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<title>GREENSPLINE</title>
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<body bgcolor="#ffffff">
<h1 align="center">GREENSPLINE</h1>
<a href="#NAME">NAME</a><br>
<a href="#SYNOPSIS">SYNOPSIS</a><br>
<a href="#DESCRIPTION">DESCRIPTION</a><br>
<a href="#OPTIONS">OPTIONS</a><br>
<a href="#1-D EXAMPLES">1-D EXAMPLES</a><br>
<a href="#2-D EXAMPLES">2-D EXAMPLES</a><br>
<a href="#3-D EXAMPLES">3-D EXAMPLES</a><br>
<a href="#2-D SPHERICAL SURFACE EXAMPLES">2-D SPHERICAL SURFACE EXAMPLES</a><br>
<a href="#CONSIDERATIONS">CONSIDERATIONS</a><br>
<a href="#TENSION">TENSION</a><br>
<a href="#REFERENCES">REFERENCES</a><br>
<a href="#SEE ALSO">SEE ALSO</a><br>
<hr>
<h2>NAME
<a name="NAME"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em">greenspline
− Interpolate 1-D, 2-D, 3-D Cartesian or spherical
surface data using Green’s function splines.</p>
<h2>SYNOPSIS
<a name="SYNOPSIS"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em"><b>greenspline</b>
[ <i>datafile(s)</i> ] [
<b>−A</b>[<b>1</b>|<b>2</b>|<b>3</b>|<b>4</b>|<b>5</b>,]<i>gradfile</i>
] [ <b>−C</b><i>cut</i>[/<i>file</i>] ] [
<b>−D</b><i>mode</i> ] [ <b>−F</b> ] [
<b>−G</b><i>grdfile</i> ] [
<b>−H</b>[<b>i</b>][<i>nrec</i>] ] [
<b>−I</b><i>xinc</i>[<i>yinc</i>[<i>zinc</i>]] ] [
<b>−L</b> ] [ <b>−N</b><i>nodefile</i> ] [
<b>−Q</b><i>az</i>|<i>x/y/z</i> ] [
<b>−R</b><i>xmin</i>/<i>xmax</i>[/<i>ymin</i>/<i>ymax</i>[/<i>zminzmax</i>]]
] [ <b>−Sc|t|g|p|q</b>[<i>pars</i>] ] [
<b>−T</b><i>maskgrid</i> ] [ <b>−V</b> ] [
<b>−:</b>[<b>i</b>|<b>o</b>] ] [
<b>−bi</b>[<b>s</b>|<b>S</b>|<b>d</b>|<b>D</b>[<i>ncol</i>]|<b>c</b>[<i>var1</i><b>/</b><i>...</i>]]
] [ <b>−f</b>[<b>i</b>|<b>o</b>]<i>colinfo</i> ] [
<b>−bo</b>[<b>s</b>|<b>S</b>|<b>d</b>|<b>D</b>[<i>ncol</i>]|<b>c</b>[<i>var1</i><b>/</b><i>...</i>]]
]</p>
<h2>DESCRIPTION
<a name="DESCRIPTION"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em"><b>greenspline</b>
uses the Green’s function G(<b>x</b>; <b>x’</b>)
for the chosen spline and geometry to interpolate data at
regular [or arbitrary] output locations. Mathematically, the
solution is composed as <i>w</i>(<b>x</b>) = sum
{<i>c</i>(<i>i</i>) G(<b>x</b>; <b>x</b>(<i>i</i>))}, for
<i>i</i> = 1, <i>n</i>, the number of data points
{<b>x</b>(<i>i</i>), <i>w</i>(<i>i</i>)}. Once the <i>n</i>
coefficients <i>c</i>(<i>i</i>) have been found then the sum
can be evaluated at any output point <b>x</b>. Choose
between ten minimum curvature, regularized, or continuous
curvature splines in tension for either 1-D, 2-D, or 3-D
Cartesian coordinates or spherical surface coordinates.
