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.

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