Statistics is hard to get right

What factors influence the probability of a dive leading to DCS? Many of us would love to know the answer to that question. Wouldn’t it be great if we had a huge database of dives with depth profiles and lots of additional data of many different divers, together with information if those dives led to various levels of decompression sickness? Then we could run all kinds of different statistics and see what factors of a dive make undesirable outcomes more or less likely.

Recently, a paper by Alessandro Marroni, Jacek Kot, Massimo Pieri, Riccardo Pelliccia and Costantino Balestra (four of which are associated with the DAN Europe Research Division) in the journal “International Maritime Health” made the rounds on social media which aims to do exactly this. Specifically, DAN keeps a database of more than 100,000 dives together with information about the almost 6000 divers and circumstances of the specific dive that they ran various statistics on. And they have 628 profiles of dives that lead to decompression sickness. The authors looked at various factors and determined if they were different between dives with and without DCS.

Some of the reported results were expected: Dives with DCS tend to have higher gas loadings in the divers’ tissues (more on that below) and this seems to be the factor with the strongest influence. For repetitive dives, longer surface intervals lead to a lower rate of DCS. Higher workload during the dive leads to more DCS.

But some other results were quite surprising: According to the study, women have a three to four times higher risk to get bent than men. Body mass index has an influence, but the lowest rate of DCS was amongst divers that are moderately to severely obese; moderately underweight divers appear to have a five times higher DCS risk than severely obese divers! Exercise before the dive doubles the DCS risk. Negative feelings before the dive reduce the DCS risk by 70%, also feeling tired before the dive lowers the risk of DCS. In a sequence of repetitive dives, the later dives are less and less likely to lead to DCS.

https://xkcd.com/552/ CC-BY-NC

Of course, correlation should not be confused with causation. You should not acquire a beer belly and leave your deco gases on the shore to lower your decompression risk. There are some cases where it seems rather clear how an initially unexpected result can be readily explained: As an example, using more deco gases was correlated with a higher chance of getting DCS. The authors rightly note that there might be confounding factors, in this case dives with more deco gases tending to be more advanced tec dives that might therefore have an inherently higher risk of DCS. But still, many of the findings of this study would run counter to what we believed before. In the words of Carl Sagan: Extraordinary claims require extraordinary evidence! So let’s look at the details more carefully.

How does DAN collect data?

The most important factor is unfortunately not really discussed in the paper: How do the dives actually get into DAN’s database? For the statistics to be meaningful, they should be representative of the dives done by the general population. Furthermore, the outcome (DCS or not) must not influence the likelihood of a dive to be included in the database. Otherwise there would be a bias that would be very hard or impossible to correct for.

And it seems, this aspect is already where the study gets into difficult waters. Apparently, there were very different ways via which dives could end up in DAN’s database. Unfortunately, there is no clear statement to be found anywhere about how exactly the data was collected, but the DAN research site gives some hints. There exists an earlier publication from a team with a large overlap with the one of the 2026 study, Cialoni et al. 2017, and this serves as a reference for explanations, as it treats a smaller, older sample of dives from seemingly the same database. The profiles from the 2017 study thus can be assumed to be also a part of the present-day data.

DAN does invite divers to submit dives to them for statistical analysis, e.g. via via a tool called the Diver Safety Guardian (DSG), and recently also via a successor, DANA Health (which is free for members, but on a subscription plan for non-members), since around 2014. Before this, they did collect profile data in different events directly from the divers – no information about how exactly this happened is available, but the participation was always voluntary.

With using the DSG, an advantage is offered to the divers for submitting their data: They can see an analysis of their dive, including some indicators for the estimated risk arising from the dive. And this, to be honest, attractive and well-intended as it is, already should ring some alarm bells: Unless you make divers submit all their dives, or at least a truly random sample, you have to expect people to in particular submit dives they found “remarkable” or “interesting”, the opposite of typical. Maybe divers are especially keen to submit dives that were particularly deep or long or where they encountered some sort of near-miss. People almost inevitably will report less of the “boring” dives they do on any given weekend. This will already present a nearly insurmountable problem for extracting meaningful statistical numbers. But it gets worse.

