A Discussion of Various Measures of Altitude
MJ Mahoney
Created: Oct 22, 2001
Last Revision: Nov 17, 2008
It is common in dealing with airborne research data to encounter many
different altitude terms. These include geometric altitude, GPS
altitude, INS altitude, pressure altitude, geopotential height, and so
on. Despite the nomenclature, there are only two altitude scales
involved: geometric altitude and geopotential altitude or height. Geometric
altitude is the scale we are most
familiar with; it is what we would measure with a tape measure. Radiosondes and
rawindsondes
(collectively, RAOBs), on the other hand, generally report geopotential
height -- a scale which relates height to gravitational
equipotentials, or surfaces of constant gravitational potential energy
per unit mass. Although geopotential height approximates geometric
height, they are not the same. An important type of geopotential height
is pressure altitude, which is based
on a standard atmospheric model
for temperature as a function of pressure. One particular model, the
International Standard Atmosphere (ISA), is what all aircraft
altimeters use to relate static pressure
measurements on an aircraft to a corresponding pressure altitude scale. There are also a number of
additional altitude terms
related to flying airplanes such as true altitude, indicated altitude,
absolute altitude and density altitude, which I discuss on another
page for completeness.
Geometric and GPS Altitude
First we discuss geometric altitude, since it is the altitude scale
that we are most familiar with. It is the height that we would measure
with a tape measure in some length unit, say meters .
Obviously it would be difficult to measure the height of an airplane
with a tape measure, but other instruments are viable such as radar
altimeters. However, altimeter measurements are relative to the local
terrain, which could range from sea level to a mountain top. If
airborne data are to be compared, it is necessary to introduce a
reference surface to which all measurements are compared. This gets a
little complicated because there are many possible reference surfaces,
or vertical datums . Historically, mean sea level (MSL) has
been used as the zero of elevation, and this is closely approximated by
an equipotential surface for the earth's gravity field called the
geoid. Although much smoother than the topographic surface of the
earth, the geoid has significant vertical undulations due to
the large scale distribution of the earth's mass. A much smoother
reference surface is the reference ellipsoid, which is used to
the approximate the shape of the earth. The reference ellipsoid
is convenient because three dimensional coordinates (latitude,
longitude, and altitude) are easily defined with respect to it.
This is a convenient point to introduce the Global Positioning
System (GPS) of satellites. The reference ellipsoid for GPS
is the so-called WGS-84 ( World Geodetic System
1984 ) system, which is defined by four parameters:
Semi-major axis (a):
|
6378.1370 km
|
Reciprocal of flattening (1/f):
|
298.257223563
|
Angular velocity of Earth (w):
|
7292115.0 x 10-11 rad/s
|
Earth's Gravitational Constant (Atmosphere
incl.) (GM):
|
3986004.418 x 108 m3/s
2
|
Flattening (f) is a term used in geodesy instead of either the
semi-minor axis or eccentricity; it is related to the semi-major and
semi-minor axes by the expression: f = (a - b) / a. Using
this expression, the semi-minor axis (b) is: 6356.7523 km, so the
equatorial
radius of the earth is 21.4 km greater than the polar radius. Note that
the angular velocity is important because gravity has two
components: the gravitational force between the Earth and another mass,
and the centrifugal force due to the fact that the Earth is a
non-inertial
reference frame.
Using GPS satellite data, the WGS-84 system has been refined several
times, most recently in 1999. However, because of the number of GPS
receivers in use and the cost to replace them, the refinements are only
used for satellite orbit
determinations. Time corrections are transmitted via the GPS
satellites which work with the WGS-84 model software
inside the GPS receivers. GPS measurements made relative to this
reference ellipsoid are generally converted to different vertical
datums by the software internal to the GPS receiver. This involves
a geoid model, the current one being EGM96 ( Earth
Gravitational Model 1996 ). EGM96 is a spherical harmonic expansion
of the gravitational potential of the Earth through degree (n) and
order (m) 360, and is comprised of 130,317 coefficients!
At the risk
of over-complicating this discussion, but in the interest of
completeness, some more altitude terminology needs to be introduced.
The height at or above the earth's surface can be measured with respect
to the reference ellipsoid or the geoid , and these
can differ by as much as 100 meters. While these reference surfaces
differ in an absolute sense, they nevertheless lie on
the same equipotential surface -- the surface defined by mean sea
level. Referring to the figure to the right, the height normal to the geoid
is called the orthometric height (H). The normal distance,
which can be positive or negative, from the reference ellipsoid
to the geoid is called geoid height (N). Finally the
normal distance from the reference ellipsoid to some point
above the earth's surface is termed the ellipsoid height ,
or geodetic height (h); it is the sum of the geoid height
and the orthometric height . For further discussion go to
the National
Imagery and Mapping Agency or the National
Geodetic Survey .
