Unfortunately, some divers are confused or irritated by the fact that SAUL is a probabilistic algorithm, particularly those looking to compare it directly with other existing algorithms in terms of what dive profiles are permitted.   The comparisons they would like to see are, essentially, between NDLs.  SAUL doesn’t have a set of NDLs to compare with.  Here’s why not.

First, a clarification:  It’s not that SAUL doesn’t use NDLs at all, just that they’re not pre-set ones.   In effect, the diver calls up an individualized set of NDLs by setting an acceptable degree of risk (of  decompression sickness).  One “size” does not fit all NDL needs.  Reasonable people can – and do – differ on how much risk they are prepared to take, both in general, and in specific situations. SAUL is sufficiently accurate that it can viably use a probabilistic algorithm, and it chooses to do so for the following reasons.

1.   In diving – as in other aspects of life – adults with access to adequate information are presumed responsible enough to make decisions for themselves (and for any children under their supervision).

2.  One important piece of the information divers need to keep in mind is that there is always some degree of risk in diving.  Over-reliance on fixed NDLs may mislead some divers into forgetting that fact.   An NDL does not mean that divers are always safe under that limit, always in danger above it.  Actively choosing an acceptable level of risk helps divers keep that in mind.

3.  The most important reason, though, is that a probabilistic algorithm is the most realistic way to deal with the occurrence of decompression sickness in diving.  This is true even just considering the general reasons outlined in an earlier posting   (NDLs, etc.). But it is the chaotic behaviour of bubbles themselves that makes a probabilistic algorithm particularly appropriate for any serious attempt to deal with the risks in diving.

Bubble behaviour can vary drastically depending on exactly where, and under what circumstances, the bubble arose, its size, whether it has entered the bloodstream and, if so, where it’s carried from there.  Most of these factors are matters of chance and unpredictable in advance.

I mention “where the bubble arose” as a factor because my research found that bubbles embedded in a soft elastic solid (i.e., muscle or cartilage tissue) will behave very differently from bubbles in a more liquid environment like blood.  In some cases, a bubble in a soft elastic solid may persist for long times, or grow, even when the tissue it’s in is undersaturated.

Luckily, predicting the behaviour of individual bubbles is not necessary in order to manage the danger of decompression sickness.  In general, algorithms try to do this by  calculating, in accordance with their particular underlying models, the accumulation and dispersal of excess nitrogen in the body.  Most algorithms then set NDLs based on the calculated nitrogen load.  It’s obvious that the frequency of decompression sickness is strongly correlated with the presence of excess nitrogen in the body.  And it’s also true that the presence of excess nitrogen in the body is correlated with an increase in the number and size of bubbles and, therefore, an increase in the cumulative  probability that one or more individual bubbles will cause problems.  This cumulative effect of bubbles on decompression sickness is already taken into account by calibrating algorithms with known data.  A more accurate algorithm (like SAUL) will do it better.

But it’s equally obvious that decompression sickness can, and sometimes does, occur when it seems to be “undeserved”.   Since bubbles are believed to be the initiating cause of decompression sickness, certain algorithms have purported to account for bubble behaviour in their calculations.  For reasons described above, this does not, and cannot work.  In my view the only proper way to take into account individual bubble behaviour is to recognize and work with its chaotic and therefore probabilistic nature.

(For anyone interested, my latest paper, Gas bubble dynamics in soft materials, has just been published by The Royal Society of Chemistry and has now been posted here under Articles.)

Once the degree of nitrogen saturation is accounted for, the additional effect caused by bubbles is effectively summarized by the bit of verse below.

on probability


Halloween is almost upon us, so it seems like a good time to talk about some seriously scary stuff:  Neurological DCS and inner ear DCS – the really bad cases of the “bends”- that can sometimes result in permanent  paralysis, deafness, even death.

You may be aware that many cases of neurological DCS are the result of a PFO (Patent Foramen Ovale) which is, essentially, a hole in the septum of the heart that divides the right side (which receives blood from the veins) from the left side (which pumps out oxygenated blood to the body).   Venous blood passing through a PFO into the arterial system is not in itself a problem, except in the case of “medically significant” ( i.e. large) PFOs.   PFOs are not uncommon in otherwise healthy individuals and usually go unnoticed.   These individuals may also naturally develop small bubbles in their circulatory system, some of which may occasionally pass through even small PFO’s.

Even when bubbles do pass through the PFO, these AGEs, or Arterial Gas Emboli (which is what they are known as once they reach the arterial system) are usually harmless – until we consider diving.   This is because  AGEs coming through PFOs are generally small, which makes them thermodynamically unstable.  Larger AGEs, if they were to occur, would be more stable, as both they and arterial blood would be pretty much equally saturated with gas.  But with a small AGE, the surface tension increases the pressure on the gas within it, which causes it to dissolve rapidly.  No more bubble, no problem.

