<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <!--Converted with LaTeX2HTML 98.1p1 release (March 2nd, 1998) originally by Nikos Drakos (nikos@cbl.leeds.ac.uk), 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>Getting the Dispersion Solution</TITLE> <META NAME="description" CONTENT="Getting the Dispersion Solution"> <META NAME="keywords" CONTENT="vol2"> <META NAME="resource-type" CONTENT="document"> <META NAME="distribution" CONTENT="global"> <META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1"> <LINK REL="STYLESHEET" HREF="vol2.css"> <LINK REL="next" HREF="node130.html"> <LINK REL="previous" HREF="node128.html"> <LINK REL="up" HREF="node127.html"> <LINK REL="next" HREF="node130.html"> </HEAD> <BODY > <!--Navigation Panel--> <A NAME="tex2html3060" HREF="node130.html"> <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="icons.gif/next_motif.gif"></A> <A NAME="tex2html3057" HREF="node127.html"> <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="icons.gif/up_motif.gif"></A> <A NAME="tex2html3051" HREF="node128.html"> <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="icons.gif/previous_motif.gif"></A> <A NAME="tex2html3059" HREF="node1.html"> <IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="icons.gif/contents_motif.gif"></A> <BR> <B> Next:</B> <A NAME="tex2html3061" HREF="node130.html">Distortion Along the Slit</A> <B> Up:</B> <A NAME="tex2html3058" HREF="node127.html">Geometric Correction</A> <B> Previous:</B> <A NAME="tex2html3052" HREF="node128.html">Detecting and Identifying Arc</A> <BR> <BR> <!--End of Navigation Panel--> <H2><A NAME="SECTION001032000000000000000"> Getting the Dispersion Solution</A> </H2> <P> The command <TT>CALIBRATE/LONG</TT> approximates the dispersion relation<A NAME="5787"> </A> for each searched row of the arc spectrum. The algorithm can be activated in different modes, controlled by the parameter <TT>WLCMTD</TT>: <P> <UL> <LI>The mode <TT>IDENT</TT> must be used if the lines have been identified interactively using <TT>IDENTIFY/LONG</TT>. <P> <LI>The mode <TT>GUESS</TT> allows a previously saved session to be used (command <TT>SAVE/LONG</TT>) for the determination of the dispersion relation. The two observation sets must in principle correspond to the same instrumental set-up. Limited shifts (up to 5 pixels) are taken into account by a cross-correlation algorithm. The name of the reference session must be indicated by the parameter <TT>GUESS</TT>. <P> <LI>It is also noteworthy to indicate the presence of a mode <TT>LINEAR</TT> introduced in the package for the purpose of on-line calibration. This mode is still <B>in evaluation phase</B> and allows the results of the command <TT>ESTIMATE/DISPERSION</TT> to be taken into account in the calibration process. </UL> <P> The command <TT>CALIBRATE/LONG</TT> provides the following results: <P> <UL> <LI>The coefficients of the dispersion relation for each searched row of the two-dimensional spectrum are computed as a series of polynomials <!-- MATH: $\lambda = p_y (x)$ --> <IMG WIDTH="53" HEIGHT="42" ALIGN="MIDDLE" BORDER="0" SRC="img247.gif" ALT="$\lambda = p_y (x)$">. The results are written in the table <TT>coerbr.tbl</TT>. <P> <LI>The starting, final and average step wavelength<A NAME="5802"> </A> of the spectrum are estimated and written in the keywords <TT>REBSTRT, REBEND, REBSTP</TT> used by the commands <TT>REBIN/LONG</TT> and <TT> RECTIFY/LONG</TT>. <P> <LI>A two-dimensional dispersion relation of the form <!-- MATH: $\lambda = \lambda(X, Y)$ --> <IMG WIDTH="78" HEIGHT="44" ALIGN="MIDDLE" BORDER="0" SRC="img248.gif" ALT="$\lambda = \lambda(X, Y)$"> is estimated if the parameter <TT>TWODOPT</TT> is set to <TT>YES</TT>. This bivariate dispersion relation is necessary to use the command <TT>RECTIFY/LONG</TT>. The resulting dispersion coefficients are stored in the keyword <TT>KEYLONGD</TT>. <P> </UL> <P> The command <TT>CALIBRATE/TWICE</TT> performs a two-pass determination of the dispersion relation. In a first pass, the lines are identified by a standard <TT>CALIBRATE/LONG</TT>. Only the lines which have consistently identified at all rows are selected for the second