DAN does collect profile data plus more medical data on “laboratory – events”, and they regularly conduct dedicated research, where specific profiles are tested. That in itself of course is great. But again those dives will not be truly representative of “random dives, done by random divers on an random day”. Which of these data did or did not enter the Database is not really clear. But in the 2017 paper, there were 970 dive profiles with subsequent bubble measurements. Obviously, those were not randomly reported, and are a part of the nearly 40.000 profiles analysed in this first batch.

We have to guess that at least the dives that were voluntarily submitted online via the DSG without further information were generally counted as “no DCS”. This guess is based on the direct observation that the questionnaire that accompanies the dive upload at least in the DSG today does not ask about symptoms of decompression sickness. So even if someone did upload dives there that ended with some symptoms, maybe hoping to find out what was “wrong” with their profile, this would likely not automatically lead to their dive being counted as a DCS dive. We don’t know which fraction of the dive profiles comes from this collection, and if other approaches did do a follow-up to check if the dive ended with DCS or not, but what is actually visible is this.

The 628 dives that were marked as “DCS” thus seem to have come to the database via a different pathway. One can imagine, DAN of course being one of the best known medical assistance organisations in diving, that at least part of those cases where reported directly as DCS cases, potentially seeking medical assistance or advice of various types. Of course it can well be that there were more and different ways via which DCS cases were collected, but this does not become clear from the paper. Anyhow, the demographics of the DCS dives are thus inevitably different from the “no DCS” dives. Some of the divers that incurred DCS may of course have reported more profiles than only the dive that went wrong, but this is still not “random divers sharing their dives” – those divers do it because they have, or suspect, symptoms and seek assistance.
The paper does not explain how exactly the DCS data was collected, nor how “DCS” is defined, and what precisely the criteria are for a dive to be counted as a “DCS dive” – is it a diagnosis by a medical professional, is it self reported symptoms, is it a confirmed insurance claim? The first database analysis from 2017 does treat the DCS cases as a different set of data, not as a part of all profiles. And that to us still seems the more correct thing to do. The 2026 paper seemingly does treat both datasets as comparable. The DCS incidence quoted in the abstract then however is just the fraction of studied dives which were marked as DCS, and will not be representative of the incidence of DCS throughout the diving population.

Why the data collection style matters

There is the big elephant in the room with that database: It seemingly collects data from fundamentally different sources. Those profiles thus cannot at all represent “dives as done by recreational divers”, but are a mixture of recreational dives, technical dives, dives that ended with DCS, dives where the outcome is not known, dives that were reported for curiosity to see the tool, large amounts of dives that were reported from some divers specifically interested in contributing.

If – as we have to assume – the DCS dives were indeed collected in a different way from the more general profile data collection, both subsets become really hard to compare. In fact, this means it is not possible to calculate a real incidence from such a comparison. Both datasets would by contrast have to be treated as being fundamentally different.

Furthermore, self-reporting is never a really good idea for statistics. We have already seen that the profiles that are shared may be substantially different from the profiles that are done throughout the diving population. And it is not clear if men and women are equally likely to report their dives. It is easy to imagine that maybe women do more than ca. 13% of all dives worldwide (which would be the fraction used in the study), but are on average less likely than men to participate in any DAN data collection. But women might at the same time be more likely to attribute light symptoms to DCS and consequently report a diving accident to DAN. At least a part of this effect honestly seems almost unavoidable, and unless this can be rigorously accounted for, the datasets thus do not make it possible to really infer any dependence of the DCS risk on gender. We have to treat the result of a three to four times higher risks for females with strong caution at the very least, and by contrast have to say it is simply not possible to make a definite statement if there even exists a significant difference on the basis of present-day data alone.

Similar surprises may lurk regarding the dependence on the BMI: We could imagine that a relatively large number of reported DCS cases were from more advanced technical dives (the dependence of the risk on tissue loadings seems to be a quite stable result). But the demographics of technical divers could well differ from the general diving population with respect to body fat. And thus an obese diver might well be found in the general diving population (and count in the “no DCS” group) but might be much less likely in an inherently risk-prone technical diving subgroup. If, however, this obese diver would indeed attempt the same more ambitious dives, their DCS rate might in fact be as large (or even larger?) than that of the people actually doing those dives. This could be like the statement that musical directors have an above the average life expectancy (due to zero child mortality amongst them).

