Provided by: gmt_4.5.11-1build1_amd64 
NAME
greenspline - Interpolate 1-D, 2-D, 3-D Cartesian or spherical surface data using Green's function
splines.
SYNOPSIS
greenspline [ datafile(s) ] [ -A[1|2|3|4|5,]gradfile ] [ -Ccut[/file] ] [ -Dmode ] [ -F ] [ -Ggrdfile ] [
-H[i][nrec] ] [ -Ixinc[yinc[zinc]] ] [ -L ] [ -Nnodefile ] [ -Qaz|x/y/z ] [
-Rxmin/xmax[/ymin/ymax[/zminzmax]] ] [ -Sc|t|g|p|q[pars] ] [ -Tmaskgrid ] [ -V ] [ -:[i|o] ] [
-bi[s|S|d|D[ncol]|c[var1/...]] ] [ -f[i|o]colinfo ] [ -bo[s|S|d|D[ncol]|c[var1/...]] ]
DESCRIPTION
greenspline uses the Green's function G(x; x') for the chosen spline and geometry to interpolate data at
regular [or arbitrary] output locations. Mathematically, the solution is composed as w(x) = sum {c(i)
G(x; x(i))}, for i = 1, n, the number of data points {x(i), w(i)}. Once the n coefficients c(i) have
been found then the sum can be evaluated at any output point x. 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 c(i) is found by solving the n by n linear system w(j) = sum-over-i {c(i) G(x(j);
x(i))}, for j = 1, n; 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.
OPTIONS
datafile(s)
The name of one or more ASCII [or binary, see -bi] files holding the x, w data points. If no file
is given then we read standard input instead.
-A The solution will partly be constrained by surface gradients v = v*n, where v is the gradient
magnitude and n its unit vector direction. The gradient direction may be specified either by
Cartesian components (either unit vector n and magnitude v separately or gradient components v
directly) or angles w.r.t. the coordinate axes. Specify one of five input formats: 0: For 1-D
data there is no direction, just gradient magnitude (slope) so the input format is x, gradient.
Options 1-2 are for 2-D data sets: 1: records contain x, y, azimuth, gradient (azimuth in degrees
is measured clockwise from the vertical (north) [Default]). 2: records contain x, y, gradient,
azimuth (azimuth in degrees is measured clockwise from the vertical (north)). Options 3-5 are for
either 2-D or 3-D data: 3: records contain x, direction(s), v (direction(s) in degrees are
measured counter-clockwise from the horizontal (and for 3-D the vertical axis). 4: records
contain x, v. 5: records contain x, n, v. Append name of ASCII file with the surface gradients
(following a comma if a format is specified).
-C 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
cut [Default uses Gauss-Jordan elimination to solve the linear system and fit the data exactly].
Optionally, append /file to save the eigenvalue ratios to the specified file for further analysis.
Finally, if a negative cut is given then /file is required and execution will stop after saving
the eigenvalues, i.e., no surface output is produced.
-D Sets the distance flag that determines how we calculate distances between data points. Select
mode 0 for Cartesian 1-D spline interpolation: -D 0 means (x) in user units, Cartesian distances,
Select mode 1-3 for Cartesian 2-D surface spline interpolation: -D 1 means (x,y) in user units,
Cartesian distances, -D 2 for (x,y) in degrees, flat Earth distances, and -D 3 for (x,y) in
degrees, spherical distances in km. Then, if ELLIPSOID is spherical, we compute great circle
arcs, otherwise geodesics. Option mode = 4 applies to spherical surface spline interpolation
only: -D 4 for (x,y) in degrees, use cosine of great circle (or geodesic) arcs. Select mode 5 for
Cartesian 3-D surface spline interpolation: -D 5 means (x,y,z) in user units, Cartesian distances.
-F Force pixel registration. [Default is gridline registration].
-G Name of resulting output file. (1) If options -R, -I, and possibly -F are set we produce an
equidistant output table. This will be written to stdout unless -G is specified. Note: for 2-D
grids the -G option is required. (2) If option -T is selected then -G is required and the output
file is a 2-D binary grid file. Applies to 2-D interpolation only. (3) If -N is selected then
the output is an ASCII (or binary; see -bo) table; if -G is not given then this table is written
to standard output. Ignored if -C or -C 0 is given.
