Evolución de las tareas de MT en Medicina

The generation of ocean surface maps was carried out using a technique based on filtering a white Gaussian process with a specified wave spectrum. This is a well-established tech-nique, commonly found in literature [Rino et al.(1991), Corsini et al.(1999), Toporkov and Brown(2000),Arnold-Bos et al.(2007a)], which preserves the Gaussian statistics of sea surface elevations and slopes, while allowing to specify particular spectral properties of the wave field. Several theoretical models for wave spectra are available in literature [Pierson and Moskowitz (1964), Hasselmann et al. (1973), Fung and Lee (1982), Apel (1994), Elfouhaily et al. (1997)]. The surfaces here are generated through filtering a 2D realization (a matrix) of white noise with the directional wave slope spectrum de-veloped by [Elfouhaily et al. (1997)]. From now on, we will simply call such spectrum the Elfouhaily spectrum, and will refer to the surfaces generated using this spectrum as Elfouhaily surfaces. A schematic of this approach is shown in figure 5.1.

This produces a wave field entirely defined by the wind speed and the wind direction.

Chapter 5 The Effect of Sea State in Spatial Maps of Scattering 95

2D-FFT S k, ( ) ! 2D-IFFT

! ~ N 0, ( ) I

Wind Direction Wind Speed (!)

(U

)

Figure 5.1: An illustration of the approach to generate a realization of a sea surface.

In principle, the Elfouhaily spectrum also requires the wave fetch¹ to be specified, how-ever we assume fully developed seas for our simulations, and therefore used a constant value for the inverse wave age, equal to 0.84 [Elfouhaily et al. (1997)]. The surfaces generated through this method have gaussian statistics, and are described as linear, as they can be interpreted as superpositions of 2D sinusoidal components, whose amplitude is proportional to the square root of the wave spectrum at the specified wavenumber [Corsini et al. (1999)]. The distribution of surface heights is Gaussian, and it can be demonstrated that the distribution of slopes is also Gaussian [Ruffini et al.(1999),Soulat (2003)]. It is worth recalling here that our objective is to simulate the scattering from large-scale roughness of the ocean only, using the FA simulator. Here we are not con-sidering the scattering from small-scale features, for which the diffusive EM scattering would need to be calculated using a different scattering model. This means that a wave number cut-off needs to be chosen to identify the surface components that constitute the large-scale roughness. Therefore only the spectral components of the spectrum for wave numbers below that cut-off will contribute to the generation of the sea surface.

Several attempts exist in the literature to propose an objective method to choose this cut-off [Garrison et al. (1998)], [Zavorotny and Voronovich (2000)], [Thompson et al.

(2005)] but there is no consensus. In our case, we determined the cut-off experimentally by simulating the sea surfaces for a small enough cut-off wavenumber, and by evaluating numerically the radius of curvature of the surface to verify a posteriori if it satisfies the KA criterion, shown in equation (4.1) in chapter 4. All the sea surfaces shown in this and the following chapters are illustrated as maps of sea surface heights, expressed in

1The wave fetch is the wave generation region, namely the length of water over which wind is blowing in a particular direction, and thus generating waves (fetch is expressed in meters).

meters. Three examples of wind-generated surfaces are shown in Fig. 5.2(a)-(c). Figure 5.2 (a) and (b) show surfaces generated with wind speeds of 5 m/s and 10 m/s, and a 0^◦ wind direction with respect to the x-axis. Figure 5.2 (c) corresponds to a wind speed of 10 m/s, and wind direction of 60^◦.

(a) (b)

(c)

Figure 5.2: (a) Sea surface (in meters) generated from the Elfouhaily spectrum, for a patch of 500 m x 500 m, at 0.2 m resolution, for a wind speed of 5 m/s and a wind direction of 0^◦; (b) Elfouhaily sea surface for a wind speed of 10 m/s and 0^◦ wind

direction; (c) Elfouhaily sea surface for a wind speed of 10 m/s and 60^◦ direction.