Analysing the dives as independent events

The study seemingly compares dives from fundamentally different sources and starts an analysis. While this leaves a lot of questions about what exactly the sought-after effects were, how the method was calibrated and validated (e.g. was there a split into train and test samples done, or attempted?), and why the numbers used are seemingly inconsistent in some places, one of the biggest problems is that all the collected profiles are treated as independent events. But in fact, an important part of the analysis, everything related to the body of the diver, is not independent at all. A lot of divers reported only one dive, while others did report way more, up to 1,432 dives come from one single diver, with the same body and probably a comparable diving style, leading to similar profiles as well. A diver who reports more does count a lot more in the analysis than a diver who reports only a small part of his dives.

And more problems …

There are further issues in this paper. For example, there is the criterion “purpose of dive” with the possibilities “recreational, instructional, guidance, student, technical and other” which were assigned the (arbitrary) numerical values 1 to 6 and these ad hoc numerical values were used in a formula to fit the probability of DCS. This honestly makes no sense at all. If at all, those six possibilities should have been used as binary variables (0 meaning “no” and 1 “yes”).

As a measure of tissue loading (which ended up being the strongest predictor of DCS outcome, and this in itself is largely uncontested) the authors of the study use a numerical value (the “DAN Surface Supersaturation Gradient” DSSG), which is not really defined well in the paper. The authors do refer to the 2017 study, which however states to use classical gradient factors (GFs).
The name “DSSG” could indeed be interpreted to mean the gradient factor at the end of the dive (i.e. when depth 0m is reached), but on inspection this is not what it is. If we assume that in the 2017 study the GFs were calculated according to the prescription by Baker, the DSSGs can actually not be GFs without making the data incompatible, or without assuming that the 320 DCS cases from the 2017 study are not included in the 2026 study (but why would they not be?). This because the numbers of the DCS cases simply would not match. There were in total 320 cases in 2017, and now there are 628 cases – a plus of 308 cases, which seems not unplausible for that time interval. The issue is, there were 46 (or 59, the presentation of the data in table 4 of that paper is not absolutely unambiguous at that point) cases of DCS with a maximum GF under 0.7 (70%) in the 2017 study, but there are only 29 cases with a DSSG of 0.7 or lower in the new paper. So, cases would have to disappear if the DSSG was a gradient factor…
Without the full dataset, we cannot determine what exactly was calculated in both papers, but seemingly the interpretation of the DSSG being a GF is in conflict with the reported numbers under the assumption that the data from the 2017 study would be part of the 2026 one.

Of course the DSSG could be the fraction of the plain vanilla 100/100 Bühlmann M-value. Without doubt, both metrics will work for a description of the tissue loading, but they are numerically not the same, and cannot be mixed. In any case, it should be clearly communicated to the avid diver what exactly is used. This is really important for the diving community, as the most common number to evaluate tissue supersaturation for most divers is the GF, a number divers know, manage and think they understand. We at this point want to caution against an idea that divers may have (although it is also not encouraged in the paper): One should be very careful to draw conclusions from the numbers reported in this study about what gradient factors should be considered “safe”; note that the gradient factor is generally smaller than the percentage of the M-value if we stay in the regime generally considered safe!

So, unfortunately, without sufficient control of the various biases in the dives that were considered in the analysis and in particular different selection criteria in the two groups (DCS/no DCS), one has to be extremely careful to draw qualitative conclusions, let alone quantitative ones. To be honest, one would have hoped that these questions would have come up in the peer-review of this paper and could have been addressed already before it appeared in the journal.

Is there a brighter outlook?

The pessimistic outlook on the reliability of some of the results we have to take here is especially sad given that DAN is almost uniquely positioned to collect such large data volumes from dives over a wide cross section of the diving population, and given also the eagerness of the diving community to learn about risk factors, to digest results, and to potentially adapt diving habits. The large number of collected dives can be seen as an indicator of that, and is a heartening result of the efforts! We sincerely hope that ways can be found to construct suitable pathways for the collection of truly representative datasets, and for more robust analyses. The diving world is definitely waiting for that, and it is our hope that the criticism raised here may not be perceived negatively, but rather as a constructive input for future work. We absolutely do hope that our readers do not take the points raised in this post as a general discouragement of data collection and statistical analysis. To the contrary, science needs all of your participation! Statistics is hard to do in diving, and maybe it is no surprise that the road to results is bumpy. But in the meantime, please do not draw definite conclusions, especially not about individual real-life cases you may encounter, on the basis of the numbers from this study alone.

