The Theoretical Diver

In a comment, Rick suggested that given that in the end, VPM-B computes depth dependent maximal gradients, one could compare those to the depth dependent gradients computed from the Buehlmann model and expressed as gradient factors (i.e. as percentage of the gradient of a plain vanilla Buehlmann gradient). Here is what I found when I did this.

 

For concreteness, I computed deco schedules with Subsurface for a dive to 120m leaving the bottom at runtime 20min with TMX18/50 (yes, I know this is far too much pO2, but for deco that does not matter), EAN50 and EAN100. For some reason, my ascent speeds are set to 9m/min up to 75% average depth, then 6m/min up to 6m and 1m/min for the final ascent. With Bühlmann with GF 30/80, I obtain a plan that reaches the surface at a runtime of 188min while for VPM-B+1 it takes only 141min.

Actually, I patched subsurface a little bit (see this branch on github) that, during deco, it prints out the allowed gradient for all tissues at all deco stops. This output I ran thru a mathematica notebook  to produce these plots:

 

On the x-axis, we have depth in meters while on the y-axis, we have the current gradient factor. The different colored dots are the different tissues (red being those with the shortest halft-time) with the thick dot indicating the leading tissue, i.e. the tissue that is causing the current stop.

The first diagram is Bühlmann while the second is VPM-B.

A few remarks are in order: First, why is Bühlmann not a single line from GF 30 to GF 40? This is because of the way I compute things: The exact Bühlmann a and b coefficients depend on the gas mixture currently in the tissue. In addition, what is actually computed is the maximal ambient pressure at the ceiling (which is the current depth only for the leading tissue) for that tissue and the gradient I plot is the difference between that and the current ambient pressure. So this is not exactly the right thing for tissues that are not leading. But still we see, it is a nice band of values and they all nicely move between 30% and 80%.

For the VPM-B plot, on the other hand, we see that the variation within a tissue is much less but the gradient factor varies a lot more between tissues. After all, the actual variation in gradient factor with depth comes about mainly because the leading tissue is changing and much less from the cubic curve that implements the Boyle compensation.

The other thing that is striking is the scale: At the last deco stop, the effective gradient factor is 124% for the leading tissue (while for some slower ones it is as high 180% (which never matters as that tissue never leads). In fact, a Bühlmann plan with GF 20/125 leads to a very similar profile.

Here are the actual profiles: First, Bühlmann GF 30/80

Depth	Duration	Runtime	Gas
➘	120m	7min	7min	(18/50)
➙	120m	13min	20min
➚	51m	9min	29min
–	51m	2min	31min
➚	48m	1min	32min
–	48m	1min	32min
➚	45m	1min	33min
–	45m	2min	34min
➚	42m	1min	35min
–	42m	2min	36min
➚	39m	1min	37min
–	39m	4min	40min
➚	36m	1min	41min
–	36m	3min	43min
➚	33m	1min	44min
–	33m	4min	47min
➚	30m	1min	48min
–	30m	6min	53min
➚	27m	1min	54min
–	27m	6min	59min
➚	24m	1min	60min
–	24m	10min	69min
➚	21m	1min	70min
–	21m	5min	74min	EAN50
➚	18m	1min	75min
–	18m	7min	81min
➚	15m	1min	82min
–	15m	8min	89min
➚	12m	1min	90min
–	12m	13min	102min
➚	9m	1min	103min
–	9m	20min	122min
➚	6m	1min	123min
–	6m	20min	142min	EAN100
➚	3m	3min	145min
–	3m	40min	185min
➚	0m	3min	188min

Then VPM-B+1: 

	Depth	Duration	Runtime	Gas
➘	120m	7min	7min	(18/50)
➙	120m	13min	20min
➚	63m	7min	27min
–	63m	1min	28min
➚	57m	1min	29min
–	57m	1min	30min
➚	54m	1min	31min
–	54m	1min	31min
➚	51m	1min	32min
–	51m	2min	33min
➚	48m	1min	34min
–	48m	1min	34min
➚	45m	1min	35min
–	45m	2min	36min
➚	42m	1min	37min
–	42m	2min	38min
➚	39m	1min	39min
–	39m	3min	41min
➚	36m	1min	42min
–	36m	3min	44min
➚	33m	1min	45min
–	33m	3min	47min
➚	30m	1min	48min
–	30m	4min	51min
➚	27m	1min	52min
–	27m	5min	56min
➚	24m	1min	57min
–	24m	7min	63min
➚	21m	1min	64min
–	21m	4min	67min	EAN50
➚	18m	1min	68min
–	18m	5min	72min
➚	15m	1min	73min
–	15m	6min	78min
➚	12m	1min	79min
–	12m	8min	86min
➚	9m	1min	87min
–	9m	12min	98min
➚	6m	1min	99min
–	6m	15min	113min	EAN100
➚	3m	3min	116min
–	3m	22min	138min
➚	0m	3min	141min

