<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <!--Converted with LaTeX2HTML 2002-2-1 (1.71) original version by: Nikos Drakos, CBLU, University of Leeds * revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan * with significant contributions from: Jens Lippmann, Marek Rouchal, Martin Wilck and others --> <HTML> <HEAD> <TITLE>3.2.2 Gridding with Splines in Tension</TITLE> <META NAME="description" CONTENT="3.2.2 Gridding with Splines in Tension"> <META NAME="keywords" CONTENT="GMT_Tutorial"> <META NAME="resource-type" CONTENT="document"> <META NAME="distribution" CONTENT="global"> <META NAME="Generator" CONTENT="LaTeX2HTML v2002-2-1"> <META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css"> <LINK REL="STYLESHEET" HREF="GMT_Tutorial.css"> <LINK REL="next" HREF="node47.html"> <LINK REL="previous" HREF="node44.html"> <LINK REL="up" HREF="node43.html"> <LINK REL="next" HREF="node47.html"> </HEAD> <BODY bgcolor="#ffffff"> <!--Navigation Panel--> <A NAME="tex2html1047" HREF="node47.html"> <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A> <A NAME="tex2html1041" HREF="node43.html"> <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A> <A NAME="tex2html1035" HREF="node45.html"> <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.png"></A> <A NAME="tex2html1043" HREF="node1.html"> <IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A> <A NAME="tex2html1045" HREF="node63.html"> <IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index" SRC="index.png"></A> <BR> <B> Next:</B> <A NAME="tex2html1048" HREF="node47.html">3.2.3 Preprocessing</A> <B> Up:</B> <A NAME="tex2html1042" HREF="node43.html">3.2 Gridding of arbitrarily</A> <B> Previous:</B> <A NAME="tex2html1036" HREF="node45.html">3.2.1.1 Exercises</A> <B> <A NAME="tex2html1044" HREF="node1.html">Contents</A></B> <B> <A NAME="tex2html1046" HREF="node63.html">Index</A></B> <BR> <BR> <!--End of Navigation Panel--> <H2><A NAME="SECTION00522000000000000000"> 3.2.2 Gridding with Splines in Tension</A> </H2> <P> As an alternative, we may use a global procedure to grid our data. This approach, implemented in the program <A NAME="tex2html188" HREF="../man/surface.html"><I><B>surface</B></I></A><A NAME="2391"></A>, represents an improvement over standard minimum curvature algorithms by allowing users to introduce some tension into the surface. Physically, we are trying to force a thin elastic plate to go through all our data points; the values of this surface at the grid points become the gridded data. Mathematically, we want to find the function <IMG WIDTH="43" HEIGHT="31" ALIGN="MIDDLE" BORDER="0" SRC="img36.png" ALT="$z(x, y)$"> that satisfies the following constraints: <BR><A NAME="1199"></A> <P> <!-- MATH $\begin{array}{ll} z(x_k, y_k) = z_k, & \mbox{for all data $(x_k, y_k, z_k), k =1,n$} \\ (1-t)\nabla^4 z - t \nabla^2 z = 0 & \mbox{elsewhere} \end{array}$ --> <IMG WIDTH="367" HEIGHT="54" ALIGN="MIDDLE" BORDER="0" SRC="img37.png" ALT="\( \begin{array}{ll} z(x_k, y_k) = z_k, & \mbox{for all data $(x_k, y_k, z_k), k =1,n$} \\ (1-t)\nabla^4 z - t \nabla^2 z = 0 & \mbox{elsewhere} \end{array} \)"> <BR> <P> where <IMG WIDTH="9" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img38.png" ALT="$t$"> is the ``tension'', <!-- MATH $0 \leq t \leq 1$ --> <IMG WIDTH="64" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img39.png" ALT="$0 \leq t \leq 1$">. Basically, as <!-- MATH $t \rightarrow 0$ --> <IMG WIDTH="40" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img40.png" ALT="$t \rightarrow 0$"> we obtain the minimum curvature solution, while as <!-- MATH $t \rightarrow \infty$ --> <IMG WIDTH="44" HEIGHT="29" ALIGN="MIDDLE" BORDER="0" SRC="img41.png" ALT="$t \rightarrow \infty$"> we go towards a harmonic solution (which is linear in cross-section). The theory behind all this is quite involved and we do not have the time to explain it all here, please see <I>Smith and Wessel</I> [1990] for details. Some of the most important switches for this program are indicated in Table <A HREF="#tbl:surface">3.3</A><A NAME="tex2html184" HREF="footnode.html#foot1611"><SUP>3.1</SUP></A>. <P> <BR><P></P> <DIV ALIGN="CENTER"><A NAME="1614"></A> <TABLE> <CAPTION><STRONG>Table 3.3:</STRONG> Some of the options in <A NAME="tex2html185" HREF="../man/surface.html"><I><B>surface</B></I></A><A NAME="2400"></A>.</CAPTION> <TR><TD> <DIV ALIGN="CENTER"><TABLE CELLPADDING=3 BORDER="1"> <TR><TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> <I>Option</I></FONT></TD> <TD ALIGN="CENTER" COLSPAN=1><FONT SIZE="-1"> <I>Purpose</I></FONT></TD> </TR> <TR><TH ALIGN="LEFT"><FONT SIZE="-1"> <B>-A</B></FONT><FONT SIZE="-1"><I>aspect</I> </FONT></TH> <TD ALIGN="LEFT"><FONT SIZE="-1"> Sets aspect ratio for anisotropic grids. </FONT></TD> </TR> <TR><TH ALIGN="LEFT"><FONT SIZE="-1"> <B>-C</B></FONT><FONT SIZE="-1"><I>limit</I> </FONT></TH> <TD ALIGN="LEFT"><FONT SIZE="-1"> Sets convergence limit. Default is 1/1000 of data range. </FONT></TD> </TR> <TR><TH ALIGN="LEFT"><FONT SIZE="-1"> <B>-T</B></FONT><FONT SIZE="-1"><I>tension</I> </FONT></TH> <TD ALIGN="LEFT"><FONT SIZE="-1"> Sets the tension [Default is 0] </FONT></TD> </TR> </TABLE><FONT SIZE="-1"> <A NAME="tbl:surface"></A></FONT></DIV></TD></TR> </TABLE> </DIV><P></P> <BR> <P> <HR> <!--Navigation Panel--> <A NAME="tex2html1047" HREF="node47.html"> <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next.png"></A> <A NAME="tex2html1041" HREF="node43.html"> <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up.png"></A> <A NAME="tex2html1035" HREF="node45.html"> <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="prev.png"></A> <A NAME="tex2html1043" HREF="node1.html"> <IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents.png"></A> <A NAME="tex2html1045" HREF="node63.html"> <IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index" SRC="index.png"></A> <BR> <B> Next:</B> <A NAME="tex2html1048" HREF="node47.html">3.2.3 Preprocessing</A> <B> Up:</B> <A NAME="tex2html1042" HREF="node43.html">3.2 Gridding of arbitrarily</A> <B> Previous:</B> <A NAME="tex2html1036" HREF="node45.html">3.2.1.1 Exercises</A> <B> <A NAME="tex2html1044" HREF="node1.html">Contents</A></B> <B> <A NAME="tex2html1046" HREF="node63.html">Index</A></B> <!--End of Navigation Panel--> <ADDRESS> Paul Wessel 2010-07-14 </ADDRESS> </BODY> </HTML>