# 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.

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