Equation for Radiative Transfer

I have no idea on what to post this week. Not exactly because I do not have a theme, but it is because I could not get even close to understand enough of the subject to post anything very conclusive about it.

As I said before, I have just started taking PhD Classes at INPE. I'm the only students on this cycle (3rd/2008) attending the Computational Optics course. I have decided to go for Applied Computing as the Area for my PhD. I was a little tired on doing research only in Informatics. I'm glad to say that I found an area that I know absolutely nothing about :-).

On these classes, we will study a single equation. Yeap, I also thought it would be easy when the professor mentioned. The trick is that the equation looks like this:

This is the base form of the equation for Radiative Transfer . It is going to be my baby for the next three months. I'm not going to even try explaining it to you. However, its purpose is really nice.

In general, this equation allows scientists to evaluate the interactions of a flow of particles when they move around the space and go from one medium to the other. In my particular case, I'm learning how to understand the interactions that take place when a flow of photons (light) hits the ocean.

Behind the algorithm, there is a very noble cause. By understanding the amount of light that reaches the ocean and the amount of light that returns (refracts) from the ocean, scientists can estimate the characteristics of the water on that particular area. For example, it would be possible to evaluate the amount of chlorophyll dissolved into the water what would allow additional calculation in order to identify how much life (fishes, etc) is also present/supported on such area. This would be a nice feature for Google Earth, isn't it? Checking out the fish populations on the move from one spot of the earth to the other, how cool would it be?

The "real" scientists, not students like me, use the most of the equation. However, I'm glad to say that we have defined several simplifications that will allow even an IT guy like me to understand (or at least try understanding) how to solve the problem. Here are some of the assumed simplifications:

1. The flow of photons will not vary on time. It is the same as assuming that the Sun would always have the same intensity of light, does not matter what time of the day.
2. We are not evaluating a particular spot on the ocean. We assume that we have an infinite geometry. We assume that all photons emitted by the light source will always reach our ocean, does not matter where.
3. Our light source is isotropic; it means that photons are equally emitted in all directions. There is no preference(different orientation) for any single spot.
4. The photons will reach a single surface. In this case, the equation will not consider the flow of photons going through clouds before they reach the ocean, for example.
5. There is also a set of frequencies(light colors) being studied. In our case, we will consider all frequencies of the visual spectrum (red to violet) as a single band. Using the average values for the entire band also makes the equation simpler. We trade accuracy by simplification on this one.
6. We also consider that there is no internal source of light. On another words, we assume that no luminous fish were swimming on our ocean during the satellite scanning.
   

There are a few other simplifications that I cannot really remember now. They make the equation "much" simpler than its base form. Here its simplified version is:



Well, after the initial 30 second that you have been cursing yourself because you have no clue on what this means, I should tell that there are a few methods already tested and that can solve this equation. Here are the ones we will implements as/if time permits:

Method Sn and Method Monte Carlo.

At this point, I'm still working on understanding the problem. I will post something about its resolution as soon as I get there.

-Luciano - Don't worry, I'm completely freaked out too :-)