pass, which then performs a new calibration on a stable set of arc lines. If after selection a good spectral coverage of the arc spectrum is secured, this method provides very stable estimates of the dispersion relation. <P> The command <TT>PLOT/CALIBRATE</TT> visualizes the lines found by the calibration process. The dispersion curve and the lines that were used to determine it are presented by <TT>PLOT/DELTA</TT>. Residuals to the dispersion curve are plotted by <TT>PLOT/RESIDUAL</TT>. For two-dimensional spectra, the command <TT>PLOT/DISTORTION</TT> can be used to check the stability of the dispersion relation along the slit. <P> The iterative identification loop consists of estimating the wavelength of all lines in the arc spectrum and associate them to laboratory wavelengths to refine the estimates of the dispersion relation. The line identification criterion will associate a computed wavelength <IMG WIDTH="52" HEIGHT="72" ALIGN="MIDDLE" BORDER="0" SRC="img250.gif" ALT="$\lambda_c$"> to the nearest catalog wavelength <!-- MATH: $\lambda_{cat}$ --> <IMG WIDTH="66" HEIGHT="72" ALIGN="MIDDLE" BORDER="0" SRC="img251.gif" ALT="$\lambda_{cat}$"> if the residual: <P> <BR><P></P> <DIV ALIGN="CENTER"> <!-- MATH: \begin{displaymath} \delta \lambda = | \lambda_c - \lambda_{cat} | \end{displaymath} --> <IMG WIDTH="146" HEIGHT="40" SRC="img252.gif" ALT="\begin{displaymath}\delta \lambda = \vert \lambda_c - \lambda_{cat} \vert \end{displaymath}"> </DIV> <BR CLEAR="ALL"> <P></P> <P> is small compared to the distance of the next neighbours in both the arc spectrum and the catalog: <P> <BR><P></P> <DIV ALIGN="CENTER"> <!-- MATH: \begin{displaymath} \delta \lambda < min(\delta \lambda_{cat}, \delta \lambda_c)*\alpha \end{displaymath} --> <IMG WIDTH="230" HEIGHT="23" SRC="img253.gif" ALT="\begin{displaymath}\delta \lambda < min(\delta \lambda_{cat}, \delta \lambda_c)*\alpha \end{displaymath}"> </DIV> <BR CLEAR="ALL"> <P></P> <P> where <!-- MATH: $\delta \lambda_{cat}$ --> <IMG WIDTH="6" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="img254.gif" ALT="$\delta \lambda_{cat}$"> is the distance to the next neighbour in the line catalog, <!-- MATH: $\delta \lambda_c$ --> <IMG WIDTH="62" HEIGHT="72" ALIGN="MIDDLE" BORDER="0" SRC="img255.gif" ALT="$\delta \lambda_c$"> the distance to the next neighbour in the arc spectrum and <I>alpha</I> the tolerance parameter. Optimal values of <IMG WIDTH="44" HEIGHT="54" ALIGN="BOTTOM" BORDER="0" SRC="img256.gif" ALT="$\alpha$"> are in the range <!-- MATH: $0 < \alpha < 0.5$ --> <IMG WIDTH="72" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="img257.gif" ALT="$ 0 < \alpha < 0.5$">. The tolerance value is controlled by the parameter <TT>ALPHA</TT>. <P> Lines are identified in a first pass without consideration of the rms of the residual values by an iterative loop controlled by the parameter <TT>WLCNITER</TT>. The residuals for each line are then checked in order to reject outliers which residual is above the value specified by the final tolerance parameter <TT>TOL</TT>. The degree of the polynomials is controlled by the parameter <TT>DCX</TT> and the iterative loop is stopped if residuals are found to be larger than <TT>MAXDEV</TT>. <P> <HR> <!--Navigation Panel--> <A NAME="tex2html3060" HREF="node130.html"> <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="icons.gif/next_motif.gif"></A> <A NAME="tex2html3057" HREF="node127.html"> <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="icons.gif/up_motif.gif"></A> <A NAME="tex2html3051" HREF="node128.html"> <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="icons.gif/previous_motif.gif"></A> <A NAME="tex2html3059" HREF="node1.html"> <IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="icons.gif/contents_motif.gif"></A> <BR> <B> Next:</B> <A NAME="tex2html3061" HREF="node130.html">Distortion Along the Slit</A> <B> Up:</B> <A NAME="tex2html3058" HREF="node127.html">Geometric Correction</A> <B> Previous:</B> <A NAME="tex2html3052" HREF="node128.html">Detecting and Identifying Arc</A> <!--End of Navigation Panel--> <ADDRESS> <I>Petra Nass</I> <BR><I>1999-06-15</I> </ADDRESS> </BODY> </HTML>