VPM-B Gradients as Gradient factors

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

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