For the purpose of the present discussion, we are interested in both
the orthometric height and the
ellipsoid height ; the former is needed since it is in the
direction of normal gravity measured when surveying, and the latter is
needed because it is what a GPS measures.They are related by the
excellent
approximation expression: h = N + H.
Pressure
Altitude
Allthough
pressure altitude is a type of geopotential height, I discuss it
separately because of it's importance in atmospheric research. In fact
the initial motivation for this web page was to show other scientists
how to relate pressure altitude and geometric altitude. The motivation
was simple: some instruments like lidars measure distance on a
geometric altitude scale, while others, like my Microwave Temperature Profiler (MTP),
measure distance on a pressure altitude scale. The pressure
altitude scale is based on the
International Civil Aviation Organization's (ICAO) International
Standard Atmosphere
(ISA), which is the same as the US
Standard Atmosphere (1976) to an altitude of 32 km. The US
Standard atmosphere (1976) is an average, piece-wise
continuous, mid-latitude temperature profile of the earth's atmosphere.
It can be used to establish -- via the hydrostatic equation and the
ideal gas law -- a
relationship between pressure and pressure altitude, using geopotential height. It differs from "normal"
geopotentail height in that it is based on a model and it assumes that
the humidity is zero. The model seldom looks like the actual atmosphere
a plane is flying in, and real atmopsheres never have zero humidity.
The US Standard Atmosphere is
one of several reference or
standard atmospheres. I only use it instead of the ISA because when
I started doing this it was easier for me to get my hands on
information about it than ISA. My apologies to aviation purists.
Pressure altitude is used so that aircraft, which use static pressure
to determine altitude, can agree upon what "altitude" they are flying
at without having to continually update their altimeters with local
pressure corrections. Technically, this is only true above 18,000 feet
(FL180). Below this altitude in North America aircraft make local
altimeter
corrections
to ensure that they are flying at the correct altitude. Using this
definition for pressure altitude, a pilot can say "I'm at Flight Level 330." (that is 33,000 feet), instead of "I'm at 262 hPa." Pressure just isn't very intuitive
since it's logarithmic with altitude, and it also decreases with
altitude. In addition to pressure altitude, there are five additional altitude
scales relevant to aviation.
Pressure altitude is also
useful to scientists involved with radiative transfer through the
atmosphere since the absorption is normally expressed in terms of
pressure (and other parameters such as temperature, frequency, and
humidity). So
pressure altitude is a "natural" co-ordinate for such calculations. On
the other hand, absorption coefficients are often expressed per unit
geometric distance, so it is often necessary to convert between the
different distance/altitude scales depending on the application.
Geopotential Height
Having described one type of geopotential height with the cart in front
of the horse, we now describe how to calculate geopotential height. The
gravitational potential energy (
) of a unit
mass of anything is simply the integral from mean sea level (z = 0
meters) to the height of the mass (z = Z). It is given by the equation:
(1) 
,
where 
is the normal gravity above the geoid. It
is a function of both geometric altitude (z) and geodetic latitude (
). Note
that normal gravity is what would be measured by a
plumb line and it includes contributions from both gravitational and
centrifugal forces. Now while the geopotential (potential energy per
unit mass) is useful for atmospheric dynamics studies (since it is a
convenient
way to compare meteorological data from different locations), it would
be more convenient if it could be expressed as a height above the
geoid. To this end, the geopotential ( ) was divided by
the normal gravity (
45) at a latitude of 45 degrees to
obtain the geopotential height scale:
(2) 
A latitude of 45 degrees was chosen because it was the latitude used by
the World Meteorological
Organization (WMO) to calibrate barometers.
Also, since surface gravity is greatest at the poles and least at the
equator, this "splits the difference" and results in geopotential
height being close to geometric height at mid-latitudes, since only the
weak
altitude dependence of remains. The difference between
geometric altitude and geopotential height can be significant (~120
meters) near the equator at ER-2 cruise
altitudes (20 km). Also, since
decreases with height (except near the poles), the geopotential height
is generally less than the geometric height. The original definition of
45
was 9.8 m/s2; however, this was changed in the
US in 1935 to the current value: 45=9.80665 m/s2.
Finally, the earth's gravity model has been modified slightly since
this most recent definition of 45, so 45
is now the gravity at 45.542 degrees instead of 45.0 degrees. However,
the definition of the geopotential height has not changed.