After first removing a linear or planar trend (Cartesian
geometries) or mean value (spherical surface) and
normalizing these residuals, the least-squares matrix
solution for the spline coefficients <i>c</i>(<i>i</i>) is
found by solving the <i>n</i> by <i>n</i> linear system
<i>w</i>(<i>j</i>) = sum-over-<i>i</i> {<i>c</i>(<i>i</i>)
G(<b>x</b>(<i>j</i>); <b>x</b>(<i>i</i>))}, for <i>j</i> =
1, <i>n</i>; this solution yields an exact interpolation of
the supplied data points. Alternatively, you may choose to
perform a singular value decomposition (SVD) and eliminate
the contribution from the smallest eigenvalues; this
approach yields an approximate solution. Trends and scales
are restored when evaluating the output.</p>
<h2>OPTIONS
<a name="OPTIONS"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em"><i>datafile(s)</i></p>
<p style="margin-left:22%;">The name of one or more ASCII
[or binary, see <b>−bi</b>] files holding the
<b>x</b>, <i>w</i> data points. If no file is given then we
read standard input instead.</p>
<table width="100%" border="0" rules="none" frame="void"
cellspacing="0" cellpadding="0">
<tr valign="top" align="left">
<td width="11%"></td>
<td width="3%">
<p><b>−A</b></p></td>
<td width="8%"></td>
<td width="78%">
<p>The solution will partly be constrained by surface
gradients <b>v</b> = <i>v</i>*<b>n</b>, where <i>v</i> is
the gradient magnitude and <b>n</b> its unit vector
direction. The gradient direction may be specified either by
Cartesian components (either unit vector <b>n</b> and
magnitude <i>v</i> separately or gradient components
<b>v</b> directly) or angles w.r.t. the coordinate axes.
Specify one of five input formats: <b>0</b>: For 1-D data
there is no direction, just gradient magnitude (slope) so
the input format is <i>x, gradient</i>. Options 1-2 are for
2-D data sets: <b>1</b>: records contain <i>x, y, azimuth,
gradient</i> (<i>azimuth</i> in degrees is measured
clockwise from the vertical (north) [Default]). <b>2</b>:
records contain <i>x, y, gradient, azimuth</i>
(<i>azimuth</i> in degrees is measured clockwise from the
vertical (north)). Options 3-5 are for either 2-D or 3-D
data: <b>3</b>: records contain <b>x</b>, <i>direction(s),
v</i> (<i>direction(s)</i> in degrees are measured
counter-clockwise from the horizontal (and for 3-D the
vertical axis). <b>4</b>: records contain <b>x</b>,
<b>v</b>. <b>5</b>: records contain <b>x</b>, <b>n</b>,
<i>v</i>. Append name of ASCII file with the surface
gradients (following a comma if a format is specified).</p></td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="3%">
<p><b>−C</b></p></td>
<td width="8%"></td>
<td width="78%">
<p>Find an approximate surface fit: Solve the linear system
for the spline coefficients by SVD and eliminate the
contribution from all eigenvalues whose ratio to the largest
eigenvalue is less than <i>cut</i> [Default uses
Gauss-Jordan elimination to solve the linear system and fit
the data exactly]. Optionally, append /<i>file</i> to save
the eigenvalue ratios to the specified file for further
analysis. Finally, if a negative <i>cut</i> is given then
/<i>file</i> is required and execution will stop after
saving the eigenvalues, i.e., no surface output is
produced.</p> </td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="3%">
<p><b>−D</b></p></td>
<td width="8%"></td>
<td width="78%">
<p>Sets the distance flag that determines how we calculate
distances between data points. Select <i>mode</i> 0 for
Cartesian 1-D spline interpolation: <b>−D</b> 0 means
(<i>x</i>) in user units, Cartesian distances, Select
<i>mode</i> 1-3 for Cartesian 2-D surface spline
interpolation: <b>−D</b> 1 means (<i>x,y</i>) in user
units, Cartesian distances, <b>−D</b> 2 for
(<i>x,y</i>) in degrees, flat Earth distances, and
<b>−D</b> 3 for (<i>x,y</i>) in degrees, spherical
distances in km. Then, if <b><A HREF="gmtdefaults.html#ELLIPSOID">ELLIPSOID</A></b> is spherical, we
compute great circle arcs, otherwise geodesics. Option
<i>mode</i> = 4 applies to spherical surface spline
interpolation only: <b>−D</b> 4 for (<i>x,y</i>) in
degrees, use cosine of great circle (or geodesic) arcs.
