Cream of the Crop 1/Cream of the Crop 1.iso/EDUCATE/DIVECOMP.ARJ/DIVECOMP.TXT

Internet Message Format | 1991-01-14 | 52KB

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Date: Tue, 8 Jan 91 13:03:46 est
From: Dan Schwartz <[e-mail]>
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Subject: Re: divecomp for DOS now available

The following shell archive contains an MSDOS EXE file which is a
port of Dave Waller's divecomp program (version 2.5).  It was compiled
with Microsoft Quick C 2.0, and uses the PC Curses library.

The program simulates the operation of a dive computer, and shows
nitrogen loading in all of the compartments during the course of
simulated dives.

Anyone interested in the source code should consult rec.scuba, or the
scubasearch archive, for Dave's original posting.

Dan Schwartz      [e-mail]
Hewlett-Packard, New Jersey Division

-------------------------------------------------------------------------------------------
 * Revision 2.5  90/12/12  15:30:02  15:30:02  dave (Dave Waller)

Changes as of revision 2.5:

New features (lots!):

Now checks for "LINES" or "WIDTH" environment variable set, truncating the
compartment bar graph appropriately (for those of you who simulate DEEP
dives :-))

New option, -P, displays compartments as absolute pressure instead of
tissue loading %age. The pressure is in <units> sea water absolute, where
<units> is whatever units have been specified with the -u option. Default
is feet. -P also causes the compartment values logged to an optional log
file (specified with -l) to be in absolute pressure instead of loading
%age.

Graph can be toggled between loading display and absolute pressure
display by pressing the 't' key WHILE THE PROGRAM IS CALCULATING, not
while it is wating for input. For example, if you are running interactive
and you enter a depth of 100 ft for 20 minutes, 't' can be used while
the 20 minutes are elapsing to toggle the screen. 't' does not affect

initial -P option.

Changed the transition depth between bottom time and surface interval
counting to 3 feet from 5 feet. This was necessary for decompression
stuff (more later).

'q' can be pressed during calculations to abort the calculated run.
For instance, when calculating a long decompression in autodecompression
mode (more on this later), you can press 'q' to return to the input
prompt. Similarly when calculating a long depth/duration combination
entered from the keyboard.

No Decompression Limits are now displayed and continuously updated along
side the dive profile for each depth displayed.

When NDLs are exceeded, the stats line now displays a ceiling, and a
time to surface assuming proper following of the ceiling in 5 foot stops.
For example, after 20 minutes at 200 feet, the stats line might display
"Ceiling: 75.0 ft Time to surface: 46 min", indicating that the diver
should not go shallower than 75 feet. If the diver makes decompression
stops every 5 feet (stopping at each stop until the ceiling goes up to
the next 5' stop), it will take 46 minutes to make it safely to the surface.

Depths deeper than the deepest depth in the dive profile display now
show a 'v' in the deepest depth rather than not being displayed at all.

Bar graph now automatically reverts to a 2:1 scale in both loading and
absolute display modes. For instance, if any compartment exceeds 100%
loading, the graph will rescale to 0-200% (similarly for absolute
pressure display).

Autodecompression: When NDLs have been exceeded, the diver can
autodecompress by entering 'd' at the depth prompt. This will cause the
program to advance to the nearest 5' deeper than the indocated ceiling,
and follow the ceiling up in 5' increments. This will produce a profile
exactly equal to that used in calculating the 'time to surface' value.

