# column number density derivations

1. total column number density of air component from partial column number density profile

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(c_{x}\) | total column number density of air component x (e.g. \(c_{O_{3}}\)) | \(\frac{molec}{m^2}\) | <species>_column_number_density {:}
   \(c_{x}(i)\) | column number density profile of air component x (e.g. \(c_{O_{3}}(i)\)) | \(\frac{molec}{m^2}\) | <species>_column_number_density {:,vertical}

   The pattern : for the first dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

   \[c_{x} = \sum_{i}{c_{x}(i)}\]

2. total column number density of total air from partial column number density profile

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(c\) | total column number density of total air | \(\frac{molec}{m^2}\) | column_number_density {:}
   \(c(i)\) | column number density profile of total air | \(\frac{molec}{m^2}\) | column_number_density {:,vertical}

   The pattern : for the first dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

   \[c_{x} = \sum_{i}{c_{x}(i)}\]

3. tropospheric column number density of air component from partial column number density profile and altitude

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(c_{x}\) | tropospheric column number density of air component x (e.g. \(c_{O_{3}}\)) | \(\frac{molec}{m^2}\) | tropospheric_<species>_column_number_density {:}
   \(c_{x}(i)\) | column number density profile of air component x (e.g. \(c_{O_{3}}(i)\)) | \(\frac{molec}{m^2}\) | <species>_column_number_density {:,vertical}
   \(z_{TP}\) | tropopause altitude | \(m\) | tropopause_altitude {:}
   \(z^{B}(l)\) | altitude boundaries (\(l \in \{1,2\}\)) | \(m\) | altitude_bounds {:,2}

   The pattern : for the first dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

   \[\begin{split}c_{x} = \sum_{i}{\begin{cases} z^{B}(2) \leq z_{TP}, & c_{x}(i) \\ z^{B}(1) < z_{TP} < z^{B}(2), & c_{x}(i) \frac{z_{TP} - z^{B}(1)}{z^{B}(2) - z^{B}(1)} \\ z_{TP} \leq z^{B}(1), & 0 \end{cases}}\end{split}\]

4. stratospheric column number density of air component from partial column number density profile and altitude

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(c_{x}\) | stratospheric column number density of air component x (e.g. \(c_{O_{3}}\)) | \(\frac{molec}{m^2}\) | stratospheric_<species>_column_number_density {:}
   \(c_{x}(i)\) | column number density profile of air component x (e.g. \(c_{O_{3}}(i)\)) | \(\frac{molec}{m^2}\) | <species>_column_number_density {:,vertical}
   \(z_{TP}\) | tropopause altitude | \(m\) | tropopause_altitude {:}
   \(z^{B}(l)\) | altitude boundaries (\(l \in \{1,2\}\)) | \(m\) | altitude_bounds {:,2}

   The pattern : for the first dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

   \[\begin{split}c_{x} = \sum_{i}{\begin{cases} z^{B}(2) \leq z_{TP}, & 0 \\ z^{B}(1) < z_{TP} < z^{B}(2), & c_{x}(i) \frac{z^{B}(2) - z_{TP}}{z^{B}(2) - z^{B}(1)} \\ z_{TP} \leq z^{B}(1), & c_{x}(i) \end{cases}}\end{split}\]

5. tropospheric column number density of air component from partial column number density profile and pressure

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(c_{x}\) | tropospheric column number density of air component x (e.g. \(c_{O_{3}}\)) | \(\frac{molec}{m^2}\) | tropospheric_<species>_column_number_density {:}
   \(c_{x}(i)\) | column number density profile of air component x (e.g. \(c_{O_{3}}(i)\)) | \(\frac{molec}{m^2}\) | <species>_column_number_density {:,vertical}
   \(p_{TP}\) | tropopause pressure | \(Pa\) | tropopause_pressure {:}
   \(p^{B}(l)\) | pressure boundaries (\(l \in \{1,2\}\)) | \(Pa\) | pressure_bounds {:,2}

   The pattern : for the first dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

   \[\begin{split}c_{x} = \sum_{i}{\begin{cases} p^{B}(2) \geq p_{TP}, & c_{x}(i) \\ p^{B}(1) > p_{TP} > p^{B}(2), & c_{x}(i) \frac{\ln(p^{B}(1)) - \ln(p_{TP})}{\ln(p^{B}(1)) - \ln(p^{B}(2))} \\ p_{TP} \geq p^{B}(1), & 0 \end{cases}}\end{split}\]

6. stratospheric column number density of air component from partial column number density profile and pressure

