Altitude types: pressure, indicated, true, density

The concept of altitude as it relates to skydiving is not as straightforward as it is often made out to be. It would be nice in certain respects for skydivers if it was as simple as “altitude is how high up you are.” And in a sense it is: what we care most about is how high we are above the point we are currently directly positioned over above the earth. This point is constantly shifting in reality, whether in freefall or under canopy—in most cases not meaningfully, but above mountainous or hilly terrain it can. This is the concept of true altitude AGL (above ground level) most of us think of when we as skydivers think of the word altitude. This is what we would really like to know, as well as that which would technically be most useful for an AAD (automatic activation device).

The irony is that nothing particularly close to true altitude has ever been displayed on a skydiver’s altimeter nor used in an AAD. The reasons for this go beyond the aforementioned drift and hilly terrain; a very specific and cumbersome set of technologies (e.g. a base station receiver on the ground, RTK GPS, LiDAR for high-resolution terrain mapping) would be required to make this happen. It isn’t even remotely feasible on a mass scale. The radar altimeters used in general aviation have problems of their own and are not feasible for skydivers either. We therefore approach the problem from other angles.

The most common approach to measuring altitude is to use a barometer and measure air pressure, which drops as altitude increases. This air pressure reading is then converted to an altitude value by using a specific mathematical formula and making certain assumptions (see below). The value displayed on the altimeter is what is known as the indicated altitude. Even in the case of a true RTK GPS altimeter (or a regular GPS altimeter for that matter), the displayed altitude is still always referred to as indicated altitude and would naturally have some deviation from true altitude, which can be approached but never known.

This of course raises the question of how exactly pressure relates to altitude and what we lose by using it and converting it with a set of assumptions.

Converting air pressure to altitude

In converting air pressure to an altitude value, we use a standardized ISA (International Standard Atmosphere) formula that uses MSL (mean sea level) as a reference point. The standard formula is metric, and assumes a specific, dry atmosphere—15 °C at sea level, cooling at 6.5 °C per 1,000 meters of climb. The cooling rate is referred to as the lapse rate: how fast temperature decreases with increasing altitude. Specifically, this is the formula to get the pressure altitude h, in meters MSL, from pressure p in pascals:

So the input is solely pressure and the output is altitude—nothing else is measured. Temperature is in the formula, but in the form of assumptions: the standard 15 °C and its lapse rate are frozen into the constants we will explain in detail below, and current actual temperature is never part of the equation. Humidity isn’t in there at all—even though both of these genuinely affect air pressure in a real atmosphere. One consequence is worth stating plainly: on the assumed day itself—15 °C at sea level, standard cooling, bone dry—a barometric altimeter is perfect. Every error discussed on this page is a real day disagreeing with the assumed one.

Taking the formula apart, p / 101,325 represents your measured pressure p as a fraction of standard sea-level pressure (defined as 101,325) in pascals: p = 0 is a vacuum, p = 101,325 is the standard atmosphere’s sea level and any higher pressure reads as negative altitude. The number 0.190263 is derived by combining three pieces of physics, and each piece brings its own constants to the result: the law that the pressure at any height is simply the weight of all the air still above you (which brings gravity g₀); the ideal gas law, which ties together a gas’s pressure, density, and temperature (which brings the gas constant R and the molar mass of air M); and the assumed steady cooling with height (which brings the lapse rate L). Combine them and this number results—a formula of its own:

where R is the universal gas constant expressed in J mol⁻¹ K⁻¹—read (joules, per mole, per kelvin): the exchange rate between temperature and energy. Warm one mole of a gas by one kelvin and its pressure-times-volume energy rises by exactly this many joules; R is what lets an equation set an amount of warm gas equal to an amount of mechanical work. Then L the lapse rate in K/m (the 6.5 °C per 1,000 m from above, written per meter), g₀ standard gravity in m/s², and M the molar mass of air in kg/mol.

So the two laws supplied the three natural constants (R, g₀, M), and one assumption was decided upon by a committee (L). The exponent is also the reason the curve is a power law rather than a simple exponential: an atmosphere that didn’t cool with height would decay exponentially instead. And it has a hidden meaning: the pressure ratio raised to this exponent equals the temperature ratio in the assumed atmosphere; the bracket is computing how far your temperature has fallen toward absolute zero.

