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<div class="section" id="column-number-density-averaging-kernel-derivations">
<h1>column number density averaging kernel derivations</h1>
<ol class="arabic">
<li><p class="first">column number density column AVK for air component from column number density AVK:</p>
<table border="1" class="docutils">
<colgroup>
<col width="13%" />
<col width="29%" />
<col width="21%" />
<col width="36%" />
</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">\(A^{c}_{x}(i)\)</span></td>
<td>column AVK for column number density profile for
air component x (e.g. <span class="math notranslate nohighlight">\(A^{c}_{O_{3}}(i)\)</span>)</td>
<td><span class="math notranslate nohighlight">\(\frac{molec/m^2}{molec/m^2}\)</span></td>
<td><cite><species>_column_number_density_avk {:,vertical}</cite></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(A^{c}_{x}(i,j)\)</span></td>
<td>AVK for column number density profile for air
component x (e.g. <span class="math notranslate nohighlight">\(A^{c}_{O_{3}}(i,j)\)</span>)</td>
<td><span class="math notranslate nohighlight">\(\frac{molec/m^2}{molec/m^2}\)</span></td>
<td><cite><species>_column_number_density_avk {:,vertical,vertical}</cite></td>
</tr>
</tbody>
</table>
<p>The pattern <cite>:</cite> for the first dimensions can represent <cite>{latitude,longitude}</cite>, <cite>{time}</cite>, <cite>{time,latitude,longitude}</cite>,
or no dimensions at all.</p>
<div class="math notranslate nohighlight">
\[A^{c}_{x}(i) = \sum_{j}{A^{c}_{x}(j,i)}\]</div>
</li>
<li><p class="first">column number density AVK for air component from number density AVK:</p>
<table border="1" class="docutils">
<colgroup>
<col width="14%" />
<col width="28%" />
<col width="22%" />
<col width="37%" />
</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">\(A^{c}_{x}(i,j)\)</span></td>
<td>AVK for column number density profile for air
component x (e.g. <span class="math notranslate nohighlight">\(A^{c}_{O_{3}}(i,j)\)</span>)</td>
<td><span class="math notranslate nohighlight">\(\frac{molec/m^2}{molec/m^2}\)</span></td>
<td><cite><species>_column_number_density_avk {:,vertical,vertical}</cite></td>
</tr>
<tr class="row-odd"><td><span class="math notranslate nohighlight">\(A^{n}_{x}(i,j)\)</span></td>
<td>AVK for number density profile for air
component x (e.g. <span class="math notranslate nohighlight">\(A^{n}_{O_{3}}(i,j)\)</span>)</td>
<td><span class="math notranslate nohighlight">\(\frac{molec/m^3}{molec/m^3}\)</span></td>
<td><cite><species>_number_density_avk {:,vertical,vertical}</cite></td>
</tr>
<tr class="row-even"><td><span class="math notranslate nohighlight">\(z^{B}(i,l)\)</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 {:,vertical,2}</cite></td>
</tr>
</tbody>
</table>
<p>The pattern <cite>:</cite> for the first dimensions can represent <cite>{latitude,longitude}</cite>, <cite>{time}</cite>, <cite>{time,latitude,longitude}</cite>,
or no dimensions at all.</p>
<div class="math notranslate nohighlight">
\[\begin{split}A^{c}_{x}(i,j) = \begin{cases}
z^{B}(j,1) \neq z^{B}(j,2), & A^{n}_{x}(i,j) \frac{\lvert z^{B}(i,2) - z^{B}(i,1) \rvert}{\lvert z^{B}(j,2) - z^{B}(j,1) \rvert} \\
z^{B}(j,1) \neq z^{B}(j,2), & 0
\end{cases}\end{split}\]</div>
</li>
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