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43
doi:10.3723/ut.31.043 International Journal of the Society for Underwater Technology, Vol 31, No 1, pp 43–47, 2012
Technical Briefing
Decompression calculations for trimix dives with
PC software: variations in the time-to-surface:
where do they come from?
Albrecht Salm
SubMarine Consulting Group, Obertorstrasse 19, DE-73728 Esslingen, Germany
Abstract
Dive computers for mixed gas diving and PC software for
decompression calculations are often considered as ‘black
boxes’ to the diver: they perform par t of their function – the
calculation of a decompression schedule – but leave the user
in a somewhat nebulous state about the relative safety of this
schedule. This is because, in reality, the technology, under-
lying algorithms and utilised constants are not clearly docu-
mented, especially if the so-called gradient factors come into
play. Gradient factors are sometimes praised as safety knobs
for the decompression schedules, or as a unique selling
proposition for these black boxes. This paper discusses the
impact of gradient factors on the calculation of decompres-
sion times, as well as how the different implementations of
dive profile data can influence these calculations.
With one inert gas in the breathing mixture, the analytical
expression for the decompression time is td. However, if there
is more than one inert gas present, the decompression time
must be calculated numerically. Therefore 480 square dive-
profiles were analysed in the technical/recreational diving
range using one freeware, two commercially available soft-
ware packages and one private software with numerical
methods. There are significant differences in the calculation
of the decompression times with trimix gases, depending on
the helium percentage. In the present analysis, these differ-
ences do not come from variations in the decompression
algorithms but rather from different implementations of these
numerical methods. Presently, a definitive answer cannot be
given about the origin of these variations but the user should
be aware that these exist.
Keywords: decompression, diving theory, mixed gas, mod-
els, simulation, technical diving, trimix
1. Introduction
Time to surface (TTS) is normally the sum of the
stop times over all decompression stops, plus the
ascent time. The algorithm accounting for inert gas
loading during an exposure to overpressure is
implemented using software for a dive computer or
desktop-based decompression software. A gradient
factor is normally used to manipulate the tolerated
inert gas partial pressures in the various theoretical
body tissues. Therefore, a decompression method
with prolonged stops can be forced using pure
mathematics but is not directly related to any physi-
ological issues. Perfusion decompression models
exist where a theoretical blood perfusion element
defines the boundary conditions. These deal mainly
with the dissolved gas phase: inert gas bubbles are
not considered within these models but are described
in other literature (see Boycott et al., 1908; Workman,
1965; Müller and Ruf, 1966, 1971; Schreiner and
Kelley, 1971; Bühlmann, 1983, 1993; Hahn, 1995;
Bühlmann et al., 2002). Other terms used for
this paper are REC for recreational diving (i.e.
SCUBA-diving with air and normally within no-
decompression limits), and TEC for technical div-
ing with a lot of equipment and usually using mixed
gases. The mixed gas employed usually contains
helium (in a trimix: oxygen, nitrogen, helium) and
decompression stops where oxygen enriched air
(EAN, Nitrox) and/or pure oxygen can be used.
Classical, perfusion-limited decompression algo-
rithms were first described by Boycott et al. (1908)
but tend now to be termed Haldane models after
one of the co-authors, JS Haldane. The Haldane
models describe the absorption of one inert gas per
compartment through a mono-exponential func-
tion; the classic Haldane equation is:
Pt(t) = Palv0 + [Pt0 - Palv0] e-kt (1)
where Pt(t) is the artial pressure of the gas in the
tissue, Pt0 is the initial partial pressure of the gas in
the tissue at t = 0, Palv0 is the constant partial pres-
sure of the gas in the breathing mix in the alveoli,
k is a constant depending on the type of tissue, and
t is time.
One mainstream source for these perfusion
algorithms is in Hills (1977), which gives Equation 1
and discusses the relationship between the tissue
half-times and the perfusion rate. The decompres-
sion time (td) for unary mixes (i.e. only one inert
E-mail address: director@divetable.de
Salm. Decompression calculations for trimix dives with PC software: Variations in the time-to-surface: where do they come from?
44
gas, e.g. enriched air, nitrox, EAN or heliox) can
be calculated directly with the Haldane equation.
The analytic expression for the decompression
time (t = td) is:
t = -τ/ln2* ln[(Pt(t) - Palv0)/(Pt0 - Palv0)] (2)
This is the analytic solution for Equation 1 and is
only possible with one inert gas – in this case N2.
With more than one inert gas, the calculation of
td has to be done numerically, via an approximation
procedure, that is by trial-and-error.
