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TON IIa:
on the use of the
TONAWANDA IIa
coefficient-sets with a
shareware desktop decompression tool
Technical Report TR 2024_09
Tonawanda IIa www.SMC-de.com 1
TR 2024_09:
TON IIa –
on the use of the
TONAWANDA IIa
coefficient-sets with a
shareware desktop decompression tool
Abstract
The TONAWANDA IIa method of calculating a decompression obligation, cornerstone in the
DCAP software, utilises the so-called MMF11F6 set of coeffcients. In order to apply this set
to a standard desktop decompression tool, we suggest a simple linear transformation. The
feasibility of this linear transformation is demonstrated along 3 published test dives, two on
air and one with Trimix. Also the DCAP output for one NOAA Trimix 18/50 table is recovered.
Contents:
Abstract ................................................................................................................................................... 1
Movaon ............................................................................................................................................... 2
Public resources ....................................................................................................................................... 2
„Gradient Factors“ ................................................................................................................................... 4
The linear transformaon ........................................................................................................................ 5
The Air and Trimix test dives ................................................................................................................... 7
History of DCAP & Tonawanda .............................................................................................................. 14
References ............................................................................................................................................. 20
Tonawanda IIa www.SMC-de.com 2
Movaon
In the Tonawanda IIa philosophy (TON IIa, the second version) of the famous DCAP
framework, the „Decompression Computation and Analysis Program“ from HRL (Hamilton
Research, Ltd) the M-values are „target depth M-values“; i.e. there is an offset of 3 msw or
10 fsw to the published standard, ubiquitous M-values from Bob Workman, Bühlmann or
others.
In order to compare for eg. with the coefficient sets for N2 and Helium from Bühlmann or to
use the TON IIa set in standard desktop decompression tools, the published MM11F6 matrix
has to be adapted in the pressure units and transformed to the offset.
For details concerning the other, usual, M-values, examples with them and their units, pls. cf.
[1].
For technical information concerning the DIVE framework, pls. cf. [2]; for the accompanying
manual pls. cf. [3].
Public resources
Publicly available is a lot of information, for eg. from the NAUTILUS dive planner, from ca.
2002 – 2008, then with support from HRL:
Tonawanda IIa www.SMC-de.com 3
The NAUTILUS dive planner offered 5, then state-of-the-art, computational methods:
This manual outlines the 5 methods on p. 37 – 41.
From 2006 on or so, further development and bug-fixing was stalled in the DCAP and the
features were freezed to a version 6.xx.
All the now following are in-depth details taken from ref. [4]:
[4] Hamilton, R.W., Muren, A., Röckert, H., Örnhagen, H. (09/1988)
Proposed New Swedish Air Decompression Tables; Paper # 10, EUBS Meeting
The MM11F6 matrix:
This matrix contains 11 compartments (cmpt # 1 → 11), the nitrogen half-times from 5 to 670
min, the BASE value in [msw], which is not an M0 and the SLOPE.
Tonawanda IIa www.SMC-de.com 4
The part of the Helium matrix looks similar:
C Matrix MM111E.DCP 88MAR21 For deep air/heliox tables, combines 1101, 11F6
DB=0 BASE=26.0 20.3 15.0 12.6 11.7 11.2 10.7 10.6 10.5 10.4 10.3
DS=0 SLOPE=1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
With the corresponding Helium half-times matrix:
COMPARTMENTS=11
C CMPT # 1 2 3 4 5 6 7 8 9 10 11
HT.OF.HE=5 10 20 40 60 80 100 130 160 200 240
(rem.: the „C“ as the first character is a comment sign for the required FORTRAN compilers)
„Gradient Factors“
DCAP / TON IIa has intrinsic factors to scale down the M-values according to:
i.e. a multiplikation of the M-values with the M.FACTOR [4]. Nowadays, one could call these
M.FACTORS happily „Gradient Factors“ (GF).
Tonawanda IIa www.SMC-de.com 5
The linear transformaon
All types of M-values are following the usual Haldane-Workman-Schreiner idea of a linear
equation for allowed / tolerated supersaturations of parallel perfused compartments, pls. cf.
[1]:
M = M0 + ΔM * d (0)
where the allowed / tolerated supersaturation M depends of
M0 : the intercept, a surfacing value, i.e. for a diving depth of 0 [units]
ΔM : the slope, i.e. change of M per unit depth
d : diving depth
The units are either psi, fsw or msw.
Other seasoned systems use the ambient pressure Pamb instead of the diving depth in order
to facilitate diving at reduced ambient pressure, i.e. a mountain lake:
Pt,tol = (Pamb / b) + a (1)
where:
Pamb : P0 + Phydrostatic
b : the slope
a : the intercept
of the linear equation, units = [bar].
For examples, details & units pls. cf.: „Introduction to Decompression Calculation“; [1].
