Calculating average depth

[Spectre]

sheck33 once bubbled...
If i make a 40 min dive and i spend 35 minutes at 40 ft with one short bounce to 100 ft then these few 100 ft readings will influence the average you get significantly and you would get an average that would not be too useful.

Huh? 35 * 40 = 1400 + 5 * 100 = 1900 /40 = 47.5 ft average

In actuality it would be more like:

35 * 40 = 1400 + 1 * 70 + 2 * 100 + 2 * 70 = 45.25

Yes, you loose data with zig-zagging profiles, as the average doesn't take into account travel time [e.g. if you are at 40 ft on one data point, and 100 at the next data point, you obviously travelled, and the 'depth' from 40 to 100 would be 70].

However the subtle points missed doesn't really matter as your only looking for an idea of your SAC. When you use your SAC for gas management planning, you're going to be using your bad SAC rates, and you'll be calculating based on depth floors. So if it's actually .55 or .57 instead of .56, it don't matter, you'd use .6 or .65 for planning anyway.

Now the only difference between the calculus method and the 'easy' method is that the calculus method is calculating the SAC rate for each data point and then averaging them together througout the series. The other method is averaging the depth and then calculating the SAC rate from that point, which is a lot easier.

If you _have_ the data to have the pressure reading for every time T, then the calculus method works well as you could then calculate intermediate SAC rates. e.g. T=3 to T=10 I was working, so my working SAC rate is... 1-2 and 11-20 I was resting, and that SAC rate is...

But calculating the SAC rate for the whole dive... you'll get the same result with the calculus method as you get with the average depth method.

 

[rcohn]

Using the average depth method to compute SAC assumes a constant SAC throughout the dive. In general this a good assumption, the integral calculation exactly defines the average SAC. If most of your dives have a have a consistent SAC value, and that is assumption is the reason for computing the SAC in the first place, then using the average depth method will provide a reliable estimate of your SAC over time.

Changing depth is not a source of error if SAC remains constant, we can break any dive into arbitrarily short time intervals and have a reasonable assumption of constant depth over each interval.

I'll use a fairly extreme example to illustrate the error caused by a non constant SAC. Assume a simple two level dive with 10 minutes at 33 ft and 10 minutes at 99 ft. At one depth I'll use a SAC of 0.5 cf/min and at the other 1.0 cf/min. The average depth is 66 ft and the average SAC is 0.75 cf/min.

For the case where the low 0.5 cf/min SAC occurs during the 33 ft level (1.0 cf/min at 99ft), we have: Gas consumed 50 cf, SAC estimate 0.833 cf/min

For the case where the low 0.5 cf/min SAC occurs during the 99 ft level (1.0 cf/min at 33ft), we have: Gas consumed 40 cf, SAC estimate 0.667 cf/min

The error in the SAC estimate for this extreme example is +-0.083 cf/min equally distributed around the true SAC value.. Typical estimate errors for real dives would be less.

Over a series of randomly selected dives, the average of the SAC values estimated using average depth will converge on the true average SAC based on the Central Limit Theorem.

Ralph

 

[donacheson]

Rick Murchison once bubbled...
However - it doesn't appear to be that complicated to me. Let's plug some actual numbers and see how it works out, ok?
For my example, I'm going to spend 10 minutes at 99 FSW, 10 at 66 FSW and 10 at 33 FSW. I note that I've used 2250 psi. Now, that's an average depth (whether you sample every 30 seconds, one second or one minute makes no difference for this profile) of 66 FSW, or 3 ATM... 2250/3 = 750; 750/30 = 25. So, using a straight average and dividing by the average ATM and the total dive time, I come up with 25 psi/min SAC.
Now, if my SAC is in fact 25 psi/min, I would expect to use 1000 psi during my 10 minutes at 99 FSW (4X25X10); 750 psi during my 10 minutes at 66 FSW (3X25X10), and 500 psi during my 10 minutes at 33 FSW (2X25X10)... 1000+750+500=2250... just what I did use, so the 25 psi/min SAC is correct.
Now, why do I need to use calculus again?
Rick

True, there's no need for calculus in your example because you've assumed that the diver's surface air consumption rate is constant throughout the dive. In the integral, this is equivalent to assuming dp/dt = R(33+z(t))/33, so cancelling the term 33/(33+z(t)).