-H Input file(s) has header record(s). If used, the default number of header records is
N_HEADER_RECS. Use -Hi 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.
-I Specify equidistant sampling intervals, on for each dimension, separated by slashes.
-L Do not remove a linear (1-D) or planer (2-D) trend when -D 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).
-N ASCII file with coordinates of desired output locations x in the first column(s). The resulting w
values are appended to each record and written to the file given in -G [or stdout if not
specified]; see -bo for binary output instead. This option eliminates the need to specify options
-R, -I, and -F.
-Q Rather than evaluate the surface, take the directional derivative in the az 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).
-R Specify the domain for an equidistant lattice where output predictions are required. Requires -I
and optionally -F.
1-D: Give xmin/xmax, the minimum and maximum x coordinates.
2-D: Give xmin/xmax/ymin/ymax, the minimum and maximum x and y coordinates. These may be
Cartesian or geographical. If geographical, then west, east, south, and north 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 -Rg and -Rd stand for global domain (0/360 and -180/+180 in longitude
respectively, with -90/+90 in latitude).
3-D: Give xmin/xmax/ymin/ymax/zmin/zmax, the minimum and maximum x, y and z coordinates. See the
2-D section if your horizontal coordinates are geographical; note the shorthands -Rg and -Rd
cannot be used if a 3-D domain is specified.
-S Select one of five different splines. The first two are used for 1-D, 2-D, or 3-D Cartesian
splines (see -D for discussion). Note that all tension values are expected to be normalized
tension in the range 0 < t < 1: (c) Minimum curvature spline [Sandwell, 1987], (t) Continuous
curvature spline in tension [Wessel and Bercovici, 1998]; append tension[/scale] with tension 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: (r) Regularized spline in tension [Mitasova and Mitas, 1993]; again,
append tension and optional scale. The last two are spherical surface splines and both imply -D 4
and geographic data: (p) Minimum curvature spline [Parker, 1994], (q) Continuous curvature spline
in tension [Wessel and Becker, 2008]; append tension. The G(x; x') for the last method is slower
to compute; by specifying -SQ you can speed up calculations by first pre-calculating G(x; x') for
a dense set of x values (e.g., 100,001 nodes between -1 to +1) and store them in look-up tables.
Optionally append /N (an odd integer) to specify how many points in the spline to set [100001]
-T For 2-D interpolation only. Only evaluate the solution at the nodes in the maskgrid that are not
equal to NaN. This option eliminates the need to specify options -R, -I, and -F.
-V Selects verbose mode, which will send progress reports to stderr [Default runs "silently"].
-bi Selects binary input. Append s for single precision [Default is d (double)]. Uppercase S or D
will force byte-swapping. Optionally, append ncol, the number of columns in your binary input
file if it exceeds the columns needed by the program. Or append c if the input file is netCDF.
Optionally, append var1/var2/... to specify the variables to be read. [Default is 2-4 input
columns (x,w); the number depends on the chosen dimension].
-f Special formatting of input and/or output columns (time or geographical data). Specify i or o 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 T (absolute calendar time), t (relative time in chosen
TIME_UNIT since TIME_EPOCH), x (longitude), y (latitude), or f (floating point) to each column or
column range item. Shorthand -f[i|o]g means -f[i|o]0x,1y (geographic coordinates).
-bo Selects binary output. Append s for single precision [Default is d (double)]. Uppercase S or D
will force byte-swapping. Optionally, append ncol, the number of desired columns in your binary
output file.
