At-Altitude Arithmetic

by

Larry "Harris" Taylor, Ph.D.

 

This is an electronic reprint of an article that appeared in SOURCES (Sept/Oct. 1994). This material is copyrighted and all rights retained by the author. This article is made available as a service to the diving community by the author and may be distributed for any non-commercial or Not-For-Profit use.  

All rights reserved.

 

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Depth gauges DO NOT MEASURE water depth! They measure pressure. Inside the device, a mechanical mechanism, coupled with a printed scale on the face of the instrument converts a measured pressure into an equivalent scale reading for water depth. The gauge will be accurate only if it is used in the environment for which it has been calibrated. When the device is taken to a different environment, such as high altitude, the reading of water depth on the gauge may be substantially different from the actual measured water depth. This is most often a problem when depth gauges calibrated at sea level are taken to altitude, as illustrated by the following numerical example.  

EXAMPLE: You are diving at a high altitude mountain lake. The barometer reads 24.61 inches (625 mm) Hg. Thus, at this altitude, 24.61 inches (625 mm) Hg (not 29.92 inches (760 mm) Hg) is the atmospheric pressure! Consider also that high mountain lakes usually are filled with fresh water (density about 62.4 lbs/cubic foot; 1.00 g/cc), not salt water (density of 64 pounds/cubic foot; 1.03 g/cc). What will the depth gauge read at an actual depth of 60 ffw (18.29 m) in this lake?

ANSWER (Using barometer in English units): First calculate the depth of water (x) that corresponds to one atmosphere at the observed barometric pressure. Remember that the height of the fluid in a  barometer is inversely proportional to the density of the fluid (1.0 g/cc for water and 13.6 g/cc for mercury) being used to measure the atmospheric pressure: 

    

NOTE: This means that one atmosphere of pressure at this altitude corresponds to a water column of about 334 inches of water. In feet:

 

Thus, every 27.9 feet of fresh water (not 33 fsw or 34  ffw) at this altitude corresponds to one atmosphere of pressure at this altitude. At this altitude, a depth measured by a lead line (not gauge) of 60 feet will be: 

In terms of "at-altitude" atmospheres, the absolute pressure would be:

2.2 atm + 1 atm = 3.2 ata  

This corresponds to a pressure of:  

NOTE: The depth gauge "senses" a pressure corresponding to 78.75 in Hg. The mechanism inside the device converts this pressure to:

  

This would then correspond to a hydrostatic sea level pressure of:

2.6 ata - 1 atm =  1.6 atm

Which would be read on the sea level calibrated scale as:

     

So, for a measured depth was 60 feet, at this altitude, the sea level calibrated gauge reads 53 feet.

Answer (Using metric units for the barometer heights):   

Determine the water equivalent of one atmosphere at this altitude:

This converts to:

Thus, at this altitude, 8.5 m corresponds to 1 ata pressure.

At depth of 18.29 mfw, the hydrostatic pressure is:

 This is an absolute "at altitude" pressure of:  

              2.2 atm + 1 atm = 3.2 ata

This means the gauge at this altitude is responding to a pressure of:

This corresponds to a sea level pressure of: 

This would then correspond to a hydrostatic sea level pressure of:

   2.6 ata - 1 atm =  1.6 atm

Which would be read on the sea level calibrated scale as:

So, the measured depth was 18.29 meters; the sea level depth gauge at this altitude would read 16.2 m.

If the sea level calibrated gauge were to be used for extended diving, then a series of corrections (generally at 10 foot (3 m) increments) could be calculated to be added to in-water depth readings for use at this altitude. True depth could then be determined by adding this "correction factor" to the observed sea-level calibrated depth gauge reading. Tables of these correction factors are available. (See, for example: ALTITUDE PROCEDURES FOR THE DIVER, by C.L. Smith.)   

BOTTOM LINE: Depth gauges measure pressure, not depth! The water depth indicated on the gauge dial reflects the actual depth ONLY if used in the environment for which the gauge was calibrated.

   OCEAN EQUIVALENT DEPTH (FOR DECOMPRESSION OBLIGATION)

Decompression obligation (Dive Table) calculations are based on pressure ratios, not actual measured in-water depths. Thus, when a diver changes altitude, the diver must be careful about the decompression tables and procedures used. Unless the dive table/computer specifically states that it has procedures for varying altitudes, divers should assume that the table/computer is only valid at sea level.  

Comment: The following is a physics discussion on the method used to obtain Ocean Equivalent Depth for use with sea level based tables. Such conversions are not as desirable as using tables or computers specifically designed for use at altitude.   