By changing just a few features of an essentially harmless situation, compression and decompression during diving turn it into a  potentially dangerous one.  These are the changes that matter:  1) During compression, body tissues accumulate increasing amounts of nitrogen so that, during decompression, they become super-saturated with nitrogen; 2)  During decompression, the number and size of bubbles in the veins increase;  3)  The size of bubbles decreases during compression.

Here’s what happens then.

The first result of these changes during a dive is a much greater probability that a bubble will pass through the PFO, simply because there are so many more of them.  (While the increase in number of bubbles occurs during decompression, remember that decompression includes, roughly speaking, all time spent after ascending from the deepest point of the dive.)

The second result is that a bubble with more gas in it could get in at depth than could happen when not diving.  How can this be, when the size of the PFO hasn’t changed?   When the size of bubbles decreases during compression, there are actually two separate things happening: One is that gas is escaping from bubbles under pressure – less gas in the bubble makes it smaller and some bubbles will disappear.  The other thing that happens is that bubbles under pressure become smaller even without losing any gas.  The radius of the bubble gets smaller, but it contains the same amount of gas.  The effect of this is that a bubble small enough to pass through a small PFO will contain more gas at depth than will be in a bubble with the same radius at the surface.  Or, to put it another way, a bubble that might be too big to pass through a PFO under normal (surface) conditions might be compressed enough at, for example, 100 fsw. to pass through.

So, we’re more likely to have a bubble pass through a PFO, and that bubble, now an AGE,  will be slower to dissolve than an AGE of the same radius formed at the surface.  This makes it more likely that the AGE will survive long enough to exit the artery into the capillaries of tissues that are supersaturated.

If the AGE does reach supersaturated tissues, it will grow, taking on nitrogen from the supersaturated tissues, and, if it gets large enough, can damage the tissues by blocking blood from reaching them (producing what’s known as ischemia) and/or by directly damaging some very sensitive tissues, like those in the inner ear, as just the pressure exerted by the growing bubble may be enough to cause them to tear.   When this damage takes place in brain, spinal column or inner ear, the damage is often permanent.

While PFOs are the most common route for bubbles to become AGEs, they can also access the arteries through AVAs (Arterio-Venous Anastomoses, which are remnants of fetal pulmonary shunts bypassing the lungs that didn’t fully close at birth), or through the lungs themselves, when alveoli of the lungs fail to completely filter out the bubbles from the venous blood.

I have a scientific interest in AGEs, and have recently published a paper on “The lifetimes of small arterial gas emboli, and their possible connection to Inner Ear Decompression Sickness” which looks at the time required for an AGE of a particular size to dissolve and the time required for it to reach the inner ear.  (I used the inner ear, because the arterial route to it is more amenable to calculation.)  I’ve posted it under Articles, just in case anyone is interested.

But, like all of you, I have a personal interest, as a diver, in avoiding DCS in general and these particularly nasty manifestations of it in particular.  Here are some precautions you may want to consider:

1.  If you have a specific reason to suspect an PFO, you definitely should see a doctor, preferably one experienced in diving medicine, for further investigation.   One example: you should suspect a PFO if you have had undeserved skin bends on more than one occasion. (And, by “undeserved” I mean, not just that you were technically within the limits set out by whatever NDL or dive computer you were using, but that you were far away from any such limits.)    In some cases where a PFO is found to exist but is not “medically significant” a doctor may suggest that, if you continue to dive you should dive “conservatively”.

2.  Unless you know otherwise, it’s safest to assume that you do have a PFO, an AVA or that the alveoli in your lungs miss filtering some bubbles.

Let’s suppose, then, that either you’ve been diagnosed with a PFO (but still permitted to dive), or that you’re being prudent by assuming you may have one.  What does diving “conservatively” mean in this context?

To start with, it means doing essentially the same things you already do to avoid any form of DCS.   (Once SAUL is available in computers, I would also suggest setting it for a lower acceptable probability of DCS than you might otherwise be inclined to do.)    Beyond, this, to dive conservatively, you might particularly want to:

a)  Allow a long surface interval between dives.  Never do a second dive less than an hour after the first.  Waiting more than an hour, if feasible, is even better.

b)  If you’re a smoker, consider stopping.  Smoking damages the lungs, which means it may increase the likelihood of the alveoli letting more and/or larger bubbles into arterial circulation.

c)  Where feasible, do all your diving on Nitrox.  If you aren’t already certified for Nitrox, get certified.

d)  Wait at least 24 hours after diving before flying (or sightseeing, etc. that involves altitudes over 6,000 ft.)

e)  Remain well hydrated.

Update – October 2014

It’s time for an update on when SAUL will be out in a dive computer.

The bad news is – there’s nothing definite yet.

