# greenspline¶

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

## Synopsis¶

gmt greenspline [ table ] [ -Agradfile+f1|2|3|4|5 ] [ -C[n]value[%][+ffile][+m|M] ] [ -D[+xxname][+yyname][+zzname][+vvname][+sscale][+ooffset][+ninvalid][+ttitle][+rremark] ] [ -E[misfitfile] ] [ -Ggrdfile ] [ -Ixinc[/yinc[/zinc]] ] [ -L ] [ -Nnodefile ] [ -Qaz|x/y/z ] [ -Rxmin/xmax[/ymin/ymax[/zmin/zmax]] ] [ -Sc|t|l|r|p|q[pars] ] [ -Tmaskgrid ] [ -V[level] ] [ -W[w]] [ -Zmode ] [ -bbinary ] [ -dnodata ] [ -eregexp ] [ -fflags ] [ -hheaders ] [ -oflags ] [ -qflags ] [ -wflags ] [ -x[[-]n] ] [ -:[i|o] ] [ --PAR=value ]

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

## Description¶

greenspline uses the Green’s function $$g(\mathbf{x}; \mathbf{x}')$$ for the chosen spline and geometry to interpolate data at regular [or arbitrary] output locations. 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. Mathematically, the solution is composed as

$w(\mathbf{x}) = T(\mathbf{x}) + \sum_{j=1}^{n} \alpha_j g(\mathbf{x}; \mathbf{x}'),$

where $$\mathbf{x}$$ is the output location, $$n$$ is the number of points, $$T(\mathbf{x})$$ is a trend function, and $$\alpha_j$$ are the n unknown weights we must solve for. Typically, $$T(\mathbf{x})$$ is a linear or planar trend (Cartesian geometries) or mean value (spherical surface) and a least-squares solution is determined and removed from the data, yielding data residuals ($$\Delta w_i = w_i - T(\mathbf{x}_i)$$); these are then normalized for numerical stability. The unknown coefficients $$\alpha_j$$ are determined by requiring the solution to fit the observed residual data exactly:

$\Delta w(\mathbf{x}_i) = \sum_{j=1}^{n} \alpha_j g(\mathbf{x}_i; \mathbf{x}_j), \quad i = 1,n$

yielding a $$n \times n$$ linear system to be solved for the coefficients. Finally, away from the data constraints the Green’s function must satisfy

$\nabla^2 \left [ \nabla^2 - p^2 \right ] g(\mathbf{x}; \mathbf{x}') = \delta (\mathbf{x} - \mathbf{x}'),$

where $$\nabla^2$$ is the Laplacian operator, $$\delta$$ is the Dirac Delta function, and $$p$$ is the tension (if desired). 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 normalization scales are restored when evaluating the output.

## Required Arguments¶

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.

## Optional Arguments¶

The solution will partly be constrained by surface gradients $$\mathbf{v} = v \hat{\mathbf{n}}$$, where $$v$$ is the gradient magnitude and $$\hat{\mathbf{n}}$$ its unit vector direction. The gradient direction may be specified either by Cartesian components (either unit vector $$\hat{\mathbf{n}}$$ and magnitude $$v$$ separately or gradient components $$\mathbf{v}$$ directly) or angles w.r.t. the coordinate axes. Append name of ASCII file with the surface gradients. Use +f to select one of five input formats: 0: For 1-D data there is no direction, just gradient magnitude (slope) so the input format is x, $$v$$. Options 1-2 are for 2-D data sets: 1: records contain x, y, azimuth, $$v$$ (azimuth in degrees is measured clockwise from the vertical (north) [Default]). 2: records contain x, y, $$v$$, 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, $$\mathbf{v}$$. 5: records contain x, $$\hat{\mathbf{n}}$$, $$v$$.

-C[n]value[%][+ffile][+m|M]

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 value [Default uses Gauss-Jordan elimination to solve the linear system and fit the data exactly]. Optionally, append +ffile to save the eigenvalues to the specified file for further analysis. If a negative value is given then +ffile is required and execution will stop after saving the eigenvalues, i.e., no surface output is produced. Specify -Cn to retain only the value largest eigenvalues; append % if value is the percentage of eigenvalues to use instead. The two last modifiers (+m|M) are only available for 2-D gridding and can be used to write intermediate grids, one per eigenvalue, and thus require a file name template with a C-format integer specification to be given via -G. The +m modifier will write the contributions to the grid for each eigenvalue, while +M will instead produce the cumulative sum of these contributions.

-D[+xxname][+yyname][+zzname][+dvname][+sscale][+ooffset][+ninvalid][+ttitle][+rremark][+vvarname]

Give one or more combinations for values xname, yname, zname (3rd dimension in cube), and dname (data value name) and give the names of those variables and in square bracket their units, e.g., “distance [km]”), scale (to multiply data values after read [normally 1]), offset (to add to data after scaling [normally 0]), invalid (a value to represent missing data [NaN]), title (anything you like), and remark (anything you like). Items not listed will remain untouched. Give a blank name to completely reset a particular string. Use quotes to group texts with more than one word. If any of your text contains plus symbols you need to escape them (place a backslash before each plus-sign) so they are not confused with the option modifiers. Alternatively, you can place the entire double-quoted string inside single quotes. If you have shell variables that contain plus symbols you cannot use single quotes but you can escape the plus symbols in a variable using constructs like \${variable/+/\+}. Note that for geographic grids and cubes (-fg) xname and yname are set automatically. Normally, the data netCDF variable is called “z” (grid) or “cube” (data cube). You can name this netCDF variable via +vvarname.