With the definitions from DCAP / TON IIa the linear equation from (0) looks like:
M = S * D + B [msw]
the SLOPE S = ΔM [msw / m]
the BASE B = „M0“ [msw]
For a conversion, the described DCAP / TON II offset has to be integrated:
M = S * (D + 3) + B [units in msw] (2)
In order to use the standard coefficient matrix, we identify:
M = Pt,tol
Pt,tol = (Pamb / b) + a = 1 / b * (P0 + g * ρ * D) + a
Re-grouping in constant & variable terms and comparing (1) with (2) yields:
1 / b * (P0 + g * ρ * D) + a = [(P0 / b ) + a] + 1/b * (g * ρ * D) = S * D + 3 * S + B (3)
i.e.:
Tonawanda IIa www.SMC-de.com 6
a = 3 * S + B – 1/b * P0 (3‘)
1 / b = S / (g * ρ) (3‘‘)
Eliminating b in (3‘) with the help of (3‘‘) yields:
a = 3 * S + B – S / (g * ρ) * P0 (4)
We want to get a first, superficial grip on the new a- & b values: thus we just take the diver
ballparks in SI-units of
P0 = ca. 1 Bar = 105 Pa
g = 10 m s-2
ρ = 103 kg m-3
thus, only ca.: 1 mws = 0.1 bar
Thus the factor P0 / (g * ρ) equates to roughly 10, and S / (g * ρ) is approx. unity in SI units
and S in [msw] / [m]. With b [dimensionless] and a [bar] we now produce a standard
coefficients matrix with (3‘‘) and (4):
These M0, the b and the a-factors in [bar] could now directly compared to the ZH-L16 set of
coefficients from Bühlmann. The same procedure yields for the Helium coefficients:
Tonawanda IIa www.SMC-de.com 7
The Air and Trimix test dives
In [4] are two test scenarios with dives on air:
We load our matrices into DIVE and the 1st. test dive 42 m / 30 min yields without any further
ado the following:
This was done with the quick-and-dirty „ballpark“ transformation and all the other parameters
in DIVE Version 3_11 have been the defaults: a perfect match!
As the DCAP M_FACTORS have not been implemented in our framework, you have to do it
in DIVE by yourself. This is demonstrated with the 2nd. test dive on air to 51 m / 30 min:
Tonawanda IIa www.SMC-de.com 8
top part: unmodified „M“-values, i.e.: GF = 1.0,
lower part: GF High = 0.80, GF Low = 1.00: a nearly perfect match!
A finetuning could be done via the:
➔ ascent rate
➔ water temperature & - density
➔ respiratory coefficient
➔ surface air pressure at the beginning of the dive, and, of course with the gradient factors:
➔ GF High & GF Low
Especially with using the appropriate values instead of the diver ballparks for:
➔ g * ρ and the proper values for the
➔ conversion: msw → bar.
Now, the supreme discipline, a Trimix dive with accelerated decompression, data taken from
[5], p. 159:
➔bottom gas Tmx 18 / 52
➔bottom depth 100 psi = ca. 58 m
➔bottom time: 60 min
Tonawanda IIa www.SMC-de.com 10
By using the two new coefficients matrices, either with the hand-made TXT files (F10 for N2
and F5 for He or with the built-in functions of the DIVE 3_11 BETA test version for selection
of the sets) from above, we get immediately, without GF, but with the „AD“ command for
accelerated decompression, without and with air-breaks resp. without and with the oxygen
correction factors (OC) from NMRI (if pO2 > 1.3 atm):
Tonawanda IIa www.SMC-de.com 11
The Tmx dive 58 m / 60 min: voilà, another nearly perfect match!!!
Also here the match could be made really prefect, if the procedures for accelerated
decompression would be aligned: i.e. length & frequency of air breaks and the MOD resp.
the max pO2!
The same remarks concerning fine-tuning are yielding for the now following, final comparison
of the DCAP output from 1998 for the NOAA Trimix 18/50 table. These tables have been
created in 1993 by HRL and finally revised in 1998 for NOAA deep research projects:
Tonawanda IIa www.SMC-de.com 14
History of DCAP & Tonawanda
From a presentation of HRL in 2003:
Fore more 1st. hand background information and a little bit of history of the Tonawanda II
model (the place near NY. where the LINDE / OSI office was): all of the now following
material is taken from a Keynotes Session from Dave at the Annual Scientific Meeting (ASM)
of the UHMS in 2013, the ref. [5]:
[5] Kenyon, D. (2013) UHMS ASM 2013, Keynotes
Tonawanda IIa www.SMC-de.com 20
References
[1] Rosenblat, Miri & Salm, Albi. (2024) Introduction to Decompression Calculation.
available @:
https://www.researchgate.net/publication/378653647_Introduction_to_Decompression_Calcu
lation/
DIVE framework:
[2] Rosenblat, Miri & Vered, Nurit. (2021) Synopsis & Fact Sheet: update per 11/2021 for
DIVE Version 3_11. 10.13140/RG.2.2.17024.56326.
available @:
https://dx.doi.org/10.13140/RG.2.2.17024.56326
[3] DIVE Version 3_11 software manual:
https://www.divetable.info/DIVE_V3/V3e/DOXV3_0.pdf
[4] Hamilton, R.W., Muren, A., Röckert, H., Örnhagen, H. (09/1988)
Proposed New Swedish Air Decompression Tables; Paper # 10, EUBS Meeting
[5] Kenyon, D. (2013) UHMS ASM 2013, Keynotes