For this blog post, I collaborated with Veronika Sievers and Dominik Elsässer, the authors of a highly recommended book on decompression theory and main contributors to the Punkfish Academy.

New decompression theory book

Let’s interrupt the regular program for a short commercial break: Veronika Sievers and Dominik Elsässer have turned their Punkfish online seminars into writing and finished a book on decompression theory:

So far only in German, it covers everything from the basics to the latest findings. It should be accessible to all divers and different from this blog does not assume you already know the basics. It’s a great read and I am not only saying that as I could contribute one or two of my nitpicks in the writing process. I like in particular that it contains a lot of references to the original research literature and pointers for further reading. It is available directly from the Nina Zschiesche’s (of Wetnotes fame) publishing house. Strong recommendation!

CNS O2 Limit Updates

In the past, I have talked about the poor empirical data underlying the NOAA limits for preventing CNS oxygen toxicity. Of course, those are tricky, since having a seizure at depth has a high likelihood of resulting in fatal outcomes while there are good chances that the diver has no way to see it coming.

On the other hand, the limits have been quite restrictive: At a pO2 of 1.3 bar, you were supposed not to exceed 180 minutes in a single dive and 210 minutes in a 24 hour period. Sticking to such a limit (with 1.3 bar being a typical rebreather set-point) will make certain dives impossible as the decompression requirements exceed three hours. This had the consequence that many technical divers have been ignoring the limits coming from this “CNS clock”. Turns out, this has not led to many dead rebreather divers due to oxygen seizures at depth.

Turning this around, there is now more data to justify relaxing those limits while controlling the risk. In March 2025, at the American Academy of Underwater Sciences Annual Meeting, there was an expert workshop that resulted in new recommendations authored by Joseph Hoyt, F Gregory Murphy, Neal W Pollock, Dawn Kernagis, Nicholas Bird, Michael Menduno, John Bright, and Simon J Mitchell. They are still very conservative with their recommendations. For example, they only recommend new limits for pO2 of 1.3 bar although mentioning that it is highly likely that those times considered save would also be save at lower partial pressure. They make no statement about higher partial pressures like 1.4 bar or even 1.6 bar quoting insufficient controlled dive date in these ranges.

So, for 1.3 bar, they say, four hours of exposure during the “working part” of the dive plus an additional four hours during decompression at rest can be considered save.

They note that this is only about CNS toxicity as this has more potentially harmful consequences and pulmonary toxicity might have to be managed in addition. Furthermore they recommend considering additional safety measure like O2 breaks, breathing low density gases and using full face masks or habitats to mitigate consequences of a seizure.

PS: A few days ago, I had to learn that Michael Menduno, one of the authors of the paper mentioned above and editor of inDepth magazine has passed away. In the limited interactions I had with him he surprised me with is generous and welcoming attitude. Rest in peace!

Questions after MI8: Final Reckoning

On Thursday, I took my daughter to the movies and we watched “Mission Impossible 8: Final Reckoning” which as my dear readers will be aware of involves an extensive under water chapter.

This leaves the theoretical diver with a number of questions regarding in particular the physics and diving shown. WARNING: Spoilers ahead!

Tom Cruise on the unter water set.

  • Why did they give him a dry suit and a prototype rebreather but forgot to give him fins? This makes him almost helpless in free water. He was lucky he basically fell on the submarine and did not miss it by a few meters which would have ended the mission immediately.
  • What is the depth at which he finds the submarine? I would think the Arctic Ocean is a few kilometres deep which makes it totally unsuitable for diving. On the other hand, in the final total it looks like not more than 20-30m which would make this whole decompression thing (and the prototype special rebreather) pretty much pointless.
  • Why did they put lights inside the full face mask to make sure he is constantly blinded? Why is the visibility so good and where does all the ambient light come from? (OK, a movie taking place for 15 minutes in complete darkness does not seem to be too promising)
  • The submarine supposedly exploded in 2012. Why is there still air inside? And once Agent Hawk opens the latch, why is the water only slowly tickling in? At any reasonable depth, I would expect the thing to be filled with water in the blink of an eye and crushing any human on the way.
  • To enter the torpedo tube, why does he have to take off his dry suit? And why does he use his knife rather than the zipper? Why can’t he take the rebreather with him? I am sure, the marine built it to fit into a torpedo tube.
  • The inflatable decompression chamber does not look like it could hold any significant pressure. And why is it built so big it can accommodate a romantic tete a tete? Why is he only one jump cut away from flying again on a plane?