As a simple counter-example to yours, consider a diver who spends 20 minutes at 99 feet and 10 at 33 feet and consumes 2400 psi, but is working against a current at 99 feet and his SAC rate there is twice that at 33 feet. Let R' be the SAC ate at 33 feet. Then, 20x4x2R' + 10x2xR' = 2400, or R' = 13.33 psi/min. The average SAC rate for the dive, R, is given by R = (26.66x20 + 13.33x10)/30 = 22.2 psi/min.

On the other hand The average depth is 77 feet, so 2400 (33/(33+77))/30 = 24 psi/min! Not a big difference, but not the same.

 

[Uncle Pug]

these small differences in figures may provide something for you to do other that the boring work sitting there on your desk but in real life they are useless.

Keep it simple...

Figure two SACs: one for moderate work & one for rest...
Figure your gas management using them...
Use working SAC for the dive...
Resting SAC for deco...
Rounded up...

 

[Spectre]

donacheson once bubbled...
As a simple counter-example to yours, consider a diver who spends 20 minutes at 99 feet and 10 at 33 feet and consumes 2400 psi, but is working against a current at 99 feet and his SAC rate there is twice that at 33 feet. Let R' be the SAC ate at 33 feet. Then, 20x4x2R' + 10x2xR' = 2400, or R' = 13.33 psi/min. The average SAC rate for the dive, R, is given by R = (26.66x20 + 13.33x10)/30 = 22.2 psi/min.

On the other hand The average depth is 77 feet, so 2400 (33/(33+77))/30 = 24 psi/min! Not a big difference, but not the same.

Donacheson's method is more accurate for finding an average SAC, however in order to use that method, you have to be able to calculate the SAC rates for the individual periods.

In that case, the individual SAC rates are better used seperately, to calculate both a rest and a work rate for better gas management planning. However without the granularity of being able to calculate SAC rates for individual periods, you have no choice but to use the average depth method.

Personally I've been trying to record my intermediate PSI at the ascent point so that I can better calculate a rest and a work SAC rate. Until that time, I just have to be aware that the SAC rate I use is somewhere in between both my rest and my work SAC, and if the dive I'm planning incorporates more 'work' time then normal, I have to plan accordingly.

It all comes down to what you're planning to use the rates for. If it's just curiosity, then the calculus method is more accurate providing you have the data available, but for gas management planning, if you have the data to calculate a work and a rest SAC rate, you're better off with that than just an average of all the SAC rates.

 

[Rick Murchison]

Uncle Pug once bubbled...
these small differences in figures may provide something for you to do other that the boring work sitting there on your desk but in real life they are useless.

Keep it simple...

Figure two SACs: one for moderate work & one for rest...
Figure your gas management using them...
Use working SAC for the dive...
Resting SAC for deco...
Rounded up...

Bingo!
Rick

 

[donacheson]

Uncle Pug once bubbled...
these small differences in figures may provide something for you to do other that the boring work sitting there on your desk but in real life they are useless.

Keep it simple...

Figure two SACs: one for moderate work & one for rest...
Figure your gas management using them...
Use working SAC for the dive...
Resting SAC for deco...
Rounded up...

Yeah, you're probably right in this case.

My proclivities toward exactness are a consequence of a career of watching foolish engineers use cookbook formulas inappropriately, sometimes reaching absurd conclusions and occasionally, disasterous consequences.

 

[tchil01]

Thanks everyone for the math lessons and for Uncle Pug for putting everything in perspective.

Since Oceanic's Oceanlog data file is sored in a MDB file, I decided to take a look at it with Microsoft Access. I figured I could link to the info, import it into excel and do my calculation. Lo and Behold, right there in one of the tables was my SAC for all of my dives. I'm going to play with it a little to see how they calculated it, but for anyone else who has this software all is not lost.


Thanks again
Ty