44,330.77 is where the assumed 15 °C at sea level is buried. This constant is exactly the sea-level temperature expressed in kelvin divided by the lapse rate: 288.15 K (which is 15 °C) / 0.0065 K per meter—in other words, the altitude at which the assumed atmosphere, starting at 15 °C and cooling 6.5 °C every 1,000 meters, would reach absolute zero. (The real atmosphere stops cooling near 11,000 meters and the formula is only used below that.)

In such a way, the formula represents your altitude as the fraction of the way you’ve cooled toward absolute zero, times the height where absolute zero would be reached if you started at 15 °C at mean sea level and the temperature dropped by 6.5 °C per kilometer going up.

One more number of particular interest results from the same four constants. Multiply the ceiling by the exponent—44,330.77 × 0.190263—and you get about 8,434 meters: the atmosphere’s scale height. If you climb one scale height, pressure falls to exactly 1/e of what it was—36.8% left, 63.2% gone—at least in air held at one temperature; in the reality of the cooling atmosphere the drop-off is a few points steeper, because the scale height itself shrinks with the cold. In that theoretical 15 °C world the whole pressure curve can be expressed simply as:

Look at what the exponent is doing in this equation. An exponent has to be a pure number (and not include a unit)—you cannot raise e to “four thousand meters”—so your height is first divided by the scale height. The number the exponent actually receives is the answer to a simple question—how many scale heights up am I?—with a minus sign attached, because pressure shrinks rather than grows as you climb. Concretely: a jumper at 13,500 feet is at 4,115 meters, and 4,115 / 8,434 = 0.49—about half a scale height up. The formula then computes e^−0.49 ≈ 0.61: at exit, roughly 61% of sea-level pressure remains. One scale height up, 1/e of the pressure remains; two up, 1/e of that; and so on. 1/e appears throughout nature wherever a quantity decays at a rate proportional to its own current size: after one characteristic interval—here, one scale height—exactly 1/e of it remains. The scale height itself has a plainer but also interesting meaning: it is how deep the atmosphere would be if it kept its sea-level density all the way up. If you squashed the entire atmosphere into one uniform layer, it would reach up to 8,434 meters—just shy of the height of Mount Everest. Half of the atmosphere’s mass sits below roughly 18,000 feet MSL; a jumper exiting at 13,500 feet is above 40% of the air on Earth.

Density altitude: air characteristics

Whereas the other three types of altitude we have covered refer to a height or elevation, density altitude refers to the characteristics of air when taking into account its pressure, temperature, and humidity. It essentially says where the air you are actually in would fit in the standard atmosphere (described above). The process goes as follows: measure the air’s pressure, temperature, and humidity; compute the density they imply; then find the altitude at which the standard atmosphere holds air of that density and report that altitude. This is not what barometric altimeters report: they are blind to air temperature and humidity. Density altitude is most often used by skydivers on the ground when considering canopy performance, and it can have truly dramatic effects that are important to understand.

An example. At Skydive City, which sits at 90 ft, on a 95 °F day with a 75 °F dewpoint (about 53% relative humidity), the density altitude would be about 2,760 ft. In performance terms, this makes a skydiver’s canopy, body, and jump plane all behave more like they would at Skydive Las Vegas—located at 2,832 ft field elevation—than they would at Skydive Santa Barbara, which has a field elevation literally only 2 ft lower than Skydive City at 88 ft. True airspeeds run about 4% higher for the same indicated numbers— the ground approaches quicker and a longer, more drawn out flare is required. The climb to altitude takes noticeably more time as well; of that 2,760 feet, roughly 370 is the humidity’s share alone—the rest is temperature and field elevation.

Humidity’s role is the one that surprises people, as it runs contrary to our physical sense of it: humid air is lighter than dry air, not heavier. A water molecule weighs 18 units; the nitrogen and oxygen it displaces weigh 28 and 32. Adding moisture therefore replaces heavy molecules with light ones. Muggy air is thin air.