Finally Bühlmann 20/125:

	Depth	Duration	Runtime	Gas
➘	120m	7min	7min	(18/50)
➙	120m	13min	20min
➚	54m	8min	28min
–	54m	1min	29min
➚	51m	1min	30min
–	51m	2min	31min
➚	48m	1min	32min
–	48m	1min	32min
➚	45m	1min	33min
–	45m	2min	34min
➚	42m	1min	35min
–	42m	2min	36min
➚	39m	1min	37min
–	39m	2min	38min
➚	36m	1min	39min
–	36m	3min	41min
➚	33m	1min	42min
–	33m	3min	44min
➚	30m	1min	45min
–	30m	5min	49min
➚	27m	1min	50min
–	27m	5min	54min
➚	24m	1min	55min
–	24m	6min	60min
➚	21m	1min	61min
–	21m	4min	64min	EAN50
➚	18m	1min	65min
–	18m	4min	68min
➚	15m	1min	69min
–	15m	7min	75min
➚	12m	1min	76min
–	12m	8min	83min
➚	9m	1min	84min
–	9m	13min	96min
➚	6m	1min	97min
–	6m	12min	108min	EAN100
➚	3m	3min	111min
–	3m	21min	132min
➚	0m	3min	135min

Dies ist der erste technische Post darüber, wie VPM-B funktioniert. Ich werde nur den Algorithmus beschrieben (mit allen wichtigen Formeln, einige davon vielleicht  in ungewohnter Form), während die physikalische Herleitung in einem späteren Post behandelt werden wird. 

Ich werde nicht versuchen, die ursprüngliche Herleitung nachzubeten (auch schon alleine, weil ich sie nicht wirklich verstehe), sondern vielmehr beschreiben, was ich durch langes Starren auf unsere Implementierung des Modells in Subsurface gelernt habe. Wie vorher schon erklärt haben wir versucht eine unabhängige Neuimplementierung zu machen, haben aber natürlich sicher gestellt, dass die Austauschpläne des ursprünglichen FORTRAN-Programms reproduziert werden. Dieses haben wir letztendlich als die Definition von VPM-B hergenommen.

Zunächst ist dieses dem Bühlmannmodell sehr ähnlich und baut darauf auf. Für diesen Post nehme ich an, dass dieses bekannt ist (falls nicht habe ich hier mal eine kurze Zusammenfassung aufgeschrieben). Insbesondere werden die Sättigungen der verschiedenen Gewebe mit Inertgasen mit der gleichen einfachen Diffusionsgleichung beschrieben:

$$\frac{dp_i}{dt} = \alpha_i(p_{amb}-p_i)$$

wobei die Zeitkonstante \(\alpha_i\) im wesentlichen der Kehrwert der Halbwertzeit des Gewebes ist. 

Der einzige Unterschied besteht darin, wie die Ceiling (die momentan minimal erlaubte Tiefe) aus den Gewebesaettigungen \(p_i\) berechnet wird: Bühlmann verwendet einen affine Zusammenhang mit dem Umgebungsdruck (die Geradengleichung wird durch die gekannten \(a\) und \(b\) Koeffizienten parametrisiert. Verwendet man Gradientenfaktoren, die wird Geradengleichung noch weiter modifiziert (abhängig von der maximalen Ceiling, aber die Ceiling, geplottet als Funktion von  \(p_i\) bleibt eine Gerade, ist also affin).

Für VPM-B ist dies anders: Nur in der nullten Näherung bleibt es eine Gerade: In der Tat wird die Ceiling einfach durch einen konstanten Gradienten (die Differenz zwischen Umgebungsdruck und Gewebedruck) gegeben, dieser ist sogar von der Tiefe unabhängig:

$$p_{amb} – p_i \le G_{0,i}.$$

Dieser Gradient wird als Anfangsgradient  \(G_0\) bezeichnet. In dieser Näherung verhält sich VPM-B genau wie ein Buehlmannmodell mit modifizierten a und b Koeffizienten.

