altitude derivations

  1. altitude from geopotential height

    symbol description unit variable name
    \(g_{0}\) mean earth gravity \(\frac{m}{s^2}\)  
    \(g_{wgs84}\) gravity at WGS84 ellipsoid \(\frac{m}{s^2}\)  
    \(R_{wgs84}\) local earth curvature radius at WGS84 ellipsoid \(m\)  
    \(z\) 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.

    This equation approximates the mean sea level gravity and radius by that of the reference ellipsoid.

    \begin{eqnarray} g_{wgs84} & = & 9.7803253359 \frac{1 + 0.00193185265241{\sin}^2(\frac{\pi}{180}\phi)} {\sqrt{1 - 0.00669437999013{\sin}^2(\frac{\pi}{180}\phi)}} \\ R_{wgs84} & = & \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}R_{wgs84}z_{g}}{g_{wgs84}R_{wgs84} - 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 \(m\)  
    \(g\) gravity \(\frac{m}{s^2}\)  
    \(g_{0}\) mean earth gravity \(\frac{m}{s^2}\)  
    \(g_{surf}\) gravity at surface \(\frac{m}{s^2}\)  
    \(GM\) 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_{surf} & = & 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(1) & = & g_{surf} \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(i) & = & g_{surf} \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(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(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
    \(g_{0}\) mean earth gravity \(\frac{m}{s^2}\)  
    \(g_{wgs84}\) gravity at WGS84 ellipsoid \(\frac{m}{s^2}\)  
    \(R_{wgs84}\) local earth curvature radius at WGS84 ellipsoid \(m\)  
    \(z_{surf}\) surface altitude (relative to mean sea level) \(m\) surface_altitude {:}
    \(z_{g,surf}\) surface geopotential height (relative to mean sea level) \(m\) 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.

    This equation approximates the mean sea level gravity and radius by that of the reference ellipsoid.

    \begin{eqnarray} g_{wgs84} & = & 9.7803253359 \frac{1 + 0.00193185265241{\sin}^2(\frac{\pi}{180}\phi)} {\sqrt{1 - 0.00669437999013{\sin}^2(\frac{\pi}{180}\phi)}} \\ R_{wgs84} & = & \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}R_{wgs84}z_{g,surf}}{g_{wgs84}R_{wgs84} - 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 \\ & \forall_{j, i < j <= N \wedge z(j)-z(i) <= 2000} \frac{T(i)-T(j)}{z(j)-z(i)} <= 0.002 & \end{eqnarray}

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