Disciplinas relacionadas con la MT

In this section, we present the EM model used in our GNSS-R Simulator to compute the scattering, and referred to as the Facet Approach (FA). As mentioned in paragraph 4.2, the FA represents a novel implementation of the Kirchhoff Approximation (KA). We will show that it preserves the advantages of KA over the high frequency GO by providing a better representation of the scattering and polarization properties, but reduces the computational cost compared to KA, making it more practical to simulate scattering from larger surfaces. The FA is a new approach to calculate the scattering from large-scale surface roughness components based on solving the Kirchhoff integral in (4.16), for a surface S represented by an ensemble of n planar facets, each of them tilted and oriented by the waves. An example of a 1D surface approximated by planar facets is shown in figure 4.4.

Figure 4.4: A portion of a 100 m 1D sea surface, with its associated facets.

In principle, the Far-Zone approximation can be applied both with respect to the entire surface S, and with respect to each facet. For the FA, we apply the Far-Zone approxi-mation with respect to each facet, and this causes some changes in some of the terms in equation (4.16). The terms R1 and R2 become the ranges from the transmitter and re-ceiver to the central point of the facet, and are therefore dependent upon the particular

Chapter 4 A New Scattering Model for GNSS-R: the Facet Approach 85 position of the facet considered. The incident and scattering unit vectors ˆn_i and ˆn_s no longer refer to the specular point on the surface, but to the central point of the facet, causing also the terms p and q to be facet-dependent. The integral (4.16) can therefore be split into the sum of integrals over each facet as follows:

E^s=�� transmitter/receiver to the central point of the m-th facet;

• pm = ˆn^m_s × [ˆn^m× E^s− ηsˆn^m_s × (ˆn^m× H^s)] is the vector term for the m-th facet, and ˆn^m is its local normal;

• qm= k0(ˆn^m_s − ˆn^m_i ) is the scattering vector relative to the m-th facet.

We assume that each facet has projections of its sides along the x and y direction (L_x and L_y respectively) equal to each other, and that these projections are also the same for all facets. We now focus on the scattered field from a single facet, which can be written in an analytical form. If the facet is sufficiently large compared to the wavelength of the incident radiation, the EM scattering from the facet can be assimilated to that from an infinite plane. This makes the vector term p_m constant across a facet, and allows it to be taken outside the integral, as follows:

E^f_s^m = K_mpm

� �

fm

e^jqm·rmdS_m (4.29)

The vector term p_mcan be evaluated by applying the KA through (4.14) and (4.15). At this point, the rest of the integral simply becomes the integral of an exponential term over a facet, which tilt along x and y can be known. In order to solve the integral, we first need to convert it from dS_m to dx_mdy_m. The integration variable is given by

rm = [x_m, y_m, z_m(x_m, y_m)], but for a single facet, which is an inclined finite plane, the analytical expression for zm(xm, ym) is well-known, and given by:

z_m = αmx_m+ βmy_m (4.30)

where αm and βm represent the inclinations of the plane along x and y, given by:

α_m = ∂z_m ˆn^m. The differential dSm is linked to dxmdym through the Jacobian as follows:

dS_m=

Chapter 4 A New Scattering Model for GNSS-R: the Facet Approach 87 At this point, we can solve the integral of the exponential in a closed-form, as follows:

E^f_s^m =g_m in brackets for Lx/2 and L_y/2 respectively, to write them as sinc functions. The final result for the scattered field from a single facet will be [Clarizia et al.(2012)]:

E^f_s^m = −gm(LxLysinc [(qx,m+ qz,mαm)Lx/2] sinc [(qy,m+ qz,mβm)Ly/2]) (4.36)

where we have defined sinc(x) = ^sin(x)_x . The total scattered field in (4.28) can be then evaluated by summing coherently (accounting for the phase term) the scattered fields from each facet, given by (4.36). The sinc terms that appear in equation (4.36) clearly indicates that the FA treats the facets as radiating antennas, with a specific non-zero width main lobe, which allows some scattered power in directions away from the specular direction. The width of the sinc lobe decreases with increasing facet size, so that large facets have scattered power concentrated around the specular direction in a narrow lobe.

