# Validation of the Numerical FEM Approach for a Single Layer

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## Passive and Active Remote Sensing

Methods for observing the environment by remote sensing can be divided in two distinct categories: active and passive. In active methods an artificially created electromagnetic wave is sent to the object to be sensed and the returned signal analysed. In passive methods it is the environment’s natural thermal emission that is detected and analysed. Thus for the active case we focus on scattering from the material and in the passive case we focus on emission. The theory behind active remote sensing is based on the theory of electromagnetism whereas the theory behind passive remote sensing is based on radiation theory. In the following sections we present these two theories, focusing on areas that are important for remote sensing. As we will see electromagnetic theory relating to scattering can be linked to the concept of emission found in thermal radiation, by Peake’s theorem (1959): we can calculate an object’s emission by integrating the scattering resulting from an incident wave. Thus, although experimentally passive and active remote sensing techniques each provide different information about an object, theoretically we can calculate one from the other. In this thesis we are interested in the emission of soil-litter systems. However in theoretical modelling approaches it is usual to calculate the emission from the scattering, including the approach we develop and apply. We will therefore present the theory behind both electromagnetic scattering and thermal radiation as both are relevant to this thesis.

Electromagnetism

The theory of electromagnetism was formulated by Maxwell. It describes the behaviour of magnetic and electric fields for a given system of electric currents and charges. It is a macroscopic theory and so does not consider the microscopic processes in a medium in the presence of an electromagnetic field. Electric and magnetic properties of a specific medium are described by three macroscopic quantities: the magnetic permeability, , the conductivity σ and the permittivity ε. Electromagnetic theory rests on Maxwell’s four equations. These four equations are very powerful since they are simple, yet fully describe an electromagnetic field problem: all electromagnetic theory can be derived from them.

Maxwell’s Equations

Maxwell’s four equations can be expressed in differential or integral form. In differential form they can be written as: Ñ.D = r.

### Plane Waves and Polarisation

There are numerous possible solutions to the wave equation and correspondingly numerous types of wave, of which the most basic form is the plane wave. In remote sensing we deal extensively with plane waves since antennas emit waves that may be considered plane far from the emitting antenna and also the earth’s thermal radiation may be considered plane when measured at a distance far from the ground.
A plane wave propagating in an arbitrary direction given by the wavevector k can be described by the following equations:
E(r, t) = E0 exp(ik × r – w.t) (2.15a).
H(r, t) = H0 exp(ik ×r – w.t) (2.15b).
where ω is the phase velocity. Note we take the real part in the above equations but it is common practice to write wave equations in complex form and the real part is implied. A wave is considered to be plane when its electric field remains in the same plane with respect to its propagation. Substituting these wave equations into Maxwell’s equations (2.1a) and (2.1b) we find that k is perpendicular to both H and E as follows: Ñ × H = ik × H = 0 and Ñ × E = ik × E = 0 .
Since the electric and magnetic fields must also be perpendicular these two fields form a plane orthogonal to the direction of propagation, called the polarisation plane.
Inserting the above equations into the wave equation we deduce the following relationship:   1  1 (2.16).
ω is calculated as 2π/λ, where λ is the wavelength. This equation for k is the dispersion relation of electromagnetic waves in unbounded space.

### Total Reflection and the Brewster Angle

Two phenomena of interest relating to plane wave reflection and transmission across a plane boundary between lossless media are total reflection and total transmission. These two can be derived from Snell’s law.
Total reflection occurs when a wave is incident from a more optically dense to a less optically dense medium (k1>k2) and the incident angle is greater than the critical angle θc, such that: sinc  k 2 k1 . (2.29).
Inserting this value into Snell’s law we find that cosθ2 is entirely imaginary for θ1≥θc and thus the wave is completely reflected; no average energy can be transmitted into the lower medium. This phenomenon is true for both H and V polarised waves.
Total transmission occurs for V polarised waves at an incident angle equal to the Brewster angle, θB, where:  1 tanB 2 2  1 (2.30).
This follows directly from Snell’s law if we let R=0.
This effect can be understood qualitatively by considering electric dipoles in the medium. The incident field is absorbed by the medium and then reradiated by oscillating electric dipoles at the interface. The dipoles oscillate in the polarisation direction of the transmitted wave, the same oscillation producing the reflected beam. However dipoles cannot radiate any energy along their direction of oscillation. Therefore when the direction of the refracted beam is perpendicular to the direction of the reflected beam, as is the case at the Brewster angle, the dipoles cannot radiate any energy in the reflected direction and total transmission occurs.