If available on your keyboard, the up and down arrow keys can now be
used to change depth by one depth increment (as shown in the profile
display) with each press during calculation. This is mainly useful while
plotting a portion of a profile (i.e. 60' for 50 minutes), and you want
to create a more realistic profile by moving up or down while
calculating.
 *
 * Revision 2.4  90/11/27  15:30:03  15:30:03  dave (Dave Waller)
 * new features:
 *
 * - tissue loading graph now compresses when tissue loading exceeds
 *   100%, to a scale of 0-200%. Rescales to 0-100% when all compartments
 *   are <= 100%.
 *
 * - Compartments can now be displayed as absolute pressure in (units) sea
 *   water, instead of % loading. Default is loading graph, absolute
 *   pressure can be selected at runtime with -P option. While the program
 *   is running, user can toggle between the two modes by typing 't' (this
 *   only works while calculations are taking place, not while waiting for
 *   input). Scale along the top is calibrated in (units) sea water absolute.
 *
 * - During calculation, depth can be adjusted up or down using the up and
 *   down arrow keys on the keyboard. For example, if the program is calculating
 *   a dive level of 60 feet for 50 minutes, the depth can be adjusted up or
 *   down by pressing the arrow keys. Adjustments are made in increments
 *   equal to the depth increment in the dive profile display.
 *
 * - If compartments are overloaded, the program can "autodecompress" by
 *   entering a depth of 'd' in interactive mode. The program will then
 *   move up to the ceiling value and "ride" it until all the compartments
 *   are <=100% loading.
 *
 * - When displaying compartment loading, pressure values are in (units) SWA
 *   instead of FSWA. (Of course, if the units are feet, then the values
 *   *are* FSWA).
 *
 * - Stats line during overloading now displays a more meaningful message
 *   regarding ceiling and decompression time.
 *
 * Revision 2.1  90/11/21  11:33:57  11:33:57  dave (Dave Waller)
 * Changed SUN ifdefs to be macro SIMPLE instead... Makefile has been
 * modified as well. Since Sun users can now compile with full SVID
 * curses functionality, it seemed unnecessary to remove the reverse
 * video bells and whistles for for them. However, this functionality
 * is retained as a "simple" version of the program for those that do not
 * have reverse video and underline capabilities on their terminals.
 *
 * Revision 2.0  90/11/21  11:11:30  11:11:30  dave (Dave Waller)
 * Bug fixes:
 *
 * - incompatible typecasting in several places made the program fail
 *   with certain model files.
 *
 * - Logical error in the sample period determination algorithm.
 *
 * Enhancements:
 *
 * - added capability to display depth in arbitrary units. Default is
 *   feet. Units are specified in either a profile file on the first line
 *   or with the -u option. To use a different unit system, the user must
 *   supply the unit name and a unit conversion factor that represents
 *   units/ft. For example, to display in fathoms, the user would type
 *
 *   divecomp -mEdge -ufathoms:0.1667
 *
 *   There are 6 feet in a fathom, or 1/6=0.1667 fathoms per foot. For meters,
 *   the specification would be -umeters:0.3048, or -um:0.3, etc.
 *
 *   The unit specification is written into an output profile, so that the
 *   correct units are used when using the profile. Units specified in a
 *   profile file override command line specification.
 *
 * - added logging capability. A log file can be specified with the -l
 *   option, and the program will write the allowed nitrogen %age for
 *   each compartment at each sample period into the log file. A header
 *   containing information regarding the model, sample period, and depth
 *   unit used is initially written into the file before the simulation
 *   begins. To make a log of a computer run, type
 *
 *   divecomp -mBuhlman -umeters:0.3 -llog.buhlman
 *
 *
 * Revision 1.9  90/11/20  11:24:48  11:24:48  dave (Dave Waller)
 * Bug fixes:
 *      - logical errors in reading sample period from specified sources
 *      - update_dive_profile() routine hada rounding error in the
 *        code that compresses the display.
 *
 * Enhancements:
 *
 *      - new profile file format. Profiles are now stored as depth-duration
 *        pairs, instead of discrete samples. This eliminates the need for
 *        sample period specification in the profile file, and makes the
 *        file more compact and easier to read. I have written a filter
 *        'cvtprof' to convert from the old format to the new format, so
 *        any existing profiles that people have can be easily converted.
 *
 *        Profiles created with the -o option in interactive mode are written
 *        in the new format.
 *
 * Revision 1.8  90/11/16  16:33:45  16:33:45  dave (Dave Waller)
 * Cleaned up rounding algorithm for depth profile; added round up to
 * compartment bar graph display, so that it reads 100% properly;
 * Fixed scale graphic at top of compartment bar graph (it was off
 * by one character position to the right, contributing tothe bars not
 * being at 100% visually when they actually were).
 *
 * Revision 1.7  90/11/16  10:34:26  10:34:26  dave (Dave Waller)
 * Added ability to specify different sample periods. Sample period ('s' in
 * the source) is specified in the following manner (decreasing precedence):
 *
 *            1) via command line option
 *               Example: divecomp -mEdge -omyprof -s2.5
 *
 *               Units are in minutes
 *
 *            2) From within a profile file. The first line can optionally
 *               specifiy a sample rate, if it has the form "s:<rate>".
 *               Example: A profile that contains 2 minute samples would
 *               look like
 *
 *               s:2.0
 *               60
 *               60
 *               60
 *               60
 *               .
 *               .
 *               .
 *
 *               (the ellipses [...] are not part of the file). 'divecomp'
 *               now automatically writes the sample period into every profile
 *               it generates via the -o option.
 *
 *             3) From a model file. Format is the same as a profile file.
 *                The most basic command, "divecomp -mEdge", for instance,
 *                will cause the computer to use the proper sampling rate
 *                for the Edge. Also, any profiles generates with this model
 *                will have the correct sampling value when used in different
 *                models.
 *
 *             4) Default: 1 minute.
 *
 * Revision 1.6  90/11/14  17:36:52  17:36:52  dave (Dave Waller)
 * Added compile time checks to modify curses behavior to accomodate
 * incomplete curses library on Sun/OS. Controlling tissue is highlighted
 * with a preceding '*' instead of reverse video, deiling message does not
 * appear in reverse video when tissue M values are exceeded, and tissue
 * bar graph is composed of '#' characters instead of reverse video
 * spaces. Not quite as pretty as the full curses version, but that's what
 * you get if you're running on a Sun.
 *
 * Original graphical functionality is maintained for other platforms.
 *
 * Revision 1.5  90/11/14  13:58:43  13:58:43  dave (Dave Waller)
 * Added the following features:
 *
 * - Decompression calculation and indication via ndl()
 * - ingas/outgas indicator nxt to each compartment number
 * - full "bottom timer" functions (i.e. dive #, bottom time, surface int)
 *
 * Fixed the following bugs:
 *
 * - would run forever if duration of 0 entered in interactive mode.
 *   Program now rejects such an entry, and asks for depth and duration
 *   again.
 *
 * - Controlling tissue indication lagged one sample behind actual value;
 *   Fixed.
 *
 * - Dive profile graph now rounds up to the nearest depth increment.
 *
 * Revision 1.4  90/11/13  16:38:49  16:38:49  dave (Dave Waller)
 * Fixed up the comments.
 *
 * Revision 1.3  90/11/13  15:11:48  15:11:48  dave (Dave Waller)
 * Initial checkin; started using RCS to keep track of revisions. This
 * revision also fixes a bug with the code that reads in half times from model
 * files -- they were stored as type 'int', which created problems for
 * fscanf if the half times were floats. Changed the array 'half[]' to
 * type 'float'.
 *  */

/* dive computer simulator, by Dave Waller ([e-mail])

=============================================================Acknowlegments:

Much thanks to Eric Williams ([e-mail]) for his
outstanding contribution to the ndl() routine to incorporate ceilings
and decompression times.  His support and timely reviews/bug reports
have been extremely helpful.  Thanks, Eric!
============================================================

This program simulates a multi-compartment model dive computer, based
upon modified Haldanean tissue absoption models.  The number of
compartments and their corresponding constants are not built into the
program, but must be loaded at runtime, allowing the program to
simulate most dive computers on the market today.

