In the use of self-contained or supplemental breathing systems which draw from a fixed volume source, it is often desirable for the user to be able to predict how long the remaining air or gas supply will last.  In SCUBA (Self Contained Underwater Breathing Apparatus) diving, for example, it is particularly useful to know how long a diver may continue at his present task and still have enough air to make a safe ascent to the surface.  For systems supplying oxygen gas to people in unpressurized aircraft, it is useful to know how long the supply of oxygen will last, particularly if the aircraft changes altitude. 
Present practice makes use of pressure gauges to tell the user how much air or gas pressure he has left, and charts which list schedules showing how much time can be expected for a given tank volume and pressure.  In SCUBA diving, for example, divers are trained to know that a typical 72 cubic foot tank pressurized to 2250 psi will last about 1 hour at a depth of 33 feet (10 meters).  The diver also knows that that same amount of air will last him only half as long at a depth of 99 feet (30 meters).  Clearly, estimating how much time remains on a given supply of air is exceedingly difficult for a diver changing depths or an aviator changing altitudes. 
The reason that endurance time varies with depth or altitude is due to a combination of human physiology and Boyle's Law of gases.  A typical person can inhale and exhale about 4 liters of air with one breath.  This breath volume stays the same whether he is in an unpressurized airplane at 18,000 feet or in a pressurized diving bell 200 feet below the surface of the sea.  However, the actual amount of gas represented by each breath at these two extremes is very different.  According to Boyle's Law, for any given volume, the number of gas molecules is directly proportional to the absolute pressure (all other things being unchanged).  At 18,000 feet, the absolute pressure is about one half an atmosphere (1/2 Bar).  At a depth of 200 feet of sea water, the absolute pressure is about 7 atmospheres (7 Bars).  This means that a person breathing 4 liter breaths at a depth of 200 feet is using about 14 times as much air per breath as a person at 18,000 feet.  If the person is breathing from a fixed supply such as a tank, ambient pressure has a tremendous effect on how long that tank will last.  For example, a 72 cubic foot SCUBA tank will provide air for about 3 hours to an aviator at 18,000 feet altitude and about 13 minutes to a diver 200 feet under the surface of the sea. 
In the following discussion, the term "air" is used for simplicity, but "air" should be taken to mean any breathable gas or mixture of breathable gasses.  Air supply levels and consumption rates are referred to in terms of pressure rather than volume or mass because pressure and pressure changes are easy to measure and work well for the present purpose.  To illustrate the invention, the SCUBA model will be used although the principles apply equally to other situations where gas is being breathed from a fixed volume container. 
Estimating the amount of time a SCUBA diver's compressed air supply will last is exceedingly important.  As might be expected, such estimates are very difficult to make accurately when the diver operates at many different depths.  Further, the diver must leave the depths before his air runs out; he must have enough air left in his tanks to make the surface safely. 
The prior art has attempted to estimate remaining gas consumption time in various ways.  The most common method has been to measure the gradual reduction in gas supply pressure and then to calculate remaining time by extrapolating that reduction rate over the remaining gas supply pressure.  Another method has been to estimate a breathing consumption rate at the surface and then adjust that rate for ambient diving pressure to estimate gas consumption during the dive.  Both methods ignore the gas required for ascent and so have no way of providing the diver with accurate information about how much longer he can safely stay without running out of gas on his ascent.  Also, both methods mislead the diver because they make no allowance in their projections for the fact that the diver's consumption rate will change when he changes depth, as the diver must do when he ascends to the surface, or the fact that the consumption rate measurements may have been made unreliable by significant non-breathing events. 
The prior art approach does not take into account other potential causes for changes in tank air pressure.  Temperature, for example, can have as large an effect on tank air pressure as breathing consumption.  For example, when a SCUBA diver descends from the warm surface to the cold depths, he may see a water temperature change of 15 degrees centigrade or more.  For a tank pressurized to 3000 psi, the drop in pressure caused by the chilling could be greater than the drop caused by 5 minutes of breathing.  Clearly, any estimate and projection of breathing consumption would be very misleading.  Also, a diver occasionally uses air for non-breathing purposes such as clearing his regulator, inflating his buoyancy compensator, or inflating a lift bag.  Obviously, it is desirable for a device that is to make accurate calculations of consumption not to be fooled by factors other than breathing. 
