# altitude derivations

1. altitude from geopotential height

symbol

description

unit

variable name

$$g$$

normal gravity at sea level

$$\frac{m}{s^2}$$

$$g_{0}$$

mean earth gravity

$$\frac{m}{s^2}$$

$$R$$

$$m$$

$$z$$

altitude

$$m$$

altitude {:}

$$z_{g}$$

geopotential height

$$m$$

geopotential_height {:}

$$\phi$$

latitude

$$degN$$

latitude {:}

The pattern : for the dimensions can represent {vertical}, {time}, {time,vertical}, or no dimensions at all.

\begin{eqnarray} g & = & 9.7803253359 \frac{1 + 0.00193185265241{\sin}^2(\frac{\pi}{180}\phi)} {\sqrt{1 - 0.00669437999013{\sin}^2(\frac{\pi}{180}\phi)}} \\ R & = & \frac{1}{\sqrt{\left(\frac{\cos(\frac{\pi}{180}\phi)}{6356752.0}\right)^2 + \left(\frac{\sin(\frac{\pi}{180}\phi)}{6378137.0}\right)^2}} \\ z & = & \frac{g_{0}Rz_{g}}{gR - g_{0}z_{g}} \end{eqnarray}
2. altitude from bounds

symbol

description

unit

variable name

$$z$$

altitude

$$m$$

altitude {:}

$$z^{B}(l)$$

altitude boundaries ($$l \in \{1,2\}$$)

$$m$$

altitude_bounds {:,2}

The pattern : for the dimensions can represent {vertical}, or {time,vertical}.

$z = \frac{z^{B}(2) + z^{B}(1)}{2}$
3. altitude from sensor altitude

symbol

description

unit

variable name

$$z$$

altitude

$$m$$

altitude {:}

$$z_{instr}$$

altitude of the sensor

$$m$$

sensor_altitude {:}

The pattern : for the dimensions can represent {time}, or no dimensions at all.

$z = z_{instr}$
4. altitude from pressure

symbol

description

unit

variable name

$$a$$

WGS84 semi-major axis

$$m$$

$$b$$

WGS84 semi-minor axis

$$m$$

$$f$$

WGS84 flattening

$$m$$

$$g$$

normal gravity at sea level

$$\frac{m}{s^2}$$

$$g_{0}$$

mean earth gravity

$$\frac{m}{s^2}$$

$$g_{h}$$

gravity at specific height

$$\frac{m}{s^2}$$

$$GM$$

WGS84 earth’s gravitational constant

$$\frac{m^3}{s^2}$$

$$M_{air}(i)$$

molar mass of total air

$$\frac{g}{mol}$$

molar_mass {:,vertical}

$$p(i)$$

pressure

$$Pa$$

pressure {:,vertical}

$$p_{surf}$$

surface pressure

$$Pa$$

surface_pressure {:}

$$R$$

universal gas constant

$$\frac{kg m^2}{K mol s^2}$$

$$T(i)$$

temperature

$$K$$

temperature {:,vertical}

$$z(i)$$

altitude

$$m$$

altitude {:,vertical}

$$z_{surf}$$

surface height

$$m$$

surface_altitude {:}

$$\phi$$

latitude

$$degN$$

latitude {:}

$$\omega$$

WGS84 earth angular velocity

$$rad/s$$

The pattern : for the dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

The surface pressure $$p_{surf}$$ and surface height $$z_{surf}$$ need to use the same definition of ‘surface’.

