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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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