Let me apologize for asking, but I've been trying to understand this topic. What would be represented in the dive profile? The actual times at depth ascending and descending plus what? What do you mean by normalized? Do you mean averaged, and if so based on what data?
One possible way to think of it can be, that a dive profile is a function : time -> pressures that, for each point in time, returns a set of gas pressures for all the gases the diver is breathing at that time. I think what dmaziuk@ is referring to as "normalized" here is the assumption that a dive always starts at time 0, and that time and gas pressures are expressed in the same units, such as minutes and ATM. For example, if we model 3 gases: oxygen, nitrogen, and helium, then for a dive at the sea level, profile(0) = ( 0.21, 0.79, 0 ), to reflect the fact that at the beginning of the dive (time 0), you are diving 0.21 ATM oxygen and 0.79 ATM nitrogen. If after 60 minutes the diver is breathing O2 at 20 feet, then profile(60) = ( 1.6, 0, 0 ), to reflect the fact that the O2 pressure is 1.6 ATM, and there are no other gases.
What would actually be measured, and subsequently provided as adjunct information along with the physical/depth/time positions in the profile? In other words, what are you trying to determine and what data would be gathered and subsequently added in the process of elaborating and expanding on the basic physical positioning involved in a current dive profile?
I'd think one generally desirable objective is that for profiles X and Y, the smaller dissimilarity(X, Y), the closer the distribution of expected outcomes of dives with such profiles X and Y. In simple terms: if the same person P dives both profiles, they'd be expected to experience similar outcomes (in terms of chances of getting bent, or whatever), and if a group of N people dives such profiles, you'd expect the various possible outcomes to have occurred with similar frequency for X and for Y. You'd probably also want to require that dissimilarity is large if profiles are "mathematically" very different (e.g., one dive is to 100 feet and another to 150 feet), even if the expected outcomes are the same (e.g., because the second dive always includes a generously padded amount of deco that renders the differences in outcome insignificant).
I realize there is not enough data to derive such metric purely from the statistics of how often people get bent. I would expect, however, that rather than using physical outcomes (such as getting bent), you can perform the same analysis based on physiological parameters or outcomes as predicted by various models.
For example, what has already been mentioned in various threads, you could compute parameters such as "max tissue supersaturation" or "integral tissue supersaturation", as a function of a dive profile, and then express dissimilarity of the two profiles in terms of how much such values differ. That would be, essentially, reducing each profile into a number, and then comparing two numbers. If there are N such values of interest, you could take them all into account by expressing dissimilarity between profiles as a distance in N-dimensional space.
Or maybe, rather than reducing each profile into a number, you could stack them side by side, and look at the absolute difference in tissue supersaturation at various points in time, then compute an integral of such a difference.
Or perhaps there are other parameters of interest that you can compute from yet some other model, and here, I have no idea what they might possibly be, I thought something else would come up.
