# aerosol optical depth derivations

1. total aerosol optical depth from partial aerosol optical depth profile

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(\tau\) | total aerosol optical depth | \(-\) | aerosol_optical_depth {:}
   \(\tau(i)\) | aerosol optical depth | \(-\) | aerosol_optical_depth {:,vertical}

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

   \[\tau = \sum_{i}{\tau(i)}\]

2. aerosol optical depth from aerosol extinction coefficient

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(z^{B}(l)\) | altitude boundaries (\(l \in \{1,2\}\)) | \(m\) | altitude_bounds {:,2}
   \(\sigma\) | aerosol extinction coefficient | \(\frac{1}{m}\) | aerosol_extinction_coefficient {:}
   \(\tau\) | aerosol optical depth | \(-\) | aerosol_optical_depth {:}

   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.

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