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#
7.29 Gridding spherical surface data using splines

Finally, we demonstrate how gridding on a spherical surface can be accomplished using Green's functions of surface splines, with or without tension. Global gridding does not work particularly well in Cartesian coordinates hence the chosen approach. We use greenspline to produce a crude topography grid for Mars based on radii estimates from the Mariner 9 and Viking Orbiter spacecrafts. This data comes from Smith and Zuber [Science, 1996] and is used here as a small (N = 370) data set we can use to demonstrate spherical surface gridding. Since greenspline must solve a N by N matrix system your system memory may impose limits on how large data sets you can handle; also note that the spherical surface spline in tension is particularly slow to compute.

    #!/bin/sh
    #               GMT EXAMPLE 29
    #
    # Purpose:      Illustrates spherical surface gridding with Green's function of splines
    # GMT progs:    makecpt, grdcontour, grdgradient, grdimage, grdmath greenspline, psscale, pstext
    # Unix progs:   rm, echo
    #
    ps=example_29.ps

    # This example uses 370 radio occultation data for Mars to grid the topography.
    # Data and information from Smith, D. E., and M. T. Zuber (1996), The shape of
    # Mars and the topographic signature of the hemispheric dichotomy, Science, 271, 184–187.

    # Make Mars ellipsoid given their three best-fitting axes:
    a=3399.472
    b=3394.329
    c=3376.502
    grdmath -Rg -I4 -F X COSD $a DIV DUP MUL X SIND $b DIV DUP MUL ADD Y COSD DUP MUL MUL Y SIND $c DIV \
            DUP MUL ADD SQRT INV = ellipsoid.nc

    #  Do both Parker and Wessel/Becker solutions (tension = 0.9975)
    greenspline -Rellipsoid.nc mars370.in -D4 -Sp -Gmars.nc
    greenspline -Rellipsoid.nc mars370.in -D4 -SQ0.9975/5001 -Gmars2.nc
    # Scale to km and remove ellipsoid
    grdmath mars.nc 1000 DIV ellipsoid.nc SUB = mars.nc
    grdmath mars2.nc 1000 DIV ellipsoid.nc SUB = mars2.nc
    makecpt -Crainbow -T-7/15/1 -Z > mars.cpt
    grdgradient mars2.nc -M -Ne0.75 -A45 -Gmars2_i.nc
    grdimage mars2.nc -Imars2_i.nc -Cmars.cpt -B30g30Wsne -JH0/6i -P -K -Ei \
            -U"Example 29 in Cookbook" --ANNOT_FONT_SIZE=12 > $ps
    grdcontour mars2.nc -J -O -K -C1 -A5 -Glz+/z- >> $ps
    psxy -Rg -J -O -K -Sc0.045i -Gblack mars370.in  >> $ps
    echo "0 90 14 0 1 LB b)" | pstext -R -J -O -K -N -D-3i/-0.2i >> $ps
    grdgradient mars.nc -M -Ne0.75 -A45 -Gmars_i.nc
    grdimage mars.nc -Imars_i.nc -Cmars.cpt -B30g30Wsne -J -O -K -Y3.6i -Ei --ANNOT_FONT_SIZE=12 >> $ps
    grdcontour mars.nc -J -O -K -C1 -A5 -Glz+/z- >> $ps
    psxy -Rg -J -O -K -Sc0.045i -Gblack mars370.in  >> $ps
    psscale -Cmars.cpt -O -K -D3i/-0.1i/5i/0.1ih -I --ANNOT_FONT_SIZE=12 -B2f1/:km: >> $ps
    echo "0 90 14 0 1 LB a)" | pstext -R -J -O -N -D-3i/-0.2i >> $ps
    # Clean up
    rm -f *.nc mars.cpt .gmt*

Our script must first estimate the ellipsoidal shape of Mars from the parameters given by Smith and Zuber so that we can remove this reference surface from the gridded radii. We run the gridding twice: First with no tension using Parker's [1990] method and then with tension using the Wessel and Becker [2008] method. The grids are then imaged with grdimage and grdcontour and a color scale is placed between them.

Figure 7.29: Gridding of spherical surface data using Green's function splines.

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