Subsequent derivation of equations as found in the comments below were
done by myself, as I sit here at work, using only my memory;
therefore, this initial pass at the simulation may contain some errors
that I will have to fix later, after I have had time to review the
pertinent material at home.  Surprisingly enough, it seems to work
pretty accurately, based upon my experience with my ORCA Delpi.  Have
fun!

       Tissue halftimes in general express the time it takes for a
       tissue to either absorb or release nitrogen such that the
       tissue is 50% equalized to the pressure differential between
       the ambient nitrogen pressure and the tissue nitrogen
       pressure. This uptake or outgassing obeys an asymptotic
       exponential, namely if ambient pressure is PA, and initial tissue
       pressure is PT0, then the tissue pressure as a function of
       time is expressed as,

       PT = PT0 + (PA - PT0)(1 - e^(-kt))

       where k = -ln(0.5)/T     = 0.6931/T
                    half           half

       This formula holds true for static conditions; i.e. PA does
       not change. In reality, a diver will probably be continuously
       changing depth, and therefore PA is changing also. A
       reasonable approximation of continous PT values can be
       calculated by sampling PA at discrete intervals, and applying
       the above formula over the preceding sample interval, then
       starting over with a new PA and PT0 for the next interval
       based upon the calculation.

       In this simulated computer, we assume datapoints are recorded
       one per minute. Since the halftimes are expressed in minutes,
       the above formula can be reduce to an adjustment at each
       sample that consists of the following:

       PTnew = PTlast + K(PA - PTlast)

       where K = (1 - e^(-k*1)) = 1-e^(-k) = 1 - (1/e^k) =

                   1 - (0.5)^(1/T    )
                         half

       Note that the K value is dependant upon consistency between
       the units of the compartment half time and the computer
       sample period. Specifically, the above formula only applies
       if the sample period is 1 unit of the half time unit (in this
       case minutes). For an arbitrary sample period s, the formula
       for the K value is

       K = 1 - (0.5)^(s/T    )
                 half

       The dive computer simulator calculates the K values on
       startup, with the sample period 's' being taken from the
       following sources, in order of decreasing priority:

       1) a sample period specified on the command line with the
          -s option

       2) A sample period specified in a profile file. This must
          take precedence over a sample period in a model file,
          otherwise the number of samples in the file will not
          match the dive profile in real time. In order to compare
          different models, the same sampling period must be used
          for a single profile.

       3) A sample period specified in the model file

       4) The default sample period of 1 second

       */

----------------------------------------------------------------------------------------------------
From: [e-mail] (Dave_Waller)
Newsgroups: rec.scuba
Subject: Dive computer program is here!
Date: 10 Nov 90 01:41:02 GMT
Organization: HP Pacific Technology Park - Sunnyvale, Ca.

Okay, folks, you asked for it. Here is my first pass at a divecomputer
program that I hacked together during lunch. As far as I can tell, it
seems to be working pretty accurately, based upon my experience with the
Delphi (although this version uses half-times and M-values as posted for
the Edge by Eric Williams).

The first response to this posting is a shell archive that contains 3
files: "divecomp.c", "makefile", and "data".

To build your dive computer, unwrap the shar file as specified in its
instructions, and type "make divecomp". This should wourk on just about
any UNIX system, as the program is really rather simple.

The file "data" contains made-up data for a simgle multi-level dive. All
it is a set of depth values, one minute apart. The program assumes that
depth is sampled every minute.

The program uses the 'curses' library, so you must have a terminal
cabable of supporting cursor addressing and reverse video. Current
features at this early quick-hack release are:

    * A continuously updated bar graph of all 12 tissue saturations
      (relative to M values), with the actual N2 tissue partial
      pressure displayed in FSW at the end of each bar. If a tissue
      exceeds 100% of the allowable nitrogen, the bar changes to
      asterisks.

    * Continuous display of depth.

    * Continuous display of bottom time, and surface interval (after
      the dive).

    * A continuously updated graph of the dive profile.

To see all of this, you need a display with about 50 lines; X-window
terminal emulators work pretty well. If you have only a 24 line
terminal, you'll have to comment out the dive profile display so it
doesn't (potentially) mess up the tissue graph.

Please post any questions/problems as followups to this note. I'll try
to get to them as quickly as I can, and it would be helpful for anyone
else having problems. Have fun!

Tomorrow: A list of planned enhancements!

Dave Waller  \  The opinions expressed are solely my own, and in no way
Hewlett-Packard Co.  \  represent those of my employer (but we all know
[e-mail] | hplabs!hpdstma!dave  \  they should!)




From: [e-mail] (Dave_Waller)
Newsgroups: rec.scuba
Subject: Latest version of divecomputer program
Date: 13 Nov 90 18:06:49 GMT
Organization: HP Pacific Technology Park - Sunnyvale, Ca.

Here is the latest version, which is MUCH improved over the version you
were running. As we all bug fixes, it incorporates many new features:

    -mmodel        specifies the file "model" as the file
            containing half-time Mvalue pairs. Must be
            specified.

    -pprofile    Specifies a file containing a dive profile,
            in one minute samples. If not specified, the
            program is in interactive mode, allowing the
            user to specify depths and durations.

    -ooutfile    Optional output filename to record interactively
            entered dive profile. Useful for saving profiles
            to be used for comparison between different
            computers using different model files.

    Computer now computes and displays NDL, and highlights the
    controlling tissue.

    Adapts to different screen sizes automagically now. Looks for
    environment variable "LINES", and if not found, assumes 24.

So, I have included the new version with this message, along with the
Edge and Navy model files, and a sample profile. Check it out and let me
know what you think.


