Ice deformation (vorticity, divergence, shear) is computed
from the RGPS ice motion products - see the
background section on ice motion
for a brief review.
Each ice motion product gives you (essentially) a list
of numbers: x,y,u,v where (x,y) are the coordinates of
points on a regular 5-km grid, and (u,v) are the displacements
of those points over the time interval between the two images.
Since the (x,y) lie on a regular 5-km grid, we can think of them
as defining a set of square cells. For each cell, we have
the displacements of the corners. Using standard finite
difference formulas we can compute an estimate of du/dx
(partial derivative) at the center of the cell using the
corner displacements. Similarly we can compute the other
partial derivatives du/dy, dv/dx, dv/dy for the cell.
Averaging the du/dx values over all the cells gives a single
large-scale value for du/dx.
It turns out that with this simple averaging procedure, the contributions
from interior grid points cancel, and what remains is equivalent to
an integration of u (or v) around the boundary of the region, in
accord with the divergence theorem. The advantage of this averaging
method is that we don't have to identify the outer boundary
of grid points explicitly.
Once the large-scale partial derivatives have been computed,
the deformation invariants follow easily:
divergence = du/dx + dv/dy
etc. Notice that du/dx and the invariants are dimensionless.
We have not divided by the time interval over which the
By restricting our attention to those cells that lie within
a certain distance of the ship, we can compute deformation
invariants on different spatial scales. The scales we
have chosen to work with are 50 x 50 km, 100 x 100 km, 150 x 150 km,
and 200 x 200 km, which is the maximum extent of the ice motion data.