Technical Documentation

Spatial Reduction Operations on the ‘lonlat’ grid

The lonlat grid describes a geographic grid. Coordinates are longitude (\(\lambda\)) and latitude (\(\phi\)), and the angular spacing between gridpoints is constant (although it can be different for each axis).

Let \(F(\lambda, \phi, t)\) be a field and \(M(\lambda, \phi)\) a gridlist mask, defined by:

\[\begin{split}M(\lambda, \phi) = \begin{cases} 1\text{ if } (\lambda, \phi) \text{ is a gridlist point;}\\ 0\text{ if } (\lambda, \phi) \text{ is not a gridlist point.} \end{cases}\end{split}\]

For example, lakes and oceans are typically excluded from LPJ-GUESS gridlists, so the value of the mask at those points is \(0\). In the formulae below, coordinates are in radians, and \(R_\oplus\) is the radius of Earth in \(\mathrm{m}\). In this system, the area of a gridcell is \(A(\lambda,\phi)=R_\oplus^2 \Delta\lambda\Delta\phi\cos(\phi)\).

Latitudinal Average

\[\mathrm{Av}_\phi(\lambda,t) = \frac{\sum_\phi F(\lambda, \phi, t) M(\lambda, \phi)}{\sum_\phi M(\lambda, \phi)}\]

Longitudinal Average

\[\mathrm{Av}_\lambda(\phi,t) = \frac{\sum_\lambda F(\lambda, \phi, t) M(\lambda, \phi)}{\sum_\lambda M(\lambda, \phi)}\]

Spatial Average

\[\mathrm{Av}_{\lambda,\phi}(t) = \frac{\sum_{\lambda,\phi} F(\lambda, \phi, t) M(\lambda, \phi) \cos(\lambda)}{\sum_{\lambda,\phi} M(\lambda, \phi) \cos(\lambda)}\]

Longitudinal Sum

\[\mathrm{Sum}_\lambda(\phi,t) = R_\oplus \Delta\lambda \cos\phi \sum_\lambda F(\lambda, \phi, t) M(\lambda, \phi)\]

Latitudinal Sum

\[\mathrm{Sum}_\phi(\lambda,t) = R_\oplus \Delta\phi \sum_\phi F(\lambda, \phi, t) M(\lambda, \phi)\]

Spatial Sum

\[\mathrm{Sum}_{\lambda,\phi}(t) = R_\oplus^2 \Delta\lambda\Delta\phi \sum_{\lambda,\phi} F(\lambda, \phi, t) M(\lambda, \phi) \cos(\phi)\]