Dave Waller  \  The opinions expressed are solely my own, and in no way
Hewlett-Packard Co.  \  represent those of my employer (but we all know
[e-mail] | hplabs!hpdstma!dave  \  they should!)


--------------------------------------------------------------------
From: [e-mail] (Eric Williams)
Newsgroups: rec.scuba
Subject: Discussion of Decompression Models (long)
Keywords: decompression, models, history, M values, dive computer
Date: 15 Nov 90 02:02:38 GMT
Organization: AIR, Stanford University

Those of you who asked for this stuff, sorry I left you hanging.
It just took me a while.  This is a discussion about decompression models.
It starts with history, then gets into math, and finally specifics
about things like diving at altitude.  (It's a *really* long message.
You may want to save it and read it in pieces.)


I.  A Little History of Decompression Models.

Our knowledge of DCS (decompression sickness) goes back over three
hundred years to 1670 when Robert Boyle described his observations of
animals in lowered pressure environments.  In the 1800's that
knowledge expanded with the frequent use of caissons and other
pressurized work environments.  The first human cases of DCS were
reported in 1841 in coal miners working in mines pressurized to keep
out water.  In 1857, Hoppe-Seyler suggested that DCS was caused by
bubbles in the tissues and suggested treatment by recompression.
In 1878, Bert discovered that it is nitrogen that is responsible for
DCS.

The British Navy commissioned J. S. Haldane to investigate the causes
of DCS and to propose a solution.  In the early 1900's by working on
goats, he made several observations which still influence our thinking
on DCS today.  His procedure was to bring the goats to a higher than
atmospheric pressure and leave them there for "a while"--I think it
was about 4 hours [EW].  Then he would lower the pressure and look for
DCS.  (Animal rights activists, where were you then?  :-)

He reported that saturated tissues could be decompressed to one half
their pressure without the onset of DCS, e.g. from 2 atm to 1 atm.
Bringing a tissue back from more than 2 atm could be done in stages,
allowing time for the tissue to resaturate to the new pressure at each
stage, e.g. from 4 atm to 2 atm then waiting, and then from 2 atm to
1 atm.

Haldane developed a model of compression and decompression which is
still used today as the basis of many decompression models.  The
Haldanean model can by summariezed by the following prinicples:

+ Absorbtion and elimination of nitrogen by the body is at an
    expontential rate.  That is, the rate at which a tissue
    takes in or releases gases is proportional to the pressure
    difference between the tissue and the surroundings (see
    math section)

+ There is a continuous spectrum of tissue types in the body with
    various rates of absorbtion and elimination.

+ The rate of absorbtion is the same as the rate of elimination for
    any given tissue group.

+ The continuous spectrum of tissue types in the body can be
    approximated by a finite number of groups from within the
    range.

In 1908, the Royal Navy adopted the tables that Haldane developed using
his model.  In 1915 the US Navy published its first set of tables (the
C & R tables).

In the 1930's Hwkins, Shilling and Hansen (and later Yarborough)
determined that the allowable supersaturation is not a constant
two as Haldane had said, but depends on the tissue half time and
the depths and duration of the dive.  In 1937, Yarborough made
tables for the US Navy based on a model that had three tissue
compartments with half times of 20 minutes, 40 minutes and 75
minutes respectively.

In 1957, the US Navy tables were corrected and improved and the
repetitive dive tables were added.  The tables were based on six
compartments with half times of 5, 10, 20, 40, 80, and 120
minutes.  The concept of "M-values" originiated with these tables.
"M-values" specify the maximum supersaturation a compartment can
withstand without getting bent.  The M values were modelled as
a constant plus an increment proportion to depth (M = M0 + dM*D).

In 1976, the Pennsylvania Analysis of Decompression from Undersea
and Aerospace (PADUA) model was introduced.  It consisted of 10
compartments, with half times all the way up 480 minutes.  More
conservative M0 and delta-M values were used.  Although the model
is more conservative than the US Navy model, no tables have been
produced for publication.

Also in 1976, Spencer made his famous Doppler Ultrasound measurements
looking for venous gas emboli (VGE -- fancy name for bubbles in the
blood).  If bubbles were the cause of DCS then why not look for bubbles
directly.  He found that at the Navy's no decompression limits there
were already significant numbers of VGE present in the blood.  He
suggested reduced limits which virtually eliminated the presence of
VGE upon surfacing.

In 1981, Huggins published new repetitive dive tables based on a model
similar to the US Navy's, but with M0 values based on the new no
decompression times from Spencer.  Furthermore, the tables considered
all 6 compartments in assigning repetitive groups, not just the 120
minute group as the US Navy did.  The tables had no provision for
decompression.

In 1983, Buhlmann developed another model for decompression (also
called the Swiss model).  It is still based on Haldane's assumptions
and on tissue compartments, but uses a different specification of
maximum compartment pressure that takes into account the ambient
pressure at the surface.  (Other tables assume 1 atm.)  They are unique
in that from the outset they were developed to be used at altitude
in Swiss lakes.  This also makes the tables useful for determining
the safety of flying after diving, due to reduced pressure in the
airplane cabin.

In 1984, Thalmann at the Navy Experimental Diving Unit (NEDU) reported
on an algorithm that the Navy had been developing for use in a closed-
circuit mixed gas system.  The algorithm is called E-L (as opposed to
E-E in the usual) because the compartment in-gassing occurs at an
exponential rate just as in the Haldanean models, but the outgassing
occurs at a slower, linear rate.

In 1985, two new models were proposed, both based on novel new
approaches.  At the University of Hawaii, the Varying Permeability
Model was proposed.  It attempts to model the number and size of
gas bubbles in the blood, and bases the ascent criteria on the
total volume of bubbles in the body.