Decompression procedures are based on some maximum theoretical pressure ratio that can be tolerated within the tissue compartments without injury to the diver. This amount of pressure may vary with the depth of the diver and the particular mathematical simulation being used. The important consideration is that the PRESSURE DIFFERENCE (i.e., ratio between the current pressure and the pressure at some more shallow depth reached on ascent), not the actual water depth, controls the decompression obligation. This is best illustrated with a numerical example:   

EXAMPLE: At the altitude above, one atmosphere of pressure corresponds to 27.9 feet (8.5 m) of fresh water. Thus, the pressure at this altitude would increase by 1 at-attitude-atm every 27.9 feet (8.5 m) of descent/ascent (as opposed to every 33 feet (10.1 m) of sea water) at sea level. This means every 27.9 feet (8.5 m) at this altitude would correspond to a pressure (in terms of atmospheres) equivalent of 33 feet (10.1 m) of seawater at sea level. So, to maintain approximately the same pressure ratios as the U.S. Navy tables (or equivalent sea level derived tables) for determining decompression obligations, one needs to determine the actual number of "atmospheres pressure" at altitude and convert this to a sea level salt-water depth. For the high altitude dive at 60 feet (18.29 m) (2.16 "altitude" atmospheres) example above:

NOTE: In the above high altitude example. our actual in-water depth was 60 feet (18.3 m). The depth gauge indicated a depth of 53 fsw (16.2 msw). The equivalent sea level depth to maintain the same pressure differential as the U.S. Navy Table between bottom depth and safe ascent depth was 71.3 fsw (21.7 msw). Thus, using gauge pressure measured depth at altitude to enter the sea level computed decompression tables would allow the diver far more bottom time (increase risk to DCS) at depth since the diver would be entering the table at too shallow a depth. 

EQUIVALENT ASCENT RATES  

Finally, ascent rates are part of the decompression calculations. US Navy sea level tables ASSUME a rate of 60 fsw per minute. The BSAC tables recommend an ascent rate of 15 m/min. This ascent rate is part of the calculations used to derive the decompression schedules. Since, at altitude, the actual amount of water column that "defines" one at-altitude-atmosphere is less than 33 feet (10.1 m) of sea water, an ascent in a high altitude mountain lake must be slower than an ascent from the corresponding depth at sea level to maintain the same rate of pressure change with time. Again, this is best illustrated with numbers. For the example above:

At sea level; recommended ascent rate is:

At this altitude; corresponding at-altitude ascent rate:

  

Thus, while diving to a measured depth of 60 feet (18.29 m) in this high altitude mountain lake, your pressure gauge would read 53 fsw (16.2 msw) and your No-Stop decompression obligation would be determined by the 80 foot (24 m) sea level schedule using a recommended ascent rate of either 50.8 ffw/min or 12.7 mfw/min.

BOTTOM LINE: Sea level based dive procedures (tables or calculators) are inadequate for determining  decompression obligations at high altitude dive sites. Divers at high altitudes (above 1000 feet; 300 meters) should consider high altitude conversion tables (The Cross Tables) based on the above technique, dive tables with variable altitude entries (Swiss, DCIEM, or BSAC air tables) or altitude compensating dive computers. Also, there is a high altitude ocean depth calculator available from NAUI for determining ocean equivalent depths to use sea level tables at altitude. In general, these methods are considered theoretical, without extensive experimental validation. There is more discussion in the altitude diving section of this textbook. However, those who wish to dive at altitude should obtain specialty training in high altitude diving procedures.   

Problem

To test your understanding of the above discussion, consider this scenario:

You have been commissioned by Indiana Jones to help recover a sacred gold headpiece covered with emeralds. Indy’s map indicates that the headpiece is located at the bottom of a high mountain lake. Indy will get you to the lake. Your responsibility is to help plan the dive to the resting place of the scared treasure.  

By lead line, you determine that the actual physical depth of the lake is 93 feet.  

Unfortunately, while you were measuring the depth of the lake, one of the equipment bearers set fire to the supplies. The altitude adjusting dive computers, high altitude dive tables and Wienke’s text on Altitude Diving have been reduced to charred rumble. Interestingly enough, all that remains is a calibrated mercury barometer and remnants of a US Navy dive table. Since you do not wish Indy to get bent on his dive, you must use the barometer, the actual depth of the lake and the heat discolored dive table to plan Indy’s dive.