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(c_{x}\) | stratospheric column number density of air component x (e.g. \(c_{O_{3}}\)) | \(\frac{molec}{m^2}\) | stratospheric_<species>_column_number_density {:}
   \(c_{x}(i)\) | column number density profile of air component x (e.g. \(c_{O_{3}}(i)\)) | \(\frac{molec}{m^2}\) | <species>_column_number_density {:,vertical}
   \(p_{TP}\) | tropopause pressure | \(Pa\) | tropopause_pressure {:}
   \(p^{B}(l)\) | pressure boundaries (\(l \in \{1,2\}\)) | \(Pa\) | pressure_bounds {:,2}

   The pattern : for the first dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

   \[\begin{split}c_{x} = \sum_{i}{\begin{cases} p^{B}(2) \geq p_{TP}, & 0 \\ p^{B}(1) > p_{TP} > p^{B}(2), & c_{x}(i) \frac{\ln(p_{TP}) - \ln(p^{B}(2))}{\ln(p^{B}(1)) - \ln(p^{B}(2))} \\ p_{TP} \geq p^{B}(1), & c_{x}(i) \end{cases}}\end{split}\]

7. column number density of total air from dry air column number density and H2O column number density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(c\) | column number density | \(\frac{molec}{m^2}\) | column_number_density {:}
   \(c_{dry\_air}\) | column number density of dry air | \(\frac{molec}{m^2}\) | dry_air_column_number_density {:}
   \(c_{H_{2}O}\) | column number density of H2O | \(\frac{molec}{m^2}\) | H2O_column_number_density {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[c = c_{dry\_air} + c_{H_{2}O}\]

8. column number density of dry air from total air column number density and H2O column number density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(c\) | column number density | \(\frac{molec}{m^2}\) | column_number_density {:}
   \(c_{dry\_air}\) | column number density of dry air | \(\frac{molec}{m^2}\) | dry_air_column_number_density {:}
   \(c_{H_{2}O}\) | column number density of H2O | \(\frac{molec}{m^2}\) | H2O_column_number_density {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[c_{dry\_air} = c - c_{H_{2}O}\]

9. column number density of H2O from total air column number density and dry air column number density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(c\) | column number density | \(\frac{molec}{m^2}\) | column_number_density {:}
   \(c_{dry\_air}\) | column number density of dry air | \(\frac{molec}{m^2}\) | dry_air_column_number_density {:}
   \(c_{H_{2}O}\) | column number density of H2O | \(\frac{molec}{m^2}\) | H2O_column_number_density {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[c_{H_{2}O} = c - c_{dry\_air}\]

10. column number density of air component from number density

    symbol | description | unit | variable name
    --- | --- | --- | ---
    \(c_{x}\) | column number density of air component x (e.g. \(c_{O_{3}}\)) | \(\frac{molec}{m^2}\) | <species>_column_number_density {:}
    \(n_{x}\) | number density of air component x (e.g. \(n_{O_{3}}\)) | \(\frac{molec}{m^3}\) | <species>_number_density {:}
    \(z^{B}(l)\) | altitude boundaries (\(l \in \{1,2\}\)) | \(m\) | altitude_bounds {:,2}

    The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

    \[c_{x} = n_{x} \lvert z^{B}(2) - z^{B}(1) \rvert\]

11. column number density of total air from number density

    symbol | description | unit | variable name
    --- | --- | --- | ---
    \(c\) | column number density of total air | \(\frac{molec}{m^2}\) | column_number_density {:}
    \(n\) | number density of total air | \(\frac{molec}{m^3}\) | number_density {:}
    \(z^{B}(l)\) | altitude boundaries (\(l \in \{1,2\}\)) | \(m\) | altitude_bounds {:,2}

    The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

    \[c = n \lvert z^{B}(2) - z^{B}(1) \rvert\]

12. column number density of air component from column mass density

    This conversion applies to both total columns as well as partial column profiles.

    symbol | description | unit | variable name
    --- | --- | --- | ---
    \(c_{x}\) | column number density of air component x (e.g. \(n_{O_{3}}\)) | \(\frac{molec}{m^2}\) | <species>_column_number_density {:}
    \(M_{x}\) | molar mass of air component x | \(\frac{g}{mol}\) |
    \(N_A\) | Avogadro constant | \(\frac{1}{mol}\) |
    \(\sigma_{x}\) | column mass density of air component x (e.g. \(\sigma_{O_{3}}\)) | \(\frac{kg}{m^2}\) | <species>_column_density {:}

    The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

    \[c_{x} = \frac{\sigma_{x}N_{A}}{10^{-3}M_{x}}\]

13. column number density of total air from column mass density

    This conversion applies to both total columns as well as partial column profiles.