At the Naval Medical Research Institute (NMRI) a statistical
approach to DCS was developed, called the Maximum Likelihood
Statistical Method.  Using a database of over 1700 individual
exposures, model parameters were determined, and then several
tables were developed associating sets of no-decompression times
with various likelihoods of DCS (e.g. 1% or 5%, etc.)

[I'm sure there is more out there, and probably more recent findings
but this is what my research a year ago turned up. --EW]


PROBLEMS

The problem with all of these is that they are only models.  Factors
such as temperature, individual characteristics, exertion level,
ascent rate and more complicated effects of bottom time are all
known to influence susceptibility to DCS, but none are included in
the models.  Attempts to model bubble formation rather than just
tissue saturation are IMHO a step in the right direction.  But as
someone pointed out a while ago on the net, there isn't even a
great correlation between observed bubbles and DCS!

A classic example where the models break down is bounce diving.
A recipe for DCS:  Something like 50% (vague recollection giving
ballpark magnitude) of divers following this profile get bent:
Doing everything by the book (e.g. proper ascent rate) dive to a
relatively deep depth ( > 60 ft) for a short time then return to
the surface.  Repeat a few times with a short surface interval
in between dives.

For the Haldanean models there is no fundamental difference between
this and regular dive profiles.  Tissues in-gas then out-gas, and if
the times are right you won't go over their limits.  But the incidence
of DCS says bounce diving defintely does something funny to your body.

One theory I've heard is that on a first dive (even relatively benign
ones) microscopic bubbles are formed.  Since it's easier for bubbles
to grow than to form in the first place, on subsequent dives they can
get much worse.  The bounce diving just multiplies this.

I believe the rational behind the safety stop is to try to get rid of
most of these bubbles (they go away better under pressure), or maybe
to prevent their formation in the first place.  I'm not really sure.
The point is the Haldanean models don't really take any of this into
account.

The moral is:  Safety stops (especially) and longer surface intervals
(somewhat) are good things!  :-)  Never trust your dive computer
completely.  It's only a model and the models have limitations.

NOTE that because of these problems with the model and the general
feeling that the model is _only_ a model, people tend to use the
term 'compartment' rather than 'tissue' when talking about Haldanean
models in order to highlight that the mathematical model may have
no physical connection to actual tissues in the body.  I sometimes
accidentally use the two interchangably (as does the literature)
but the tendency is towards using 'compartment'.  So if I should
slip and say tissue where I mean compartment, you'll know what i mean.



SOME NOTES ABOUT THE EXPONENTIAL MODEL

The number that characterizes how quickly a compartment takes in or
releases pressure is called the HALF-TIME.  It is analogous to a half-
life in radiation.  It is defined as the time it takes for the pressure
to drop (or rise) from its initial value HALFWAY to ambient pressure.
In theory it would take an infinite amount of time to actually get to
ambient pressure (because as you get closer it slows down).  But in
practice a few half-times is all that's needed (after 3.5 half times
you're at 90% and after six half times you're at 98.4%).  Six half-
times is generally considered to be saturation.

ASIDE:  The 12 hour time after diving before you're considered to be
"clean" comes from this.  The US Navy tables had a 120 minute slowest
compartment, so after 6*2 hours all of your compartments would be
back to normal surface pressure.  Current thinking is that this isn't
long enough when you do extensive diving over a long period or before
going to altitude (e.g. flying).  Thus they incorporate slower compartments
to give the model more "memory."


II.  The Math

Go back and look at the basis of the Haldanean model.  That's where this
picks up.  (Dave Waller ([e-mail]) gives much of this in
the openning comments of the dive computer program.)

The change in a compartment's pressure is proportional to the difference
between the compartment's current pressure and ambient pressure.
Translation:

    dP = k * (PA - P) * dt        where P is compartment pressure
                      and PA is ambient pressure

Solving this and using the initial condition P = PT0 at t=0 gives the
following (math in between deleted :-)

    P = PT0 + (PA - PT0)*(1 - e^(-kt))

If you want to do an iterative computation to keep track of the
compartment pressures, the computation is what you would expect:

    P_new = P_old + (PA - P_old)*(1 - e^(-kt))

Where do we get k?  Well, we have another condition.  We know that
at t=HALF-TIME, (1 - e^(-kt)) = 0.5 by definition.  Thus:

    k = ln(2) / HALF-TIME


M-VALUES

So we know how to model the compartments now.  How do we know when
they are at their limits, etc?

The models specify M-values which are the maximum compartment pressures
for which it is still safe to return to the surface (assuming 1 atm at
the surface).  More concisely M-values are the maximum absolute PARTIAL
PRESSURE of nitrogen measured in feet of seawater (FSW).  (Absolute,
meaning that 0 is a total vacuum.)

In numbers, at sea level (1 atm =~= 33 fsw) the partial pressure of
nitrogen is 0.79 * 33 = 26 fsw.  The US Navy table uses an M-value of
104 fsw for the 5 minute compartment.  Thus for returning to the
surface a 4:1 supersaturation is permissible without [symptoms of]
DCS.

[ASIDE: IMHO, It's not exactly clear when to say DCS actually begins.
Damage caused by small bubbles could be so minor as to be asymptomatic]

Note that since M-values are given the the nitrogen partial pressure
you would want to do the calculations of compartment pressures with
just the nitrogen.  i.e.  PA is ambient nitrogen partial pressure.
The program calls this apn2.  At depth (D >= 0)

    apn2 = (D + P0) * 0.79        where D is depth
                          P0 is surface ambient pressure
                        (usually 33 fsw)


NO DECOMPRESSION TIME REMAINING

So you want to compute how much time you have left at a given depth
before you're compartments saturate.  Just solve the P equation for t
given the pressures.

