Have I understood the basics of decompression theory, GF99 and SurfGF?

[lowwall]

I disagree. As you know we all have different risk factors determined by age, physical shape, hydration, previous decompression stress to name a few. Baker's gradient factors allowed divers to adjust their risk factors by raising or lowering the maximum allowed pressure in the tissue compartments throughout the dive. The fact that it mimics VPM in creating lower first stops is coincidental.

It wasn't coincidental at all. The entire reason for developing GF Low was to allow software based on Haldanean dissolved gas models like ZHL-16 to produce deeper stops than you would get by simply adjusting the maximum allowable M value (aka GF High) which is what was normally done at the time to adjust for conservatism.

And the reason Baker wanted to do that was that he had accepted the tenets of VPM and was actually working on refining it into what would eventually become VPM-B.

I don't know if he ever came out and said this directly, but you can clearly see in his writing the assumption that incorporating deeper stops than would be required by using the same M-value throughout the ascent makes for a less risky profile.

I'm attaching a copy of Baker's paper Clearing Up The Confusion About "Deep Stops".

 

[lowwall]

It's not been explicitly studied. However, in the Navy's current probabilistic algorithms, the slope of the M-value line is approximately 1. This is why Doolette sets his GFLow at 0.83*GFHigh, as that results in a slope of around 1 for ZHL-16C. Basically, Buhlmann felt the fast tissues could tolerate greater tension. VPM wanted them to have less tension. The Navy is in the middle.

According to Short comment on Doolette’s “GRADIENT FACTORS IN A POST-DEEP STOPS WORLD” – The Theoretical Diver , .83x doesn't actually do a very good job of matching an M-value slope of 1.

Now I have to go find what the Navy is up to

 

[David Carron]

First, there is no time dependence on this ceiling calculation as described. If P is a function of time (via Schreiner), then so is the stop depth from this equation. You would need to find where this D(t) equals the ascent D(t). While you could do this in closed form for nitrox, the equation would necessarily include the starting depth, which is not present in the above.

Second, a and b are not constants when Trimix is involved. You could assume they are for the ascent, but it's much easier to ignore the ascent for the ceiling/stop calculation, as that's a conservative choice.

It's not a huge deal showing a stop that's deeper than necessary (because it used the tension at the start of the ascent). The off-gassing during ascent is certainly included during the actual ascent, and the ceiling & stop is recomputed accordingly during the ascent. In other words, the stop will clear on the way up.

My primary point is that some computers today have the horsepower to find the intersection of those two time-dependent depth curves (the ceiling depth and the diver depth). They skip showing a stop that will clear when ascending at the assumed rate. This avoids the discontinuity mentioned when NDL hits 0.

You are absolutely correct that on and off-gassing during ascent must be taken into account to calculate both the NDL and stop durations, and also that a closed-form solution only exists for the special case of a single inert gas.

The symptom of not taking ascent into account for the NDL calculation is that either NDL will drop to zero but no stops will be calculated for another minute or so or that several minutes of stops can be reported immediately after NDL drops to zero.

In principle at least, even older computers like the original OSTC have the computing power to use an iterative solution, it's just a question of how much time you want to spend optimising your code: you can still see traces of some of these optimisations in subsurface's code which uses cached versions of common expressions evaluated at common time intervals.

 

[David Carron]

calculate the stop at the current GF using the formula:
D = ((P - GF * a) / (GF / b - GF + 1)) - Psb

'where: D = depth, P = total gas pressure, GF = gradient factor,
'a = coefficient a, b = coefficient b, Psb = ambient surface pressure
'the first part within parentheses calculates a pressure. Subtracting
'the surface pressure converts the result to a gauge depth.

It seems to me that there may slight scope for concern here since this expression calculates a pressure, not a depth.

 

[inquis]

It seems to me that there may slight scope for concern here since this expression calculates a pressure, not a depth.

It's actually a relative pressure due to the final subtraction of the ambient surface pressure. This relative pressure -- in units of feet of seawater -- is numerically equal to the depth (also units of feet of seawater).

 

[David Carron]

It's actually a relative pressure due to the final subtraction of the ambient surface pressure. This relative pressure -- in units of feet of seawater -- is numerically equal to the depth (also units of feet of seawater).

Well spotted, although measuring depth in units of pressure needs to be (and probably was, now that I come to think of it) stated up-front.

 

[dmaziuk]

According to Short comment on Doolette’s “GRADIENT FACTORS IN A POST-DEEP STOPS WORLD” – The Theoretical Diver , .83x doesn't actually do a very good job of matching an M-value slope of 1.

Now I have to go find what the Navy is up to

I assume he's averaging over all 17 compartments, as they all have their own M0 and dM since Workmann:

(Edit) which is another can of worms because on most recreational dives you could ignore the slow compartments, and that will change the slope of your "average" quite a bit.

 

[lowwall]

Well spotted, although measuring depth in units of pressure needs to be (and probably was, now that I come to think of it) stated up-front.

Everything related to depth in scuba is measured in units of pressure. Even what looks like a depth on your display is in feet/metres of water which is actually a pressure unit, just like the mmHG (millimetres of mercury) used in medicine and meteorology.

 

[David Carron]

Everything related to depth in scuba is measured in units of pressure. Even what looks like a depth on your display is in feet/metres of water which is actually a pressure unit, just like the mmHG (millimetres of mercury) used in medicine and meteorology.

All of the dive computer firmware that I've worked on and with define depth in units of depth and pressure in units of pressure and converts between pressures and depths for display (so when you're diving in fresh water 100m displayed is actually 100m and not 97.5m )

Even subsurface stores depths as mm.

 

[lowwall]

All of the dive computer firmware that I've worked on and with define depth in units of depth and pressure in units of pressure and converts between pressures and depths for display.

Even subsurface stores depths as mm.

I think we are agreeing here. Except for the technical distinction that they are actually converting between two pressure units for display, rather than between pressure and depth units. This is because the display is always some variation of feet/metres of salt/fresh/EN 13319* water, which is a pressure unit, rather than feet or metres which is a length measurement.

* extract from European Standard 13319:

4.1 Depth measurement

4.1.1 Gauge factor for the transformation from pressure to depth
The gauge factor shall be such that an increase of pressure of 1 bar would cause an increase in the depth displayed
of 10 m.

NOTE: This rule assumes a water density of 1,0197 kg/l, i.e., in fresh water of 1,00 kg/l the geometric depth is
102 % of the display while in sea water of a density of 1,03 kg/l the geometric depth is 99 % of the display. Since
the physiological relevant figure is the environmental pressure only, the geometric depth is of much inferior relevance
for the diver.