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#### Wave Propagation in a lossy medium

If medium 2 in figure 2.2 has a relative permittivity constant with a non-zero imaginary part, then the transmitted wave experiences a loss in energy as it travels through this medium, associated with the imaginary part of the permittivity. To illustrate this, let us consider a wave propagating in the z direction through an isotropic medium with complex relative permittivity εr and complex relative permeability µ r. Inserting (2.16) into (2.15a) (and settingr=1) we find that the electric field can be expressed as:
E x E x ez (2.31a).
 ez (2.31b).
E y E y Ez  Ez ez (2.31c).
The electric and magnetic fields are both perpendicular to each other and to the direction of travel of the wave. Let us therefore take the E field to be entirely in the y direction and the H field in the x direction.
γ is the propagation constant and is complex since it depends on εr and r, which are both complex. It can be written as:  i ‘ (2.32).
where α is the attenuation constant and β’ is the phase constant. Thus a wave travelling in a lossy medium, i.e. one with non-zero values ofr” and/orr”, will be attenuated by a factor of e-α. Figure 2.3 shows a representation of this. A wave passing from a vacuum to a medium will also undergo a phase change of +β’.

1. Introduction
2. Background Theory
2.1 Passive and Active Remote Sensing
2.2 Electromagnetism
2.2.1 Maxwell’s Equations
2.2.2 The Wave Equation
2.2.3 Plane Waves and Polarisation
2.2.4 A Superposition of Waves
2.2.5 The Poynting Vector
2.2.6 Waves at boundaries
2.2.6.1 Reflection and Transmission coefficients for H and V polarization
2.2.6.2 Total Reflection and the Brewster Angle
2.2.6.3 Wave Propagation in a lossy medium
2.2.7 Layered media
2.2.8.1 Radiation from current sources: the Hertzian Dipole
2.3 Dielectric Properties of Mixtures: Effective Media
2.3.1 Physical mixing formulas
2.3.2 The Semi-empirical Refractive Mixing Formula
2.4.1 Important Quantities in Radiation and their definitions
2.4.2.2 Non-black body Radiation and Emissivity
2.4.3.2 The Simplified Radiative Transfer Equation
2.5 Passive Microwave Remote Sensing of Land
2.5.1 Emission of Bare Soil Surfaces
2.5.1.1 The soil dielectric permittivity constant in the microwave region
2.5.1.2 Non-uniform Temperature and Dielectric profiles
2.5.1.3 Rough Surface Scattering and Emission
2.5.2 Modelling the emission of the ground covered by vegetation
2.5.3 Note
3. Modelling the Emission of the Soil-Litter system
3.1 Microwave Emission of Soil: modelling the effects of a rough surface
3.1.1 A Semi-Empirical model for soil emission at L-Band
3.1.1.1 Q-h Model Formulation
3.1.1.2 Model Development
3.1.1.3 Comparing semi-empirical models to theoretical models
3.1.2 Analytical Models
3.1.2.1 General Approach
3.1.2.2 Different Analytical Models
3.1.3 Numerical Models
3.1.3.1 Methodology
3.1.3.2 Numerical Methods
3.1.3.3 Selection of the appropriate numerical approach
3.1.3.4 Validation of Numerical Methods
3.1.3.5 Problems of current interest
3.2 Modelling the contribution of the litter layer to forest emission at L-band
3.2.1 The Forest structure
3.2.2 Remote Sensing of Forests
3.2.2.1 Experimental data
3.2.2.2 Theoretical Models
3.2.3 Discussion
3.3 Choice of approach for modelling the soil-litter L-band emission in this thesis
4. The Numerical FEM Approach developed for Calculating the L-band Scattering and Emission of the Soil-Litter system found in Forests
4.1 Model Description
4.1.1 Ansoft’s HFSS software
4.1.2 Calculating the scattering and emission of forest multilayer structures using a numerical FEM approach
4.1.2.1 Building the structure
4.1.2.2 HFSS Simulations
4.1.2.3 Analysing Results calculated by HFSS
4.1.3 Note
4.2 A sensitivity analysis to set model parameters
4.2.1 Model Parameters and calculation conditions
4.2.2 Method and Results
4.2.2.1 Number of Rough Surfaces
4.2.2.2 Integration step, s
4.2.2.3 Type of incident beam
4.2.2.4 Surface size
4.2.2.5 Calculation Cost
4.2.3 Conclusions: Values determined for model parameters
5. Validation of the Numerical FEM Approach for a Single Layer
5.1 Comparison with Fresnel for a flat surface
5.1.1 HFSS calculation set up
5.1.2 Results and Conclusions
5.2 Comparison with the Method of Moments for a rough surface
5.2.1 Method of Moments data
5.2.2 Method
5.2.3 Results
5.2.4 Conclusion
5.3 Comparison between the numerical approach, experimental data and the AIEM model
5.3.1 SMOSREX 2006 dataset
5.3.2 Method
5.3.3 Results and Discussion
5.3.4 Conclusions
6. Emissivity of the Soil-Litter system: comparison with Experimental Data and the Schwank Model
6.1 The Bray 2009 Experimental campaign and the Schwank model predictions
6.2 Method
6.3 Results and Discussion
6.4 Conclusions and Perspectives
7. Conclusions
8. Perspectives .

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