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B.2.3 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 ( ) and thus repeat values after every . 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 and , 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 and . If y is periodic all the same holds for y.

Geographical
  conditions indicate the following:

  1. If and also 180 modulo then 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 then a ``north pole condition'' is used at , else a default condition is used there.

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

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

Default
  boundary conditions are

  on the boundary, where is represented by the values in the grid file, and is the derivative in the direction normal to a boundary, and

  is the two-dimensional Laplacian operator.

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