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<H2><A NAME="SECTION001723000000000000000"></A>
<A NAME="13297"></A>
<BR>
B.2.3 Boundary Conditions for operations on grids
</H2>
<P>
<A NAME="tex2html588"
HREF="http://www.soest.hawaii.edu/gmt"><B>GMT</B></A> has the option to specify boundary conditions in some programs
that operate on grids (<A NAME="tex2html589"
HREF="../grdsample.html"><I><B>grdsample</B></I></A><A NAME="13537"></A> <B>-L</B>; <A NAME="tex2html590"
HREF="../grdgradient.html"><I><B>grdgradient</B></I></A><A NAME="13547"></A>
<B>-L</B>; <A NAME="tex2html591"
HREF="../grdtrack.html"><I><B>grdtrack</B></I></A><A NAME="13557"></A> <B>-L</B>; <A NAME="tex2html592"
HREF="../nearneighbor.html"><I><B>nearneighbor</B></I></A><A NAME="13567"></A> <B>-L</B>;
<A NAME="tex2html593"
HREF="../grdview.html"><I><B>grdview</B></I></A><A NAME="13577"></A> <B>-L</B>). The boundary conditions come into play
when interpolating or computing derivatives near the limits of the
region covered by the grid. The <I>default</I> 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 <I>x</I> (and/or
<I>y</I>), or that the data cover a sphere with <I>x</I>,<I>y</I>
representing <I>longitude</I>,<I>latitude</I>, then there are better
choices for the boundary conditions.
Periodic conditions on <I>x</I> (and/or <I>y</I>) are chosen by
specifying <I>x</I> (and/or <I>y</I>) as the boundary condition flags;
global spherical cases are specified using the <I>g</I> (geographical)
flag. Behavior of these conditions is as follows:
<P>
<DL>
<DD><A NAME="13321"></A>
<P>
</DD>
<DT><STRONG>Periodic</STRONG></DT>
<DD>conditions on <I>x</I> indicate that the data are
periodic in the distance (<!-- MATH
$x_{max} - x_{min}$
-->
<IMG
WIDTH="74" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img149.gif"
ALT="$x_{max} - x_{min}$">) and thus repeat
values after every <!-- MATH
$N = (x_{max} - x_{min})/x_{inc}$
-->
<IMG
WIDTH="147" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img150.gif"
ALT="$N = (x_{max} - x_{min})/x_{inc}$">. 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 <IMG
WIDTH="56" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img138.gif"
ALT="$x = x_{min}$">
and <IMG
WIDTH="58" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img139.gif"
ALT="$x = x_{max}$">, and there are <I>N</I> + 1 columns in the file.
This is not the case in a pixel-registered file, where there are only
<I>N</I> and the first and last columns are located at
<!-- MATH
$x_{min} + x_{inc}/2$
-->
<IMG
WIDTH="85" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img151.gif"
ALT="$x_{min} + x_{inc}/2$"> and <!-- MATH
$x_{max} - x_{inc}/2$
-->
<IMG
WIDTH="87" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img152.gif"
ALT="$x_{max} - x_{inc}/2$">. If <I>y</I> is
periodic all the same holds for <I>y</I>.
<P>
</DD>
<DT><STRONG>Geographical</STRONG></DT>
<DD>conditions indicate the following:
<A NAME="13338"></A>
<P>
<OL>
<LI>If <!-- MATH
$(x_{max} - x_{min}) \geq 360$
-->
<IMG
WIDTH="130" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img153.gif"
ALT="$(x_{max} - x_{min}) \geq 360$"> and also 180 modulo <IMG
WIDTH="54" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img154.gif"
ALT="$x_{inc} = 0$">
then a periodic condition is used on <I>x</I> with a period of 360;
else a default condition is used on the <I>x</I> boundaries.
<P>
</LI>
<LI>If condition 1 is true and also <IMG
WIDTH="67" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img155.gif"
ALT="$y_{max} = 90$"> then a
``north pole condition'' is used at <IMG
WIDTH="32" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img156.gif"
ALT="$y_{max}$">, else a default
condition is used there.
<P>
</LI>
<LI>If condition 1 is true and also <IMG
WIDTH="77" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img157.gif"
ALT="$y_{min} = -90$"> then a
``south pole condition'' is used at <IMG
WIDTH="30" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img158.gif"
ALT="$y_{min}$">, else a default
condition is used there.
<P>
</LI>
</OL>
<P>
``Pole conditions'' use a 180<IMG
WIDTH="11" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
SRC="img61.gif"
ALT="$^{o}$"> phase-shift of the data,
requiring 180 modulo <IMG
WIDTH="54" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img154.gif"
ALT="$x_{inc} = 0$">.
<P>
</DD>
<DT><STRONG>Default</STRONG></DT>
<DD>boundary conditions are
<A NAME="13351"></A>
<P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\nabla^2 f = \frac{\partial}{\partial n} \nabla^2 f = 0
\end{displaymath}
-->
<IMG
WIDTH="120" HEIGHT="40" BORDER="0"
SRC="img159.gif"
ALT="\begin{displaymath}\nabla^2 f = \frac{\partial}{\partial n} \nabla^2 f = 0 \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<P>
on the boundary, where <IMG
WIDTH="46" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img160.gif"
ALT="$f(x, y)$"> is represented by the values in
the grid file, and <!-- MATH
$\partial/\partial n$
-->
<IMG
WIDTH="36" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img161.gif"
ALT="$\partial/\partial n$"> is the derivative in the direction normal to a
boundary, and
<P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\nabla^2 = \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right)
\end{displaymath}
-->
<IMG
WIDTH="125" HEIGHT="46" BORDER="0"
SRC="img162.gif"
ALT="\begin{displaymath}\nabla^2 = \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right) \end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<P>
is the two-dimensional Laplacian operator.
<P>
</DD>
</DL>
<A NAME="13359"></A>
<P>
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<ADDRESS>
Paul Wessel
2001-04-18
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