# At-Altitude Arithmetic

By Larry "Harris" Taylor
November 1, 1998
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This is an electronic reprint of an article that appeared in SOURCES (Sept/Oct. 1994). It is part of the diving physics chapter of NAUi's Advanced text, Mastering Advanced Diving. This material is copyrighted and all rights are retained by the author. This material 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.

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?

ENGLISH ANSWER: First calculate the depth of water (x) that corresponds to one atmosphere at the observed barometric pressure. Remember that atmospheric equivalent height is inversely proportional to the density of the fluid being used to measure pressure:

```
24.61 in Hg             1.0 g/cc
-----------   =    --------------
x   in H20           13.6  g/cc

x = 334.7 inches water
```

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

```334.7 in   x   1 ft   =  27.9 feet
------
12 in
```

Thus, every 27.9 feet of fresh water (not 33 fsw) 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:

```60 ffw       =   2.15 atm
------------
27.9 ffw/atm
```

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:

```24.61 in Hg   x  3.2 ata  =    78.75 in Hg
-----------
ata
```

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

```
78.75 in Hg                =   2.63 sea level ata
--------------------------
29.92 in Hg/sea level ata
```

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:

```1.6 atm x  33 fsw    =  52.8  =  53 fsw
------
atm
```

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

METRIC SOLUTION:br> Determine the water equivalent of one atmosphere at this altitude:

```625 mm Hg         1.00 g/cc
-----------   =   --------------
x  mm H20        13.6  g/cc

x = 8500 mm H20
```

This converts to:

```8500 mm  x       1 m  =  8.5 m
-------
1000 mm
```

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

At depth of 18.29 mfw, the hydrostatic pressure is:

```18.29 m        =    2.15 atm
----------
8.5 m/atm
```

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:

```3.2 atm x   624 mm Hg   =  1996.8 mm Hg
---------
atm
```

This corresponds to a sea level pressure of:

```1996.8   mm Hg                =   2.63 sea level ata
--------------------------
760  mm Hg/sea level ata
```

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:

```1.6 atm  x   10.1 m   = 16.2 m
------
atm
```

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 sea water 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:

```ENGLISH:          2.16 atm   x     33 fsw     =   71.3 fsw
------
atm

METRIC:           2.16 atm   x   10.1 msw     =   21.8 msw
--------
atm
```

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:

```ENGLISH:        60 fsw  x     1 atm     =    1.82 atm
------      -------               ----
min       33 fsw                min

METRIC:         15 m    x     1 atm     =    1.49 atm
----         ------               ---
min          10.1 m               min
```

At this altitude; corresponding at-altitude ascent rate:

```ENGLISH:      1.82 atm  x   27.9 ffw    =    50.8 ffw
--------      --------              ----
min           atm              min

METRIC:       1.49 atm  x    8.5  m     =    12.7   m
---           ---              ----
min           atm               min
```

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.

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