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 for number density profile for 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 for volume mixing ratio profile for 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 for number density profile for 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 for volume mixing ratio profile for 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.

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