Let T = time to reach saturation
    M = the M-value
    C = current compartment pressure
    A = ambient pressure (at depth)

Then

    M = C + (A - C)*(1-e^(-kT))

Solving for T:

    T = (1/k) * ln((A - C) / (A - M))

or in terms of half-time, H:

    T = H * log-base-2 ((A - C) / (A - M))

Note that using the properties of logarithms you can manipulate the
equation around a bit, but the form is the same.

Also note that the solution of this equation is dependent on there
being a solution!  The three variables A, C and M have six possible
relations (e.g. A > C > M, etc)  Only two of these have meaningful
solutions, but all of them give you some information about the
compartment.

They are:

A > M > C    The compartment is still below its limit (what I might call
        normal, as opposed to being in a decompression mode,
        M > C) but is eventually going to exceed its limit
        because A > M and A > C.  The equation _does_ have a
        meaningful solution for T, it's the no decompresion
        time remaining.

M > A > C    The compartment is 'normal' (M > C) and furthermore it
        is what I might call 'safe' i.e. (M > A) so not only
        is it not over the limit, it will never go over the
        limit at this ambient pressure.  (A > C) so the
        compartment is in-gassing (increasing in pressure).
        Since it is 'safe' the T equation has no solution.
        You can say T = +INFINITY for this compartment.

M > C > A    The compartment is 'normal' (M > C).  It is 'safe' (M > A)
        it is outgassing (decreasing in pressure) (C > A).
        Again, T = +INFINITY.

A > C > M    The compartment is over is limit (C > M), so it's no
        longer safe to return to the surface.  Your dive computer
        would be in 'decompression mode' now, if it had one.
        Furthermore it is still in-gassing (A > C).  At this
        ambient pressure the compartment will never get back below
        its limit (A > M).

        In a decompression mode, the no decompression limit no
        longer has meaning, but might like to know how long it
        take before we're out of decompression.  I'll call this
        TDC.

        For this condition, TDC = +INFINITY

C > A > M    The compartment is in a decompression mode (C > M).
        It is out-gassing (C > A) but it will still never get
        back below the limit because (A > M).  TDC = +INFINITY

C > M > A    The compartment is in a decompression mode (C > M).
        It is out-gassing (C > A) and will eventually get back
        to a safe state.  The equation for T also solves for
        this condition.  Look at the LOG(stuff) to convince
        yourself that it will return a positive, finite solution
        that _is_ what we are looking for.  Thus, TDC = equation
        gives to time left to decompress.


Interesting relations to note are:

    (C > M)   over/under M-value limit
    (C > A)   out/in gassing
    (M > A)   'safe' vs 'unsafe'**

** This term I coined myself for lack of a better mneumonic.  I call it
safe because in 'normal' mode a 'safe' compartment will never go over
its M-value.  In decompression mode, when the compartment is already
over its M-value, only a 'safe' one will eventually get back back below
the limit.


THE REAL COMPUTATION

The above discussion only covers a single compartment.  The overall
state of the model is a composite of the state of the individual
compartments.

- If any compartment is in decompression, then it will dominate over
    compartments that are not (i.e. are 'normal').

- Among 'normal' compartments the one with the lowest NDL is the
    controlling compartment.

- Among 'decompressing' compartments the one with the highest TDC
    is the controlling compartment.


Note that it can be very tough to give an absolutely correct answer
for TDC, because you can have a mixture of compartments some over-
saturated (in decompression) and some under the limit but still
in-gassing.

At a particular depth, you might be counting down the minutes for
one compartment to get back below the limit and mean while another
one goes over.  Granted it's slightly pathalogical, but definitely
_can_ happen.


CEILING

Given that you have decided you are in a decompression mode and
can't go up to the surface, what depth _is_ safe?  The smallest
depth (shallowest) you can ascend to when in a decompression mode
is called your ceiling.

To compute this we need to know how M changes with depth.  We may
be over the M-value for the surface, but at some depth we'll be
o.k.

The original Haldanean principles just specified a ratio that is
safe.  e.g. 3:1 (compartment to ambient) is the limit.  You could
use this to figure out what your ceiling.

For some reason (maybe this way works better, less DCS, maybe someone
was just lazy, I dunno) this isn't the way most model do it.  Dating
back to 1957 and the US Navy tables, the following method is used:

    M_at_depth = M_at_surface + delta-M * depth
or
    M = M0 + dM * depth

(the 'd' of dM is actually supposed to be the capital greek letter
'delta', but the net doesn't seem to support greek!  :-)

Each compartment has an M0 and dM value.  Frankly, I'm not sure exactly
how the dM's were derived for the Navy Model.  The ORCA Edge model just
sets them all [somewhat arbitrarily --EW] to 1.0 (which is less than
the Navy values and therefore more conservative).

For a particular compartment, the ceiling is simply:

    ceiling = (C - M0) / dM

where C = compartment pressure
      M0 = what I used to call M, max pressure allowed at surface
      dM = the incremental change in M with depth

To compute the composite ceiling for the whole model, just take
the worst case (deepest ceiling).


STAGED-DECOMPRESSION

In order to decompress you must be at an ambient pressure that's
lower than the compartment value (so that you out-gas) and yet
deeper than your ceiling.  This can lead to staged decompression
for some situations.

Note that the slow compartments also have lower M-values.  If you've
gone way over on your faster tissues and slightly over on your slow
tissues, you would have to do a staged-decompression.

The ceiling would be determined by the fact that you were way over
on the faster tissues.  But at the depth specified by the ceiling,
the slower tissues may not be outgassing, since ambient pressure
could be greater than the compartment pressure.

As you waited below your ceiling, the fast compartment would
outgas and the ceiling would slowly move up.  You could then move
up to a shallower depth.  You would repeat this until you were at
a depth sufficiently shallow for the slower tissues to outgas.

