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<div class="section" id="altitude-derivations">
<h1>altitude derivations</h1>
<ol class="arabic">
<li><p class="first">altitude from geopotential height</p>
<table border="1" class="docutils">
<colgroup>
<col width="19%" />
<col width="31%" />
<col width="23%" />
<col width="27%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">description</th>
<th class="head">unit</th>
<th class="head">variable name</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(g_{0}\)</span></td>
<td>mean earth gravity</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
<td> </td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(g_{wgs84}\)</span></td>
<td>gravity at WGS84 ellipsoid</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
<td> </td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(R_{wgs84}\)</span></td>
<td>local earth curvature radius
at WGS84 ellipsoid</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
<td> </td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(z\)</span></td>
<td>altitude</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
<td><cite>altitude {:}</cite></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(z_{g}\)</span></td>
<td>geopotential height</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
<td><cite>geopotential_height {:}</cite></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(\phi\)</span></td>
<td>latitude</td>
<td><span class="math notranslate nohighlight">\(degN\)</span></td>
<td><cite>latitude {:}</cite></td>
</tr>
</tbody>
</table>
<p>The pattern <cite>:</cite> for the dimensions can represent <cite>{vertical}</cite>, <cite>{time}</cite>, <cite>{time,vertical}</cite>, or no dimensions at all.</p>
<p>This equation approximates the mean sea level gravity and radius by that of the reference ellipsoid.</p>
<div class="math notranslate nohighlight">
\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}</div></li>
<li><p class="first">altitude from bounds</p>
<table border="1" class="docutils">
<colgroup>
<col width="18%" />
<col width="47%" />
<col width="10%" />
<col width="25%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">description</th>
<th class="head">unit</th>
<th class="head">variable name</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(z\)</span></td>
<td>altitude</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
<td><cite>altitude {:}</cite></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(z^{B}(l)\)</span></td>
<td>altitude boundaries (<span class="math notranslate nohighlight">\(l \in \{1,2\}\)</span>)</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
<td><cite>altitude_bounds {:,2}</cite></td>
</tr>
</tbody>
</table>
<p>The pattern <cite>:</cite> for the dimensions can represent <cite>{vertical}</cite>, or <cite>{time,vertical}</cite>.</p>
<div class="math notranslate nohighlight">
\[z = \frac{z^{B}(2) + z^{B}(1)}{2}\]</div>
</li>
<li><p class="first">altitude from sensor altitude</p>
<table border="1" class="docutils">
<colgroup>
<col width="25%" />
<col width="32%" />
<col width="13%" />
<col width="30%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">description</th>
<th class="head">unit</th>
<th class="head">variable name</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(z\)</span></td>
<td>altitude</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
<td><cite>altitude {:}</cite></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(z_{instr}\)</span></td>
<td>altitude of the sensor</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
<td><cite>sensor_altitude {:}</cite></td>
</tr>
</tbody>
</table>
<p>The pattern <cite>:</cite> for the dimensions can represent <cite>{time}</cite>, or no dimensions at all.</p>
<div class="math notranslate nohighlight">
\[z = z_{instr}\]</div>
</li>
<li><p class="first">altitude from pressure</p>
<table border="1" class="docutils">
<colgroup>
<col width="17%" />
<col width="27%" />
<col width="31%" />
<col width="25%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">description</th>
<th class="head">unit</th>
<th class="head">variable name</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(a\)</span></td>
<td>WGS84 semi-major axis</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
<td> </td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(b\)</span></td>
<td>WGS84 semi-minor axis</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
<td> </td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(f\)</span></td>
<td>WGS84 flattening</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
<td> </td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(g\)</span></td>
<td>gravity</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
<td> </td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(g_{0}\)</span></td>
<td>mean earth gravity</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
<td> </td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(g_{surf}\)</span></td>
<td>gravity at surface</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
<td> </td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(GM\)</span></td>
<td>WGS84 earth’s gravitational
constant</td>
<td><span class="math notranslate nohighlight">\(\frac{m^3}{s^2}\)</span></td>
<td> </td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(M_{air}(i)\)</span></td>
<td>molar mass of total air</td>
<td><span class="math notranslate nohighlight">\(\frac{g}{mol}\)</span></td>
<td><cite>molar_mass {:,vertical}</cite></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(p(i)\)</span></td>
<td>pressure</td>
<td><span class="math notranslate nohighlight">\(Pa\)</span></td>
<td><cite>pressure {:,vertical}</cite></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(p_{surf}\)</span></td>
<td>surface pressure</td>
<td><span class="math notranslate nohighlight">\(Pa\)</span></td>
<td><cite>surface_pressure {:}</cite></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(R\)</span></td>
<td>universal gas constant</td>
<td><span class="math notranslate nohighlight">\(\frac{kg m^2}{K mol s^2}\)</span></td>
