  Nowadays, climate models rely on couplers. Each complete climate model is
broken into different sub-models (oceanic, atmospheric,...), each one working
on a different grid. The coupler brings these models together and interpolates
the physical quantities between the grids. However, neither the coupler nor
sometimes the sub-models themselves know precisely the grid cell edges. They
only know the grid cell corners (vertices) and the true grid cell areas. Thus,
the coupler has to make assumptions about the grid cell edges in order to
compute the grid cell intersections. For first-order schemes, the most
straightforward way to interpolate scalar quantities is to directly use these
approximate grid cell intersections, that don' take the true grid cell areas
into account. It is the method used in the "conservative" regridding option
implemented in the widely used spherical interpolation package SCRIP. We show
that is doesn't preserve integrals in the general case, whether the coupler
using the SCRIP-generated weights transmits directly the intensive quantity, or
its extensive counterpart (that is the same quantity multiplied by the true
area of the individual source grid cell). We show how to modify the
interpolation scheme to preserve integrals in the general case.