This should give you an idea of why it's hard to compute the
actual TDC (time to decompress).  In addition, while you were stuck
at the deeper ceiling, other slow tissues with low M-values could
be in-gassing and could go over the limit.

In order to give the right number from the start you would have to
consider all these possibilities.


III.  Other neat stuff, like adjusting for altitude

When you dive at altitude the surface ambient pressure is lower
than normal (something like 2/3 atm at 10,000 ft is what I remember
hearing).  In order for the tissues to see the same oversaturation
*ratio* which is what we believe to be the controlling factor, you
must reduce the M values.

For example, the Navy 5 minute compartment M0 of 104 fsw gives a
4:1 supersaturation ratio (104 / (33*0.79)).  If we went to 10,000 ft
(assuming my 2/3 atm number is right) the same 104 fsw would give
a ratio of 6:1.  Some compensation must be done.

ASIDE:  This is also relevant to flying after diving.  Since the
compartment of commercial jet-liners is typically kept at 8,000 ft
what your dive computer says is o.k. to do at sea level could
cause problems if you immediately hopped on a plane for home.
In fact this make air-evacuation after a diving accident a little
tricky (and expensive !!!).

The question is how to do the compensation.  I don't have a solid
answer for this, but we can still think about it.

The obvious way to handle it is to scale the M-values to preserve
the same supersaturation ratios that you had on the surface.  e.g.
in the previous case 104 fsw would scale to 69.3 fsw (104 * 2/3).
IMHO this is a very reasonable way to do it.

The only model that I have info on that was designed to handle
altitude is the Buhlmann model.  This model specified the maximum
pressures a little differently than the M-value model started by
the US Navy.  In this model:

    Pmax = a + Pamb/b

instead of:

    M = M0 + dM * depth

where 'a' and 'b' are the constants for the compartment.  Pamb is the
absolute pressure at depth (e.g. 0 is a vacuum).  Thus it includes both
depth and surface pressure together.  Pmax is the same as M -- the
maximum pressure that a compartment can have without problem.

The two models are equivalent with a little manipulation:

    Pamb = P0 + P_depth

if you use fsw as your unit of pressure, then P_depth is just the depth.
Thus:

    Pmax = (a + P0/b) + (P_depth) * (1/b)

If you convert the units of 'a' to fsw (b is unitless) then:

    M0 = a' + P0/b
    dM = 1/b

where P0 is ambient pressure at the surface.

The Buhlmann model was designed for and has been tested at altitude
in Swiss lakes.  Therefore, we should also consider this approach to
altitude compensation.

In the first model (straight ratios) if the pressure dropped by 1/3
then 1/3 of the M0 was subtracted.  In the Buhlmann model only a
fraction of the pressure drop is subtracted, with the actual
fraction depending on the a's, b's and surface pressure.

In practice, the fraction varies from 35% for the fastest compartments
to 80% for the slowest (with most compartments above 60%).  Thus we
might subtract only 35% of the 1/3 of the normal M0.

Unfortunately I don't know how they decided on the a's and b's which
determine this.  Since I don't have good information on how these
numbers came about it's hard to adapt this to other models.

Does anyone out there know how dive computer X compensates for
altitude?  Some I know of use the Buhlmann model.  The Delphi is
one I'm curious about because I'd bet Orca used their same Edge
model but scaled the values somehow.

[Is anyone out there still reading at this monster of a posting!! :-)]



REFERENCES

<see my previous post about M-values for various models.>

The pointer to the original Buhlmann paper is in the
Proceedings of the AAUS Dive Computer Workshop, 1989.

A coulple of people asked me for more info on how to get the
publications I cited.  (The American Academy of Underwater
Sciences and Michigan Sea Grant Publications).

I gave out the addresses on an individual basis, but I figured
I should wait until I had contacted them to find out their policy/
price/scoop before posting to the net-at-large.

If you want the addresses, mail me and I'll send them out as long
as the volume isn't overwhelming.  If you can wait a bit, I just
dashed off a couple of letters to them explaining what I want and
will post to the net when I get a response.


--Eric Williams. [e-mail]

P.S.  If you haven't already done so, try out the dive computer
program it's really neat!  (and helps you get a feel for how all
this works)

--------------------------------------------------------------------

From: [e-mail] (Eric Williams)
Newsgroups: rec.scuba
Subject: Re: Feature Request: Let's design our own dive computer
Summary: decompression, m-values, dive computer
Date: 8 Nov 90 21:42:43 GMT
Organization: AIR, Stanford University

(This is a little late, I tried posting yesterday, but had problems)

About a year ago I did a comparison report of different dive computer
models for my Open Water I course.  A dive computer was (and still is)
on my project list, so I've been following this discussion with
interest.

I've got values for four different models, including the Navy.  They've
been transcribed about three different times (source to notes to report to
here) so if you have any of these, please compare, I could have garbled them.

First, the numbers, then some references on decompression models.  Some
discussion about M values and delta M's should probably accompany this,
but I'll assume we're all on the same wavelength and save it for
another posting.  (The M values are most important, the delta M's are
only used in decompression algorithms.)

============
US NAVY

T 1/2    M0    dM
(min)    (fsw)    (--)
------    -----    -----
5    104    2.27
10    88    2.00
20    72    1.71
40    58    1.40
80    52    1.29
120    51    1.27

<< Sanity check: Do we all still agree?  Everybody should have the
   same numbers here. >>

<<< Now that this is late, I've checked this against duis@sgi and they
    agree.  I don't think there are other compartments in the Navy model.
    (Like 180, 240, etc.) >>>
=============
Huggins Model.  These numbers were derived and used by Karl Huggins
  to generate the no-bubbles tables.

