# number density averaging kernel derivations

1. number density AVK of air component from column 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.

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

2. number density AVK from volume mixing ratio AVK

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(A^{n}_{x}(i,j)\) | 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}
   \(A^{\nu}_{x}(i,j)\) | AVK of volume mixing ratio profile of air component x (e.g. \(A^{\nu}_{O_{3}}(i,j)\)) | \(\frac{ppv}{ppv}\) | <species>_volume_mixing_ratio_avk {:,vertical,vertical}
   \(n(i)\) | number density profile of total air | \(\frac{molec}{m^3}\) | number_density {:,vertical}

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

   \[\begin{split}A^{n}_{x}(i,j) = \begin{cases} n(j) \neq 0, & A^{\nu}_{x}(i,j) \frac{n(i)}{n(j)} \\ n(j) = 0, & 0 \end{cases}\end{split}\]

3. number density AVK from volume mixing ratio dry air AVK

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(A^{n}_{x}(i,j)\) | 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}
   \(A^{\bar{\nu}}_{x}(i,j)\) | AVK of volume mixing ratio profile of air component x (e.g. \(A^{\bar{\nu}}_{O_{3}}(i,j)\)) | \(\frac{ppv}{ppv}\) | <species>_volume_mixing_ratio_dry_air_avk {:,vertical,vertical}
   \(n_{dry_air}(i)\) | number density profile of dry air | \(\frac{molec}{m^3}\) | dry_air_number_density {:,vertical}

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

   \[\begin{split}A^{n}_{x}(i,j) = \begin{cases} n_{dry_air}(j) \neq 0, & A^{\bar{\nu}}_{x}(i,j) \frac{n_{dry_air}(i)}{n_{dry_air}(j)} \\ n_{dry_air}(j) = 0, & 0 \end{cases}\end{split}\]