AI Real Gas Blending

I understand the kids today use generative AI for everything. So, it is just natural to use it to blend some diving gases as well. I already had my CGI script to compute how to top up your gases using Subsurface’s real gas approximation.

Now, I played around with some large language models and Claude.ai turned out to be able to rewrite this in JavaScript (which I don’t really speak but enough to put some finishing touches). As a bonus, there is no “Calculate” button but as in the Subsurface planner, you get instant results. You can find it here.

Everything in a stand alone html file that you can of course also install locally. You find in in the RealBlender Github Repository.

Don’t measure gas use as pressure drop per time

Here is an email I wrote to the Subsurface mailing list in response to the suggestion to offer the option to display gas use (SAC or RMV) in units of pressure drop per time even when you use only a single cylinder size:

There are two separate issues:

One is if you measure the gas consumption in units of

a) pressure drop per time

or

b) volume of gas at some reference pressure (typically surface pressure) consumed per time

the other is which term or abbreviation you use for a) or b).

Both discussions are very old already and Subsurface decided to call the measurement of b) „SAC“ or surface air consumption. It seems some other people call that RMV (respiratory minute volume) and maybe the latter ones are in a majority and Subsurface took a poor choice. I don’t know, maybe. In German, it is called AMV (Atemminutenvolumen which translates to breathing minute volume which is as stupid in German as it sounds in English but at least everybody uses it so there is no discussion). I don’t really care what you want to call it, maybe ABC is also an option or XYZ and I am open to changing the name in Subsurface if a large majority of English speakers say that one is the preferred term (but so far my impression is that there are people in both camps).

But the first issue is not up to discussion, the correct way do express it is b)! The only reason some people might think a) is a viable option is because in the water you have a gauge that measures pressure but you don’t measure amount of gas (as mols or volume at reference pressure) directly. So if you know how much your pressure gauge dropped in the last five minutes (assuming constant depth) you have a rough idea how many more minutes you can stay at the current depth and not running into gas problems. It gives you a rule of thumbs estimate but not more since future consumption depends on many other factors.

Once you are out of the water and write your log you can let your computer do the calculation to convert this into volume (amount of gas) units which is a much more meaningful way of expressing things. Others have mentioned that it makes no sense to compare pressure units if you you have different cylinder sizes. But even if you say you only have a single cylinder and will always use the same you should realise the conversion between pressure drop and amount of gas used is only simple in a world that knows only about ideal gases. Subsurface takes great proud in the fact it is aware this is only an approximation and takes into account the pressure dependent compressibility of breathing gases. And if you do that a pressure drop only translates to an amount of gas when you also know the starting pressure.

Let me show you in an example that this difference is actually relevant: I am diving in a world that uses metric units so my numbers come from common sizes an pressures in metric units but for your convenience I will translate those to imperial units (I guess you are using those because only in imperial units people have the idea that a) might be a meaningful thing):

Let’s do an air dive to 66ft (20m) for 22 minutes (i computed this in the planner) using D12 tanks (two 12 litre cylinder) with a total volume of 167cft. 

If the pressure drops from 2900psi to 2320psi (200bar to 160bar)  the SAC is 0.48cft/min (13.6 l/min).

If however the pressure drops from 1450psi to 870psi (100bar to 60bar), obviously the same difference in pressure, the SAC is 0.54cft/min (15.3 l/min).

So the difference in SAC (amount of gas used) is more than 10% if you start from a full cylinder or from a half empty cylinder and drop by the same amount of pressure. This difference in invisible if you measure it in pressure units but it is probably much higher than what you worry about when you look in trends of your gas consumption over time. (The reason is that at higher pressure air is significantly less compressible than at lower pressures).