T 1/2    M0    dM
(min)    (fsw)    (--)
-----    -----    -----
5    102.0    0
10    85.0    0
20    67.5    0
40    54.5    0
80    47.5    0
120    43.0    0

<< There are no dM values for this model because it is not intended to
   be used as for decompression. >>

=================
ORCA Edge (and Skinny Dipper, I think)  Also by Huggins
  These were published in 1987.  There are no guarantees
  that these are the numbers actually used, or that they
  haven't been tweaked slightly.

T 1/2    M0    dM
(min)    (fsw)    (--)
-----    -----    -----
5    100.0    1.0
11    81.8    1.0
17    71.5    1.0
24    63.7    1.0
37    55.9    1.0
61    50.7    1.0
87    46.8    1.0
125    43.0    1.0
197    39.1    1.0
271    36.5    1.0
392    33.9    1.0
480    33.0    1.0

<< These are from the Sea Grant publication by Huggins, see references >>

=========================
Buhlman (Swiss) Model.  Some explanation is in order here.  The
  Buhlman model was developed in Switzerland and (I think) been
  tested at alititude in Swiss lakes.  It is based on a different
  calculation of maximum compartment pressure than the M value one
  (Pmax = a + Pamb/b, Pamb is absolute ambient pressure (in Pascals!)
  where a and b are the numbers specified for each compartment.)

  It was my observation, (I've never seen this in print)
  that it was equivalent to the M value calculation (M = M0 + dM*D)
  with a little manipulation and units conversion:
        M0 = a + P0/b, dM = 1/b, 'a' must be converted to fsw,
        b is unitless, P0 is air pressure at the surface.

  The following are my calculations of what M values would be for
  this model.  Unfortunately, I've don't seem to have the original
  a's and b's around, but they're in the references.

T 1/2    M0    dM
(min)    (fsw)    (--)
-----    -----    -----
2.65    114    1.22
7.94    91    1.22
12.2    77    1.21
18.5    70    1.20
26.5    65    1.18
37    58    1.16
53    54    1.15
79    53    1.12
114    53    1.12
146    51    1.07
185    51    1.07
238    48    1.06
304    43    1.04
397    43    1.04
503    43    1.04
635    43    1.04

<< Again, note that these are unchecked and unpublished. >>

==================================

REFERENCES

Edmunds, Carl, "Dive Computers -- The Australian Experience"
    Proceedings of the AAUS Dive Computer Workshop, AAUS
    Costa Mesa, CA, USCSG-TR-01-89, 1989, pp. 59-68.

Egstrom, Glen H., "Dive Computers, Dive Tables and Decompression"
    same as above, pp. 163-168.

Huggins, Karl E., Micropressor Applications of Multi-Level Air
    Decompression Problems.  Michigan Sea Grant Program,
    NA86AA-D-SG043, 1987.

Lewis, John E., "The Datamaster II -- A Fundamentally Different
    Dive Computer," Proceedings ... AAUS... 1989, pp. 43-47.

Loyst, Ken and Michael Steidley, Diving with Dive Computers,
    Watersport Publishing, San Diego, 1989.

Somers, Lee H. "Dive Computers in Scientific Diving Programs"
    Proceedings ... AAUS...1989, pp. 203-205.

Short, Donald R. and C. Mark Flahan, "Reconstructing the Navy
    Tables," Proceedings .. AAUS ... 1989, pp. 181 - 187.


Unfortunately, I don't have any of these references myself.  I
borrowed them from my instructor.

Let's keep this discussion going.  I think it's very interesting.
I'd be happy to port another article with background on M values
and compartments and how they're used to compute, no decompression
time remaining and decompression ceiling, etc. etc.

--Eric Williams. [e-mail], uucp? thru decwrl?


------------------------------------------------------------------
From: [e-mail] (David Duis)
Newsgroups: rec.scuba
Subject: Re: Feature Request: Let's design our own dive computer
Summary: Extended Navy M-values table
Keywords: dive computers, algorithms, features, US Navy Tables, M values
Date: 9 Nov 90 01:32:23 GMT
Reply-To: [e-mail]
Organization: Silicon Graphics, Inc., Mountain View, CA

In article <[e-mail]> [e-mail] (David Duis) writes:
>As promised, here are the US Navy M-values, as used to compute the
>current Navy tables.
>
>These figures represent the maximum allowable no-decompression
>nitrogen-loading.  Figures for the 160, 200, and 240 minute
>compartments are not listed in the NAUI book -- I'll look them up in
>_Physiology & Medicine of Diving_ RSN (Real Soon Now :-).

As promised, here is the extended table.  Please note that the
160,200, and 240 minute compartments are used only for the Navy
Exceptional Exposure tables.  However, I consider a multi-day
multi-dive-per-day dive trip to be an "exceptional exposure," thus I
have included these values.


    Compartment         Surfacing ratios:
    Half time      TOTAL   NITROGEN  M-VALUE (fsw)
        5             4:1    3.15:1     104
       10          3.4:1    2.67:1      88
       20          2.75:1    2.18:1      72
       40         2.22:1    1.76:1      58
       80            2:1    1.58:1      52
      120         1.96:1    1.55:1      51
      160         1.96:1    1.55:1      51
      200           1.96:1    1.55:1      51
      240         1.92:1    1.52:1      50


I look forward to ANY simulation, now that we have these values.
While it would be nice to model every computer, a simple multi-level
simulation of the Navy Tables would be nice.

Anyone have the M-values for the DSAT table, which uses a 60-minute
controlling half-time?  I believe these are the same values as those
used in the Sherwood Source/Oceanic Sport, so we'd be doubly served.


Cheers,

Dave Duis                             NAUI AI Z9588, PADI DM 43922
[e-mail]      opinions my own, and subject to dispute

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