Perfusion models for air, nitrox, EAN and
heliox as breathing gases are based on extensive
records of well-documented dives, whereas those
for trimix diving are not. For one inert gas perfusion
models are mathematically straightforward and
have enjoyed popular implementations in many dive
computers and PC programs (Bühlmann, 1983,
1993; Bühlmann et al., 2002). Technical divers want
to dive deeper and longer, and many of their dives
are outside the trusted envelope. Nonetheless,
studies on this envelope have been already pub-
lished at length (e.g. Brubakk and Neuman, 2003)
and, in summary, consider:
• only inert gas loading;
• mono-exponential relationships for one
compartment – such compartments are all in a
parallel circuit, while the linear connections
(e.g. spleen to liver, or bowel to liver) are not
considered; and
• mono-calculation of inert gas saturation and
de-saturation in a symmetrical manner, i.e. with
the identical coefficient in the exponential terms
of the Haldane equation (equation 1).
However, some of the potential drawbacks when
modifying these models for use for decompression
modelling of trimix diving are:
• that user-dependent physiology and adaption are
not reflected at all in the algorithms;
• inconsistent consideration of metabolic gases
such as oxygen, carbon dioxide and water;
• the influence of ‘uneventful’ decompression
exists where only the gas in solution may be con-
sidered and not the free gas phase (bubbles);
• that no allowance is made for short-term pres-
sure changes and their relative influence against
the fastest half-times;
• the effects of workload, temperature and excessive
oxygen partial pressures; and
• consideration of the second inert gas and repeti-
tive dives.
Another critical point is that the mapping of the
compartment half-times from nitrogen to helium is
normally done according to Graham’s law using
the square root of the proportion of the molecular
weights (i.e. ca. 2.65); this factor is uniform to all
compartments. This has been met with criticism
from serious researchers in the field (Lightfoot
et al., 1978; D’Aoust et al., 1979; Rodchenkov and
Skudin, 1992). Especially in newer experiments,
the perfusion rates are viewed quite differently
(Doolette et al., 2005). The pivotal 2.65 seems to be
valid only for saturation exposures (Berghage et al.,
1979) which are not pertinent to technical diving.
With a so-called trimix there are two inert gases:
N2 (nitrogen) and He (helium) along with oxygen.
This generates two exponential functions with dif-
ferent exponents for the same compartment, one for
N2 and one for He. The inert gas saturation (or the
de-saturation) for these two has to be calculated sep-
arately, but the criteria for safe ascent are the same
regarding length of time. This is where problems
arise with the numerical calculation but for commer-
cial applications in oilfield settings, the numerical
approximation of a TTS is standard procedure. The
present study presents a methodology for examin-
ing the performance of decompression models
employed in the management of trimix diving.
2. Methods
There are at least three simple methods to evaluate
decompression times (td):
1. Trial and error method: for small increments in
time, e.g. 1sec or 0.1min, all relevant terms are
calculated and checked to see if the ascent crite-
ria are met. This is called a classical numerical
solution.
2. Quasi-analytical method: an error is tacitly
accepted by using Equation 2. Thus the two dif-
ferent tolerated overpressures are considered as
independent constants for each phase of the
decompression.
3. Approximation method: all the exponential terms
are approximated via a polynomial expression,
i.e. Taylor Expansion (Bronstein and Semend-
jajew, 1979).
For commercially available off-the-shelf (COTS)
desktop decompression software, method 1 should
be used because the computing power of topical PC
hardware does not impose any waiting time for the
users, unlike standard mix gas diving computers.
The relatively high costs incurred during the devel-
opment for waterproof hardware combined with
low sale volumes means that the industry tends to
use standard chips rather than full-custom micro-
chips (ASIC) in diving computers. However, in
comparison, standard chips are somewhat slower
and have high energy consumption.
45
Vol 31, No 1, 2012
Method 1, in comparison to method 2, consumes
more computing power, time and memory, and
includes more variables. All of these factors can gen-
erate limitations in equipment that is being designed
for use under water and so there is a tendency to
employ method 2 where costs are at premium and
the results from the calculations are needed rapidly.
Unfortunately, the actual methods used in com-
mercial products are rarely known because the
manufacturers of dive computer hardware tend to
cite commercial confidentiality in reply to any
enquiries.
