4 Atmosphere models for aircraft altimetry
The elements presented in this section are mainly taken from the ICAO (International Civil Aviation Organization) Standard Atmosphere Manual, and the revised atmosphere model ATMORev? used in the Eurocontrol BADA (Base of Aircraft Data) performance model.
4.1 The International Standard Atmosphere (ISA)
4.1.1 The Hydrostatic Equation
\[ -dp= \rho g dh \tag{4.1}\]
where \(\rho\) is the air density.
4.1.2 The Ideal Gaz Law
\[ p= \rho R T \tag{4.2}\]
In this equation, \(R\) is the specific constant for dry air (\(R= 287.05287\quad m^2/K.s^2\)), and \(T\) is the air temperature.
4.1.3 The Geopotential Altitude
In Equation 4.1, the acceleration \(g=g(h)\) due to the combined effects of Earth gravitation and rotation varies with the altitude \(h\). Depending on the chosen Earth model, the expression of \(g(h)\) can be more or less complex.
To simplify the expression of the atmospheric model equations, we introduce a new quantity, the geopotential altitude \(H\), defined by Equation 4.3, where \(g_0 = 9.80665\quad m/s^2\) is the reference value for the gravity of Earth, taken at the mean sea level at a reference latitude.
\[ gdh = g_0dH \tag{4.3}\]
4.1.4 Characteristics of the Standard Atmosphere at Mean Sea Level
The main characteristics of the ISA atmosphere at mean sea level are shown in Table 4.1.
| Earth gravity | \(g_0= 9,80665\) | [m/s\(^2\)] |
|---|---|---|
| Atmospheric pressure | \(p_0= 101325\) | [Pa] |
| Temperature | \(T_0= 288,15\) | [K] |
| Air density | \(\rho_0= 1,225\) | [kg/m\(^3\)] |
| Speed of sound | \(a_0=340,294\) | [m/s] |
4.1.5 Temperature as a Function of Geopotential Altitude
The temperature is a piecewise linear function of the geopotential altitude.
\[ T= T_b + \beta_b(H-H_b) \tag{4.4}\]
The different atmospheric layers, with the values of the temperature gradient up to the altitude of 80 km, are described in Table 4.2. Note that Equation 4.4 can be used for negative altitudes, with the parameters of the layer \(0\).
| N\(^\circ\) couche | Altitude géopotentielle | Limite inf. | Gradient | Nom |
| \(b\) | \(H_p\), [km] | \(T_b\), [K] | \(\beta_b\), [K/km] | |
| 0 | 0 | 288.15 | -6.5 | troposphère |
| 1 | 11 | 216.65 | 0 | stratosphère |
| 2 | 20 | 216.65 | +1.0 | stratosphère |
| 3 | 32 | 228.65 | +2.8 | stratosphère |
| 4 | 47 | 270.65 | 0 | stratosphère |
| 5 | 51 | 270.65 | -2.8 | mésosphère |
| 6 | 71 | 214.65 | -2.0 | mésosphère |
Commercial aviation is concerned by the first two layers, in the troposphere and the beginning of the stratosphere. The troposphere and the stratosphere are separated by an isobaric surface, the tropopause, at \(11\) km altitude.
4.1.6 Atmospheric pressure
Equation 4.2 gives us an expression for the air density \(\rho= \frac{p}{RT}\) which can be replaced in Equation 4.1, by introducing the geopotential altitude (see Equation 4.3).