Für die nächste Näherung, die “Boyle-Kompensation” müssen wir etwas über Blasen wissen. Alles, was wir brauchen, ist, dass diese eine Oberfläche haben und dass die Oberfläche zur Gesamtenergie proportional zu ihrer Fläche beiträgt:

$$E_{surf}= \gamma r^2$$

für eine Blase mit Radius \(r\) (wir können numerische Faktoren wie \(4\pi\) in der Proportionalitätskonstantante \(\gamma\) absorbieren). Die Kraft auf die Oberfläche, die diese zu verkleinern sucht, ist dann

$$F_{surf} = \frac{\partial E_{surf}}{\partial r} = 2\gamma r.$$

Diese Kraft wirkt auf die gesamte Fläche, teilen wir durch den Flächeninhalte erhalten wir einen Beitrag zum Druck in der Blase:

$$G=\frac{2\gamma}r.$$

Derartige Formeln, bei denen eine Blasenoberfläche  \(2\gamma/r\) zum Druck beiträgt, werden im Folgenden immer wieder auftauchen.

Insbesondere zu dem Zeitpunkt im Tauchgang mit der tiefsten Ceiling ist der Gradient \(G_0\) genau dies. “Boyle-Kompensation” bedeutet, dass der von nun an tiefenabhängige Gradient (also der Umgebungsdruck minus Gewebedruck) dadurch gegeben ist, dass man eine Blase nach dem Boyle-Marriott-Gesetz expandieren lässt, also dass man annimmt, dass \(pV\) konstant ist. \(p\) im Inneren der Blase (was nach dem Modell mit dem Gewebedruck gleich ist) wird dann berechnet als der Umgebungsdruck plus dem momentanen maximalen Gradienten. Das Volumen der Blase ist proportional zu \(r^3\) und dies ist nach der obigen Formel proportional zu \(1/G^3\). Alles in allem ist die erlaubte Differenz zwischen Gewebe- und Umgebungsdruck in der Tiefe \(d\) gegeben durch die Lösung von

$$\frac{G+p(d)}{G^3} = \frac {G_0+p_{1st\ ceiling}}{G_0^3}.$$

Folglich ist der der maximale Gradient keine Gerade mehr, sondern die Lösung einer kubischen Gleichung (für die wir in Subsurface auch einen analytischen Ausdruck verwenden statt wie im Originalcode eine Nullstellensuche mittels Newtonverfahren zu veranstalten).

 

Ich habe noch nicht erklärt, wie der Anfangsgradient \(G_0\) berechnet wird. Für VPM-B lautet der Ausdruck dafür

$$G_0 = 2 \frac\gamma{\gamma_c} \frac {\gamma_c – \gamma}{r_{reg}}.$$

Hier ist  \(r_{reg}\) der “Crushing-Radius”, der sich mittels

$$\frac{2(\gamma_c-\gamma)}{r_{reg}} = p_{max} + \frac{2(\gamma_c-\gamma)}{r_{crit}}$$

aus der Oberflächenspannung der Blase berechnen lässt.

Hier wiederum sind  \(\gamma\) und \(\gamma_c\) Oberflächenspannungen (am Ende: Modellparameter), ebenso wie \(r_{crit}\). Letzteres ist der Parameter, an dem man den Konservatismus des Modells einstellen kann: Jede zusätzliche Konservatismusstufe (zum Beispiel VPM-B+3) erhöht \(r_{crit}\) um 5 Prozent.  \(p_{max}\) ist das Maximum des Umgebungsdrucks während des Tauchgangs. Im Prinzip soll man diesen auch wieder exponentiell  an  \(r_{crit}\) angleichen; dies kann man aber getrost ignorieren, da die Halbwertzeit von zwei Wochen viel länger als alle relevanten Zeitskalen eines Tauchgangs sind.

Jetzt haben wir alles, was wir brauchen, um eine Deko-Profil zu berechnen: Für jede Tiefe kennen wir wie im Buhlmann-Modell den maximal erlaubten Überdruck.

Es stellt sich heraus, dass die so erhaltenen Dekoprofile viel zu lange dauern. Hier kommt der “Kritische-Volumen-Algorithmus” ins Spiel. Wir erhöhen leicht den Anfangsgradienten (bisher war das \(G_0\) und unterwerfen den dann ebenso der Boyle-Korrektur.