On the plus side, I have contacted some computer manufacturers and am continuing to identify and contact others whose computers would benefit from incorporating SAUL.  Still on the plus side, SAUL is completely dive-computer-ready, lacking only the specific software connection between it and any given computer (a user interface that will work with SAUL).  For this reason, I haven’t given up on a potential 2015 release of SAUL.  It’s a bit of a long-shot at this point, but not out of the question.

If you have a favourite mid-range dive computer in mind that you’d like to see SAUL in, feel free to suggest it to the manufacturer.  I’d welcome inquiries, whether or not it’s from one I have already contacted.

On to other matters.  Part of the reason I haven’t posted for a while (aside from efforts re SAUL) is an active research program.  I have a very capable young researcher from Mexico in my lab who’s now approaching the end of his 3 year term.  Together, we’ve produced a sizeable body of interesting research.  The downside to that is the somewhat tedious and time-consuming process of transforming that research into published papers.  With one substantial paper published this summer, there are still another 4 or 5 papers at various stages of their evolution towards a published state.

The focus of the research was on the fundamentals of bubble behaviour.  Along the way, some of what we’ve found has interesting implications for diving, including PFO’s, inner-ear DCS, muscle and joint DCS, and DCS in breath-hold diving.  I will be discussing some of this in future posts, as each relevant paper is published.  (The reason for waiting is so I can post the published paper in the Articles section at the same time.)



This is the first in a series of comparisons between SAUL and other dive planners.   For obvious reasons, I can’t do a direct NDL to NDL comparison.  (SAUL, being a probability-based model, doesn’t actually have NDLs.)    Instead, single NDL profiles from the PADI tables will be paired with their expected probability of DCS according to SAUL.


Dives with air, including 3 min safety stop at 15 fsw


PADI  NDL                                        SAUL                                       SAUL  at 75% PADI NDL BT

Depth(fsw)    BT(min)                    Prob. of DCS (as a %)                   Prob. of DCS (as a %)     

 35                  205                               0.1750                                        0.0637

 40                  140                               0.1560                                        0.0645

 50                    80                               0.2090                                        0.1020

 60                    55                               0.3210                                        0.1510

 70                    40                               0.4030                                        0.1720

 80                    30                               0.4140                                        0.1490

 90                    25                               0.4740                                        0.1670

100                   20                               0.4080                                        0.1130

110                   16                               0.3000                                        0.0640

120                   13                               0.2020                                        0.0300

130                   10                               0.0796                                        0.0014

140                     8                               0.0250                                        0.0000

To put it another way, your likelihood of getting bent, if you dive profiles right at the PADI no-decompression limits, averages out at just over 1 in 400.   But your likelihood of getting bent on any particular NDL profile ranges from a low of about 1 in 4000 (140 fsw, 8 min) to a high of almost 1 in 200 (90 fsw, 25 min).   Of course, most of us don’t usually dive right at the limits.  If you limit your bottom time to three-quarters of the PADI no-decompression limits, your  likelihood of getting bent (shown in the right-hand column) averages out at just over 1 in 700 and ranges between a low of pretty near zero (less than 1 in 1,000,000 for 140 fsw, 6 min) to a high of about 1 in 550 (70 fsw, 30 min).   If you were diving with SAUL, your own personal “NDL” would depend on what you chose as an acceptable level of risk.  If, for example, you input 0.5000 (1 in 200) as your risk level – not really advisable – you could dive any of the profiles in the PADI NDLs and even increase your bottom times on many of them.  If you input a more sensible 0.2500 (1 in 400), you could dive the PADI NDLs at 35, 40, 50, 120, 130, or 140 fsw but would be held to varying shorter bottom times between 60 and 120 fsw.


Dives with “32 NITROX” (32% O2), including 3 min safety stop at 15 fsw

PADI  NDL                                                     SAUL

Depth(fsw)    BT(min)                                    Prob. of DCS (as a %)

  45                  220                                           0.0219

 50                  155                                           0.0122

 55                  110                                           0.0073

 60                    90                                           0.0194

 70                    60                                           0.0444

 80                    45                                           0.0845

 90                    35                                           0.1100

100                   30                                           0.1690

110                   25                                           0.1730

120                   20                                           0.1060

130                   18                                           0.1260


Dives with “36 NITROX” (36% O2), including 3 min safety stop at 15 fsw

PADI  NDL                                                     SAUL

Depth(fsw)    BT(min)                                    Prob. of DCS (as a %)

  50                  220                                           0.0004

 55                  155                                           0.0000

 60                  115                                           0.0000

 65                    90                                           0.0000

 70                    75                                           0.000008

 80                    55                                           0.0150

 90                    40                                           0.0198

100                   35                                           0.0688

110                   29                                           0.0822


SAUL indicates that diving with either form of Nitrox is safer than PADI NDL tables would suggest.  The “riskiest” dive in the lot – 32 NITROX at 110 fsw , 25 min –  has just slightly more than a 1 in 600 chance of resulting in DCS.  The safest for 32 NITROX – 55 fsw, 110 min – runs a DCS risk of less than 1 in 14,000.  The 36 NITROX in the PADI NDL tables, as a group, are safer still, with almost half of them bearing DCS risks of less than 1 in 1,000,000.  The “riskiest” 36 NITROX dive – 110 fsw, 29 min – still has a DCS risk of less than 1 in 1,200.  While I did calculate the probabilities of DCS for dives at 75% of the PADI NDL bottom times for both forms of Nitrox, it’s not really worth printing them out – they’re all pretty close to zero, the highest probability there being just over 1 in 10,000 (32 NITROX, 110 fsw, 18 3/4   min).    