Historically, the motivation for introducing the concept of
geopotential height was to come up with a scheme by which the
pressure, temperature, and relative humidity measured by radiosondes
around the world could be compared on a common altitude scale. To
this end, we begin with the ideal gas law:
(3) 
,
where p(z) is the pressure, 
(z) is the density,
Rd is the gas
constant for dry air (287.05307 J /kg K), and Tv(z) is the
virtual temperature. T v is derived by re-writing the ideal
gas law for dry air in a manner which accounts for water vapor, thus
allowing the gas constant for dry air to be used. The resulting
expression is:
(4) 
.
Tv is a function of pressure (p), temperature
(T), and relative humidity (RH), and in the form given here also
involves the saturation vapor pressure for water vapor (es
), which a function of only T, and the ratio of the molecular weights
of wet and dry air (
= 0.622). Tv
accounts for the fact that air
containing water vapor is less dense than dry air, and it is always
greater than the actual physical temperature. If the virtual
temperature
were not used, the gas constant would be a function of the vapor
content (that is, it wouldn't be constant).
Now the differential form of the hydrostatic equation is:
(5) 
,
where
(z) is the density and is gravity. If we use the ideal
gas law (equation (3))
to
eliminate (z), we obtain the
differential form:
(6) dz = -Rd T
v(z) d [ln(p(z))]
Now referring to equation (1) for the gravitational potential energy,
it is immediately recognized that the
integral of the left hand side of equation (6) is the gravitational
potential energy per unit mass of air. On dividing equation (6) by
45
and integrating from z = Z1 to z = Z2
, we arrive at an expression for calculating the geopotential height
between any two radiosonde levels without knowing anything about the
local gravity.
(7) 
,
The right-most expression is known as the hypsometric equation
; it is important since it relates the thickness, H12(Z,), between two pressure
surfaces (p1 and p2) directly to the average
virtual temperature (Tv avg) (and two constants: Rd
and 45). The hypsometric equation
can be summed between radiosonde pressure levels from the surface to
the highest measurement to obtain the geopotential height at any
radiosonde level. This of course can only be done from the surface,
which generally will not be at sea level. However, the WMO provides the
geopotential height for every radiosonde launch site, so this offset
can be added to the sum to get the geopotential height relative to the
geoid, or mean sea level.
Finally, we address the issue of converting between geopotential height
and geometric height. This can be done by writing the expression for
the geopotential height in differential form:
(8) 45 dh = (z,) dz.
In fact, this expression is often used as the definition
of the geopotential height. Denoting for simplicity R as the radius of
the earth at latitude,
and using the inverse square law for gravity, we can write for any
latitude ( ) and geometric
altitude (z):
(9) (R + z,)
= (R, ) (R/ (R + z))2.
Substituting equation (9) into equation (8) and integrating from z = 0
to z = Z -- assuming (z, )
does not vary with z -- gives:
(10) 
Now of course (z, ) does vary with geometric altitude (z) so this
must be
dealt with. A trick suggested by W. D. Lambert in 1949 in the Smithsonian
Meteorological Tables (SMT), which is a frequently cited
source for converting between geometric altitude and geopotential
height, was to compensate for the variation of (z, ) by
letting R assume
an appropriate value that satisfied certain boundary conditions. This
would account for the fact that
equation (9) only applied to a non-rotating sphere composed of
spherical
shells of equal density, which is not true for earth, and for the fact
that the centrifugal component of gravity increases linearly with
radius, rather
than inversely with the square of the radius as shown in this equation.
Taking the derivative of equation (9) with respect to z, evaluating the
resulting expression at z = 0, and then solving for R, we obtain:
(11) 
Note the minus sign in the denominator of equation (11), and remember
that the derivative is evaluated at z=0 (which is why the z dependence
is dropped on SMT). The
notation RSMT
is used because this technique was originally described in the Smithsonian
Meteorological Tables (R. J. List, Editor, 1968). Note that RSMT
is not the radius of the Earth at any particular latitude; rather it is
a value that
is needed to account for the combined effect of gravitational and
centrifugal
forces with altitude z. Using this notation, equation (10) becomes:
(12) 
,
or on inverting to solve for Z
(13) 
.
Since the later equation is often used in calculations to convert
radiosonde geopotential height to geometric altitude, a Taylor series
expansion can
be used over radiosonde altitudes to obtain:
(14) 
where H is in km and we used 45 = 9.80665 m/s2.
This equation would agreed exactly with the Smithsonian Meteorological Tables if
we
had used 45 = 9.80 m/s2. There are some
issues however with
using the Smithsonian Meteorological Tables that go beyond the
fact that we have computers now that can easily calculate the
conversion
between geometric and geopotential height. The tables use the International
Ellipsoid 1935 (which is different from WGS-84) and they assume
45
= 9.8 m s-2 exactly. But there are other
issues. The value of normal gravity is stated to be:
(15) 
and is based on a 1949 report of the International Association of
Geodesy titled: "Gravity Formulas for Meteorological Purposes "
by W. D. Lambert. The tables also use the following expression for the
negated derivative of normal gravity:
(16) 
Unfortunately, the origin of this expression has not been published (W.
