I'm having trouble understanding/deriving Bühlmann coefficients "a" and "b".

[inquis]

FYI, Chapman also talks about flying after diving, but you should also ignore that. Testing by Duke University in 2002 unequivocally showed it's more complicated than just an "equivalent depth". The US Navy ascent tables (which predated the Duke study) indicate the same.

 

[dmaziuk]

...

Calculating b for tissue compartment 2
Baker's paper says b is just the inverse of the "ΔM slope" column. b = 1 / ΔM = 1/1.5352 = .65138...
Chapman's paper says b = 1.005 - ( tht ^ -1/2 ) =1.005 - (1/sqrt(8)) = .65144

These values for b are similar, but not identical.

They're both 0.65139 +/- 0.00001. I have not seen any evidence that the precision to 4 decimal digits has anything to do with the accuracy of the actual model; having seen a lot of scientific data I very much suspect they come from %5.4f printf mask that makes the table columns align nicely when printed in fixed-width font.

Measure with a micrometer...

 

[tridacna]

I am trying to derive a and b coefficients for ZH-L16 using the Nitrogen M values in "Understanding M-values" by Erik Baker. I can find already computed coefficients published here and in An Explanation of Buehlmann's ZH-L16 Algorithm by Paul Chapman, so I am able to check my work.

I will take tissue compartment 2 for my examples (just to avoid the potential compartment 1/1b confusion and because M_0 is the same for ZH-L16A, B, and C).

Calculating b for tissue compartment 2
Baker's paper says b is just the inverse of the "ΔM slope" column. b = 1 / ΔM = 1/1.5352 = .65138...
Chapman's paper says b = 1.005 - ( tht ^ -1/2 ) =1.005 - (1/sqrt(8)) = .65144

These values for b are similar, but not identical. I computed the values both ways for all 16 compartments and all values for b are close but not identical, with the largest error being .02 in compartment 1, followed by .01 in compartment 4. Chapmans Which value or derivation of b is more correct?

Calculating a for tissue compartment 2
Chapman's paper says a =2 * ( tht ^ (-1/3) ) = 2 / cuberoot(8) = 1
I used Chapman's formula to compute the other 15 tissue compartments' a values and all matched his published values so I am comfortable with this.

But Baker's paper says a = M_0 - ΔM * P_amb = 83.2 - 1.5352 * P_amb.
I am confused about what P_amb is. Baker says "The Coefficient a is the intercept at zero ambient pressure(absolute)", so I would expect P_amb = 0, but then you get a = M_0 and that is obviously incorrect.

Since I have Chapman's published values, I can actually solve for P_amb since I know what a should be.

P_amb = (a - M_0) / -ΔM = (1 - 83.2)/-1.5352 = 53.5435...

But P_amb = 53.45 is only a solution for compartment 2. When computed the same way, the other tissue compartments have solutions ranging from 40 to 55.1.

How can I properly understand P_amb, and what values belong in Baker's formula for calculating coefficient a?

Thanks in advance.

Curious. What is the reason behind the question? Academic musing or real purpose?

 

[ThisSurfaceIntervalSucks]

They're both 0.65139 +/- 0.00001. I have not seen any evidence that the precision to 4 decimal digits has anything to do with the accuracy of the actual model; having seen a lot of scientific data I very much suspect they come from %5.4f printf mask that makes the table columns align nicely when printed in fixed-width font.

Measure with a micrometer...

You ended your quote before you got to the next sentence where I call out the larger differences in tissue compartments 1 and 4. A difference of .02 in compartment 1 is a difference of .02/1.2599=1.6%, which is a little bit more than a rounding error.

But ultimately I think the values I see widely published are maybe not so widely agreed upon. Subsurface has different numbers for a and b in many of the tissue compartments including tissue compartment 1.

I think Chapman's numbers are probably reasonably close but they are mathematically derived and so are not as reliable as the values specifically selected for ZH_L16A/B/C, which is probably specifically why he listed them as ZH_L16A, to subtly warn that these numbers are not conservative enough to be used in any planning or dive computer software. There are various implementations by software developers of dive planners and dive computers and each may have their own set of coefficients.