\[dp= -\rho g dh= -\rho g_0dH= - \frac{g_0}{RT}pdH \]
Taking into account the expression of Equation 4.4 in the altitude layer (numbered \(b\)) where we are located, we obtain:
\[%\begin{aligned} \frac{dp}{p} = - \frac{g_0}{RT}dH = - \left(\frac{g_0}{R}\right)\frac{dH}{T_b + \beta_b(H-H_b)} % = - \frac{g_0}{R \beta_b} \left(\frac{\beta_b dH}{T_b + \beta_b(H-H_b)}\right) %\end{aligned} \]
This leads to two possible expressions of the pressure as a function of the geopotential altitude, depending on the value of the temperature gradient:
\[\begin{aligned} \beta_b \neq 0 & \quad\quad \ln\frac{p}{p_b} = - \frac{g_0}{R \beta_b} \ln\left(\frac{T_b + \beta_b(H-H_b)}{T_b}\right)\\ \beta_b=0 & \quad\quad \ln\frac{p}{p_b} = - \frac{g_0}{R T_b}(H-H_b)\end{aligned} \tag{4.5}\]
\[\begin{aligned} \beta_b \neq 0 & \quad\quad p = p_b\left[\frac{T_b + \beta_b(H-H_b)}{T_b}\right]^{- \frac{g_0}{R \beta_b}}\\ \beta_b=0 & \quad\quad p =p_b \exp\left[- \frac{g_0}{R T_b}(H-H_b)\right]\end{aligned} \tag{4.6}\]
4.1.7 Geopotential Altitude as a Function of Atmospheric Pressure
Conversely, starting from Equation 4.5, we can easily express the geopotential altitude \(H\) as a function of the atmospheric pressure.
\[\begin{aligned} \beta_b \neq 0 & \quad\quad H= H_b + \frac{T_b}{\beta_b}\left[\left(\frac{p}{p_b}\right)^{- \frac{R \beta_b}{g_0}} -1\right]\\ \beta_b=0 & \quad\quad H =H_b - \frac{R T_b}{g_0}\ln\left(\frac{p}{p_b}\right)\end{aligned} \tag{4.7}\]
4.1.8 Air Density as a Function of Geopotential Altitude
The density of air is simply expressed from Equation 4.2.
\[ \rho= \frac{p}{RT} \tag{4.8}\]
It is expressed as a function of geopotential altitude by replacing pressure and temperature by their expressions from Equation 4.6 and Equation 4.4. We then find Equation 4.9, where \(\rho_b= \frac{p_b}{RT_b}\) is the air density at the base of the considered altitude layer.
\[\begin{aligned} \beta_b \neq 0 & \quad\quad \rho = \rho_b\left[\frac{T_b + \beta_b(H-H_b)}{T_b}\right]^{- \frac{g_0}{R \beta_b}-1}\\ \beta_b=0 & \quad\quad \rho =\rho_b \exp\left[- \frac{g_0}{R T_b}(H-H_b)\right]\end{aligned} \tag{4.9}\]
4.1.9 Speed of Sound
The speed of sound in air is given by the following equation, where \(\kappa = 1.4\) for air :
\[ a= \sqrt{\kappa RT} \tag{4.10}\]
4.2 Non-ISA atmospheres for Altimetry
In general, the real atmosphere does not satisfy the assumptions of the International Standard Atmosphere (ISA). An atmosphere can be non-ISA in many different ways. First of all, it is not always composed of dry air only. Also, the temperature and pressure conditions at sea level and/or the temperature gradient may be different from those defined by the standard atmosphere.
For the measurement of aircraft altitude, however, relatively simple assumptions about the atmosphere are made, using the notion of pressure altitude, which we will detail in the rest of this section.
4.2.1 Concept of Pressure Altitude
Let \(p\) be the pressure at geopotential altitude \(H\), in the non-ISA atmosphere modeled in this section. The geopotential pressure altitude (or simply pressure altitude) is defined as the geopotential altitude at which the pressure \(p\) would be measured if the atmosphere were standard.
The pressure altitude is denoted \(H_P\). Note that, by definition, the isobar \(p=p_0= 1013.25\) it hPa corresponds to a zero pressure altitude (\(H_p=0\)).
4.2.2 Hydrostatic Equilibrium, Law of Perfect Gases, Humidity
For altimetry purposes, the non-ISA atmosphere modeled in this section is assumed to be at hydrostatic equilibrium, and to follow the law of perfect gases. The humidity of the air is not taken into account, as in the ISA model. Equation 4.1 and Equation 4.2 which were made for the ISA model remain valid, with the same specific constant \(R\) for air.
