# geopotential derivations

1. geopotential from geopotential height

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(g_{0}\) | mean earth gravity | \(\frac{m}{s^2}\) |
   \(z_{g}\) | geopotential height | \(m\) | geopotential_height {:}
   \(\Phi\) | geopotential | \(\frac{m^2}{s^2}\) | geopotential {:}

   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.

   \[\Phi = g_{0}z_{g}\]

2. surface geopotential from surface geopotential height

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(g_{0}\) | mean earth gravity | \(\frac{m}{s^2}\) |
   \(z_{g,surf}\) | surface geopotential height | \(m\) | surface_geopotential_height {:}
   \(\Phi_{surf}\) | surface geopotential | \(\frac{m^2}{s^2}\) | surface_geopotential {:}

   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.

   \[\Phi_{surf} = g_{0}z_{g,surf}\]