It would be very desirable if the diver had a way to estimate how long his remaining air supply will last in various situations.  In particular, it would be useful for a diver to know if he has time to finish his present task before his diminishing air supply will require surfacing.  If the diver must undergo a decompression regimen, the amount of air required for ascent can represent a substantial portion of the total air supply he has available for the dive.  An accurate estimate of the air required for decompression can be extremely important to the safety of the dive. 
Conventional means for displaying information to the diver is in the form of numbers or simple bar graphs.  As the amount of information required to be displayed becomes greater, the opportunities for the diver to become confused increase dramatically.  This problem is aggravated at depth by nitrogen narcosis.  Narcosis, caused by the combined effects on the human brain of nitrogen and pressure, resembles intoxication.  Narcosis greatly increases the difficulty of understanding displays which would be simply understood on the surface. 
For the purposes of this patent, the terms below have the following meanings.  Where the word "air" is used, it should be taken to mean any breathable gas. 
Ceiling--In dives requiring decompression, the ceiling is the shallowest underwater depth to which a diver may ascend without risking decompression sickness.  During decompression, the ceiling slowly rises, finally reaching the actual water surface and allowing the diver to safely leave the water.  PA1 DCS--Decompression sickness (bends) results from formation of gas bubbles (usually nitrogen) in body tissues.  Gas absorbed by tissues when a diver is breathing pressurized air at depths greater than about 20 feet can accumulate to a point where it will create bubbles and cause DCS if the diver ascends too quickly.  PA1 Decompression--A required delay or series of delays during a diver's ascent to allow dissolved gas in the tissues time to dissipate without the formation of bubbles and thus prevent DCS.  The term may also refer to a dive that will require the diver to make decompression stops instead of ascending directly to the surface.  PA1 Remaining Air Time (RAT)--The amount of time a fixed volume of gas would last at the present gas consumption rate and the present ambient pressure.  PA1 Adjusted Remaining Air Time (ARAT)--The amount of time a fixed volume of gas would last at the present gas consumption rate (RAT) if the rate were modified according to Boyle's Law for ambient pressures other than the pressure at which the gas consumption rate was originally determined.  PA1 No-Decompression Time (NDT)--The amount of time a diver may remain at a particular depth without being required to decompress when he ascends to the surface.  PA1 No-Decompression Line (ND)--On a graph of time and depth, a curve that represents the limits to which a diver may go without being required to decompress.  PA1 Safe Ascent Profile (SAP)--A schedule of depths and times representing a diver's optimum safe route to the surface, including any stops required for Decompression or any delays needed to allow the Ceiling to reach the surface.  When calculated from Ceiling information, the SAP normally projects the diver to ascend to some predetermined depth below the Ceiling (such as 5 feet) and remain below the Ceiling until the Ceiling reaches the surface.  PA1 Ascent Air Requirement (AAR)--The amount of air that will be required for a diver's safe ascent to the surface, including air required for any delays needed for decompression.  This value may also be referred to as the air required for decompression.  PA1 Safe Air Limit (SAL)--The point in time in the future where the air consumed at the present air consumption rate plus the Ascent Air Requirement (AAR) will just equal the remaining available air supply, less any allowances for reserve.  The SAL may be adjusted to give an SAL for depths other than the present.  PA1 Safe Operating Envelope (SOE)--On a two-dimensional graph of time and depth, that area of the graph representing the times and depths where a diver may operate safely.  In a dive where decompression is not permitted, the SOE is normally bounded by the ND line and the surface (if air supply is not considered).  Where decompression is required, the SOE is normally bounded by the SAL line and the ceiling.  PA1 Event Window--In the selection of qualifying breathing events, an Event Window defines the maximum and minimum pressure drops within which an event is considered to be a qualifying event.  PA1 Pictorial Display--A means of presenting complex information in a simple form by using a two dimensional chart or graph with axes chosen so that the user's condition can be easily viewed in spatial relationship to other conditions or options.