column number density averaging kernel derivations

1. column number density column AVK for air component from column number density AVK:

   symbol	description	unit	variable name
   \(A^{c}_{x}(i)\)	column AVK for column number density profile for 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 for column number density profile for 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. column number density AVK for air component from number density AVK:

   symbol	description	unit	variable name
   \(A^{c}_{x}(i,j)\)	AVK for column number density profile for 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 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}
   \(z^{B}(i,l)\)	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.

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