Die Idee dabei ist, dass der Körper eine bestimmte zusätzliche Menge an Gas in Blasenform verträgt. Dieses erlaubte Volumen ist wiederum ein Modellparameter und wird mit einem wie folgt für einen Anfangsgradienten \(G_0′\) berechnet: Es wird angenommen, dass es vor dem Tauchgang eine bestimmte Verteilung der Blasenzahl abhängig von ihrer Größe im Körper gibt. Dadurch, dass der Anfangsgradient von \(G_0\) auf \(G_0′\) erhöht wird, erhöht sich die Zahl der wachsenden Blasen (generell wachsen grosse Blasen, da bei ihnen der Druckanteil durch die Oberflächenspannung kleiner ist, siehe oben, während kleine schrumpfen). Die Zahl der zusätzlich wachsenden Blasen ist dann in linearer Näherung proportional zu (und dies ist einer der Orte, wo ich die Herleitung nicht verstehe, aber verwenden wir den Ausdruck trotzdem):

$$\frac{G_0′-G_0}{G_0′-\alpha G_{crush}}.$$

 

Hier ist  \(\alpha\) ein weiterer Modellparameter und  \(G_{crush}\) ist der maximale Überdruck des Umgebungsdrucks zum Gewebedruck während des Tauchgangs und nach den Regeln des Modells ebenso der maximale Unterschied zwischen dem Innen- und dem Außendruck einer Blase (wenn dieser zu groß ist, kommen wir ins “nichtpermeable Regime” und man wendet eine weitere Boyle-Kompensation an).

Eine weiter Annahme ist, dass die Rate mit der Gas in die Gasphase in Blasen geht, proportional zum Gradienten \(G_0′\) ist (die erfolgte Boyle-Compensation fällt hier unter den Tisch). Da man also annimmt, dass der Gradient während der Deko (definiert als die Zeit, zwischen dem Zeitpunkt, bei dem das erste Gewebe beginnt, den Gewebedruck wieder abzubauen bis zum Erreichen der Oberfläche) konstant ist, multiplizieren wir den Gradienten mit der Dekozeit (dies ist die Zeit \(t_D\) die wir oben berechnet haben, als wir den Dekoplan aus den Gradienten berechnet haben).

Sobald wir die Oberfläche erreichen, gibt es immer noch einen Überdruck im Gewebe, zunächst \(G_0′\) (wiederum wird die Boyle-Kompensation unterschlagen) und im Folgenden fällt dieser exponentiell entsprechend der obigen Diffusionsgleichung auf 0. Somit ist das Zeit-Integral des Gradienten an der Oberfläche \(G_0’/\alpha_i\). Nimmt man alles zusammen, ist das Kriterium für den neuen Gradienten \(G_0′\)

$$V\ge \frac{G_0′-G_0}{G_0′-\alpha G_{crush}} G_0′ \left(t_D +\frac 1{\alpha_i}\right)$$

Dies löst man nach \(G_0′\) auf, was dann für einen neuen Dekoplan benutzt wird, der wiederum ein neues \(t_D\) ergibt. Dieses Verfahren wiederholt man, bis sich \(G_0′\) nicht mehr ändert.

Bis auf etwas Kleingedrucktes (wie etwa wie der Dekoplan aus dem Gradienten im Detail berechnet wird) ist dies der gesamte VPM-B Algorithmus. Mit diesen Informationen kann man ihn im Prinzip selber implementieren.

In einem zukünftigen Post plane ich dann die Physik anzusehen, die aufgerufen wird, um die hier verwendeten Gleichungen herzuleiten.

This is the first technical post about the inner workings of the VPM-B deco model. Here I will just present the algorithm (including the relevant formulas, some of them maybe in an unusual form) while the physical motivation or derivation will be the subject of a future post.

 

I try not to parrot the original explanations (which, honestly, I do not really understand) but rather put in words what I learned by staring long enough at the subsurface implementation of the model. As I explained earlier, we tried to make this an independent implementation but of course in the end we made sure it produces the identical deco schedules to the original FORTRAN code (that we took as the definition of the model).

 

First of all, it is not that different from the Bühlmann model, it rather builds on it. For the sake of presentation, I will assume you are familiar with how that model works (if not, I wrote a brief summary with some background in German, to be translated to English soon). In particular, it computes the same tissue loadings with inert gases following a simple diffusion equation

$$\frac{dp_i}{dt} = \alpha_i(p_{amb}-p_i)$$

where the time constant \(\alpha_i\) is essentially the inverse tissue half-time.

 

The only difference is how to compute the current ceiling for a snapshot of tissue loadings (“tensions” in the deco theory speak) \(p_i\): Bühlmann proposes an affine dependence on the ambient pressure (with the affine relation given in terms of the famous \(a\) and \(b\) coefficients. If you use gradient factors, this affine relation is further modified (depending on your lowest ceiling), but when you plot the ceiling as a function of \(p_i\) it is still a straight line, i.e. affine.