The barometer reads the pressure to be 546 mm Hg. Fortunately, you remember that the specific gravity of mercury is 13.6 and the specific gravity of water is 1.00.  

The charred remain of the table gives the following time/depth listing

 

So, as divemaster for Indiana Jones, you need to determine:  

a.       What will Indy’s oil filled depth gauge read on the bottom?

b.       What is the equivalent sea level ocean depth?

c.       How long will Indy have to find and recover the artifact without incurring a decompression obligation?

Solution:

Although there are faster ways to solve this problem (e.g. memorizing a specific formula), I prefer to take the solution in several logical steps so that,  regardless of a particular situation, understanding of what is known, what is unknown and  how  I convert the units of known to the unknown will direct my solution. 

  1. Water depth for atmospheric pressure at this altitude:

Depth of a 1atmosphere column of water will be inversely proportional to specific gravity:

 

             Converting to feet fresh water

       

            The water pressure at 93 feet of fresh water

   

            This corresponds as an absolute pressure of: 

3.81 atm + 1 atm = 4.81 atm total pressure (ata) at altitude

The pressure in mm Hg at this altitude would be:

 

             This hydrostatic pressure at sea level  (where 1 atm is 760 mm Hg) would be:

  2626.3 mm Hg    760 mm Hg  =  1866.3 mm Hg 

Converting to a sea level pressure:

Converting sea level atm to salt water depth:

2.46 atm x 33 fsw / atm = 81.2 fsw

  1. The ocean equivalent depth would be:

3.82 atm  x  33 fsw / atm  = 126.1 fsw

  1. The US Navy entry would be:

The 126.1 depth is rounded to the next greater 10 foot entry, so as divemaster, you recommend that Indy’s dive be controlled by the 130 fsw entry allowing him 10 minutes to find and recover the headpiece.

Thus, Indy will have to dive to 93 feet of fresh water at an altitude near 9000 feet above sea level, his depth gauge will read 81 fsw and he will treat the dive as if his decompression schedule obligation was determined by a sea level dive of 130 fsw. 

Comment: Obviously, this problem is fiction. Indy would NOT require scuba! He would do the 10 minutes dive on a single breath. The dive would end with Indy surfacing at sunset in front of a mid-western archeological museum with the sacred headpiece in the arms of the beautiful mermaid that assisted his dive. James Bond would do the dive differently, but that’s another story!

Additional Reading:

Bassett, B. "Diving And Altitude:  Can They Be Mixed," Sport Diver, Sep/Oct. 1980, 120-124.

Egi, S. & Brubakk, A. "Diving At Altitude: A Review Of Decompression Strategies," Undersea & Hyperbaric Medicine, 22(3), 1995, 282-300.   

Layman, L. The Basics of Diving High, Dive Training, August, 2000. 37-42.

Leech, J. McLean, A. & Mee, FB.  High Altitude Dives In The Nepali Himalaya, Undersea & Hyperbaric Medicine, 21(4), 1994, 459-466.

Lenihan, D. & Morgan, K. HIGH ALTITUDE DIVING, US Dept. Interior, Santa Fe, NM. 1975, 23 pages.   

Lowry, J. Scuba & Altitude, AOPA Pilot, March, 1987, 84-89.

Millar, I. Post Diving Altitude Exposure, SPUMS, 26(2), June, 1996, 135-140.    

Orr, D. An Attitude About Altitude, Alert Diver, March/April, 1999, 22-24.

Rossier, R. Altitude Diving, Dive Training, August, 1995, 38-44.

Schwankert, S. Going Up To Get Down, Discover Diving, February, 1996, .24-28

Smith, C. ALTITUDE PROCEDURES FOR THE OCEAN DIVER, NAUI, Colton, CA. 1975, 46 pages. 

Taylor, G. Diving At Altitude, Immersed, Summer, 1997, 54-55. 

Wienke, B. HIGH ALTITUDE DIVING, NAUI, Montclair, CA. 1992, 40 pages.

Wienke, B. Up & Down, ADM, Issue 7, 18-22.

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Credit:

Portions of this article  were used in my chapter on Dive Physics appearing in:

Bove and Davis' Diving Medicine (4 th Edition), published by Saunders (Elsevier)

 About The Author: 

Larry "Harris" Taylor, Ph.D. is a biochemist and Diving Safety Coordinator at the University of Michigan. He has authored more than 100 scuba related articles. His personal dive library (See Alert Diver, Mar/Apr, 1997, p. 54) is considered one of the best recreational sources of information In North America.

  Copyright 2001-2004 by Larry "Harris" Taylor

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