You will not be able to seriously monitor your gas consumption over time if you use pressure/time units unless you always use the same cylinder and always start your dive with an identical starting pressure.

For this reason I am convinced that option a) is simply wrong (or for a less aggressive term: uninformed) and I will veto using it in Subsurface or even offering it as an option to the used (just as I would veto offering to turn off the real gas corrections even though we get many complaints about Subsurface getting gas calculations „wrong“).

DIY Submarine

Not a lot of new content here recently. The last days, I attended 37c3, the annual hacker convention of the Chaos Computer Club which took place again in Hamburg . So, today, we will have some engineering content rather than the usual physics one.

You have to see the super entertaining talk by two guys who built their own submarine from scratch.

There is a video of their presentation (including English translation) they have a website and there is a documentary about their project.

Money quote: “What distinguishes submarines from other boats is that they can surface again.”

Probabilistic decompression models: Probably problematic

It has been very quiet here for a while. That is partly because there is a topic that I wanted to discuss here and since my take was going to be at least somewhat problematic, I felt for a while that I needed to do more reading to get the facts right. But now the time has come for me to talk about: Probabilistic decompression models.

The idea as such is very simple: Many models for decompression including the ones that I talk a lot about in this blog, Bühlmann, VPM-B and also DCIEM are “deterministic” models: They give you a plan and if you stick to it (or dive more conservatively) you should be fine but you must not violate the ceilings they predict at any given time. On the other hand, there are probabilistic models that aim to compute a probability of getting DCS for any possible profile and which you can then turn around and prescribe a maximal probability of DCS and then you optimise your profile (typically for the shortest ascent time) that gives you that prescribed probability of DCS.

That sounds like a great advantage: Don’t we all know that there is no black and white in decompression but there are large grey areas. Even when you stick to the ceilings of your deterministic model there is a chance that you get bent while on the other hand you do not immediately die when you stick your head above the ceiling, quite the opposite, there are chances that even with substantial violation of the decompression plan you will still be fine. That sounds pretty probabilistic to me.

Of course, things are more complicated. Also the deterministic models do not really aim at a black/white distinction. Rather, they are also intended to be probabilistic but with a fixed prescribed probability built in. For recreational diving (here as usually including technical diving but not commercial diving), the accepted rate of DCS is usually assumed to be one hit in a few thousand dives. That seems to be the sweet spot between not too often having to call the helicopter (and most recreational divers never experiencing a DCS hit) and being overly conservative (too short NDL times, too long decompression stops). So, the only difference is that for the probabilistic models, this probability is an adjustable parameter.

You can turn this around: Why would you be interested in dive profiles that have vastly different probabilities of DCS that this conventional one in a few thousand? After all, even for one possible value of p(DCS) it is very hard to collect enough empirical data to pin down the parameters of a decompression model. Why on earth would you want to do that also for probabilities that you do not intend to encounter in your actual diving? Why would I be interested in computing an ascent plan that will bend me one dive in twenty for example? Or in only one dive in a million? That sounds to be overly complicated for useless information.

The answer is exactly in your restricted ability to conduct millions of supervised test dives: Let us assume you have a probabilistic model with a number of parameters that still need to be determined empirically. But you have a priori knowledge that the general form of your model is correct (we will come back to this assumption below), you can do your test dives with depths, bottom times and ascents that your model gives you DCS probabilities in in the range of 5-50% say, depending on the parameters. Much more aggressive than you intend to dive in the end. But such test dives will give you a lot of DCS cases and you do not need too many dives to determine if the true probability is 5%, 20% or 50% and adjust the model parameters accordingly. You gain already a lot of information with just 10-20 dives while for dives with the 1/10000 rate of DCS you need many more dives to have enough DCS cases to establish the true probability.

Once you have established the parameters of your model with these high risk dives where you have to use your chamber a lot to treat your guinea pig divers after bending them you can then use your model for dives that have a much healthier conservatism where your model gives you for example p(DCS)=1/10000.

