Solution Comparison

For the purpose of this research, an appropriate RTE solution is sought from those mentioned in the previous section. As focus has been primarily on variability in the underlying optical properties and noise condition, an appropriate solution is one that is able to accurately implement a channel with variable properties. Particularly, the solution chosen must have the ability to model: attenuation changes with depth, first introduced in sec. 3.3.2; changes of refractive index with depth, mentioned in sec. 3.5; temporal changes, encapsulated by turbulence and daily or seasonal cycles; finally, changes to the background lighting condition from sec. 4.2. In the remainder of this section, the RTE solutions are tested against this specification, criterion at a time.

Varying Composition with Depth

In deep oceans, where the chlorophyll level is near zero, scattering is low and use of the BSF is acceptable because light that is re-scattered towards the receiver will be of low intensity compared

to that of light directly received. However, towards the surface of the ocean, scattering and

subsequent multi-path effects become very common, so it is important to calculate the temporal dispersion. The RTE and derived numerical solutions are all capable of doing so to a good degree of accuracy, although discrete ordinate methods will begin to fail in areas of extremely high chlorophyll level due to the VSF becoming highly-peaked.

For communication links that have a noticeable change in composition with depth, the inherent optical properties become a function of depth. Working out the power loss with the

Beer-Lambert law is done by simply averaging the attenuation coefficient over the

communications link, as was implemented in ch. 4; the BSF can also be calculated using averaged values. However, it is not known how accurate this is in comparison with considering a truly varying attenuation coefficient. Having said this, in the form of the RTE given in eq. 5.2, the RTE also does not take account of inherent property changes. If the inherent optical properties in this equation were written in vector form which depends on the position vector r, it would be possible to include attenuation gradients into subsequent numerical models. In this case, Monte Carlo methods offer the best model due to its flexibility. For solutions using discrete ordinates, there will be substantial implications on the run time, as more layers are needed to accurately describe the gradually changing attenuation. In both cases, special consideration needs to be given to the link orientation, the VSF and RTE descriptions and subsequent solutions are better suited to links where the optical property variability is perpendicular to the direction of the beam, i.e. vertical underwater communication links.

Varying Refractive Index with Depth

Changes in the bulk refractive index of the aquatic medium are built into the description of the VSF. For this reason, all models are able to take a depth-varying refractive index into account. However, for communication links which are not perpendicular or in parallel with the changes in depth, modelling the directional impact of these refractive index gradients is not possible. This is because in the derivations of the VSF and RTE, inherent optical property changes are assumed to be over a single dimension only (the direction of photon travel). Links at other orientations underwater, on the other hand, have two simultaneously changing components.

It is noted that light transmitted at an angle through a gradually changing refractive index will slowly deviate from a straight line as it travels through multiple refractive index boundaries, possibly bending a communication link away from the expected receiver location. As the impact of this is potentially significant, depending on the extent of beam deviation, ray tracing methods are used later in this chapter (in sec. 5.4) to estimate the magnitude of the beam deviation under these conditions.

Temporal Variation

Temporal variation comprises of small-scale changes in the aquatic medium from turbulence and larger scale daily and seasonal changes. The latter may be modelled by simply inputting a new set of inherent optical properties describing the medium at a particular point in time. This is not limited to any particular model type.

As the most significant effect of turbulence for a communication system is temporal dispersion, the Beer-Lambert law and BSF are severely limited in their ability to describe a link passing through turbulent waters. In the RTE, turbulence can be modelled as a statistical distribution of scattering strength, vector directions and vector location, as hinted upon in sec. 2.5.5. Monte Carlo methods can readily implement this but the subsequent statistical errors are likely to be significant, considering a statistical solution is being formed from other statistical distributions. It is possible to use invariant imbedding or discrete ordinates, but this is extremely complex and has not yet been attempted by anyone researching underwater optical communications.

Variable Background Lighting Conditions

The contribution from sunlight and other underwater light sources can be modelled in two different ways for a communication system. The first way is as a source of noise, this can be added onto any of the RTE solutions at a later stage, as was done in sec. 4.6. Modelling background light as noise means only the power implications are taken into consideration, it provides no information of spatial or temporal distribution of the background light.

The second method is to consider solar and other background sources to be a secondary, indirect source with their own associated internal radiance. The RTE and subsequent numerical methods take this into account with the final term in eq. 5.2, though Monte Carlo approximations for downwelling irradiance have been noted to be particularly prone to errors (Mueller, 2000). Additionally, if dealing with direct vector RTE solutions, such as in (Jaruwatanadilok, 2008), these background sources are added to the incoherent part of the solution, which turns a usually

first order differential equation into something much more difficult to solve. Considering

background illumination as a secondary source can also be achieved through the BSF, where the contributions from two BSFs can simply be superimposed on a desired receiver location. The additional light in this case will be modelled as a spatial noise pattern.

Having now surveyed the different modelling schemes against the specification derived by the

state of a light beam after being transmitted through a known medium by including temporal and spatial distributions of light. Of the three numerical schemes explored here, Monte Carlo was shown to be the most versatile, although it is compromised by slow run time and statistical errors, particularly in the presence of turbulence and downwelling irradiances. Other modelling schemes, such as the BSF and Beer-Lambert law, have shown to be suitable for communication links where temporal information is not required, such as in a long distance link in clear ocean where scattering is minimal. The benefit of using these schemes is their simplicity, however, as was highlighted in sec. 4.6, these schemes are somewhat limited in terms of their consideration for scattered light. For this reason, Monte Carlo methods are chosen as the appropriate advanced channel model for the purpose of this research on optical property variability.