Temperature and humidity are interrelated; heat does not amplify humidity’s effect—heat allows more humidity to exist. A given amount of water vapor has essentially the same effect whether the air around it is cold or warm. What changes with temperature is how much water the air can hold, and that ceiling roughly doubles every 10 °C. Air at 41 °F can be saturated and still be carrying half a percent water by weight; at 95 °F it can hold nearly four percent. So a hot day is not simply a hot day: it is a hot day that is also permitted to be a very wet one, and the two effects stack in the same (dangerous) direction.

Computing density altitude takes three steps. First the water vapor pressure e (meteorology’s traditional symbol—confusingly no relation to Euler’s constant e from earlier) from the dewpoint T_d in °C:

Then the actual air density, treating dry air and water vapor as two gases sharing the same space:

where T is temperature in kelvin, R_d = 287.058 and R_v = 461.495 J kg⁻¹ K⁻¹. The two constants differ because water vapor is lighter—that difference is the entire humidity effect.

Then read that density back off the standard atmosphere, which gives density altitude in kilometers:

Returning to our example: The 75 °F dewpoint gives a water-vapor pressure of about 2,957 Pa—that is how much of the day’s pressure is water. Subtracting it from the station pressure (about 100,996 Pa at 90 ft) and treating dry air and water vapor as two gases sharing the same space gives the actual density: 1.129 kg/m³. Reading that density back off the standard atmosphere—asking at what altitude the standard keeps air this thin—returns about 841 meters: the 2,760 feet above.

Our density altitude calculator pulls the current observation from any airport weather station and runs these exact formulas live.

An illustrative example

Let us again use Skydive City as an example. For the purposes of the example, let us assume it is now cooler at 90 °F, the dewpoint is still 75 °F, and this time there is a ridge of high pressure overhead, holding the day’s barometric pressure slightly above standard. The numbers break down as follows:

We thus have five readings spanning 2,328 feet. Note that while the indicated and true altitudes match on both the skydiver’s and pilot’s altimeters, this is only because the pilot’s is manually set according to the current sea-level pressure and the skydiver’s automatically calibrates to current pressure altitude and zeros. As they climb, their indicated altitude will diverge from true altitude, for reasons covered in this article.

And note the last row: the grass performed like 2,253 feet of thin air while the zeroed altimeter read 0. At that density altitude a canopy flies roughly 3% faster in true airspeed for the same indicated airspeed—the measurable reason behind everything people say about hot-day landings, and why density altitude is worth knowing even though no altimeter shows it.

The four types of altitude

Sometimes the idea of MSL and AGL is conflated with the types of altitude. MSL and AGL have the idea of a datum—a reference where something is measured from: sea level or ground, respectively. The four types of altitude instead refer to quantities.

Only indicated and true altitude can have AGL and MSL values, as their datum provides a height offset. Pressure and density altitude do not directly refer to a height and are instead a reference to the atmosphere.

Pilots are typically taught a list of five types—indicated, true, absolute, pressure, and density—where absolute altitude means height above the terrain. Our classification has clearer delineations and highlights the fact that true and absolute altitude are merely MSL and AGL variations of the same type of quantity.

Indicated altitude: what zeroing actually does

In our example above, the pressure altitude was −75 feet because the day’s pressure was higher than the standard atmosphere’s. The skydiving altimeter nonetheless read 0. Zeroing simply moves the reference level to the point that would be of relevance to a skydiver. Modern digital skydiving altimeters do this automatically on the ground; analog ones have a dial to zero them. A pilot instead dials the current reported sea-level pressure into the Kollsman window—the small setting readout on the altimeter’s face—which is why the pilot’s altimeter read +90. In terms of the formula from earlier, setting an altimeter simply replaces the 101,325 with the pressure the setting refers to—that is the whole of it.

Approximating true altitude using air pressure

The barometric method has significant limits when it comes to approximating true altitude that increase with altitude, as you would need the average temperature and humidity of the entire column of air beneath you to compute it.

The resulting gap between indicated and true is the temperature—and to a lesser extent the humidity—error. This is covered in a skydiver’s guide to how barometric altimeters work. In short: cold air makes your altimeter read high (the dangerous direction), and hot air makes it read low. On a 90 °F day at a sea-level dropzone, for example, an indicated 13,500 feet at exit can be more than 600 feet higher in true altitude AGL. The error grows with altitude and shrinks to zero at the ground.