Theory of the Thesis
2.5: Optical Properties of Seawater
2.5.3: Scattering
The second important IOP is the scattering, the dispersion of incident photons into other directions that prevents the forward on-axis transmission. Hence, the beam is spread, losing light intensity that result in angular redistribution of the optical field. Scattering in the open ocean is due to organic particles, such as phytoplankton, while in the coastal waters it is produced inorganic matter.
Scattering can be classified into three broad regimes depending on the radius of the particles [47,55]:
Molecular (Rayleigh) scattering , caused by inhomogeneous local concentrations of and particulate matter, with an intensity proportional to the sixth power of their diameter and inversely proportional to the fourth power of the wavelength;
Turbulent scattering , not strongly dependent on the wavelength but proportional to the square to particle diameter; very large fluctuations of temperature
Large particles , organic and inorganic particles approximately ten times the wavelength.
The volume scattering function (VSF), expressed in units of is a probability density function that describes the angular distribution of photons as a function of the direction of photon travel into a solid angle with the centered in . The scattering angle is described by the polar coordinates, with the nadir angle
25
And azimuthal angle , as shown in Figure 2.8 following the notations adopted by C.D Mobley [56].
Considering the case of the deflections caused by large particles of size comparable to and assuming a spatially homogeneous medium, after integrating the VSF over all angles gives the total scattering coefficient that is a um of the total scattering events regardless of the single event at a specific angle. For sake of simplicity, the scattering is considered azimuthally symmetric, i.e., the azimuth dependence is neglected.
(2.4)
It is usually assumed in the literature a spatially homogeneous medium and spherical and/or randomly oriented particles. The widely cited scattering dataset is the one published by Petzold in 1972 [57], based on the measurements in different waters: very clear in Bahama Island, coastal offshore southern California and turbid harbor in San Diego, California.
The physical meaning of VSF is the scattered intensity per unit incident irradiance per unit volume of water. Figure 2.9 shows the VSF of representative ocean waters under single scattering conditions as measured [57]. From the Log-log plot it is clearly visible that, in
26
all the water types considered, the scattering event for angle smaller than ° (forward direction) is about four orders of magnitude larger than side and backscatter. This arises from the fact that the diameter of the particles suspended in seawater is many times larger than the wavelength of transmission. Thus, the peak of forward scattering for angles could be quite beneficial from the perspective of the communications link geometry (discussed in Section 2.2.2). In fact, photons that are scattered multiple times have a higher chance of being redirected back into the receiver, increasing multipath delay. The latter results in photons reaching the photodetector by two or more paths and, because of the presence of multiple paths, more than one pulse will be received at different times. This, in turn, may result in intersymbol interference (ISI) between the received pulses and the time taken for each bit of data. The temporal dispersion between two consecutive photons is contained slowing down the bit rate, thereby decreasing the performance of the communication system [44, 58]. However, for highly turbid waters and moderate distance, it can be neglected [59].
Despite the fact that the pioneering Petzold’s measurements are a reference set of experimental data for the VSF, they represent only a few types of water in a particular time and thus they can be considered as an indicative value. When modelling the optical properties of a specific water it is then necessary to determine the VSF by the size distribution of particles. As an example, experimental data collected by Agrawal offshore of the New Jersey showed a temporal variability with increasing difference from Petzold’s measurements when increasing the depth [60].
In order to simulate the effects of increased water turbidity in a laboratory, Maalox antacid is used as a scattering agent in addition to fresh water. It consists mostly of with a very similar VSF to that of seawater, allowing to compare the taken measurements to actual ocean environments [57, 61, 62].
A commonly used technique to simulate the scattering effect within an underwater optical channel is multicasting Monte-Carlo numerical simulation (MCNS). The main result from this approach is a higher number of photons collected by the receiver, thus better performance. The reason for this difference relies on the fact that the scattering component in the Lambert-Beer’s law is considered as an optical loss only. In many situations, especially in seawaters characterized by high turbidity, a fraction of the total number of scattered photons would still be
27
able to reach the receiver. However, this situation is likely to induce ISI, as discussed above.
Recently, a modified version of the Beer’s law for a Lambertian source, such as a light-emitting diode (LED), which includes an exponential function with weighting coefficients, has been proposed by Liquet al. [36]. The exponential function is obtained by numerical fits from data in the scientific literature for typical values of various seawaters.
As anticipated in Section 2.1.2, Morel in 1991 [52] modelled the dependence of the scattering coefficient, , as a function of the chlorophyll concentration
(2.5)
where is the scattering coefficient for pure seawater. As one would expect, the value of the scattering coefficient Equation (2.5) increases as the wavelength increases. This behavior is shown in Figure 2.7. Also, similarly to Equation (2.3), there is a non-linear dependence between and .