Looking in a more general way at comparisons between SAUL and PADI, their respective conclusions on safe versus unsafe dives are not too far apart.  Nitrox is, indeed, significantly safer than air.  For air, SAUL sees the PADI NDLs as being, for the most part, of roughly equal risk and at a level of risk that is reasonable (considering that they are NDLs – i.e, limits, not necessarily preferred profiles).  SAUL diverges from PADI in finding its NDLs in the mid-depth range to be a little riskier than some divers may expect, while dives at more shallow or deeper depths are safe enough that divers who tolerate greater (but still reasonable) risks could be allowed a little more leeway.   Of course, being “allowed” to increase times at the shallowest depths means nothing on a single tank of air.  Very few, if any, divers can stretch their air to accommodate the 205 minutes PADI permits at 35 fsw or even the 140 minutes permitted at 40 fsw.  






SAUL’s progress into a dive computer is switching paths.  Saul is no longer scheduled to appear in a Liquivision computer, as we were not able to come to an agreement.  At present, I am exploring other options and will continue to post updates.

A new post, “HOW SAUL RELATES TO THE PADI DIVE TABLES” will be up within the next day or two.


The Doctor Is In

Q:  How close is SAUL to a release date?

 A:  I still can’t set a precise date (or even a precise season).   At this point, the algorithm itself is already programmed and computer-ready.   The next step will be reworking a user interface to meet SAUL’s needs, since the input and the outputs would  differ from what’s currently in use.  Then, on to some beta testing and anything that that might lead to.  Finally, manufacturing and getting it out to market.  Along the way, of course, the usual business- related necessities have to be dealt with.   So my best guess would be a release date sometime in 2015.  


 Q:  How will the inputs and outputs be different?  

A:  On the input side, you will be able to select the level of risk (of DCS) you are willing to take, not just on a “more risk/less risk” basis but on an actual percentage basis.  The outputs you will see will be: a) time remaining at current depth, based on the risk level you input and an ascent that includes a 3-minute safety stop at 15 feet; b) the expected “hit” rate if you were to make an unplanned emergency ascent, without a safety stop; c) the expected “hit” rate if you were to make an unplanned ascent with a 3 minute  m   safety stop; d) (after your ascent) your expected “hit” rate.


Q:   Won’t “d)” be the same as your input risk level (if you’ve stayed to your limit) or the same as “b)” or “d)” if you made an unplanned ascent?

 A:   No, not exactly.  Because part of the total risk can be attributed to risk that occurs during all underwater ascents, and that part would be over.


 Q:    Some divers have started doing 5 minute stops instead of 3 minute stops.  Is that a good idea?

 A:    If you want to do it, go ahead.  It won’t hurt you – provided you’re still talking about a stop at 15 feet.   On the other hand, for most low-risk recreational diving 5 minutes will provide almost no additional benefit over 3 minutes.  Some exceptions, where it might be safer, would be on dives where you’re close to the no-decompression limits, particularly if a large portion of the dive has been spent at a medium depth (say, around 80 fsw). 


Q:   Do the compartments in the SAUL model represent particular tissues – like bone, muscle, etc.?

A:   No.  The compartments in SAUL represent the way the body as a whole takes on and gives off dissolved inert gas, and the risk the body incurs as a consequence.


 Q:  Do we have to wait for a “The Doctor Is In” segment to ask a question? 

A:  Not at all.  Questions are welcome in the Comments section, not only on “The Doctor Is In” segments, but in all posts with a comments section.  Questions do not need to be related to the specific topic(s) addressed in the post.




                        Imagine you’re piloting a small boat.   Your navigational equipment and skills are not really all they should be, but you’re still good.  You’re just following the coastline or sailing around an island.  If minor corrections to your course are needed from time to time, no problem.   You’ll be in approximately the right place as the first landmark comes into view.   You adjust your course, if necessary, and then on to the next landmark.   But what if there were no landmarks?  What if you decide to set sail from California to Hawaii, with no improvement in your navigational equipment or skills?  This time you have a real problem.  Without landmarks, over that long distance, your minor navigational deviations can build up until you are so far off course, you could bypass the entire chain of Hawaiian islands without catching so much as a glimpse of them.