The pressures $$p(i)$$ are expected to be at higher levels than the surface pressure (i.e. lower values). This should normally be the case since even for pressure grids that start at the surface, $$p_{surf}$$ should equal the lower pressure boundary $$p^{B}(1,1)$$, whereas $$p(1)$$ should then be between $$p^{B}(1,1)$$ and $$p^{B}(1,2)$$ (which is generally not equal to $$p^{B}(1,1)$$).

\begin{eqnarray} g & = & 9.7803253359 \frac{1 + 0.00193185265241{\sin}^2(\frac{\pi}{180}\phi)} {\sqrt{1 - 0.00669437999013 {\sin}^2(\frac{\pi}{180}\phi)}} \\ m & = & \frac{\omega^2a^2b}{GM} \\ g_{h}(1) & = & g \left(1 - \frac{2}{a}\left(1+f+m-2f{\sin}^2(\frac{\pi}{180}\phi)\right)z_{surf} + \frac{3}{a^2}z_{surf}^2\right) \\ g_{h}(i) & = & g \left(1 - \frac{2}{a}\left(1+f+m-2f{\sin}^2(\frac{\pi}{180}\phi)\right)z(i-1) + \frac{3}{a^2}z(i-1)^2\right), 1 < i \leq N \\ z(1) & = & z_{surf} + 10^{3}\frac{T(1)}{M_{air}(1)}\frac{R}{g_{h}(1)}\ln\left(\frac{p_{surf}}{p(i)}\right) \\ z(i) & = & z(i-1) + 10^{3}\frac{T(i-1)+T(i)}{M_{air}(i-1)+M_{air}(i)}\frac{R}{g_{h}(i)}\ln\left(\frac{p(i-1)}{p(i)}\right), 1 < i \leq N \end{eqnarray}
5. surface altitude from surface geopotential height

symbol

description

unit

variable name

$$g$$

nominal gravity at sea level

$$\frac{m}{s^2}$$

$$g_{0}$$

mean earth gravity

$$\frac{m}{s^2}$$

$$R$$

$$m$$

$$z_{surf}$$

surface altitude

$$m$$

surface_altitude {:}

$$z_{g,surf}$$

surface geopotential height

$$m$$

surface_geopotential_height {:}

$$\phi$$

latitude

$$degN$$

latitude {:}

The pattern : for the dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

\begin{eqnarray} g & = & 9.7803253359 \frac{1 + 0.00193185265241{\sin}^2(\frac{\pi}{180}\phi)} {\sqrt{1 - 0.00669437999013{\sin}^2(\frac{\pi}{180}\phi)}} \\ R & = & \frac{1}{\sqrt{\left(\frac{\cos(\frac{\pi}{180}\phi)}{6356752.0}\right)^2 + \left(\frac{\sin(\frac{\pi}{180}\phi)}{6378137.0}\right)^2}} \\ z_{surf} & = & \frac{g_{0}Rz_{g,surf}}{gR - g_{0}z_{g,surf}} \end{eqnarray}
6. tropopause altitude from temperature and altitude/pressure

symbol

description

unit

variable name

$$p(i)$$

pressure

$$Pa$$

pressure {:,vertical}

$$T(i)$$

temperature

$$K$$

temperature {:,vertical}

$$z(i)$$

altitude

$$m$$

altitude {:,vertical}

$$z_{TP}$$

tropopause altitude

$$m$$

tropopause_altitude {:}

The pattern : for the dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

The tropopause altitude $$z_{TP}$$ equals the altitude $$z(i)$$ where $$i$$ is the minimum level that satisfies:

\begin{eqnarray} & 1 < i < N & \wedge \\ & 5000 <= p(i) <= 50000 & \wedge \\ & \frac{T(i-1)-T(i)}{z(i)-z(i-1)} > 0.002 \wedge \frac{T(i)-T(i+1)}{z(i+1)-z(i)} <= 0.002 & \wedge \\ & \frac{\sum_{j, i < j < N \wedge z(j+1)-z(i) <= 2000} \frac{T(j)-T(j+1)}{z(j+1)-z(j)}}{\sum_{j, i < j < N \wedge z(j+1)-z(i) <= 2000}{1}} <= 0.002 & \end{eqnarray}

If no such $$i$$ can be found then $$z_{TP}$$ is set to NaN.