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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 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

  on the boundary, where f(x, y) 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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