Select <i>mode</i> 5 for Cartesian 3-D surface spline
interpolation: <b>−D</b> 5 means (<i>x,y,z</i>) in
user units, Cartesian distances.</p></td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="3%">
<p><b>−F</b></p></td>
<td width="8%"></td>
<td width="78%">
<p>Force pixel registration. [Default is gridline
registration].</p> </td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="3%">
<p><b>−G</b></p></td>
<td width="8%"></td>
<td width="78%">
<p>Name of resulting output file. (1) If options
<b>−R</b>, <b>−I</b>, and possibly
<b>−F</b> are set we produce an equidistant output
table. This will be written to stdout unless <b>−G</b>
is specified. Note: for 2-D grids the <b>−G</b> option
is required. (2) If option <b>−T</b> is selected then
<b>−G</b> is required and the output file is a 2-D
binary grid file. Applies to 2-D interpolation only. (3) If
<b>−N</b> is selected then the output is an ASCII (or
binary; see <b>−bo</b>) table; if <b>−G</b> is
not given then this table is written to standard output.
Ignored if <b>−C</b> or <b>−C</b> 0 is
given.</p> </td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="3%">
<p><b>−H</b></p></td>
<td width="8%"></td>
<td width="78%">
<p>Input file(s) has header record(s). If used, the default
number of header records is <b><A HREF="gmtdefaults.html#N_HEADER_RECS">N_HEADER_RECS</A></b>. Use
<b>−Hi</b> if only input data should have header
records [Default will write out header records if the input
data have them]. Blank lines and lines starting with # are
always skipped.</p></td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="3%">
<p><b>−I</b></p></td>
<td width="8%"></td>
<td width="78%">
<p>Specify equidistant sampling intervals, on for each
dimension, separated by slashes.</p></td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="3%">
<p><b>−L</b></p></td>
<td width="8%"></td>
<td width="78%">
<p>Do <i>not</i> remove a linear (1-D) or planer (2-D)
trend when <b>−D</b> selects mode 0-3 [For those
Cartesian cases a least-squares line or plane is modeled and
removed, then restored after fitting a spline to the
residuals]. However, in mixed cases with both data values
and gradients, or for spherical surface data, only the mean
data value is removed (and later and restored).</p></td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="3%">
<p><b>−N</b></p></td>
<td width="8%"></td>
<td width="78%">
<p>ASCII file with coordinates of desired output locations
<b>x</b> in the first column(s). The resulting <i>w</i>
values are appended to each record and written to the file
given in <b>−G</b> [or stdout if not specified]; see
<b>−bo</b> for binary output instead. This option
eliminates the need to specify options <b>−R</b>,
<b>−I</b>, and <b>−F</b>.</p></td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="3%">
<p><b>−Q</b></p></td>
<td width="8%"></td>
<td width="78%">
<p>Rather than evaluate the surface, take the directional
derivative in the <i>az</i> azimuth and return the magnitude
of this derivative instead. For 3-D interpolation, specify
the three components of the desired vector direction (the
vector will be normalized before use).</p></td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="3%">
<p><b>−R</b></p></td>
<td width="8%"></td>
<td width="78%">
<p>Specify the domain for an equidistant lattice where
output predictions are required. Requires <b>−I</b>
and optionally <b>−F</b>.</p></td></tr>
</table>
<p style="margin-left:22%;"><i>1-D:</i> Give
<i>xmin/xmax</i>, the minimum and maximum <i>x</i>
coordinates. <i><br>
2-D:</i> Give <i>xmin/xmax/ymin/ymax</i>, the minimum and
maximum <i>x</i> and <i>y</i> coordinates. These may be
Cartesian or geographical. If geographical, then <i>west,
east, south,</i> and <i>north</i> specify the Region of
interest, and you may specify them in decimal degrees or in
[+-]dd:mm[:ss.xxx][W|E|S|N] format. The two shorthands
<b>−Rg</b> and <b>−Rd</b> stand for global
domain (0/360 and -180/+180 in longitude respectively, with
-90/+90 in latitude). <i><br>
3-D:</i> Give <i>xmin/xmax/ymin/ymax/zmin/zmax</i>, the
minimum and maximum <i>x</i>, <i>y</i> and <i>z</i>
coordinates. See the 2-D section if your horizontal
coordinates are geographical; note the shorthands
<b>−Rg</b> and <b>−Rd</b> cannot be used if a
3-D domain is specified.</p>
<table width="100%" border="0" rules="none" frame="void"
cellspacing="0" cellpadding="0">
<tr valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p style="margin-top: 1em"><b>−S</b></p></td>
<td width="7%"></td>
<td width="78%">
<p style="margin-top: 1em">Select one of five different
splines. The first two are used for 1-D, 2-D, or 3-D
Cartesian splines (see <b>−D</b> for discussion). Note
that all tension values are expected to be normalized
tension in the range 0 < <i>t</i> < 1: (<b>c</b>)
Minimum curvature spline [<i>Sandwell</i>, 1987], (<b>t</b>)
Continuous curvature spline in tension [<i>Wessel and
Bercovici</i>, 1998]; append <i>tension</i>[/<i>scale</i>]
with <i>tension</i> in the 0−1 range and optionally
supply a length scale [Default is the average grid spacing].
