On Fri, 10 Apr 1998, Tom Mount wrote:

> William
> I just did that in the examples I post and although it is just a small
> difference it is slightly shorter so the answer is true.

Because I can see where this thread is going, let me cut
to the chase and do the maths showing why depth
won't matter (bear with me, it's not too painful :)

Here's the core Buhlmann ZHL16 formula, I used slightly
different names for the terms, because Buhlmann's are
really verbose (and german), but the formula is the same.  

This is the way you calculate a single gas, N2 or He.  Trimix
is a bit trickier, but it basically works the same
way:

ptol = (prev + (pp - prev) * (1.0 - 2^(-time/ht)) -a ) * b

where:

"ptol" is the minimum tolerable pressure, in BAR for a given tissue.
"prev" is the saturation level of that tissue at the previous sample 
   interval, and is what you get if you strip out the "a" and "b" 
   references.
"pp" is the inspired partial pressure of N2
"time" is the time elapsed since the last sample interval
"ht" is the halftime of the tissue in question (supplied by Buhlmann)
"a" and "b" are constant coefficents supplied by Buhlmann.  There's
   a different set for each ofthe 16 compartments.
"^" is notation for exponentiation (two to the power of the stuff after it).

The core of this is pretty much a standard halftime decay formula.  The
key here is the inputs - partial pressure of N2 and elapsed time.
Notice there's nothing about oxygen here.

The way you run the algorithm is: 
[1] Calculate the partial pressure of the nitrogen in the
    mix by subtracting the amount of o2.
[2] Get the elapsed time since you last did a calculation.
[3] Run the above formula for each of the 16 tissue
    compartments.
[4] The compartment with the least tolerable ceiling
    is your "controlling tissue" - the deco ceiling.
[5] Keep doing steps 1-4 until all tissues can tolerate
    less than 1 BAR (less if you are diving at altitude).
    Do this every few seconds or so, stopping the
    ascent just below the ceiling until it's OK to ascend
    further.

They key to the whole thing is that absolute depth isn't
used anywhere - all you need at any given time to calculate
your ceiling (and saturation) is the current partial pressure 
of N2 in the mix you are breathing, and the amount of time since the 
last sample.  Oxygen itself isn't considered anywhere.

So you can see that once you go onto 100% O2, your "pp"
(inspired partial pressure of N2) variable goes to 
zero (or close to it, if you are including
some water vapor).  If you are at 100 feet or 10, "pp" will still 
be zero, which means the rate of offgassing will not 
change.  This means that your ultimate deco profile 
will not change.

Of course, as soon as you have a FN2 above zero
(a Nitrox), then your PPN2 will vary with depth, 
and so where you do your hang will matter, a litte.

For instance, for the 100 for 100 dive on air, with
deco on 90%, doing both the 10 and 20 gives a runtime of
158 minutes.  Combining the stops gives 159 minutes.

Notice that combining the stops actually *adds* time, 
which makes sense, because the PN2 is higher at 20ft,
so the off-gassing rate will be slower. 

There are instantaneous versions of this algorithm,
where you don't need to iterate, and different
sets of a/b coefficents for tables, dive computers,
and pure theory, but it's still the same thing
when it comes to exclusive use of partial pressures.

Not having depth involved in the input
is very convenient, because it makes for easy 
calculation for diving at altitude, and makes
it very simple to switch deco gasses.

But any way you slice it, combining your
10/20 stops at 20 while on pure O2 will
not reduce your deco time.

There are always minor variations in
implementation details between packages that 
will swing deco times a bit in either direction,
but if ProPlanner is ZHL16 based, I can't
figure out how they are getting these results. 

-Will

> If you read the two examples of the bulhman model I posted when you do the
> entire stop at 20 feet and do not go to 10 feet the deco time is shortened.
> .
> Below is the example this is a pure bulhman model on a square dive ran these
> on pro planner as it is the quickest to print out.
> 
> 1.air to 100 feet 100 minutes deco on air runtime is 243 minutes
> 2 air to 100 feet 100 minutes deco on oxygen at 20 and 10 feet RT 165
> 3.same dive deco on oxygen at 20 feet only RT is 162 minutes
> This is 3 minutes less than if deco is at 20 and 10 feet, thus oxygen at 20
> allows a shorter deco than doing a 20 and 10, thus answer is true
> another example
> 160 (air for simplicity) 30 minutes deco on air  RT105 min.
> Same dive deco on oxygen at 20 and 10 feet RT 66 minutes
> Same dive do oxygen stop at 20 feet only (no 10 foot stop) RT 64 min
> Deco on oxygen at 20 feet only is 2 minutes less than if stops were made at
> 20 and 10
> Another example trimix 21 25 160 30 minutes
> RT deco on bottom mix 159 minutes
> RT deco on oxygen at 20 and 10  74 minutes
> RT deco on oxygen at 20 feet only 72 minutes
> again we see a shorter deco by using oxygen at 20 feet only
> or
> RT deco on EAN 70 starting at 40 feet 67 minutes
> It is a small difference but you deco faster on oxygen at 20 feet than you
> do on oxygen at 20 and 10 feet. It is a fact
> Tom Mount
> 
> 
> 
> 

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