 

This is different for VPM-B: Only in the zero’s approximation it is still a straight line. In fact, there the ceiling is simply given by a maximal gradient (difference between tissue pressure and ambient pressure) that is independent of the depth:

$$p_{amb} – p_i \le G_{0,i}.$$

This is called the initial gradient \(G_0\). In this approximation, VPM-B is like Buehlmann, only with different a and b coefficients.

 

For the next approximation, the “Boyle compensation” we need to know a little bit about bubbles. in fact, the only thing one needs to know is that they have a surface and in the total energy balance of the bubble the surface contributes proportional to its area:

$$E_{surf}= \gamma r^2$$

for a bubble of radius r (we can absorb all kinds of numerical constants like \(4\pi\) in the proportionality constant \(\gamma\). Then the force on the bubble surface that wants to shrink it is

$$F_{surf} = \frac{\partial E_{surf}}{\partial r} = 2\gamma r.$$

Finally, this force acts on all of the surface, by dividing by its area again, we get the surface’s contribution to the pressure

$$G=\frac{2\gamma}r.$$

This type of formula, the a bubble surface contributing to the bubble’s internal pressure by \(2\gamma/r\) you will see over and over again.

In particular, that the deepest point in the dive, or actually when reaching the first ceiling, the gradient \(G_0\) is exactly this. And the Boyle compensation means that the (now depth dependent) maximal gradient (i.e. ambient pressure minus tissue pressure) is given by letting such a bubble expand according to Boyle’s law, i.e. by assuming that \(pV\) is constant. \(p\) inside the bubble (which is supposedly the same as the tissue pressure) is then given by the ambient pressure plus the momentary maximal gradient while the volume is proportional to \(r^3\) which according to the above formula is proportional to \(1/G^3\). So in total, the allowed difference between the tissue and the ambient pressure at depth \(d\) is given by solving

$$\frac{G+p(d)}{G^3} = \frac {G_0+p_{1st\ ceiling}}{G_0^3}$$

 

So, rather than being a straight line, the maximal gradient is now the solution of a cubic equation (for which, in Subsurface, we use an analytic expression rather than running a Newton root finder as in the original implementation).

 

So far, I did not tell you what the initial gradient \(G_0\) is. The VPM-B expression for it is

$$G_0 = 2 \frac\gamma{\gamma_c} \frac {\gamma_c – \gamma}{r_{reg}}.$$

Here, \(r_{reg}\) is the crushing radius which is again determined from an expression of a bubble surface tension:

$$\frac{2(\gamma_c-\gamma)}{r_{reg}} = p_{max} + \frac{2(\gamma_c-\gamma)}{r_{crit}}$$

Here \(\gamma\) and \(\gamma_c\) are some surface tensions (effectively parameters of the model) as is \(r_{crit}\). The latter is where one can tune the conservatism of the model: Each level of conservatism (like three levels in VPM-B+3) increases \(r_{crit}\) by 5 percent. \(p_{max}\) is the maximal ambient pressure encountered during the dive. In principle, it is also assumed that over time, \(r_{reg}\) also exponentially approaches \(r_{crit}\) again, but it is totally safe to ignore this since the decay time is supposed two weeks, far too long to play any role at the typical time scales of the dive.

 

Now, you have everything to compute a deco schedule: Like in the Buehlmann model, you know for each depth what the maximal gradient is.

 

Turns out, these schedules take far too long. This is where the “critical volume algorithm” comes into play. We now slightly increase the first gradient (the role so far played by \(G_0\)) which then is also subject to the Boyle compensation.

 

The idea here is that the body can tolerate an additional amount of gas going into bubbles. The maximal volume, again, is a model parameter and it is compared to the excess volume as follows: One assumes there is some distribution of bubble sizes. By enlarging the gradient \(G_0\) to \(G_0′\), the number of additional bubbles will be (this is one of the places where I do not understand the derivation, but let’s take the expression anyway) proportional to

$$\frac{G_0′-G_0}{G_0′-\alpha G_{crush}}.$$

 

Here, \(\alpha\) is another constant and \(G_{crush}\) is the maximal excess of the ambient pressure over the tissue pressure over the dive which also happens to be the pressure gradient over the surface of the bubble (if that is too big, we are in the “impermeable regime” and then there is another Boyle type compensation).