An example of doing your study in a regime where you expect to bend many contestants is the famous NEDU deep stop study where the divers were in quite cold water with insufficient thermal protection and in which they had to work out on ergometers while in the water all to drive up the expected number DCS cases (there not necessarily with a probabilistic model in the background but just the intend to see a difference in deeper vs shallower decompression schedules where of course 0 DCS cases for both ascents wouldn’t be very informative).

But as so often, there is no such thing as free lunch: You are extrapolating data! You make experiments in a high risk regime of your data and then hope that the same model with the same parameters also in a low risk regime. You are extrapolating your data over several orders of magnitude of probability. This can only work if you can be sure your model is correct in its form and the only thing to determine empirically are the few parameters. But in the real world, in particular in decompression science, such a priori knowledge is not at hand.

Let me illustrate this in a simplified example: Let’s assume, there is only one parameter x and your model is that the logarithm of p(DCS) is a straight line (an affine function) of x

\(\log(p_{DCS}) = m x +b\)

Then, in your experiments you do a number of test dives at various values of x, find the corresponding rates of DCS and finally fit the slope m and the intercept b by linear regression.

The red area is the high risk regime where the model is tested, the green are where it is supposed to be applied. If the reality is not a straight line (solid) but there is small quadratic component (dotted) the difference in the green are can be significant while the fit to the experiments in the red area is still good.

If however the a priori assumption of the model that a straight line does the job is not justified because there is also a small quadratic contribution, say, you can fit your parameters as closely as you want but still get very far off in the extrapolation to where you intend to use your model.

This is of course only an example and for more complicated models you will use a maximum likelihood method to find optimal values of your model parameters given the outcomes of various test dives. But what this will never be able to do is to verify the form of your model assumptions. You are always only optimising the model parameters but never the model itself. For that you need independent knowledge and let’s hope you have that.

VVAL18

To be specific, let us look at one probabilistic algorithm more concretely: VVAL18 which was developed by Edward Deforest Thalmann for the US Navy based on a database of a few thousand navy dives. For military divers, the accepted risk of DCS is much higher, if 2% of the dives result in symptoms that need to be treated that is still considered fine given that decompression chambers are typically available at the dive spot. This model is also known as the Thalmann algorithm and is described for example in short in this technical report. (Similar things could for example be said about the SAUL decompression model which is similar in spirit and also inspired by the navi data)

E.D. Thalmann (Wikipedia)

This model also uses compartments with half times but leaves open the possibility that the partial pressures do not follow the usual diffusive dynamics with the rate of change proportional to the pressure difference to the surroundings but for off-gassing also allow for the possibility of a constant rate (which leads to linear rather than exponential time dependence).

For each compartment i, there is a fixed threshold pressure pth that as an excess pressure is considered harmless and above that a relative excess pressure is calculated

\(e_i = \frac{p_i-p_{amb}-p_{thi}}{p_{amb}}\)

To compute the risk, for each second of the dive and each compartment, risk of not getting bent in the second is assumed to be

\(e^{-a_ie_i}\qquad (*)\)

for some constants ai. Finally, all these individual risks are considered to be independent, so the “survival probability”, the probability of not developing DCS is assumed to be simply the product of all the individual probabilities for compartments and seconds (since they are in the exponent, you can there integrate the ei over time and sum over tissues).

These constitute the a priori assumptions of the model that I was talking about: The exponential dependence of risk on relative overpressure (*) and the statistical independence of tissues and instances of time. According to these assumptions, your risk of DCS increases exponentially in time when you do longer decompression (assuming the excess pressure is kept constant) for deeper dives or longer bottom times (this is clearly at odds with the assumptions of for example the Bühlmann model that allows you to have arbitrary long decompression obligations as long as you do not violate a ceiling) and you are allowed to have arbitrary large excess pressures if you keep the duration of the excess short enough (tell that to your soda bottle).

With these assumptions, for which I could not find any justification in the literature for, except “we came up with them”, lack of imagination, then the parameters of the model are optimised using maximum likelihood. In the Thalmann case, there three tissues and the parameters to be fitted are the half-times, the thresholds pth and the constants ai.

The three half-times Thalmann ends up with are roughly one minute, one hour and ten hours (with large uncertainties), the thresholds are essentially 0, and the ai are in the 1/50000 range (you can find the values in an appendix of the report cited above, note that time units of minutes are used rather than the hypothetical second I used here).

