# volume mixing ratio averaging kernel derivations

1. volume mixing ratio AVK from number density 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.

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

2. volume mixing ratio dry air AVK from number density 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 with regard to dry air (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.

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