The surfaces shown in figure 5.2 clearly show the changing roughness with increasing wind speed. In this case, the wave heights increases with increasing wind speed, as does the dominant wavelength, as the peak of the Elfouhaily spectrum shifts towards smaller wavenumbers for increasing wind speeds [Elfouhaily et al. (1997)]. Note that here a larger wind speed does not mean a rougher sea in terms of small scale ripples, but rather higher and longer waves. Although a change in the direction of the waves can clearly be seen by comparing figures 5.2 (b) and 5.2 (c), the Elfouhaily spectrum does not produce a strongly directional sea, where the wavefronts are clearly defined. In the next paragraph, we will show that other spectra (like the JONSWAP wind wave spectrum) are

Chapter 5 The Effect of Sea State in Spatial Maps of Scattering 97 capable of reproducing seas with a much more pronounced directionality than Elfouhaily.

The simulated surfaces in figure 5.2 are large enough to include a sufficient number of dominant ocean wavelengths for both wind speeds considered. In the case of the 10 m/s wind speed, the surface is able to capture on the order of five dominant wavelengths.

The need to establish and capture the spectrum up to the wavenumber cut-off would set in principle an upper bound to the size of the resolution cell in space. However, it is possible to increase the resolution in space by zero-padding in the wavenumber domain, which is equivalent to taking a longer spectrum and low-pass filtering it. The cut-off wavenumber used to simulate these surfaces was equal to 2π rad/m, and it corresponds to a cut-off wavelength of 1 m, which is roughly equivalent to five times the incident radar wavelength (0.19 m) in the case of L-band. This is not a very high wave number cut-off, and it does not impose excessive filtering of the original surface. We checked a posteriori that this cutoff produces a surface that satisfies the KA roughness criterion (4.1). As seen in chapter 4, this criterion depends on the incident radar wavenumber, as well as on the incidence angle and the radius of curvature. The radii of curvature along x and y have been calculated through the following equations:

r^x_c =

Figure 5.3 shows a 10 m/s sea surface snapshot, and the corresponding maps of absolute value of radii of curvature, calculated along the x-direction and y-direction. The radii of curvature are on average slightly smaller along the x-direction, because this is the direction in which the waves are travelling, and with the largest surface curvature.

The left-hand term of the KA roughness criterion in equation (4.1) was therefore com-puted using an incidence angle equal to 20^◦, a realistic value for GNSS-R geometries, and the median radius of curvature of the surface along x, which is a more robust and preferable choice over the mean. The resulting values is about 8, which is reasonably

(a) (b)

(c)

Figure 5.3: A 10 m/s sea surface snapshot in meters (a), and the corresponding radii of curvature along x (b) and y (c), in absolute values.

larger than 1. Figure 5.4 shows a pure swell component, modeled as a simple 2-D sinu-soidal wave, with amplitude of 1 m and wavelength of 100 m, and traveling parallel to wind direction (a) and in a direction 60^◦ from the wind direction, clockwise from the x-axis (b).

(a) (b)

Figure 5.4: Swell field (in meters), modelled as a 2D sinusoidal wave with amplitude of 1 m, wavelength of 100 m, and travelling in a direction of 0^◦ (a) and 60^◦ clockwise

(b) with respect to the x-axis.

Chapter 5 The Effect of Sea State in Spatial Maps of Scattering 99 Figure 5.5 (a) and (b) show a sea surface resulting from the superposition of a wind wave surface like that shown in 5.2 (b), with the swell in figure 5.4 (a) and (b) respectively.

We notice that the effect of the swell, whose amplitude in this case is not particularly high, appears clearly visible in the final surfaces, as well as its direction. The swell can of course influence more or less the configuration of the final surface, depending on the amplitude and wavelength of the underlying wind waves, as well as those of the swell itself.

(a) (b)

Figure 5.5: Composition of the 10 m/s wind wave surface of figure 5.2(b) with the swell shown in 5.4 (a) and 5.4 (b). Heights are expressed in meters.

This last example illustrates how explicit sea surface simulations give the flexibility to consider realistic and more complex sea states, featuring co-existing wind and swell waves, which cannot be represented using a simple PDF of slopes, as in the case of GO. Note that the surfaces in figure 5.5 are still linear, as their spectrum will simply be the one of the original wind wave surface, plus an extra spectral component at the wavenumber of the swell. The surfaces we have presented are characterized by different roughness, and it is at this stage convenient to define a measure of such roughness. We define the roughness for each surface as the sum of the Mean Square Slopes (MSS) along the x- and y-direction of the x − y plane where the sea surface is situated, as follows:

RO = MSS_x+ MSS_y (5.2)

This represents an important parameter, as it changes the sea surface configuration, and its response to the scattering of GPS signals.