4.2.3 Assumptions on Temperature Gradient
The following assumptions are made about the temperature profile, with respect to the altimetry requirements:
the layers of atmosphere are defined in pressure altitude \(H_p\), and not in geopotential altitude \(H\)1. For aviation purposes, we only consider the two lowest layers : the troposphere, and the stratosphere, separated by the tropopause located at the pressure altitude given by Equation 4.11: \[ H_{p,\text{trop}}=11000\quad m \tag{4.11}\]
the temperature depends linearly on pressure altitude (i.e. the altitude that would be observed if the atmosphere were ISA), with the following gradient: \[ \begin{aligned} &\frac{dT}{dH_p} = & -6.5\quad\text{K/km}\quad\quad\text{for}\quad H_p<H_{p,\text{trop}} \\ & & 0 \quad\text{K/km} \quad\quad\text{for}\quad H_p \geq H_{p,\text{trop}}\\ \end{aligned} \tag{4.12}\]
In the following, we will denote \(\beta\) the numerical constant of the temperature gradient in the troposphere:
\[ \beta=-6.5\quad\text{K/m} \tag{4.13}\]
4.2.4 Temperature and “ISA Temperature”
Let \(T\) be the temperature at a given point in the atmosphere located at a pressure altitude \(H_p\). We will denote \(T_{\mathit{\tiny ISA}}\) the temperature that we would have observed at the same pressure altitude (and thus at the same pressure) if the atmosphere had been standard.
According to the assumptions made in this section, in each layer of the atmosphere, the temperature is a linear function of the pressure altitude, with an identical temperature gradient for \(T\) and \(T_{\mathit{\tiny ISA}}\). Consequently, the difference between \(T\) and \(T_{\mathit{\tiny ISA}}\) remains constant whatever the pressure altitude \(H_p\), at the vertical of a given geographical point. We will denote \(\Delta T\) this difference.
\[ T= T_{\mathit{\tiny ISA}} + \Delta T \tag{4.14}\]
Taking the isobar \(p=p_0\) as the troposphere base, the temperature profiles for \(T\) and \(T_{\text{\tiny ISA}}\) are expressed as follows in Equation 4.15, where the pressure altitude of the tropopause is given by Equation 4.11.
\[\begin{aligned} &T= T_0 + \Delta T + \beta H_p & \quad\quad\text{for}\quad H_p<H_{p,\mathit{trop}} \\ &T_{\mathit{trop}} = T_0 + \Delta T + \beta H_{p,{\mathit{trop}}} & \quad\quad\text{à la tropopause}\\ &T= T_{\mathit{trop}} & \quad\quad\text{for}\quad H_p \geq H_{p,\mathit{trop}}\end{aligned} \tag{4.15}\]
\[\begin{aligned} &T_{\mathit{\tiny ISA}}= T_0 + \beta H_p & \quad\quad\text{for}\quad H_p<H_{p,\mathit{trop}} \\ &T_{\mathit{\tiny ISA},\mathit{trop}} = T_0 + \beta H_{p,{\mathit{trop}}} =216,65\quad K & \quad\quad\text{à la tropopause}\\ &T_{\mathit{\tiny ISA}}= T_{\mathit{\tiny ISA},\mathit{ trop}} & \quad\quad\text{for}\quad H_p \geq H_{p,\mathit{ trop}}\end{aligned} \tag{4.16}\]
4.2.5 Reference Altitudes and Levels in Altimetry
4.2.5.1 The isobar \(p=p_0=1013.25\) it hPa (StdRef)
By definition of pressure altitude, the isobar StdRef is at pressure altitude \(H_p=0\). Its temperature (at a given geographical point) can differ from the ISA conditions by a \(\Delta T\) difference.