 

Furthermore, rate of additional out gasing is assumed to be proportional to the gradient \(G_0′\) (totally ignoring the first Boyle compensation) and the whole thing lasts (assuming a constant gradient) over all of the decompression time (defined as the time from the first tissue starting to release gas again until we reach the surface). This decompression time \(t_D\) is what we computed above where I said we can now compute a deco plan from the gradients.

 

Once we reach the surface, there is still an overpressure in the tissue compared to the ambient pressure, it starts with \(G_0′\) (again ignoring the Boyle compensation) and then, according to the diffusion equation above, approaches 0 exponentially. so the time integral of the gradient yields \(G_0’/\alpha_i\). Throwing all together, we get as a criterium for the maximal \(G_0′\)

$$V\ge \frac{G_0′-G_0}{G_0′-\alpha G_{crush}} G_0′ \left(t_D +\frac 1{\alpha_i}\right)$$

This we then solve for \(G_0′\) which we can use for computing a new deco schedule which gives a new \(t_D\) and we repeat this until \(G_0′\) does not change anymore.

 

Apart from some minor fine print (like how in detail the deco schedules are computed) is all there is to VPM-B. With this information, you should, in principle be able to implement it yourself.

 

In the next post, I will look into the physics that is invoked to argue for some of the expressions that I only quoted here.

One of the features that Subsurface users kept asking for were other deco models besides Bühlmann (including gradient factors). In particular, they wanted to see a bubble model implemented. I was reluctant to do so for some time since I believed (like many other divers) that you can produce fine decompression profiles that look like they are coming from a bubble model by using gradient factors, in particular by using settings like \({GF}_{low}=20-30\). But more important: I did not understand how they worked.

 

And this is not because I cannot understand how you can derive this or that formula from physical laws (I do that for a living), but because the text that are supposed to describe these models appeared to me to be confused. There were huge gaps in my understanding of the original texts, like this or this, those seemed to me like spending a lot of space on absolutely trivial calculations (like integrating an exponential function) while on the other hand not clearly saying what the actual strategy is and what all those symbols meant. Plus some of the derivations seems to use some slightly unorthodox algebra…  There are other texts by other authors that are clearly derived from these originals but I keep having the impression that those secondary authors did not understand the difficult parts either, since many of them are just paraphrasing the originals.

 

For RGBM there is no hope of understanding it since its inner workings are a trade secret. But the situation of VPM-B is actually better: There is a very concise definition of the model. But it comes in the form of a FORTRAN program by Eric Baker. It is about 75 pages when printed out on paper and it is FORTRAN, so again, the core of the algorithm does not directly present itself to the reader.

 

Somebody had run this FORTRAN program through f2c but of course the resulting C code is not any more readable. But at least, that allowed be to single step through it with a debugger (rather ran running my finger through 75 pages of spaghetti code). Since then, there are a few more implementation, Since then there are a few more implementations including the C version by Bryan Waite which is at least idiomatic C. But still it follows pretty much in a 1:1 fashion the original FORTRAN code including the separation into sub routines and variable names.

 

But last summer, there was the opportunity to have a Slagvi, a Google Summer of Code student work on this and do a VPM-B implementation from scratch. I was his mentor and this here is what I learned. Slagvi’s task was not just to link the existing code to Subsurface but really do a reimplementation so that in the process we could understand what is actually going on (and what part of the “explanation” is just a smoke screen).

 

You can find the result in the subsurface source code, in particular in the two files deco.c and planner.c.

This post here is a start of a series of (to be written) posts that aim at a fresh presentation of how VPM-B works. The next looks at the actual algorithm.

Welcome to my new theoretical scuba diving blog!

 

What is “theoretical diving” you will probably ask. Let me explain. By day, I work as a physicist at Ludwig-Maximilians-Universität München, as a theoretical physicist more specifically. But I also like a lot to go scuba diving. Unfortunately, I do not have as much time anymore to go diving as I used to have (for a variety of reasons). So instead, I like at least to think about diving. And I think about it as a theoretical physicist. In this blog, I would like to share what I came up with thinking about scuba diving.

Besides this blog, another theoretical diving activity of mine is contributing to Subsurface, the open source dive logging software started by Linus Torvalds and Dirk Hohndel. In particular, there I try to contribute to decompression calculations and the dive planner. In the course of working on that program, I also keep learning things (besides how to work in an open source project) that I would like to share here.

 

The plan to start this blog (besides my regular which means mainly physics blog) has been there for almost two years. So let’s see at which rate I manage to write posts. I hope I can also find some other contributors.

So, here we go…