The next is a 2-D or 3-D spline: (<b>r</b>) Regularized
spline in tension [<i>Mitasova and Mitas</i>, 1993]; again,
append <i>tension</i> and optional <i>scale</i>. The last
two are spherical surface splines and both imply
<b>−D</b> 4 and geographic data: (<b>p</b>) Minimum
curvature spline [<i>Parker</i>, 1994], (<b>q</b>)
Continuous curvature spline in tension [<i>Wessel and
Becker</i>, 2008]; append <i>tension</i>. The G(<b>x</b>;
<b>x’</b>) for the last method is slower to compute;
by specifying <b>−SQ</b> you can speed up calculations
by first pre-calculating G(<b>x</b>; <b>x’</b>) for a
dense set of <b>x</b> values (e.g., 100,001 nodes between -1
to +1) and store them in look-up tables. Optionally append
/<i>N</i> (an odd integer) to specify how many points in the
spline to set [100001]</p></td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p><b>−T</b></p></td>
<td width="7%"></td>
<td width="78%">
<p>For 2-D interpolation only. Only evaluate the solution
at the nodes in the <i>maskgrid</i> that are not equal to
NaN. This option eliminates the need to specify options
<b>−R</b>, <b>−I</b>, and <b>−F</b>.</p></td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p><b>−V</b></p></td>
<td width="7%"></td>
<td width="78%">
<p>Selects verbose mode, which will send progress reports
to stderr [Default runs "silently"].</p></td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p><b>−bi</b></p></td>
<td width="7%"></td>
<td width="78%">
<p>Selects binary input. Append <b>s</b> for single
precision [Default is <b>d</b> (double)]. Uppercase <b>S</b>
or <b>D</b> will force byte-swapping. Optionally, append
<i>ncol</i>, the number of columns in your binary input file
if it exceeds the columns needed by the program. Or append
<b>c</b> if the input file is netCDF. Optionally, append
<i>var1</i><b>/</b><i>var2</i><b>/</b><i>...</i> to specify
the variables to be read. [Default is 2-4 input columns
(<b>x</b>,<i>w</i>); the number depends on the chosen
dimension].</p> </td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p><b>−f</b></p></td>
<td width="7%"></td>
<td width="78%">
<p>Special formatting of input and/or output columns (time
or geographical data). Specify <b>i</b> or <b>o</b> to make
this apply only to input or output [Default applies to
both]. Give one or more columns (or column ranges) separated
by commas. Append <b>T</b> (absolute calendar time),
<b>t</b> (relative time in chosen <b><A HREF="gmtdefaults.html#TIME_UNIT">TIME_UNIT</A></b> since
<b><A HREF="gmtdefaults.html#TIME_EPOCH">TIME_EPOCH</A></b>), <b>x</b> (longitude), <b>y</b>
(latitude), or <b>f</b> (floating point) to each column or
column range item. Shorthand
<b>−f</b>[<b>i</b>|<b>o</b>]<b>g</b> means
<b>−f</b>[<b>i</b>|<b>o</b>]0<b>x</b>,1<b>y</b>
(geographic coordinates).</p></td></tr>
<tr valign="top" align="left">
<td width="11%"></td>
<td width="4%">
<p><b>−bo</b></p></td>
<td width="7%"></td>
<td width="78%">
<p>Selects binary output. Append <b>s</b> for single
precision [Default is <b>d</b> (double)]. Uppercase <b>S</b>
or <b>D</b> will force byte-swapping. Optionally, append
<i>ncol</i>, the number of desired columns in your binary
output file.</p></td></tr>
</table>
<h2>1-D EXAMPLES
<a name="1-D EXAMPLES"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em">To resample the
<i>x,y</i> Gaussian random data created by <b><A HREF="gmtmath.html">gmtmath</A></b>
and stored in 1D.txt, requesting output every 0.1 step from
0 to 10, and using a minimum cubic spline, try</p>
<p style="margin-left:11%; margin-top: 1em"><b>gmtmath
−T</b> 0/10/1 0 1 <b>NRAND</b> = 1D.txt <b><br>
psxy −R</b>0/10/-5/5 <b>−JX</b> 6i/3i
<b>−B</b> 2f1/1 <b>−Sc</b> 0.1 <b>−G</b>
