward the first scatterer, and the transmitted wave of the first scatterer is scattered down to the second scatterer before scattering to surface 1. While the (up, up, down) mechanism, in which the incident wave is scattered toward the first scatterer and the transmitted wave of the first scatterer is scattered upward to the second scatterer before being scattered.
Xu implemented the FEM in the calculation of the total and distributed field decomposition (TSFD) for layered sea ice (Xu, Brekke, Doulgeris and Melandsø, 2018). In addition, FDTD method was used to calculate the scattering effect of the complex shapes of ice scatterer of cloud by another researcher.
Model Development on Second-Order RT Equation using CEM Techniques
In addition, the aspect ratio of the ellipsoid can affect the results of the backscattering coefficient of the snow environment. However, this shape is used to investigate the backscattering coefficient of the snow environment.
Summary
A weak form of the integral equation is introduced, namely the variational approximation or the weighted residual approximation. The first approximation is a linear functional equation where the calculation of the electromagnetic quantity is a direct method.
Comparison between FEM and MoM
In the eigenvalue approach, the MoM technique implements the matrix eigenvalue equation to solve problems in electromagnetics. The steps for applying FEM and MoM to electromagnetic problems (Bhobe, Holloway and Piket-May, 2001) are given in Table 3.1, where the differences between the steps are summarized.
Coupled FEM/MoM
In previous research, different CEM techniques were used in the calculation of the scattering properties of the scatterers in snow medium. In FEKO software, the numerical FEM approach is integrated with the MoM technique to improve the efficiency of the calculation.
Formulation of Coupled FEM/MoM
Vector {𝑒} and {ℎ} are the unknown coefficients of the electric field of the volume and the magnetic field of the closure, respectively. The equation can be discretized in the form of the Galerkin procedure, as shown in equation 3.21.
Methodology of Coupled FEM/MoM in RT Equation
In the FEKO simulation, the geometries are constructed using CADFEKO and the far-field electric fields at various scattered angles are calculated, compiled and applied to the Stokes matrix of the RT equation to calculate the backscattering coefficient of the snow medium as shown in Figure 3.5. After all the electric far fields of the scattering are generated in the FEKO software, these data are collected and used in the Stokes matrix of the Radiative Transfer equation to calculate the backscattering coefficient for various parameters and will be compared with the Mie scattering model and satellite data.
Summary
Unfortunately, as the electrical size of the distribution increases, this mentioned technique produces a large number of unknowns due to the discretization of the distribution volume (Fu, Jiang, and Ewe, 2016). For example, FMA is responsible for speeding up the matrix-vector calculation in the integral equation of the EM problem.
Equivalent Principle Algorithm
Formulation of EPA
The electric and magnetic field equations as shown in equations 4.1 and 4.2 are decomposed from the entire domain into subdomains with the equivalent currents on the surfaces of the subdomains. Although EPA serves many advantages in terms of computational memory and execution time, since the equivalent surface is very close (Tiryaki, 2010), the level of accuracy is being reduced due to the representation of the basis function and causes difficulties in numerical calculation. (Fu, Jiang and Ewe, 2016).
Relaxed Hierarchical Equivalent Source Algorithm (RHESA)
Spherical Equivalence Surface
The domain under investigation is first cubed to create the lowest level oct-tree without any intersection. Each cube labeled with the alphabet Gi is surrounded by a child-level sphere (lower level) which is labeled ad 𝐸𝑆𝑖𝐶 and is surrounded by a parent-level sphere (high level) which is categorized as 𝐸𝑆𝑖𝑃.
Formulation of RHESA
The far field radiated by the source array is calculated from the ES of the source array for the observation array (Chan Fai, 2018). Fields radiated from the source array generate equivalent sources in the ES of the observation array, which generate electric and magnetic fields within the observation array (Fu, Jiang, and Ewe, 2016).
