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Boundary Conditions for operations on grids

GMT has the option to specify boundary conditions in some programs that operate on grids (grdsample  -–L; grdgradient  -L; grdtrack  -L; nearneighbor  -L; grdview  -–L). The boundary conditions come into play when interpolating or computing derivatives near the limits of the region covered by the grid. The default boundary conditions used are those which are ``natural'' for the boundary of a minimum curvature interpolating surface. If the user knows that the data are periodic in x (and/or y), or that the data cover a sphere with x,y representing longitude,latitude, then there are better choices for the boundary conditions. Periodic conditions on x (and/or y) are chosen by specifying x (and/or y) as the boundary condition flags; global spherical cases are specified using the g (geographical) flag. Behavior of these conditions is as follows:

 

Periodic
conditions on x indicate that the data are periodic in the distance ( xmax - xmin) and thus repeat values after every N = (xmax - xmin)/xinc. Note that this implies that in a grid-registered file the values in the first and last columns are equal, since these are located at x = xminand x = xmax, and there are N + 1 columns in the file. This is not the case in a pixel-registered file, where there are only N and the first and last columns are located at xmin + xinc/2 and xmax - xinc/2. If y is periodic all the same holds for y.

Geographical
conditions indicate the following:  

1.
If $(x_{max} - x_{min}) \geq 360$ and also 180 modulo xinc = 0then a periodic condition is used on x with a period of 360; else a default condition is used on the x boundaries.

2.
If condition 1 is true and also ymax = 90 then a ``north pole condition'' is used at ymax, else a default condition is used there.

3.
If condition 1 is true and also ymin = -90 then a ``south pole condition'' is used at ymin, else a default condition is used there.

``Pole conditions'' use a 180o phase-shift of the data, requiring 180 modulo xinc = 0.

Default
boundary conditions are  


\begin{displaymath}\nabla^2 f = \frac{\partial}{\partial n} \nabla^2 f = 0 \end{displaymath}

on the boundary, where f(x, y) is represented by the values in the grid file, and $\partial/\partial n$ is the derivative in the direction normal to a boundary, and


\begin{displaymath}\nabla^2 = \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right) \end{displaymath}

is the two-dimensional Laplacian operator.

 


next up previous contents index
Next: Sun raster files Up: 2-D grdfiles Previous: Pixel registration
Paul Wessel
1999-07-01