D. Lambert, Some notes on the calculation of the geopotential,
unpublished manuscript , 1949), so the assumptions going into it
are not clear. In any event, if the value of R found in equation (11)
is used in equation (10), it satisfies two boundary conditions: it has
the correct surface gravity, and it has the correct vertical gravity
gradient
based on the International Ellipsoid of 1935. Lest there be any
doubt that this is a fictitious radius, it is easily shown that the
radius RSMT
is greater at the pole than at the equator, and very different from any
real earth radius (see Figure 3).
Figure 3. The ficticious
"radius" ginned up by W. D. Lambert (Rsmt)
behaves in a completely different manner from the ellipsoidal radius of
the earth (Rellipsoid) and can
they can differ by >40 km, yet many papers and online discussions
mistakenly refer to it as the "radius of the earth at a particular
latitude." Rwgs is a
closed-form version of the ficticious radius, which is derived below. Russa is the ficticious radius
(based on the Smithsonian
Meteorological Tables with g = 9.80665 m/s2) at a latitude of
45.542 degrees assumed for the US
Standard Atmosphere (1976) (see discussion below). Rsmt and Rwgs differ by almost 350 m.
Rather than work with this undocumented result of W. D. Lambert, I have
followed
the same approach using the currently recognized ellipsoid of
revolution, namely WGS-84.
Before proceeding, we need to specify a
number of additional derived parameters for this earth reference
system. They include:
Semi-minor axis (b)
|
6356.7523142 km
|
Flattening ( f = (a - b) / a )
|
0.003352811
|
Linear eccentricity (  )
|
521.854008974 km
|
Eccentricity (e= E / a )
|
0.081819
|
Polar gravity ( p )
|
9.8321849378 m s -2
|
Equatorial gravity ( e )
|
9.7803253359 m s -2
|
Somigliana's Constant ( )
|
1.931853 x 10-3
|
Gravity ratio ( )
|
0.003449787
|
Armed with these values, we begin by writing Somigliana's Equation for
normal gravity on the surface of an ellipsoid of revolution ( Heiskanen
and Moritz, Physical Geodesy, 1969):
(17) 
On performing a Taylor series expansion in the vertical,
we obtain the following
expression for the normal gravity:
(18) 
,
or numerically:
(19) 
.
On taking the derivative of this expression with respect
to z and evaluating it at the surface, we obtain the following version
of equation (11):
(20) 
,
or numerically:
(21) 
This expression can then be used in equation (10) to convert from
geometric altitude to geopotential height; that is:
(22) 
H(Z,
) calculated using this expression agrees with direct integration of
the value of gravity given by equation (18) to within 2 mm at the
equator at 20 km! Furthermore, if a more
accurate expression is used for gravity , instead of the Taylor
series expansion given in equation (18), the agreement is better than
1.5 mm at all latitudes at 20 km! As noted above, it is often necessary
to invert equation (22) to solve for Z; that is:
(23) 
.
If equation (23) is approximated by a Taylor series expansion, and
optimized for altitudes from 0 to 25 km, we have:
(24) 
to an accuracy of 20 cm, the best accuracy being near the ground and at
~20 km.
As an aside, it is worth noting that the US Standard Atmosphere 1976
provides the following expressions for the geometric/geopotential
height conversion:
(25) H(Z) = Ro Z / (Ro + Z) ,
or, on solving for Z,
(26) Z(H) = Ro H / (Ro - H).
where Ro = 6356.766 km. Since the US Standard Atmosphere
is nominally defined for a latitude of 45 degrees, and no other, these
equations
are trivially derived from equations (22) and (23) by assuming s( ) = 45
= 9.80665 m/s2. As mentioned earlier, gravity is
9.80665 m/s2 at a latitude of 45.542 degrees, and the
value of Ro used in the US Standard Atmosphere is the
ficticious radius at this latitude (not 45 degrees).
Formulae are also provided in the Federal Meteorological Handbook
No. 3, which discusses requirements for rawindsonde and pibal
observations (Office of the Federal Coordinator of Meteorology, 1997).
Regrettably, this primary reference uses equation (15) for ( ),
which is based on a 1935 reference ellipsoid, and it states that the
relevant radius in equation (22) is "Re, the radius of
the earth at latitude ." As explained above, this is incorrect.
I have attempted several times to contact the
OFCM and my correspondence has not been acknowledged. Not surprisingly
I have found many web sites with conversion calculators that blindly
use this incorrect information. Caveat Emptor!
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