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### PROGRAM:

#### NAME

greenspline - Interpolate using Green's functions for splines in 1-3 dimensions

#### SYNOPSIS

greenspline [ table ] [ [1|2|3|4|5,]gradfile ] [ [n|v]cut[/file] ] [ mode ] [ grdfile ] [
xinc[/yinc[/zinc]] ] [ ] [ nodefile ] [ az|x/y/z ] [ west/east/south/north[/zmin/zmax][r]
] [ c|t|l|r|p|q[pars] ] [ maskgrid ] [ [level] ] [ ] [ -b<binary> ] [ -d<nodata> ] [
-f<flags> ] [ -h<headers> ] [ -o<flags> ] [ -x[[-]n] ] [ -:[i|o] ]

Note: No space is allowed between the option flag and the associated arguments.

#### 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 the sum can be evaluated at any
output point x. Choose between 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.

None.

#### OPTIONALARGUMENTS

table 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.

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[n|v]cut[/file]
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. Specify -Cv to use the
largest eigenvalues needed to explain cut % of the data variance. Alternatively,
use -Cn to select the cut largest eigenvalues. If a file is given with -Cv then we
save the eigenvalues instead of the ratios.

-Dmode Sets the distance flag that determines how we calculate distances between data
points. Select mode 0 for Cartesian 1-D spline interpolation: -D0 means (x) in user
units, Cartesian distances, Select mode 1-3 for Cartesian 2-D surface spline
interpolation: -D1 means (x,y) in user units, Cartesian distances, -D2 for (x,y) in
degrees, Flat Earth distances, and -D3 for (x,y) in degrees, Spherical distances in
km. Then, if PROJ_ELLIPSOID is spherical, we compute great circle arcs, otherwise
geodesics. Option mode = 4 applies to spherical surface spline interpolation only:
-D4 for (x,y) in degrees, use cosine of great circle (or geodesic) arcs. Select
mode 5 for Cartesian 3-D surface spline interpolation: -D5 means (x,y,z) in user
units, Cartesian distances.

-Ggrdfile
Name of resulting output file. (1) If options -R, -I, and possibly -r 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 -C0 is given.

-Ixinc[/yinc[/zinc]]
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).

-Nnodefile
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 -r.

-Qaz|x/y/z
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).

-Rxmin/xmax[/ymin/ymax[/zminzmax]]
Specify the domain for an equidistant lattice where output predictions are
required. Requires -I and optionally -r.

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.

-Sc|t|l|r|p|q[pars]
Select one of six 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 1-D or 2-D spline: (l) Linear (1-D) or Bilinear (2-D) spline; these produce
output that do not exceed the range of the given data. 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 -D4: (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 (a series solution) so we pre-calculate values and
use cubic spline interpolation lookup instead. Optionally append +nN (an odd
integer) to change how many points to use in the spline setup . The finite
Legendre sum has a truncation error [1e-6]; you can lower that by appending +elimit
at the expense of longer run-time.

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 -r.

-V[level] (more ...)
Select verbosity level [c].

-W Expect data weights in the final input column, typically given as weight = 1 /
sigma, the data uncertainty. This results in a weighted least squares fit. Note
that this only has an effect if -CC is used.

-bi[ncols][t] (more ...)
Select native binary input. [Default is 2-4 input columns (x,w); the number depends
on the chosen dimension].

-bo[ncols][type] (more ...)
Select native binary output.

-d[i|o]nodata (more ...)
Replace input columns that equal nodata with NaN and do the reverse on output.

-f[i|o]colinfo (more ...)
Specify data types of input and/or output columns.

-h[i|o][n][+c][+d][+rremark][+rtitle] (more ...)

-icols[l][sscale][ooffset][,...] (more ...)
Select input columns (0 is first column).

-ocols[,...] (more ...)
Select output columns (0 is first column).

-r (more ...)
Set pixel node registration [gridline].

-x[[-]n] (more ...)
Limit number of cores used in multi-threaded algorithms (OpenMP required).

-^ or just -
Print a short message about the syntax of the command, then exits (NOTE: on Windows
use just -).

-+ or just +
Print an extensive usage (help) message, including the explanation of any
module-specific option (but not the GMT common options), then exits.

-? or no arguments
Print a complete usage (help) message, including the explanation of options, then
exits.

--version
Print GMT version and exit.

Print full path to GMT share directory and exit.

#### 1-DEXAMPLES

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

gmt math -T0/10/1 0 1 NRAND = 1D.txt
gmt psxy -R0/10/-5/5 -JX6i/3i -B2f1/1 -Sc0.1 -Gblack 1D.txt -K > 1D.ps
gmt greenspline 1D.txt -R0/10 -I0.1 -Sc -V | psxy -R -J -O -Wthin >> 1D.ps

To apply a spline in tension instead, using a tension of 0.7, try

gmt psxy -R0/10/-5/5 -JX6i/3i -B2f1/1 -Sc0.1 -Gblack 1D.txt -K > 1Dt.ps
gmt greenspline 1D.txt -R0/10 -I0.1 -St0.7 -V | psxy -R -J -O -Wthin >> 1Dt.ps

#### 2-DEXAMPLES

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 Technical Reference and Cookbook example 16, try

gmt greenspline table_5.11 -R0/6.5/-0.2/6.5 -I0.1 -Sc -V -D1 -GS1987.nc
gmt psxy -R0/6.5/-0.2/6.5 -JX6i -B2f1 -Sc0.1 -Gblack table_5.11 -K > 2D.ps
gmt grdcontour -JX6i -B2f1 -O -C25 -A50 S1987.nc >> 2D.ps

To use Cartesian splines in tension but only evaluate the solution where the input mask
grid is not NaN, try

gmt greenspline table_5.11 -Tmask.nc -St0.5 -V -D1 -GWB1998.nc

To use Cartesian generalized splines in tension and return the magnitude of the surface
slope in the NW direction, try

gmt greenspline table_5.11 -R0/6.5/-0.2/6.5 -I0.1 -Sr0.95 -V -D1 -Q-45 -Gslopes.nc

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

gmt greenspline pt.d -R-3.2/3.2/-3.2/3.2 -I0.1 -Sc -V -D1 -A1,slopes.d -Gslopes.nc

#### 3-DEXAMPLES

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

gmt greenspline table_5.23 -R5/40/-5/10/5/16 -I0.25 -Sr0.85 -V -D5 -G3D_UO2.txt

#### 2-DSPHERICALSURFACEEXAMPLES

To recreate Parker's  example on a global 1x1 degree grid, assuming the data are in
file mag_obs_1990.d, try

greenspline -V -Rg -Sp -D3 -I1 -GP1994.nc mag_obs_1990.d

To do the same problem but applying tension of 0.85, use

greenspline -V -Rg -Sq0.85 -D3 -I1 -GWB2008.nc 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

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 <greenspline.html>. Pre-processing your data with blockmean
<blockmean.html>, blockmedian <blockmedian.html>, or blockmode <blockmode.html> 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 <greenspline.html> is fully 64-bit compliant if compiled as such. For
spherical data you may consider decimating using gmtspatial <gmtspatial.html> nearest
neighbor reduction.

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).

4. The series solution implemented for -Sq was developed by Robert L. Parker, Scripps
Institution of Oceanography, which we gratefully acknowledge.

5. If you need to fit a certain 1-D spline through your data points you may wish to
standard boundary conditions (such as the natural cubic spline, which sets the
curvatures at the ends to zero). In contrast, greenspline's 1-D spline, as is
explained in note 1, does not specify boundary conditions at the end of the data
domain.

#### 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.

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