\[\begin{aligned} &p_{\mathit \tiny StdRef}= p_0\\ &T_{\mathit \tiny StdRef}= T_0 + \Delta T\\ &T_{\mathit \tiny ISA, StdRef}= T_0\\ &H_{p,{\mathit \tiny StdRef}} = 0\\ \end{aligned} \]
4.2.5.2 Mean Sea Level (MSL).
By definition, mean sea level is at geopotential altitude \(H=0\), and at geodetic altitude \(h=0\). The pressure differs from the pressure \(p_0\) under ISA conditions at sea level by a difference \(\Delta p\). The pressure altitude at mean sea level \(H_{p,{\mathit{ \tiny MSL}}}\) is obtained simply by replacing the geopotential altitude \(H\) by the pressure altitude \(H_p\) in the first expression of Equation 4.7 valid for the ISA atmosphere, and applying it to the troposphere by taking as a base the isobaric \(p=p_0\).
\[\begin{aligned} &p_{\mathit{ \tiny MSL}}= p_0 + \Delta p\\ &T_{\mathit{ \tiny MSL}}= T_0 + \Delta T + \beta H_{p,{\mathit{ \tiny MSL}}} = T_{\mathit{ \tiny ISA, MSL}} + \Delta T\\ &T_{\mathit{ \tiny ISA, MSL}}= T_0 + \beta H_{p,{\mathit{ \tiny MSL}}}\\ &H_{p,{\mathit{ \tiny MSL}}} = \frac{T_0}{\beta}\left[\left(\frac{p_{\mathit{ \tiny MSL}}}{p_0}\right)^{\frac{g_0}{\beta R}} -1 \right]\\ &H_{\mathit{ \tiny MSL}} = 0\end{aligned}\]
4.2.6 Relation between Geopotential and Pressure Altitudes
Combining Equation 4.1 and Equation 4.2, we see that a variation in pressure \(dp\) in the non-ISA atmosphere corresponds to a variation in geopotential altitude \(dH\), according to the following Equation 4.17.
\[ dp= - \frac{p}{RT} g_0 dH \tag{4.17}\]
The same variation of pressure in an ISA atmosphere would correspond to a variation of pressure altitude \(dH_p\), satisfying Equation 4.18.
\[ dp= - \frac{p}{RT_{\mathit{ \tiny ISA}}} g_0 dH_P \tag{4.18}\]
Dividing the expression Equation 4.17 by Equation 4.18, we obtain the relation between geopotential altitude variation and pressure altitude variation, given by the following Equation 4.19.
\[ \frac{dH}{dH_p}=\frac{T}{T_{\mathit{ \tiny ISA}}} \tag{4.19}\]
The relation Equation 4.20 between \(H\) and \(H_p\) is obtained by integrating Equation 4.19 taking into account Equation 4.15 and Equation 4.16 for the expression of the temperatures \(T\) and \(T_{\mathit{ \tiny ISA}}\).
\[\begin{aligned} &H= H_p - H_{p,\mathit{ \tiny MSL}} + \frac{\Delta T}{\beta} \ln \left(\frac{T_0+\beta H_p}{T_{\mathit{ \tiny ISA,MSL}}}\right)& \quad\quad\text{for}\quad H_p<H_{p,\mathit{ trop}}\\ &H_{\text{trop}} = H_{p,\mathit{ trop}} - H_{p,\mathit{ \tiny MSL}} + \frac{\Delta T}{\beta} \ln \left(\frac{T_{\mathit{ \tiny ISA},trop}}{T_{\mathit{ \tiny ISA,MSL}}}\right)&\\ &H = H_{\text{trop}} + \frac{T_{\mathit{ \tiny ISA},trop}}{T_{\mathit{ \tiny ISA,MSL}}}(H_p - H_{p,\mathit{ trop}}) & \quad\quad\text{for}\quad H_p \geq H_{p,\mathit{ trop}}\end{aligned} \tag{4.20}\]
