# column number density averaging kernel derivations

1. column number density column AVK of air component from column number density AVK

symbol

description

unit

variable name

$$A^{c}_{x}(i)$$

column AVK of column number density profile of 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 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}

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. tropospheric column number density column AVK of air component from total column number density column AVK

symbol

description

unit

variable name

$$\tilde{A}^{c}_{x}(i)$$

column AVK of tropospheric part of column number density profile of air component x

$$\frac{molec/m^2}{molec/m^2}$$

tropospheric_<species>_column_number_density_avk {:,vertical}

$$A^{c}_{x}(i)$$

column AVK of column number density profile of air component x (e.g. $$A^{c}_{O_{3}}(i)$$)

$$\frac{molec/m^2}{molec/m^2}$$

<species>_column_number_density_avk {:,vertical}

$$z_{TP}$$

tropopause altitude

$$m$$

tropopause_altitude {:}

$$z^{B}(i,l)$$

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.

$\begin{split}\tilde{A}^{c}_{x}(i) = \begin{cases} z^{B}(i,1) < z_{TP}, & A^{c}_{x}(i) \\ z^{B}(j,1) \geq z_{TP}, & 0 \end{cases}\end{split}$
3. stratospheric column number density column AVK of air component from total column number density column AVK

symbol

description

unit

variable name

$$\tilde{A}^{c}_{x}(i)$$

column AVK of stratospheric part of column number density profile of air component x

$$\frac{molec/m^2}{molec/m^2}$$

tropospheric_<species>_column_number_density_avk {:,vertical}

$$A^{c}_{x}(i)$$

column AVK of column number density profile of air component x (e.g. $$A^{c}_{O_{3}}(i)$$)

$$\frac{molec/m^2}{molec/m^2}$$

<species>_column_number_density_avk {:,vertical}

$$z_{TP}$$

tropopause altitude

$$m$$

tropopause_altitude {:}

$$z^{B}(i,l)$$

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.

$\begin{split}\tilde{A}^{c}_{x}(i) = \begin{cases} z^{B}(i,2) \leq z_{TP}, & 0 \\ z^{B}(j,2) > z_{TP}, & A^{c}_{x}(i) \end{cases}\end{split}$
4. column number density AVK of air component from 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.

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