Methodology of RHESA in RT Equation
The methodology of the RT-RHESA is similar to the RT-Coupled FEM/MoM technique. The backscatter coefficient obtained from RHESA is compared with results generated from coupled FEM/MoM from FEKO software and the theoretical result from Mie to study the accuracy of the CEM techniques and investigate the suitability of ice scatterers in the form of snow.
Summary
Introduction
The other parameters needed to investigate the backscattering coefficient of the snow medium are listed in Table 5.1 (Chan Fai, 2018). The radius of the spherical scatterer used in the FEKO and RHESA simulation is the same as the one used in the Mie scattering analysis, which was 0.54mm.
Effect of Various Incident Angles on Backscattering Coefficient in FEKO and RHESA simulation
In Figure 5.3(e) and (f), the FEKO and RHESA simulated data for spherical and non-spherical scatterers follow the trend of the Mie analytical result of VH polarization. However, for HH polarization in Figure 5.4(d), only RHESA-generated droxtal-shaped scatterers follow the trend of the backscatter coefficient of Mie's theoretical results.
Effect of Various Frequencies on Backscattering Coefficient in FEKO and RHESA simulation
Effect of Various Layer Thickness on Backscattering Coefficient in FEKO and RHESA simulation
As the layer thickness increases, the difference between FEKO-generated spherical and non-spherical scatterers also increases. The difference between RHESA-generated scatterers increases with Mie analytical solution, as the layer thickness increases.
Effect of Various Volume Fraction on Backscattering Coefficient in FEKO and RHESA simulation
Mie's theoretical results, while the difference is even greater for the cases of other shapes (such as hexagonal column and cylinder). For the cross-polarized backscattering coefficient, as shown in Fig. 5.7 (f), all ice scatterers follow the trend with Mie's theoretical results.
Summary
Due to these advantages, the difference between RHESA generated results, and Mie theoretical results is low despite the increase in the layer thickness of snow medium and frequency. In the next chapter, these CEM-generated results are further compared with the ground truth measurement to validate the accuracy of the CEM-generated results and compare the effectiveness of applying two CEM methods in the second-order RT equation for snow medium.
Introduction
Comparison of CEM Techniques Generated Backscattering Coefficient of Snow Medium with CLPX Data at L-Band
All scatterer shapes simulated by FEKO and RHESA overlap with the Mie analysis result. It can be concluded that at low frequency, both CEM techniques are suitable for use in calculating the backscattering coefficient for a snow medium.
Comparison of CEM Techniques Generated Backscattering Coefficient of Snow Medium with RADARSAT Data at C-Band
For the FEKO-generated backscatter coefficients, the spherical scatterer followed by the hexagonal column mainly represents the ice scatterers for all sites. For the backscattering coefficients generated by RHESA, spherical scatterers followed by peanut, droxtal, and ellipsoid represent the ice scatterers for most sites.
Comparison of CEM Techniques Generated Backscattering Coefficient of Snow Medium with CLPX Data at Ku-Band
As described in chapter 5, the accuracy of the coupled FEM/MoM decreases as the frequency and layer thickness of snow medium increase. Thus, all the shapes are the potential shapes of the ice dispersers of snow medium.
Summary
As presented in Chapter 5, as the frequency and thickness of the snow medium layer increase, the accuracy of the backscattering coefficient produced by FEKO decreases compared to that of RHESA. The differences between the backscattering coefficient produced by FEKO and the theoretical Mie result increase as the frequency and thickness of the snow medium layer increase.
Conclusion of the Research
In Chapter 3 and 4 of the thesis, the development of the RT-coupled FEM/MoM and RHESA, respectively, is presented. In the last series of the study, the comparison was performed with the Ku band frequency CLPX data.
Advantages and Limitations of the CEM Approaches
This is because the increase in the number of unknowns and numerical approaches with low-order basis function may cause inconsistencies in the results. Therefore, the calculation of the field and currents is calculated according to the low-order basis function.
Future Improvement
Modeling active microwave remote sensing of snow using dense media radiative transfer theory (DMRT) with multiple scattering effects. Modeling active microwave remote sensing of snow using dense media radiative transfer theory (DMRT) with multiple scattering effects.