black 1D.txt <b>−K</b> > 1D.ps <b><br>
greenspline</b> 1D.txt <b>−R</b> 0/10 <b>−I</b>
0.1 <b>−Sc −V</b> | <b>psxy −R −J
−O −W</b> thin >> 1D.ps</p>
<p style="margin-left:11%; margin-top: 1em">To apply a
spline in tension instead, using a tension of 0.7, try</p>
<p style="margin-left:11%; margin-top: 1em"><b>psxy
−R</b>0/10/-5/5 <b>−JX</b> 6i/3i <b>−B</b>
2f1/1 <b>−Sc</b> 0.1 <b>−G</b> black 1D.txt
<b>−K</b> > 1Dt.ps <b><br>
greenspline</b> 1D.txt <b>−R</b> 0/10 <b>−I</b>
0.1 <b>−St</b> 0.7 <b>−V</b> | <b>psxy −R
−J −O −W</b> thin >> 1Dt.ps</p>
<h2>2-D EXAMPLES
<a name="2-D EXAMPLES"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em">To make a
uniform grid using the minimum curvature spline for the same
Cartesian data set from Davis (1986) that is used in the GMT
Cookbook example 16, try</p>
<p style="margin-left:11%; margin-top: 1em"><b>greenspline</b>
table_5.11 <b>−R</b> 0/6.5/-0.2/6.5 <b>−I</b>
0.1 <b>−Sc −V −D</b> 1 <b>−G</b>
S1987.grd <b><br>
psxy −R</b>0/6.5/-0.2/6.5 <b>−JX</b> 6i
<b>−B</b> 2f1 <b>−Sc</b> 0.1 <b>−G</b>
black table_5.11 <b>−K</b> > 2D.ps <b><br>
grdcontour −JX</b>6i <b>−B</b> 2f1 <b>−O
−C</b> 25 <b>−A</b> 50 S1987.grd >>
2D.ps</p>
<p style="margin-left:11%; margin-top: 1em">To use
Cartesian splines in tension but only evaluate the solution
where the input mask grid is not NaN, try</p>
<p style="margin-left:11%; margin-top: 1em"><b>greenspline</b>
table_5.11 <b>−T</b> mask.grd <b>−St</b> 0.5
<b>−V −D</b> 1 <b>−G</b> WB1998.grd</p>
<p style="margin-left:11%; margin-top: 1em">To use
Cartesian generalized splines in tension and return the
magnitude of the surface slope in the NW direction, try</p>
<p style="margin-left:11%; margin-top: 1em"><b>greenspline</b>
table_5.11 <b>−R</b> 0/6.5/-0.2/6.5 <b>−I</b>
0.1 <b>−Sr</b> 0.95 <b>−V −D</b> 1
<b>−Q</b>-45 <b>−G</b> slopes.grd Finally, to
use Cartesian minimum curvature splines in recovering a
surface where the input data is a single surface value
(pt.d) and the remaining constraints specify only the
surface slope and direction (slopes.d), use</p>
<p style="margin-left:11%; margin-top: 1em"><b>greenspline</b>
pt.d <b>−R</b>-3.2/3.2/-3.2/3.2 <b>−I</b> 0.1
<b>−Sc −V −D</b> 1 <b>−A</b>
1,slopes.d <b>−G</b> slopes.grd</p>
<h2>3-D EXAMPLES
<a name="3-D EXAMPLES"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em">To create a
uniform 3-D Cartesian grid table based on the data in
table_5.23 in Davis (1986) that contains <i>x,y,z</i>
locations and a measure of uranium oxide concentrations (in
percent), try</p>
<p style="margin-left:11%; margin-top: 1em"><b>greenspline</b>
table_5.23 <b>−R</b> 5/40/-5/10/5/16 <b>−I</b>
0.25 <b>−Sr</b> 0.85 <b>−V −D</b> 5
<b>−G</b> 3D_UO2.txt</p>
<h2>2-D SPHERICAL SURFACE EXAMPLES
<a name="2-D SPHERICAL SURFACE EXAMPLES"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em">To recreate
Parker’s [1994] example on a global 1x1 degree grid,
assuming the data are in file mag_obs_1990.d, try</p>
<p style="margin-left:11%; margin-top: 1em">greenspline
<b>−V −Rg −Sp −D</b> 3
<b>−I</b> 1 <b>−G</b> P1994.grd
mag_obs_1990.d</p>
<p style="margin-left:11%; margin-top: 1em">To do the same
problem but applying tension and use pre-calculated Green
functions, use</p>
<p style="margin-left:11%; margin-top: 1em">greenspline
<b>−V −Rg −SQ</b> 0.85 <b>−D</b> 3
<b>−I</b> 1 <b>−G</b> WB2008.grd
mag_obs_1990.d</p>
<h2>CONSIDERATIONS
<a name="CONSIDERATIONS"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em">(1) For the
Cartesian cases we use the free-space Green functions, hence
no boundary conditions are applied at the edges of the
specified domain. For most applications this is fine as the
region typically is arbitrarily set to reflect the extent of