To assist in answering this question for the tech-
nical diver, the following experimental method
was developed: 480 square-wave dive profiles were
generated to be representative of those regularly
observed in the TEC/REC domains, with depth
ranging between 30–80m (6 profiles at 10m incre-
ments) and with a range of bottom times (20–60min;
5 profiles in 10min increments). The profiles used
helium fractions of 5–80% (16 profiles in 5% incre-
ments), with only one normoxic mix (i.e. no travel
gases and no EAN decompression mixes). The pro-
files were evaluated with four software products and
compared to:
• two commercially available COTS decompres-
sion software products that have a very broad
user basis in the TEC community;
• one freeware/shareware version of DIVE (www.
divetable.info/dwnld_e.htm, version 2_900); and
• the commercial version 3_0 of DIVE.
All of these four products claim to have imple-
mented the Bühlmann method for calculating
decompression (Bühlmann, 1983, 1993; Bühlmann
et al., 2002) called ZHL-n (where ‘ZH’ represents
Bühlmann’s hometown of Zurich; ‘L’ is the linear
equations of the criteria for safe ascent; and n is the
number of compartments/half-times). In addition
to the standard ZHL method, it was possible to set
the above-mentioned gradient factors. During the
analyses gradient factors were set to 1.0 for all of
the products.
The version 3_0 of DIVE implemented method 1
exactly, while the freeware version 2_900 was flawed
with a problematic implementation of method 2.
For the two COTS products, the available technical
documentation was incomplete and no statements
were available from the programmers to detail what
methods were being used.
The first step, tested these four products against
each other with 40 different air and Nitrox/EAN32
profiles. The test checked the actual convergence
of the numerical method 1 with the COTS prod-
ucts. Table 1 shows one paradigm with the TTS val-
ues for a square dive to 40m, with the bottom times
ranging from 20min to 60min.
A sensitivity analysis was performed for the
numerical solution in order to ensure that minor
variations in the starting parameters did not lead to
mathematical artefacts. The four products were
compared against the ‘gold standard’, which is the
Zuerich 1986 (ZH-86) table for air dives (Bühl-
mann et al., 2002). This presented deviations of
±2min per decompression stage; sometimes the
staging began 3m deeper in comparison to the
table. This is mainly because of the different sets of
coefficients used: the ZH-86 table uses the ZHL-16
B set, whereas decompression software or dive com-
puters normally use the ZHL-16 C set (Bühlmann
et al., 2002). In addition, the printed tables treat
truncations in a completely different way to dive
computers. There are many US Navy trials that con-
firm that decompression information generated
from published diving tables rarely matches com-
puter-generated output (Joiner et al., 2007).
To force comparability, all the calculations in the
present study were based solely on the set ZHL-16
C and there was no manipulation via gradient fac-
tors (GF) – that is, GF high/GF low = 1.00 or 100%
of the original published a- and b-coefficients.
There were also slight adaptations of the dive pro-
files via ascent and descent rates, to make sure that
the bottom times and the inert gas doses matched.
3. Results and discussion
Evidently there are significant differences in the
calculation of decompression times depending on
the helium-fraction and the amount of decompres-
sion obligation as determined by the inert gas dose
(see Fig 1). These differences are not caused by
variations in the decompression algorithm, but
instead through different methods of calculation.
Table 1: TTS for EAN32 dive versus the four products (TTS, i.e. sum of all
decompression stop times + time for ascent)
40m, Nitrox/EAN 32 bottom times [min]: 20′30′40′50′60′
TTS DIVE 2_900 8 16 28 42 55
TTS DIVE 3_0: numerical solution 7 17 28 40 57
TTS COTS product 3 5 15 28 41 53
TTS COTS product 4 7 16 28 41 54
Salm. Decompression calculations for trimix dives with PC software: Variations in the time-to-surface: where do they come from?
46
Fig 1 shows the deviation of the TTS based on the
percentage of helium in the breathing mix, using
the example of a dive to 40m with a bottom time of
40min.
The x axis in Fig 1 is the percentage of helium in
the breathing mix from 10% to 80%, while the
y axis is the Delta TTS. This is a difference of the
numerical solution to an arithmetic mean out from
the three TTS according to: ∑ (td,1 + td,2 + td,3)/3,
where td,i is the calculated td of the products i =
1 - 3 (DIVE 2_900, COTS product 3, COTS
product 4).
The x axis is defined as the zero baseline of the
TTS of the numerical solution. An ‘error’ in min-
utes is the deviation (Delta TTS) of this mean
value against the TTS of the numerical solution.
