# column number density averaging kernel derivations

1. column number density column AVK of air component from column number density AVK

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(A^{c}_{x}(i)\) | column AVK of column number density profile of air component x (e.g. \(A^{c}_{O_{3}}(i)\)) | \(\frac{molec/m^2}{molec/m^2}\) | <species>_column_number_density_avk {:,vertical}
   \(A^{c}_{x}(i,j)\) | AVK of column number density profile of air component x (e.g. \(A^{c}_{O_{3}}(i,j)\)) | \(\frac{molec/m^2}{molec/m^2}\) | <species>_column_number_density_avk {:,vertical,vertical}

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

   \[A^{c}_{x}(i) = \sum_{j}{A^{c}_{x}(j,i)}\]

2. tropospheric column number density column AVK of air component from total column number density column AVK

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(\tilde{A}^{c}_{x}(i)\) | column AVK of tropospheric part of column number density profile of air component x | \(\frac{molec/m^2}{molec/m^2}\) | tropospheric_<species>_column_number_density_avk {:,vertical}
   \(A^{c}_{x}(i)\) | column AVK of column number density profile of air component x (e.g. \(A^{c}_{O_{3}}(i)\)) | \(\frac{molec/m^2}{molec/m^2}\) | <species>_column_number_density_avk {:,vertical}
   \(z_{TP}\) | tropopause altitude | \(m\) | tropopause_altitude {:}
   \(z^{B}(i,l)\) | altitude boundaries (\(l \in \{1,2\}\)) | \(m\) | altitude_bounds {:,vertical,2}

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

   \[\begin{split}\tilde{A}^{c}_{x}(i) = \begin{cases} z^{B}(i,1) < z_{TP}, & A^{c}_{x}(i) \\ z^{B}(j,1) \geq z_{TP}, & 0 \end{cases}\end{split}\]

3. stratospheric column number density column AVK of air component from total column number density column AVK

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(\tilde{A}^{c}_{x}(i)\) | column AVK of stratospheric part of column number density profile of air component x | \(\frac{molec/m^2}{molec/m^2}\) | tropospheric_<species>_column_number_density_avk {:,vertical}
   \(A^{c}_{x}(i)\) | column AVK of column number density profile of air component x (e.g. \(A^{c}_{O_{3}}(i)\)) | \(\frac{molec/m^2}{molec/m^2}\) | <species>_column_number_density_avk {:,vertical}
   \(z_{TP}\) | tropopause altitude | \(m\) | tropopause_altitude {:}
   \(z^{B}(i,l)\) | altitude boundaries (\(l \in \{1,2\}\)) | \(m\) | altitude_bounds {:,vertical,2}

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

   \[\begin{split}\tilde{A}^{c}_{x}(i) = \begin{cases} z^{B}(i,2) \leq z_{TP}, & 0 \\ z^{B}(j,2) > z_{TP}, & A^{c}_{x}(i) \end{cases}\end{split}\]

4. column number density AVK of air component from number density AVK

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(A^{c}_{x}(i,j)\) | AVK of column number density profile of air component x (e.g. \(A^{c}_{O_{3}}(i,j)\)) | \(\frac{molec/m^2}{molec/m^2}\) | <species>_column_number_density_avk {:,vertical,vertical}
   \(A^{n}_{x}(i,j)\) | AVK of number density profile of air component x (e.g. \(A^{n}_{O_{3}}(i,j)\)) | \(\frac{molec/m^3}{molec/m^3}\) | <species>_number_density_avk {:,vertical,vertical}
   \(z^{B}(i,l)\) | altitude boundaries (\(l \in \{1,2\}\)) | \(m\) | altitude_bounds {:,vertical,2}

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

   \[\begin{split}A^{c}_{x}(i,j) = \begin{cases} z^{B}(j,1) \neq z^{B}(j,2), & A^{n}_{x}(i,j) \frac{\lvert z^{B}(i,2) - z^{B}(i,1) \rvert}{\lvert z^{B}(j,2) - z^{B}(j,1) \rvert} \\ z^{B}(j,1) = z^{B}(j,2), & 0 \end{cases}\end{split}\]