 What does our imaginary boating scenario have to do with algorithms and dive computers?  As I mentioned in earlier posts, algorithms used in constructing dive tables, were primarily engaged in “smoothing” the Navy data.   Almost any algorithm could do this in some fashion; staying relatively close to existing data, it was hard to go very far wrong.  You can compare it to piloting the boat along the coastline.  The use of dive computers meant that decisions – predictions, really – were being made about a wide variety of dives, some of which were far removed from the profiles in dive tables.   An algorithm might work okay for dive tables but still be well out of its depth here.  That’s because, rather like navigating the trip to Hawaii, you have a long series of calculations where even small inaccuracies can build up to a very wrong final result.

 So, when your predictions are longer range, whenever you’re talking about a long  series of calculations – whether in navigation or in dive computer algorithms – accuracy is particularly important.  To construct a more accurate dive computer algorithm, you would need to see the full picture, or as much of it as is possible.  Ideally, you would want full and complete data sets covering all possible dive situations, particularly those where the probability of DCS is highest.  But, as mentioned in previous posts in this series, studies on humans in such high risk situations won’t, can’t, and probably shouldn’t be conducted.  How, then, do you fill in the huge missing part of the picture?

 What about existing data on submarine escapes?  Unfortunately, this data is not only very sparse, but involves scenarios so completely different from those common to the data we use (and, to some extent, completely different from each other as well) that they don’t really provide much help.

Venous bubble counts have been used, notably by DCIEM, with some success, but they too have limitations.   One limitation is that the correlation of bubble counts with DCS is not very strong (somewhat stronger for very high risk dives, weaker for low risk dives).  The greater limitation is the same one that affects the Navy data sets – you still can’t use high risk situations on humans.

 Looks like it comes down to animal studies.   There are several problems here:  The fact that animals do not, generally speaking, scuba dive is easily handled.   And physiologists are used to dealing with the scaling involved in comparing animal studies to human studies. But how DCS manifests itself in animals is a little trickier.  For one thing, they don’t discuss their symptoms.  And, it turns out, the symptoms do vary from one species to another.   Then you have the problems of which species to use and how to actually combine animal data with human data.

                                     WHY ANIMALS DON’T SCUBA DIVE

“Regulator hoses are too short”

“They never give me enough weights”









“In water? You’re kidding, right?


A paper by R.S. Lillo and others in the Journal of Applied Physiology used Hill equation dose-response models to successfully combine animal data with human data to look at DCS probability in saturation dives.   A saturation dive is one where the diver has been at the stated depth until fully saturated – in humans, a period of about 24 hours- and then does a direct ascent.  Successfully combining them means that a model using the combined data was better at predicting the results of a different series of human saturation dives (not included in either set of data used to predict it) than was the human saturation data alone.

On the graph below I’ve put in the DCS probability for saturations dives at 33 fsw, 40 fsw, and 50 fsw, that Lillo found using the Hill equation model.   On the same graph, I also show what results would be predicted for those same dives by SAUL, by a SAUL version that incorporates the effect of bubbles, by the Navy’s LE1 model, by a typical parallel (Haldane) 2-compartment model, and by the USN93 model.   (The Hill equation model and the USN93 model are each shown as points with their associated 95% confidence intervals.  Both SAUL models, the LE1 model and the Haldane-type model are shown as continuous functions.  The Navy’s LE1 model and their USN93 are essentially overlapping each other.)


One thing that I hope is immediately obvious is that both SAUL models come much closer to the Hill points than any of the other models.   What may take a few moments longer to notice is that the SAUL models are also the only ones that produce the same shape (called a sigmoid curve) as the Hill points would if they were joined.  What’s much less obvious is why this particular comparison between different models matters, since these saturation dives in no way resemble anything recreational divers might consider doing.

It matters for several reasons.  One is simply the general proposition that greater accuracy in general is good and likely to result in safer diving, even though these particular dives aren’t directly relevant.  Another reason is that the particularly high “hit” rates in these dives illustrates differences in accuracy more clearly.  But here’s what I consider the most important reason.  Saturation dives are the very simplest of dives – at least for decompression modelling.   The uptake of nitrogen is already complete.   Everything that happens now relates only to off-gassing.  (Unlike most other dives where the effects of both uptake and off-gassing must be accounted for.)    So all the DCS rates shown on the graph (both experimental and model-generated) are directly related to the off-gassing process.   And the off-gassing process appears to produce DCS rates in the form of a sigmoid curve.  With models, it is always the underlying structure of an equation that dictates what shape it will produce on a graph.   The underlying structure of an equation comes from the model the equation is trying to represent.   Because SAUL models use interconnected compartments, the rate equations representing them are multi-exponential and this will produce a sigmoid curve on the graph.  All the other models in the graph, being Haldane-based, use parallel compartments, so their rate equations are essentially single exponent equations and will produce something very close to a straight line on the graph.