The calculation of this arithmetic mean was super-
imposed by the strong closeness of the td from the
three products. The absolute errors (see the verti-
cal error margins) rise with the increase of the inert
gas dose and with the increase of the percentage of
He in the mix. The curve progression is more or
less universal for all of the 480 square profiles. In
relatively simplified and qualitative terms, the fol-
lowing can be determined:
• In the region of the helium fractions 0.05 up to
ca. 0.25, the TTS is overrated with positive error
(i.e. the TTS is too great, and the decompression
is too conservative).
• In the region of helium fractions which is rele-
vant to most technical divers, that is ca. 0.30 – ca.
0.40, the error vanishes – Delta TTS = 0.
• In the region of increasing helium fraction, the
error is negative (i.e. the TTS is too small, and
the decompression is too liberal).
The results of the two COTS products and DIVE
2_900 were very close to each other and so a similar
calculation method is assumed. However, this ‘simi-
lar’ method means that the error of DIVE 2_900
could be repeated in the implementations of the
two COTS products. In other words, the relative
identity of the absolute values and the prefix leave
room for speculation that the two COTS products
are using method 2, although there are also some
other factors that could be responsible for these
deviations. The following are a few possible factors,
although this list is not exhaustive:
• undocumented gradient factors;
• a respiratory coefficient unequal to 1;
• another weighting of other inert gases;
• another weighting of the water density;
• empirically adapted a/b coefficients, especially
for and as a consequence of the helium fraction;
• small deviations from the original helium ZHL
spectrum of half-times (i.e. a mismatch of a and
b coefficients with the half-time);
• utilisation of the so-called ‘1b’ compartment,
instead or additive to compartment ‘1’;
• ascent rates varying with depth;
• de-saturation varying with depth and ascent
rate; and
• different approach to truncations.
Restrictions in software operations caused by
hardware limitations were quite common in the
early days of dive computers. For example, there was
a product in Europe which could only interpolate
linearly between stored table values instead of cal-
culating full-scale saturation/desaturation relation-
ships. Even today, there are applications which rely
on a modified ZHL instead of the promised and
advertised bubble model.
4. Conclusions
There is a raft of constraints to be considered when
attempting to expand the largely theoretical
approach detailed in the present study into a wider
determination of how models are being imple-
mented in some dive computers. It is difficult to
develop any solid conclusions and there may be
additional legal considerations. This limits the abil-
ity to achieve some transparency in how some of the
models are being implemented. The lack of open
documentation of the ‘defaults’ and constants
leads to numerous questions: for example, is there
really a ZHL inside a computer when the label
reads ‘ZHL’?
The clear message resulting from these tests is
the following: a decompression time in a digital dis-
play, be it on a dive computer or a PC, is subject to
interpretation. This is not so much because of errors
in the measurements (e.g. pressure, time, tem-
perature) and other statistical contemplations, but
rather caused by the method of programming and
the choice of a solution for a mathematical algo-
rithm (i.e. the software technology and implemen-
tation). The range for these interpretations is not
only in volumetric terms, but also is dependent on
Fig 1: Delta TTS versus percentage of He in the breathing
mix dive to 40m with a bottom time of 40min
47
Vol 31, No 1, 2012
the inert gas dose and the helium fraction, in the
one- or two-digit percent range.
Therefore, the answer to the question in the title
(where do variations in the time-to-surface come
from?) is not straightforward. First, the wisdom of
using perfusion algorithms could be questioned, but
perfusion models work much better than the bubble
models (see above); to quote Hamilton (1978): ‘Hal-
dane works if you use it properly’. Second, with the
aforementioned gradient factors, the users could fix
the Delta TTS variations shown in Fig 1. However,
the question remains: do gradient factors then pro-
vide a safer decompression schedule or are they bet-
ter employed for user-based software manipulation,
as illustrated in the example of method 2?
This will need to be the subject of future
research, as new technology and products are
being introduced constantly.
Acknowledgments
This is an abbreviated and modified version of a
paper that appeared in CAISSON 2012 26: 4–12
(available at gtuem.praesentiert-ihnen.de/caisson_
03-11.pdf accessed on 21 October 2012). Several
parts of this paper were presented during a lecture
at the 12th scientific meeting of the GTUEM on
20 March 2011 in Regensburg, Germany.
Thanks go to the entire crew of GTUEM for
making it possible to give a lecture on this topic.
Especially to Prof Willi Welslau (president of
GTUEM, Vienna, for a constant peer review) and to
Prof Jochen D Schipke (University Medical Center
for experimental surgery, Duesseldorf) for editorial
work and for patience with my oft unorthodox
approach. Thanks go as well to a couple of my tech-
nical diving students in Elat/Israel for fiddling
about the deco software.
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