1-D EXAMPLES
To resample the x,y Gaussian random data created by gmtmath and stored in 1D.txt, requesting output every
0.1 step from 0 to 10, and using a minimum cubic spline, try
gmtmath -T 0/10/1 0 1 NRAND = 1D.txt
psxy -R0/10/-5/5 -JX 6i/3i -B 2f1/1 -Sc 0.1 -G black 1D.txt -K > 1D.ps
greenspline 1D.txt -R 0/10 -I 0.1 -Sc -V | psxy -R -J -O -W thin >> 1D.ps
To apply a spline in tension instead, using a tension of 0.7, try
psxy -R0/10/-5/5 -JX 6i/3i -B 2f1/1 -Sc 0.1 -G black 1D.txt -K > 1Dt.ps
greenspline 1D.txt -R 0/10 -I 0.1 -St 0.7 -V | psxy -R -J -O -W thin >> 1Dt.ps
2-D EXAMPLES
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
greenspline table_5.11 -R 0/6.5/-0.2/6.5 -I 0.1 -Sc -V -D 1 -G S1987.grd
psxy -R0/6.5/-0.2/6.5 -JX 6i -B 2f1 -Sc 0.1 -G black table_5.11 -K > 2D.ps
grdcontour -JX6i -B 2f1 -O -C 25 -A 50 S1987.grd >> 2D.ps
To use Cartesian splines in tension but only evaluate the solution where the input mask grid is not NaN,
try
greenspline table_5.11 -T mask.grd -St 0.5 -V -D 1 -G WB1998.grd
To use Cartesian generalized splines in tension and return the magnitude of the surface slope in the NW
direction, try
greenspline table_5.11 -R 0/6.5/-0.2/6.5 -I 0.1 -Sr 0.95 -V -D 1 -Q-45 -G 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
greenspline pt.d -R-3.2/3.2/-3.2/3.2 -I 0.1 -Sc -V -D 1 -A 1,slopes.d -G slopes.grd
3-D EXAMPLES
To create a uniform 3-D Cartesian grid table based on the data in table_5.23 in Davis (1986) that
contains x,y,z locations and a measure of uranium oxide concentrations (in percent), try
greenspline table_5.23 -R 5/40/-5/10/5/16 -I 0.25 -Sr 0.85 -V -D 5 -G 3D_UO2.txt
2-D SPHERICAL SURFACE EXAMPLES
To recreate Parker's [1994] example on a global 1x1 degree grid, assuming the data are in file
mag_obs_1990.d, try
greenspline -V -Rg -Sp -D 3 -I 1 -G P1994.grd mag_obs_1990.d
To do the same problem but applying tension and use pre-calculated Green functions, use
greenspline -V -Rg -SQ 0.85 -D 3 -I 1 -G WB2008.grd mag_obs_1990.d
CONSIDERATIONS
(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 surface instead.
(2) In all cases, the solution is obtained by inverting a n x n double precision matrix for the Green
function coefficients, where n is the number of data constraints. Hence, your computer's memory may
place restrictions on how large data sets you can process with greenspline. Pre-processing your data
with blockmean, blockmedian, or blockmode is recommended to avoid aliasing and may also control the size
of n. For information, if n = 1024 then only 8 Mb memory is needed, but for n = 10240 we need 800 Mb.
Note that greenspline is fully 64-bit compliant if compiled as such.
(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 -C).
TENSION
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 scale values; you must experiment to
find the valid range and a useful setting. For more information on tension see the references below.
REFERENCES
Davis, J. C., 1986, Statistics and Data Analysis in Geology, 2nd Edition, 646 pp., Wiley, New York,
Mitasova, H., and L. Mitas, 1993, Interpolation by regularized spline with tension: I. Theory and
implementation, Math. Geol., 25, 641-655.
Parker, R. L., 1994, Geophysical Inverse Theory, 386 pp., Princeton Univ. Press, Princeton, N.J.
Sandwell, D. T., 1987, Biharmonic spline interpolation of Geos-3 and Seasat altimeter data, Geophys. Res.
Lett., 14, 139-142.
Wessel, P., and D. Bercovici, 1998, Interpolation with splines in tension: a Green's function approach,
Math. Geol., 30, 77-93.
Wessel, P., and J. M. Becker, 2008, Interpolation using a generalized Green's function for a spherical
surface spline in tension, Geophys. J. Int, 174, 21-28.
Wessel, P., 2009, A general-purpose Green's function interpolator, Computers & Geosciences, 35,
1247-1254, doi:10.1016/j.cageo.2008.08.012.
SEE ALSO
GMT(1), gmtmath(1), nearneighbor(1), psxy(1), surface(1), triangulate(1), xyz2grd(1)
GMT 4.5.11 5 Nov 2013 GREENSPLINE(1gmt)