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.

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