The next few posts in this sequence will deal more directly with recreational diving and how SAUL relates to dive-related myths/anecdotal knowledge and to other models or decompression tables.

(Before we get to those posts, we may switch course briefly for a few “The Doctor is In” segments. )


Algorithms and Dive Computers

I’m still dealing with the question of : If different dive tables, algorithms, and dive computers all stem from essentially the same data, why are they so different?  When I left the previous post I said this one would start with the particular problems that arise when your data comes from living beings, and it will.  (I also promised this post wouldn’t be as long delayed as the previous one.  I’ve kept that promise – but just barely.  I’ll try to do better with the next post.)

The basic idea in doing studies is to have complete control over absolutely everything in the surrounding situation, so that the only differences in the outcomes will be as a direct result of the changes that you make in the situation.   (Good luck with that!)   That level of control is difficult enough with inanimate objects (coins, dice, widgets, or whatever).  Things that are alive are a large step beyond that.  There is always something else going on that you can’t control – often several things – whether or not you are directly aware of them.   If you’re lucky, these other things will have little or no effect on the outcomes.

When your studies involve people, things get even more complicated.  Not only are people less controllable – you can’t keep them under 24-hour supervision, regulate diet, select breeding stock, etc., as you can with lab rats – there are additional restrictions on acceptable outcomes.   An obvious example:  you can’t do a decompression study where 80% of the divers get the bends.   Even if your own moral compass wouldn’t preclude that, an ethics committee will.    While such restrictions are valid and necessary, they do tend to constrain the range of data that can be collected.   I’ll get back to this point a bit later (or in the next post).

Right now, let’s get back to the Navy data discussed in the previous post.   What use can be made of all that data?   The most pragmatic use is as a simple guide to the underwater workplace, both naval and commercial.   Dives that had resulted in few or no cases of DCS were deemed safe for use; dives with unacceptably high rates of DCS were deemed unsafe.   Out of this, the first Navy dive tables were born.  All the dives were essentially square profile. (I say “essentially” because, of course, dives requiring decompression were only square profile up to a point –  that point at which the first decompression stop began.)    The tables were constructed almost directly from the data. An algorithm was used mainly as a sort of “smoothing device” to keep the tables internally consistent and to fill in any points for which there was no data.

This “smoothing” is necessary because, as discussed in the previous post, the result of a study or experiment should be looked at as a range of possibilities rather than as an absolute answer, with the actual number found being merely the best estimate, in the absence of other information.   Here’s a clear example (not actual, but possible) of how other information could change your best estimate of a study’s results:   Suppose that, in a dive at a particular depth for 20 minutes, 4 out of 100 divers got bent.  Four percent would be your initial best estimate of DCS probability for that dive.   But suppose that, in another dive at that same depth, this time for 22 minutes, only 3 out of 100 divers got bent.  You would not accept 4% DCS as your best estimate for 20 minutes at that depth, and 3% DCS as your best estimate for 22 minutes at the same depth.   You would, in some fashion need to adjust your best estimates so that they made sense together.   The “smoothing” done by the algorithm is an adjustment of best estimates in the tables so that they all make sense together.

Take a break for a brief “power nap”.



fruitbat                  puffer3




Okay, break’s over; let’s get back to work.

Even before we get to dive computers, and to dives that are more varied than those in the dive tables, we begin to see some of the reasons for different results coming from the same data.  One biggie: How much risk is an acceptable risk?  Is a 4% chance of DCS okay?  2%?  Less than 1%?   The other obvious source of difference is exactly how you adjust a large number of initial best estimates so that they make sense together.

In our fictional example above, someone might end up with new “best estimates” of 3.2% for 20 minutes and 3.4%  for 22 minutes while someone else might have it as 3.4% for 20 minutes and 3.8% for 22 minutes.  (Of course, if a 2% chance of DCS was the maximum acceptable, neither 20 nor 22 minutes at that depth would be permitted.   If 3.5% risk of DCS was used as the acceptable limit, one table might allow both dives while another would allow the 20 minute dive but not the 22 minute one.  If no adjustment had been done to the original best estimates, a table allowing a 3.5% maximum risk would allow the dive for 22 minutes, but not the one for 20! ).

These two choices – maximum acceptable chance of DCS and method of adjusting or “smoothing” the results – are the primary reasons for the real life differences you can see between the U.S. Navy Tables, the DCIEM tables, and the various other tables that exist.

For most commercial diving applications, dive tables, with their essentially “square profile” approach, were reasonably appropriate.  Divers would descend to the work site and remain at that depth until time to ascend.   As recreational diving increased in popularity, tables that assumed that a diver remained at a single bottom depth until ascending became awkward to use.   That was frequently not how recreational divers wanted to dive.