I have serious doubts about the intrinsic assumptions of (*) and the statistical independence of time segments. But for Navi use where you accept DCS risks of a few percent those may be ok since any model with sufficiently many fitted parameters will reproduce dives with similar parameters. But failed assumptions will bite you when you extrapolate your model out from the high risk regime to recreational diving as failed model assumptions tend to blow up under extrapolation.

I want to mention that I am grateful to the LMU Statistics Lab that I could discuss with them some of the issues mentioned. Of course all mistakes here are my own.

Salinity and water density

This post is a copy from an answer in the Subsurface support forum but I post it here as well as this point comes up over and over. It is about the setting of some dive computer where you can set the density of water or whether you are diving in sea or fresh water. There is a similar setting in Subsurface, but the default setting in the preferences is to hide it. For a good reason.

Kevin Jones from Vancouver, Canada, CC BY 2.0 https://creativecommons.org/licenses/by/2.0, via Wikimedia Commons

Let me mention once more why this option is turned off by default: There is a good chance that by fiddling with the density you make changes whose effects are not what you intend: The density is the relevant constant that controls the conversion between depth and ambient pressure. That you probably intend. But what is easy to forget: Your dive computer displays depth and reports it in the log that is transferred to Subsurface. But it does not really measure depth, rather it measures ambient pressure and it uses the density to convert that to depth.

It does this conversion because we humans are used to think in terms of depth which is much like a length and we have an idea how much 10m is, much more so than 2bar. 

For most things diving, however, depth does not matter at all, ambient pressure does. This includes gas consumption (as your regulator regulates according to ambient pressure, not to depth) and all deco calculations (because also there partial pressures in their relation to ambient pressure dictate what is happening in your body). Depth only matters when you think about breaking a new world record or worry if the mast of the wreck sticks out of the water or if you buoyo line is long enough. For gas usage estimates and deco calculations, it would be much more honest if your computer displayed ambient pressure in bar rather than depth in m. Only that for the average Joe that would be hard to digest and we have to live with the fact that this conversion is done back and forth by the computer and by Subsurface all the time.

But what really makes zero sense is to use different values of density when translating back and forth. Also your dive computer does not measure the density, this is a setting that you have to make manually. Yes, you could change that setting on your dive computer every time you switch between sea and fresh water to get a more accurate depth display (which as I explained above most likely does not matter at all). But my guess would be you forget to change that setting for at least half of your dives. So my very strong recommendation would be to set it on your dive computer once and for all to any value (maybe according to where you do most of your diving) and set Subsurface accordingly and never ever change it again. The result can be that some of your depth readings are slightly off but at least  your gas and deco calculations will be consistent.

Blending real gases

As mentioned before, gases in diving cylinders are not only not sufficiently well approximated by the real gas equation but also the van der Waals equation, despite its prominence in thermodynamics teaching, is not doing much better.

Subsurface does better than this using a polynomial fit to table data for the compressibility of the three relevant gases. In a discussion at ScubaBoard, the question came up how to use this in gas blending. After an initial version using Mathematica, I sat down and implemented it as a perl script and hooked it up to this web page for everybody’s use and enjoyment. Here it is:

Real Blender

source code is on GitHub. Right now, it does only nitrox. But it computes instructions to up up partly filled cylinders with any pre-existing nitrox mix.

Let me know if you think extending it to trimix would be useful for actual use. I am not sure what the best user interface would be in that case: For nitrox, specifying the initial and target mix and pressure and two top up mixes, the blending problem generally has a solution. But with three gas components to get right, it is in general impossible with only two top up mixes. So you either have to use three (linearly independent) top up mixes or let one thing unspecified. That could be either the oxygen or helium fraction of the final mix or you have to leave open one of the gas fractions of the top up gases.

So what do you do in practice, which component do you leave unspecified?

Update: I updated the script so it can now also handle blending trimix (starting from a partially filled cylinder, you can specify three top up gases it will now calculated the intermediate pressures you have to fill up to). To blend nitrox, specify the target mix as containing no helium and leave one of the top up mixes empty.

Update: I discovered an error in the calculation (I calculated the mix according to pressures rather than volumes at 1 bar) that should be fixed now (April 10, 2022)