your data. However, if your application requires particular
boundary conditions then you may consider using
<b><A HREF="surface.html">surface</A></b> instead. <br>
(2) In all cases, the solution is obtained by inverting a
<i>n</i> x <i>n</i> double precision matrix for the Green
function coefficients, where <i>n</i> is the number of data
constraints. Hence, your computer’s memory may place
restrictions on how large data sets you can process with
<b>greenspline</b>. Pre-processing your data with
<b><A HREF="blockmean.html">blockmean</A></b>, <b><A HREF="blockmedian.html">blockmedian</A></b>, or <b><A HREF="blockmode.html">blockmode</A></b> is
recommended to avoid aliasing and may also control the size
of <i>n</i>. For information, if <i>n</i> = 1024 then only 8
Mb memory is needed, but for <i>n</i> = 10240 we need 800
Mb. Note that <b>greenspline</b> is fully 64-bit compliant
if compiled as such. <br>
(3) The inversion for coefficients can become numerically
unstable when data neighbors are very close compared to the
overall span of the data. You can remedy this by
pre-processing the data, e.g., by averaging closely spaced
neighbors. Alternatively, you can improve stability by using
the SVD solution and discard information associated with the
smallest eigenvalues (see <b>−C</b>).</p>
<h2>TENSION
<a name="TENSION"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em">Tension is
generally used to suppress spurious oscillations caused by
the minimum curvature requirement, in particular when rapid
gradient changes are present in the data. The proper amount
of tension can only be determined by experimentation.
Generally, very smooth data (such as potential fields) do
not require much, if any tension, while rougher data (such
as topography) will typically interpolate better with
moderate tension. Make sure you try a range of values before
choosing your final result. Note: the regularized spline in
tension is only stable for a finite range of <i>scale</i>
values; you must experiment to find the valid range and a
useful setting. For more information on tension see the
references below.</p>
<h2>REFERENCES
<a name="REFERENCES"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em">Davis, J. C.,
1986, <i>Statistics and Data Analysis in Geology</i>, 2nd
Edition, 646 pp., Wiley, New York, <br>
Mitasova, H., and L. Mitas, 1993, Interpolation by
regularized spline with tension: I. Theory and
implementation, <i>Math. Geol., 25</i>, 641−655. <br>
Parker, R. L., 1994, <i>Geophysical Inverse Theory</i>, 386
pp., Princeton Univ. Press, Princeton, N.J. <br>
Sandwell, D. T., 1987, Biharmonic spline interpolation of
Geos-3 and Seasat altimeter data, <i>Geophys. Res. Lett.,
14</i>, 139−142. <br>
Wessel, P., and D. Bercovici, 1998, Interpolation with
splines in tension: a Green’s function approach,
<i>Math. Geol., 30</i>, 77−93. <br>
Wessel, P., and J. M. Becker, 2008, Interpolation using a
generalized Green’s function for a spherical surface
spline in tension, <i>Geophys. J. Int, 174</i>, 21−28.
<br>
Wessel, P., 2009, A general-purpose Green’s function
interpolator, <i>Computers & Geosciences, 35</i>,
1247−1254, doi:10.1016/j.cageo.2008.08.012.</p>
<h2>SEE ALSO
<a name="SEE ALSO"></a>
</h2>
<p style="margin-left:11%; margin-top: 1em"><i><A HREF="GMT.html">GMT</A></i>(1),
<i><A HREF="gmtmath.html">gmtmath</A></i>(1), <i><A HREF="nearneighbor.html">nearneighbor</A></i>(1), <i><A HREF="psxy.html">psxy</A></i>(1),
<i><A HREF="surface.html">surface</A></i>(1), <i><A HREF="triangulate.html">triangulate</A></i>(1),
<i><A HREF="xyz2grd.html">xyz2grd</A></i>(1)</p>
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