NDL Repetitive Dives 
in the Reduced Gradient 
Bubble Model


Copyright 1999 By Eric Maiken

This page is a bit rough, it was part of the deco list's discussion of repetitive dives within the No Deco Limits, and in context of the "conventional" RGBM extension of the VPM:

The situation is quite different for VPM/ RGBM decompression calculations, where the hREPETITIVE is essentially equal to 1 (and hence not an issue), and hEXCESS is unlikely to be provoked in normal diving operations. On the other hand, Wienke's hREGENERATION, (essentially the same as Yount and Hofmann's) is not considered here in detail. This factor is important over multi day diving, and can reduce allowed gradients by as much as 50% over a week's time. So, on that week-long wreck/ cave trip, you can't get away on Saturday with the tricks you pulled on Monday....

Here are Wienke's definitions of the factors that reduce the allowable gradients:

hREPETITIVE = 1 - (1 - GMIN/G) Exp[-lm tSURFACE INTERVAL ]   (see Figs. 1 and 3, below).

hEXCESS = (Dn)MAX/(Dn)j   (See Fig. 2, below. Also note how the time-independent form of hEXCESS affects Fig.4).

hCOMPOSITE = hREPETITIVE x hEXCESS  (See Figs. 4-6, below).

For completeness, note that hREGENERATION = Exp[-lREGENERATION tMULTI-DAY ]. lREGENERATION ~ (1500 min)-1, should also be included as a factor in hCOMPOSITE, however, hREGENERATION ~ 1 for a single day's diving.

 

Click on Each Figure to See an Enlargement

Repetitive Gradients.gif (14263 bytes) Fig. 1   Here's a plot of the RGBM's reductions for the ZHL 16 compartments vs. surface interval. This plot is for an initial 40 ft. dive to the VPM No-Deco Limit (NDL) of 140 min, with lm set to (60 min)-1 in hREPETITIVE.

Significant reductions in the allowed gradients are predicted for surface intervals of 120 min or less.
Deeper Gradients.gif (7802 bytes) Fig. 2   The RGBM restricts deeper-than-previous dives by the factor hEXCESS, which is equal to 1 for all repetitive dives shallower than previous dives, but falls quickly for deeper dives.

This plot is for a 40 ft initial dive. A 100 ft.-deep repetitive dive reduces all of the allowed gradients by about 50%.
Composite Gradients.gif (15369 bytes) Fig. 3   This figure illustrates the somewhat trivial mathematical result of hCOMPOSITE obtained by multiplying the reductions of the repetitive dive gradients shown in Fig. 1 by the hEXCESS |100ft factor of Fig. 2.

However, these reductions are not trivial from an operational standpoint. Essentially, if a diver makes a 40 ft-deep/100 ft-deep reverse profile sequence, the ascent from the repetitive dive is drastically limited by the reduced gradients.

 

 

  

Fig. 4   A shadow plot of the surface representing the composite gradient reduction factor hCOMPOSITE for the 18.5-min compartment. Note that the upper and lower orange lines in the hCOMPOSITE vs. time projection correspond to the curves in Figs. 1 and 3, respectively.

A generalization that is clear from the plot is that hCOMPOSITE is increasingly restrictive for short surface intervals and repetitive dives that are deeper than the preceding dives.

But, is hCOMPOSITE properly formed? 

Note that as the surface interval increases, the plot levels out onto a constant at each dive depth in a hCOMPOSITE shell. Does this mean that if you dive 40 ft today, that your 100 ft dive a week later should be restricted? Clearly not. Figs. 5 and 6 illustrate an alternative.

 

Fig. 5 An alternative form for  the 18.5 min hCOMPOSITE Obtained by relaxing the restriction on hEXCESS in the time domain by an exponential factor.

The proposed time and depth dependence for each hEXCESS is:

where d1 is the first dive depth, d is the repetitive depth, the set of exponential half-times l are ln(2)/tZHL16. r0, gc, and g are the usual VPM parameters.
Fig. 6   The alternative 146-min hCOMPOSITE relaxes more slowly than the 18.5 min ZHL compartment shown in Fig. 5.

 

Fig. 7 A comparison of the time dependence of the conventional (black-colored family of curves previously displayed as Fig.3) and alternative forms (rainbow family) for hEXCESS. In this case, a longer set of half-times than Figs. 5 and 6 l = ln(2)/(3 tZHL16 ) was used for the set. The Linear-Log scale accentuates the short time scale.

Note that after one-day, most of the compartments that control ordinary diving (say, half-times less than the 146 min) have an alternative-form hEXCESS ~ 1. Also, the alternative form for the fastest compartments is plausible: on short time scales, they are most restrictive; for long time-scales they are least restrictive.