When I took my initial certification course, before diving computers were in use, we were told to use the tables in accordance with the deepest depth we descended to – even if we only stayed at that depth for a moment or two.   While it seems way too  conservative, without a known safe alternative, that was (and remains) the best practice.  At least it was simple.  The method for calculating times for repetitive dives from the tables (which you probably learned, but may have since banished from memory) involved turning over the tables after the first dive, categorizing yourself as A,B,C, etc., based on the nature of that first dive, the time elapsed since your ascent (changing, of course, as more time elapsed), then turning back to the tables, using your category to adjust the tables to get the allowable parameters for your next dive.  After a while, PADI developed a dive wheel that simplified the repetitive dive calculation somewhat.   Still, dive computers, when they finally arrived, were very welcome indeed.

The most significant change that came with dive computers – besides simplifying what the diver had to do – was the ability to continuously incorporate information from the depth gauge and the computer’s clock into deco calculations.  That made it possible to calculate multilevel dives differently from the “deepest depth” method.

But now the algorithms had a more complex job to do.  The nature of that complexity and how it’s dealt with will be covered in the next blog.


NDL and Decompression Tables, Algorithms, and Dive Computers

I realize it’s been far too long since my last blog post.   I have been working very hard,   getting SAUL ready for inclusion in a dive computer and attending to other scientific work.   I’ve also managed to get in a little relaxation and diving – Aloha from Hawaii, everyone.   I’ll try to keep the posts coming with a little more regularity now.

I promised some posts about SAUL’s validation, and these will be happening.  But before they do, we need a prequel of sorts – a little background information that every diver should know about decompression tables, NDL’s, algorithms, and dive computers.  What makes it hard, is that we divers are such a diverse lot.   Some of you will already know much of what’s in today’s post, and in greater detail.  To a very few of you, the information may seem entirely new.  Most of you will probably fall somewhere in between those two extremes.    So, let’s talk about..

NDL and Decompression Tables, Algorithms, and Dive Computers

You and your buddy are doing almost exactly the same dive.  His dive computer is telling him to surface; yours is still allowing the dive to continue.

Or you’re a commercial diver working different sites.  Some employers are using the U.S. Navy tables, others the DCIEM tables.  The tables differ in how long you’re supposed to work at a particular depth and, for decompression diving, the time spent at each decompression stop.

In recreational diving, some of you may deliberately use one dive algorithm (or dive computer) for certain types of dives and a different algorithm (or dive computer) for other types of dives, knowing they will differ in what they allow.  What’s going on here?

All dive tables, algorithms and dive computers are based on data from actual dives.  Until very recently, the largest data bank for this purpose, which will be referred to below simply as Navy data, had been produced by the U.S. Navy, in collaboration with DCIEM, and the Royal Navy.  (While the total amount of data collected in recent years under DAN’s PDE program greatly surpasses it, the Navy still holds the most systematically varied and organized databank.)   So, most dive tables, algorithms, and dive computers spring from the Navy data.   This leads to the obvious question: If they come from essentially the same data, why are they different?

We could try to finesse the question by comparing it to different artists painting the same harbor scene and producing very different paintings, but that’s not really a satisfactory comparison.  Paintings are art; dive tables and algorithms are supposed to be science.

Without getting into any esoteric questions like the exact natures of science, truth or meaning, we just want to know why the same huge data bank leads different scientists to different conclusions.   Some of the answer lies in how scientists think, in the theories they use to interpret data.    But a large part of the answer lies in the nature of all data, in the nature of data from living things, and in the particular complications involved in collecting data from people.

We’ll talk about data first.   Data in general is a collection of measurements made under specified conditions.  In the case of diving data, the measurements collected would essentially be the determination of  “bent” or “not bent”  or “niggles” (which refers to having mild symptoms of the bends, but the symptoms disappear on their own, without recompression therapy), while the specified conditions would be all the details of the dive.  For each set of specified conditions – e.g., 80 feet for 30 minutes bottom time, breathing air, coupled with a descent rate of 75 feet/min and an ascent rate for the direct ascent to the surface of 30 feet/min – you would see the total number of divers and the number who got  “bent” or “niggles”.  For each set of specified conditions, you could then work out the probability of getting “bent”.   (“Niggles”, usually considered a partial case of the bends, would be assigned an appropriate part-value:  less than the “1” reserved for being “bent” but greater than the “0” for “not bent”.   Typically, a niggle would count as one tenth of a hit.)


Okay, it’s a little slow going – but we’re getting there.

So, you’ve got your data.  What does it tell you?  In the example above you would now know what percentage of the sample of divers measured under that particular set of  specified conditions got “bent”.   But, of course, you didn’t measure an infinite number of divers.  If you did the exact same study again, you might get a somewhat different percentage.  There are statistical methods that are used to determine how close the result you got is likely to be to the hypothetical “true result” – the result you would find if it were possible to measure an infinite number of divers.  This is how the possible difference between the results of a study and a “true result” might be shown on a graph.