An illustration of the radiating lobes for facets of two different sizes is illustrated in figure 4.5.

The width of the sinc lobe also depends on the overall incidence and scattering geometry at the facet. An example of how such lobe varies for different geometries is illustrated in figure 4.6.

(a) (b)

Figure 4.5: An example of the 3D radiating lobe for a smaller facet (left) and a larger facet (right). The colors refer to the distance from each point on the surface of the lobe

to the origin of the lobe.

!

Z

ˆn_f

Facet

From Tx Specular Dir.

!

Z

ˆn_f

Facet

From Tx To Rx

!

Z

ˆn_f

Facet

From Tx Specular Dir.

!

!

!

Specular Dir

(a)

(b)

Figure 4.6: First row: schematic of geometry for three different cases, where a 1 m facet is tilted with an angle of 10^◦away from the local normal, and the incidence angle is 10^◦ (a), 30^◦ (b) and 50^◦ (c). Second row: Plot of a 1D sinc function (the first sinc

term in equation (4.36)), as a function of the scattering angle, for the three cases.

Chapter 4 A New Scattering Model for GNSS-R: the Facet Approach 89 Figure 4.6 shows in 1D, on the x − z plane, how the width of the sinc increases, particu-larly the half lobe towards the receiver, when the incidence angle increases too. Here the side of the facet has been kept equal to 1 m. This distortion of the lobe is intrinsically linked to the equation in (4.36), and it is basically caused by the x-, y- and z-component of the q vector, which form the arguments of the sinc functions. More specifically, since the scattering vector q is built as the difference between two unitary vectors, it is not a unitary vector itself, and therefore its magnitude and components can change signif-icantly, depending on the incidence and scattering angles. A simple example can help illustrate this effect: for a nadir-pointing monostatic case, we would have the unit in-cidence and scattering vector both aligned with the z-axis, in opposite directions, such that the q vector will only have a z-component, and will have a magnitude of 2. If instead the incidence and scattering angles were to be 45^◦, the resulting q vector would still have a z-component only, but with a magnitude of √

2. On the other hand, in the extreme case of unit incidence and scattering vectors aligned along x (θi = θs= 90^◦), the q vector would have only a x-component, of magnitude 2. This effect has been detected before. Figure 4.7 (a) shows the distorted sinc lobe from [Balanis (1989)], representing the modelled scattering pattern of a finite conducting plate, for an incident plane wave.

Figure 4.7 (b) shows a plot of the first sinc term in (4.36), where the incident, scattering angle and local normal have been matched to the example shown in [Balanis (1989)].

The two sinc functions in figures 4.7 (a) and (b) are the same, which also confirms the validity of equation (4.36) of the FA. This distortion of the sinc lobe could be a matter of concern, since the scattered power from a facet does not appear to drop fast enough for directions away from the specular, when the angle between the incident vector and the local normal is high (figure 4.6 (c)). The FA model could therefore produce a higher scattered power that the actual one, from areas within the glistening zone that are far from the SP. In physical terms, one might therefore expect the width of the sinc lobe to depend only on the facet size, and to be centered at the direction of reflection, which depends on the facet inclination. However, we will show in chapter 5 that the scattering results when width of the sinc lobe is fixed are not consistent with results obtained from Geometrical Optics, while those obtained from the FA model without any constraint on the sinc lobe are much more consistent with GO. For this reason, we have used the FA

model mathematically derived, and expressed by equation (4.36), as this is the correct approximation.

(a) (b)

Figure 4.7: (a) plot of the scattered field from a finite conducting plate, of side 2λ, where a plane wave hits the plate with an incidence angle of 30^◦ [Balanis(1989)]. (b)

1D plot of first sinc term in (4.36), computed for the same parameters as in (a).