Fases de un sistema basado en MT

The size of the facets is a key parameter that affects the applicability of FA. The choice of facet size has to be a trade-off between the need to comply with the KA conditions on roughness, the ability of the facets to adequately approximate the underlying sea surface, and computational expense. Therefore the choice of facet size cannot be entirely arbitrary but must be governed by some specific criteria. The criteria we adopt here to choose the size of the facet stem from considerations first presented in [Bass and Fuks(1979)] to discuss the applicability of the Kirchhoff approximation. These relate to geometrical conditions illustrated in figure 4.8, which constitute the starting point for the formulation of the criterion for KA applicability, formulated in (4.1).

With reference to figure 4.8, it is considered that the reflection of an electromagnetic wave at a point A on the surface can be taken to occur as if from a tangent plane centered at that point, if one can identify a region on the tangent plane Σr with linear dimensions which are large relative to the EM wavelength, but which also does not deviate noticeably at the edges of the region from the underlying surface. This region on the tangent plane is what we define as a facet. The argument above translates into

Chapter 4 A New Scattering Model for GNSS-R: the Facet Approach 91

Figure 4.8: Illustration of the tangent plane Σrand its local coordinate system, with the half-length AB of the facet and the distance from the underlying surface BD.

the two mathematical conditions [Bass and Fuks (1979)]:

AB� 1

k₀cos (θ) (4.37)

BD� cos (θ)

k₀ (4.38)

where AB and BD are the segments shown in figure 4.8. We can easily see that:

BD = OB− OD =�

AB²+ rc2− rc (4.39)

where rc is the local radius of curvature of the surface. Thus, we can express both (4.37) and (4.38) in terms of AB as:

In our case, AB represents half the size of our facet, and criteria (4.40) and (4.41) will be used to determine the appropriate facet size [Clarizia et al.(2012)]. It is worth noting that the inequalities (4.40) and (4.41) can be more or less difficult to satisfy, depending on the quantitative interpretation of the much greater than and much smaller than inequality signs. We could choose to interpret those as AB having to be larger (or

smaller) than the term on the right-hand side by a factor of, say, at least 10. In chapter 5, we analyse how the choice such a factor strongly influences and can ultimately limit the applicability of (4.40) and (4.41), and that a stringent condition like the choice of a factor of 10 makes the KA not applicable in most cases. In our simulations, shown in the next chapters, we have therefore relaxed these conditions by considering a factor smaller than 10. It is interesting to note that, if we combine (4.40) and (4.41) we obtain a condition similar to the standard applicability condition (4.1) of the Kirchhoff Approximation, namely:

�3

2k0rccos (θ) � �³

1 − cos⁴(θ) (4.42)

For simplicity, in our simulations we will use the standard Kirchhoff Approximation condition (4.1) to define what constitutes large-scale surface roughness compliant with KA, and the conditions (4.40) and (4.41) to determine the size of the facets.

Chapter 5

The Effect of Sea State in Spatial Maps of Scattering

5.1 Introduction

In this chapter, we exploit the FA-based scattering simulator for realistic ocean surfaces, and investigate the results through spatial maps of scattering from these surfaces. First we focus on the methodology to simulate explicit realizations of linear gaussian sea surfaces, with different sea states. We consider sea surfaces composed of wind waves only, obtained from different theoretical wave spectra (Elfouhaily, JONSWAP), and mixed sea surfaces composed of wind waves and a swell. For these surfaces, we calculate the EM scattering of a simple incident spherical wave, using the FA method, outlined in Chapter 4. We validate the FA through comparison of its scattering results with both GO and a numerical implementation of the full KA, for the same sea surface conditions.

Next, we focus on how the scattering results change for different underlying sea surfaces in the space domain, leaving those relative to the delay-Doppler domain to be discussed in Chapter 6. The results are first presented in the form of spatial maps of Normalized Radar Cross Sections (NRCS), where the NRCS is calculated individually for each facet

93

on the sea surface. In order to investigate potential polarization effects, we analyse the ratio between NRCSs of different polarization, and call it the Polarization Ratio (PR).

Subsequently, we examine results in the form of NRCS and PR curves, as a function of the scattering angle, calculated from the whole surface, as a coherent summation of the contributions from all the facets. In both cases, we show that the scattered power and polarization ratio are well-correlated with the underlying sea surface from which the scattering originated, and they are sensitive to differences of the sea surface conditions.

The last section of the chapter shows how this FA-based simulator, developed for L-Band and in a GNSS-R context, actually works well also for other incident wavelengths, and different scattering geometries (i.e. monostatic), providing a further proof of its flexibility and versatility. Some of the work and results shown here were presented in a recent publication in IEEE Transactions on Geoscience and Remote Sensing, by [Clarizia et al. (2012)].