The dot or circle indicates the actual result of the study  – in this case, some of the results from DAN’s Project Dive Exploration – while the lines sticking out from the top and bottom of each dot or circle show the range within which the “true result” is probably located.   So, while the circles are, in each case, your best guess at the “true result”, you’re 95% certain that the “true result” lies somewhere in between the top and bottom of the two lines sticking out from each circle.  The results for AIR are based on approximately103,000 single dives while the results for NITROX are based on approximately 7,000 single dives.  In general, the larger the sample size, the closer it is likely to be to the “true result” (and the shorter the lines sticking out from the circle are likely to be).

There are other ways of talking about this same issue of a sample measurement versus a “true result”.    When public opinion polls show their results, they will usually contain a statement something like: “These results are accurate to within 3 percentage points, nine times out of ten.”

The fact that the results of a study are only an estimate of some “true result” is an issue that applies to all kinds of data.


Take heart! We’re getting closer now.

Notice that, in our earlier example, we were talking about only one set of conditions where depth, bottom time, the rate of descent, the rate of ascent, and the breathing mixture were each specified.  So our data refers directly only to dives under that exact same set of conditions – almost as if it were a single question on a public opinion poll.  If you were to change any one or more of those conditions – 60 feet instead of 80, 40 minutes instead of 30, etc. – or if you were to add an additional condition – say, making the dive multi-level, or adding a safety stop – each and every change would be like adding a completely new question to the opinion poll.   (Making for a completely unmanageable opinion poll, of course.)

While the Navy data contains a large number of sets of different specified conditions, it would have to be infinitely large to cover all possible sets of specified conditions.

There are two points that should be sandwiched in here before we go on (in the next posting) to deal more specifically with data from living things, particularly people.

The first is a comment on the nature of diving data.  As you can see from the above, diving data is inherently probabilistic to begin with.  That is, you have the number of divers in a particular set of conditions, and you have the number who got “bent”, which is easily expressed as a percentage of the number of divers, and which amounts to an estimate of the probability of getting bent under those particular conditions.   Why do I bother emphasizing this rather obvious conclusion?  (And no, it’s not primarily because SAUL is a probability-based model.)

It seems that a substantial number of divers may not be aware, or may not totally accept the fact that, if you dive there is always some probability, however minute, that you will get bent.   Anecdotally, I have heard from a number of diving doctors about patients who resist a diagnosis of decompression illness, protesting that they can’t possibly be bent, as they’ve never exceeded the tables or their dive computers.  But the truth is, unless you avoid diving altogether (or, possibly, if, while diving, you never exceed a depth in the general neighborhood of 15 feet – a feat which may be even more difficult than avoiding diving altogether) there is always some non-zero probability of getting bent.  True, for most recreational diving, that probability is very small.  On the other hand, the probability of winning a major lottery prize is even smaller.  Yet, for each such prize, there is at least one winner.

The other point we need to mention here is this:  Even though data from each separate set of diving conditions must be looked at as a completely separate question, we know, both intuitively and logically, that,  inevitably, there must be some relationship or connection between these disparate “questions”.  And that, of course is where algorithms, decompression tables, etc. come in.   But before we get to those, we need to deal with the particular problems that arise when your data comes from living beings, particularly people.

So my next post will start with that issue and go on from there.  And, while I can’t say exactly when that next post will appear, I promise it won’t be as long delayed as this one was.


What’s Happening?

Those of you who have read “Coming Soon to a Dive Computer Near You” may have noticed that “Soon” has been a while in coming.  But we are now a lot closer to getting SAUL into a dive computer.  Liquivision is collaborating with us to produce a dedicated SAUL dive computer and I am hard at work adapting programs for that purpose.  We don’t have a projected release date yet so, obviously, you won’t find it under your Christmas tree this year.  As for next year, who knows?

I came across a video (on Vimeo.com) that was taken of my presentation to the International Congress of Hyperbaric Medicine last year in South Africa and posted a link to that on my Articles page.  You may need to turn up the sound to hear it properly.  I think I need to work on speaking a bit louder (or closer to the microphone).

The Articles page now has the “To Stop Or Not to Stop… And Why” article from Diver magazine.   I will be also be posting my original version of that article, with a minor update to it, because: a) I generally preferred my wording (rightly or wrongly), where a few editorial changes were made, b) I wanted to post the second cartoon that was submitted with the article (but not published) , and c) this updated version was accepted for publication by the European editon of  Alert Diver.

Now that SAUL is getting a little closer to coming out in a computer, it’s probably time to pull together the the various ways in which SAUL’s validity can be demonstrated.  I’ll try to do this in a series of relatively short pieces – not necessarily consecutive – on the blog.  Once a few have been done, I’ll put them, and subsequent ones, together for easy reference under a new heading.