where
\[\begin{aligned} &H_{p,\mathit{ \tiny MSL}} = \frac{T_0}{\beta}\left[\left(\frac{p_{\mathit{ \tiny MSL}}}{p_0}\right)^{\frac{g_0}{\beta R}}-1\right]\\ &H_{p,\mathit{ trop}} = 11000\quad\text{m}\\ &T_{\mathit{ \tiny ISA,MSL}} = T_0+ \beta H_{p,\mathit{ \tiny MSL}}\\ &T_{\mathit{ \tiny ISA},trop} = T_0+\beta H_{p,\mathit{ trop}}\\\end{aligned} \]
4.2.7 Pressure \(p\) as a Function of Pressure Altitude \(H_p\)
Equation 4.6 can be transposed directly to the non-ISA case, replacing \(H\) by \(H_p\) and \(T\) by \(T_{\mathit{ \tiny ISA}}\). For the troposphere and stratosphere, we then obtain the expressions given in Equation 4.21:
\[\begin{aligned} &p = p_0\left[\frac{T_0 + \beta H_p}{T_0}\right]^{- \frac{g_0}{R \beta}} & \quad\quad\text{for}\quad H_p<H_{p,\text{trop}}\\ &p_{\text{trop}}= p_{\mathit{ \tiny ISA},trop}= p_0\left[\frac{T_0 + \beta H_{p,\text{trop}}}{T_0}\right]^{- \frac{g_0}{R \beta}}&\\ &p =p_{\mathit{ \tiny ISA},trop} \exp\left[- \frac{g_0}{R T_{\mathit{ \tiny ISA},trop}}(H_p-H_{p,\text{trop}})\right]& \quad\quad\text{for}\quad H_p \geq H_{p,\text{trop}}\end{aligned} \tag{4.21}\]
4.2.8 Pressure Altitude \(H_p\) as a Function of Pressure \(p\)
\[\begin{aligned} &H_p= \frac{T_0}{\beta}\left[\left(\frac{p}{p_0}\right)^{- \frac{R \beta}{g_0}} -1\right] & \quad\quad\text{for}\quad p \geq p_{\mathit{ \tiny ISA},trop}\\ % \quad\quad\text{pour}\quad H_p<H_{p,\text{trop}}\\ & H_p = H_{p,\text{trop}} - \frac{R T_{\mathit{ \tiny ISA},trop}}{g_0}\ln\left(\frac{p}{p_{\mathit{ \tiny ISA},trop}}\right)& \quad\quad\text{for}\quad p < p_{\mathit{ \tiny ISA},trop}\end{aligned} \tag{4.22}\]
with \[p_{\mathit{ \tiny ISA},trop}= p_0\left[\frac{T_{\mathit{ \tiny ISA},trop}}{T_0}\right]^{- \frac{g_0}{R \beta}} =p_0\left[\frac{T_0 + \beta H_{p,\text{trop}}}{T_0}\right]^{- \frac{g_0}{R \beta}} \]
4.2.9 Air Density \(\rho\) as a Function of Pressure Altitude \(H_p\)
\[\begin{aligned} &\rho = \frac{p_0}{T_0+\Delta T + \beta H_p}\left[\frac{T_0 + \beta H}{T_0}\right]^{- \frac{g_0}{R \beta}} & \quad\text{for}\quad H_p<H_{p,\text{trop}}\\ & \rho_{\text{trop}}= \frac{p_{\mathit{ \tiny ISA},trop}}{R(T_{\mathit{ \tiny ISA},trop}+\Delta T)}&\\ & \rho = \frac{p_{\mathit{ \tiny ISA},trop}}{R(T_{\mathit{ \tiny ISA},trop}+\Delta T)} \exp\left[- \frac{g_0}{R T_{\mathit{ \tiny ISA},trop}}(H_p-H_{p,\text{trop}})\right]& \quad\text{for}\quad H_p \geq H_{p,\text{trop}}\end{aligned} \tag{4.23}\]
4.2.10 Speed of Sound
The speed of sound in the non-ISA atmosphere is given by \(a= \sqrt{\kappa RT}\) (Equation 4.10), the temperature \(T\) being given by Equation 4.15.
Note that the pressure altitude and geopotential altitude are the same, when the atmosphere is standard↩︎