 

[ThisSurfaceIntervalSucks]

Curious. What is the reason behind the question? Academic musing or real purpose?

I am toying with the idea of creating dive planner software.

 

[JMarc]

IIRC Bülhmann started with a formula and then adjusted the values (at least for B and C sets) to better fit experimental results (for the B set) and add a margin (for the C set intented to be used by computers using the real profile, tables add margins in other ways by their nature — using a "square" profile — and their construction — adding margins at places which make reproducing the exact values difficult if you don't do it in exactly the same way).

 

[dmaziuk]

You ended your quote before you got to the next sentence where I call out the larger differences in tissue compartments 1 and 4. A difference of .02 in compartment 1 is a difference of .02/1.2599=1.6%, which is a little bit more than a rounding error.

There are 2 compartments #1, the one you're quoting above is the 4-minute compartment. There is also a 5-minute one with 1.1696/0.5578 a/b. There are also 3 sets of 'a' coefficients that differ in the middle compartments. Are you certain you're comparing apples to apples? (Note that I haven't looked closely at your numbers, maybe you are.)

 

[ThisSurfaceIntervalSucks]

There are 2 compartments #1, the one you're quoting above is the 4-minute compartment. There is also a 5-minute one with 1.1696/0.5578 a/b. There are also 3 sets of 'a' coefficients that differ in the middle compartments. Are you certain you're comparing apples to apples? (Note that I haven't looked closely at your numbers, maybe you are.)

You are correct, I was not comparing apples to apples. Chapman's numbers for compartment 1 were computed from 1b (and subsurface uses 1b) and I was using 1a in my spreadsheet.

FWIW, I noticed today that Wikipedia does call out the discrepancy I noticed in compartment 4.

The b values calculated do not precisely correspond to those used by Bühlmann for tissue compartments 4 (0.7825 instead of 0.7725) and 5 (0.8126 instead of 0.8125).

 

[dmaziuk]

FWIW, I noticed today that Wikipedia does call out the discrepancy I noticed in compartment 4.

They may have been "adjusted" for sets B/C -- what's normally quoted everywhere is that a coefficients were adjusted (from TC 6 on), but maybe Herr Doktor did the b too. Without reading the Tauchmedizin we'll never know.

 

[34109411]

I am trying to derive a and b coefficients for ZH-L16 using the Nitrogen M values in "Understanding M-values" by Erik Baker. I can find already computed coefficients published here and in An Explanation of Buehlmann's ZH-L16 Algorithm by Paul Chapman, so I am able to check my work.

I will take tissue compartment 2 for my examples (just to avoid the potential compartment 1/1b confusion and because M_0 is the same for ZH-L16A, B, and C).

Calculating b for tissue compartment 2
Baker's paper says b is just the inverse of the "ΔM slope" column. b = 1 / ΔM = 1/1.5352 = .65138...
Chapman's paper says b = 1.005 - ( tht ^ -1/2 ) =1.005 - (1/sqrt(8)) = .65144

These values for b are similar, but not identical. I computed the values both ways for all 16 compartments and all values for b are close but not identical, with the largest error being .02 in compartment 1, followed by .01 in compartment 4. Chapmans Which value or derivation of b is more correct?

Calculating a for tissue compartment 2
Chapman's paper says a =2 * ( tht ^ (-1/3) ) = 2 / cuberoot(8) = 1
I used Chapman's formula to compute the other 15 tissue compartments' a values and all matched his published values so I am comfortable with this.

But Baker's paper says a = M_0 - ΔM * P_amb = 83.2 - 1.5352 * P_amb.
I am confused about what P_amb is. Baker says "The Coefficient a is the intercept at zero ambient pressure(absolute)", so I would expect P_amb = 0, but then you get a = M_0 and that is obviously incorrect.

Since I have Chapman's published values, I can actually solve for P_amb since I know what a should be.

P_amb = (a - M_0) / -ΔM = (1 - 83.2)/-1.5352 = 53.5435...

But P_amb = 53.45 is only a solution for compartment 2. When computed the same way, the other tissue compartments have solutions ranging from 40 to 55.1.

How can I properly understand P_amb, and what values belong in Baker's formula for calculating coefficient a?

Thanks in advance.

i totally agree with you.