<td> </td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(T(i)\)</span></td>
<td>temperature</td>
<td><span class="math notranslate nohighlight">\(K\)</span></td>
<td><cite>temperature {:,vertical}</cite></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(z(i)\)</span></td>
<td>altitude</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
<td><cite>altitude {:,vertical}</cite></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(z_{surf}\)</span></td>
<td>surface height</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
<td><cite>surface_altitude {:}</cite></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(\phi\)</span></td>
<td>latitude</td>
<td><span class="math notranslate nohighlight">\(degN\)</span></td>
<td><cite>latitude {:}</cite></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(\omega\)</span></td>
<td>WGS84 earth angular velocity</td>
<td><span class="math notranslate nohighlight">\(rad/s\)</span></td>
<td> </td>
</tr>
</tbody>
</table>
<p>The pattern <cite>:</cite> for the dimensions can represent <cite>{latitude,longitude}</cite>, <cite>{time}</cite>, <cite>{time,latitude,longitude}</cite>,
or no dimensions at all.</p>
<p>The surface pressure <span class="math notranslate nohighlight">\(p_{surf}\)</span> and surface height <span class="math notranslate nohighlight">\(z_{surf}\)</span> need to use the same definition of ‘surface’.</p>
<p>The pressures <span class="math notranslate nohighlight">\(p(i)\)</span> 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, <span class="math notranslate nohighlight">\(p_{surf}\)</span> should
equal the lower pressure boundary <span class="math notranslate nohighlight">\(p^{B}(1,1)\)</span>, whereas <span class="math notranslate nohighlight">\(p(1)\)</span> should then be between <span class="math notranslate nohighlight">\(p^{B}(1,1)\)</span>
and <span class="math notranslate nohighlight">\(p^{B}(1,2)\)</span> (which is generally not equal to <span class="math notranslate nohighlight">\(p^{B}(1,1)\)</span>).</p>
<div class="math notranslate nohighlight">
\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}</div></li>
<li><p class="first">surface altitude from surface geopotential height</p>
<table border="1" class="docutils">
<colgroup>
<col width="18%" />
<col width="28%" />
<col width="21%" />
<col width="33%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">description</th>
<th class="head">unit</th>
<th class="head">variable name</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(g_{0}\)</span></td>
<td>mean earth gravity</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
<td> </td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(g_{wgs84}\)</span></td>
<td>gravity at WGS84 ellipsoid</td>
<td><span class="math notranslate nohighlight">\(\frac{m}{s^2}\)</span></td>
<td> </td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(R_{wgs84}\)</span></td>
<td>local earth curvature radius
at WGS84 ellipsoid</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
<td> </td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(z_{surf}\)</span></td>
<td>surface altitude
(relative to mean sea level)</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
<td><cite>surface_altitude {:}</cite></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(z_{g,surf}\)</span></td>
<td>surface geopotential height
(relative to mean sea level)</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
<td><cite>surface_geopotential_height {:}</cite></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(\phi\)</span></td>
<td>latitude</td>
<td><span class="math notranslate nohighlight">\(degN\)</span></td>
<td><cite>latitude {:}</cite></td>
</tr>
</tbody>
</table>
<p>The pattern <cite>:</cite> for the dimensions can represent <cite>{latitude,longitude}</cite>, <cite>{time}</cite>, <cite>{time,latitude,longitude}</cite>,
or no dimensions at all.</p>
<p>This equation approximates the mean sea level gravity and radius by that of the reference ellipsoid.</p>
<div class="math notranslate nohighlight">
\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}</div></li>
<li><p class="first">tropopause altitude from temperature and altitude/pressure</p>
<table border="1" class="docutils">
<colgroup>
<col width="20%" />
<col width="28%" />
<col width="14%" />
<col width="38%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">symbol</th>
<th class="head">description</th>
<th class="head">unit</th>
<th class="head">variable name</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><span class="math notranslate nohighlight">\(p(i)\)</span></td>
<td>pressure</td>
<td><span class="math notranslate nohighlight">\(Pa\)</span></td>
<td><cite>pressure {:,vertical}</cite></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(T(i)\)</span></td>
<td>temperature</td>
<td><span class="math notranslate nohighlight">\(K\)</span></td>
<td><cite>temperature {:,vertical}</cite></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(z(i)\)</span></td>
<td>altitude</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
<td><cite>altitude {:,vertical}</cite></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(z_{TP}\)</span></td>
<td>tropopause altitude</td>
<td><span class="math notranslate nohighlight">\(m\)</span></td>
<td><cite>tropopause_altitude {:}</cite></td>
</tr>
</tbody>
</table>
<p>The pattern <cite>:</cite> for the dimensions can represent <cite>{latitude,longitude}</cite>, <cite>{time}</cite>, <cite>{time,latitude,longitude}</cite>,
or no dimensions at all.</p>
<p>The tropopause altitude <span class="math notranslate nohighlight">\(z_{TP}\)</span> equals the altitude <span class="math notranslate nohighlight">\(z(i)\)</span> where <span class="math notranslate nohighlight">\(i\)</span> is the minimum level that satisfies:</p>
<div class="math notranslate nohighlight">
\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}</div><p>If no such <span class="math notranslate nohighlight">\(i\)</span> can be found then <span class="math notranslate nohighlight">\(z_{TP}\)</span> is set to <cite>NaN</cite>.</p>
</li>
</ol>
</div>
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