    symbol | description | unit | variable name
    --- | --- | --- | ---
    \(c\) | column number density of total air | \(\frac{molec}{m^2}\) | column_number_density {:}
    \(M_{air}\) | molar mass of total air | \(\frac{g}{mol}\) | molar_mass {:}
    \(N_A\) | Avogadro constant | \(\frac{1}{mol}\) |
    \(\sigma\) | column mass density of total air | \(\frac{kg}{m^2}\) | column_density {:}

    The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

    \[c = \frac{\sigma N_{A}}{10^{-3}M_{air}}\]

14. column number density from column volume mixing ratio

    symbol | description | unit | variable name
    --- | --- | --- | ---
    \(c\) | total column number density of total air | \(\frac{molec}{m^2}\) | column_number_density {:}
    \(c_{x}\) | total column number density of air component x (e.g. \(c_{O_{3}}\)) | \(\frac{molec}{m^2}\) | <species>_column_number_density {:}
    \(\nu_{x}\) | column volume mixing ratio of quantity x with regard to total air | \(ppv\) | <species>_column_volume_mixing_ratio {:}

    The pattern : for the dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

    \[c_{x} = \nu_{x}c\]

15. column number density from column volume mixing ratio dry air

    symbol | description | unit | variable name
    --- | --- | --- | ---
    \(c_{dry\_air}\) | total column number density of dry air | \(\frac{molec}{m^2}\) | dry_air_column_number_density {:}
    \(c_{x}\) | total column number density of air component x (e.g. \(c_{O_{3}}\)) | \(\frac{molec}{m^2}\) | <species>_column_number_density {:}
    \(\bar{\nu}_{x}\) | column volume mixing ratio of quantity x with regard to dry air | \(ppv\) | <species>_column_volume_mixing_ratio_dry_air {:}

    The pattern : for the dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

    \[c_{x} = \bar{\nu}_{x}c_{dry\_air}\]

16. column number density of air component from volume mixing ratio

    symbol | description | unit | variable name
    --- | --- | --- | ---
    \(a\) | WGS84 semi-major axis | \(m\) |
    \(b\) | WGS84 semi-minor axis | \(m\) |
    \(c_{x}\) | column number density of air component x (e.g. \(c_{O_{3}}\)) | \(\frac{molec}{m^2}\) | <species>_column_number_density {:}
    \(f\) | WGS84 flattening | \(m\) |
    \(g\) | normal gravity at sea level | \(\frac{m}{s^2}\) |
    \(g_{0}\) | mean earth gravity | \(\frac{m}{s^2}\) |
    \(g_{h}\) | gravity at specific height | \(\frac{m}{s^2}\) |
    \(GM\) | WGS84 earth’s gravitational constant | \(\frac{m^3}{s^2}\) |
    \(M_{air}\) | molar mass of total air | \(\frac{g}{mol}\) | molar_mass {:}
    \(N_A\) | Avogadro constant | \(\frac{1}{mol}\) |
    \(p\) | pressure | \(Pa\) |
    \(p_{0}\) | standard pressure | \(Pa\) |
    \(p^{B}(l)\) | pressure boundaries (\(l \in \{1,2\}\)) | \(Pa\) | pressure_bounds {:,2}
    \(R\) | universal gas constant | \(\frac{kg m^2}{K mol s^2}\) |
    \(T_{0}\) | standard temperature | \(K\) |
    \(z\) | altitude | \(m\) |
    \(\nu_{x}\) | volume mixing ratio of quantity x with regard to total air | \(ppv\) | <species>_volume_mixing_ratio {:}
    \(\phi\) | latitude | \(degN\) | latitude {:}
    \(\omega\) | WGS84 earth angular velocity | \(rad/s\) |

    The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

    \begin{eqnarray} g & = & 9.7803253359 \frac{1 + 0.00193185265241{\sin}^2(\frac{\pi}{180}\phi)} {\sqrt{1 - 0.00669437999013 {\sin}^2(\frac{\pi}{180}\phi)}} \\ m & = & \frac{\omega^2a^2b}{GM} \\ p & = & e^{\frac{\ln(p^{B}(2)) + \ln(p^{B}(1))}{2}} \\ z & = & -\frac{RT_{0}}{10^{-3}M_{air}g_{0}}\ln(\frac{p}{p_{0}}) \\ g_{h} & = & g \left(1 - \frac{2}{a}\left(1+f+m-2f{\sin}^2(\frac{\pi}{180}\phi)\right)z + \frac{3}{a^2}z^2\right) \\ c_{x} & = & -\nu_{x}\frac{N_A}{10^{-3}M_{air}g_{h}}(p^{B}(2)-p^{B}(1)) \end{eqnarray}