-E[misfitfile]

Evaluate the spline exactly at the input data locations and report statistics of the misfit (mean, standard deviation, and rms). Optionally, append a filename and we will write the data table, augmented by two extra columns holding the spline estimate and the misfit.

-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) For 3-D cubes the -G option is optional. If set, it can be the name of a 3-D cube file or a filename template with a floating-point C-format identifier in it so that each layer is written to a 2-D grid file; otherwise we write (x, y, z, w) records to stdout. (4) 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 -Z 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[/zmin/zmax]]

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 -Z 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 -Z4: (p) Minimum curvature spline [Parker, 1994], (q) Continuous curvature spline in tension [Wessel and Becker, 2008]; append tension. The $$g(\mathbf{x}; \mathbf{x}')$$ 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 [10001]. 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]

Select verbosity level [w]. (See full description) (See cookbook information).

-W[w]

Data one-sigma uncertainties are provided in the last column. We then compute weights that are inversely proportional to the uncertainties squared. Append w if weights are given instead of uncertainties and then they will be used as is (no squaring). This results in a weighted least squares fit. Note that this only has an effect if -C is used. [Default uses no weights or uncertainties].

-Zmode

Sets the distance flag that determines how we calculate distances between data points. Select mode 0 for Cartesian 1-D spline interpolation: -Z0 means (x) in user units, Cartesian distances, Select mode 1-3 for Cartesian 2-D surface spline interpolation: -Z1 means (x,y) in user units, Cartesian distances, -Z2 for (x,y) in degrees, Flat Earth distances, and -Z3 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: -Z4 for (x,y) in degrees, use cosine of great circle (or geodesic) arcs. Select mode 5 for Cartesian 3-D surface spline interpolation: -Z5 means (x,y,z) in user units, Cartesian distances.

-bi[ncols][t] (more …)

Select native binary format for primary 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.

-e[~]“pattern” | -e[~]/regexp/[i] (more …)

Only accept data records that match the given pattern.

-f[i|o]colinfo (more …)

Specify data types of input and/or output columns.

-icols[+l][+ddivide][+sscale][+ooffset][,][,t[word]] (more …)

Select input columns and transformations (0 is first column, t is trailing text, append word to read one word only).

-ocols[,…][,t[word]] (more …)

Select output columns (0 is first column; t is trailing text, append word to write one word only).

-q[i|o][~]rows[+ccol][+a|f|s] (more …)

Select input or output rows or data range(s) [all].

-r[g|p] (more …)

Set node registration [gridline].

-wy|a|w|d|h|m|s|cperiod[/phase][+ccol] (more …)

Convert an input coordinate to a cyclical coordinate.

-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 exit (NOTE: on Windows just use -).

-+ or just +

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

-? or no arguments

Print a complete usage (help) message, including the explanation of all options, then exit.

--PAR=value

Temporarily override a GMT default setting; repeatable. See gmt.conf for parameters.

## 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:

gmt begin 1D
gmt math -T0/10/1 0 1 NRAND = 1D.txt
gmt plot -R0/10/-5/5 -JX6i/3i -B -Sc0.1 -Gblack 1D.txt
gmt greenspline 1D.txt -R0/10 -I0.1 -Sc | gmt plot -Wthin
gmt end show


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

gmt begin 1Dt
gmt plot -R0/10/-5/5 -JX6i/3i -B -Sc0.1 -Gblack 1D.txt
gmt greenspline 1D.txt -R0/10 -I0.1 -St0.7 | gmt plot -Wthin
gmt end show


## 2-d Examples¶

To make a uniform grid using the minimum curvature spline for the same Cartesian data set from Table 5.11 in Davis (1986) that is used in the GMT Technical Reference and Cookbook example 16, try:

gmt begin 2D
gmt greenspline @Table_5_11.txt -R0/6.5/-0.2/6.5 -I0.1 -Sc -V -Z1 -GS1987.nc
gmt plot -R0/6.5/-0.2/6.5 -JX6i -B -Sc0.1 -Gblack @Table_5_11.txt
gmt grdcontour -C25 -A50 S1987.nc
gmt end show


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.txt -Tmask.nc -St0.5 -V -Z1 -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.txt -R0/6.5/-0.2/6.5 -I0.1 -Sr0.95 -V -Z1 -Q-45 -Gslopes.nc


To use Cartesian cubic splines and evaluate the cumulative solution as a function of eigenvalue, using the output template with three digits for the eigenvalue, try:

gmt greenspline @Table_5_11.txt -R0/6.5/-0.2/6.5 -I0.1 -Gcontribution_%3.3d.nc -Sc -Z1 -C+M


Finally, to use Cartesian minimum curvature splines in recovering a surface where the input data is a single surface value (pt.txt) and the remaining constraints specify only the surface slope and direction (slopes.txt), use:

gmt greenspline pt.txt -R-3.2/3.2/-3.2/3.2 -I0.1 -Sc -V -Z1 -Aslopes.txt+f1 -Gslopes.nc


## 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:

gmt greenspline @Table_5_23.txt -R5/40/-5/10/5/16 -I0.25 -Sr0.85 -V -Z5 > 3D_UO2.txt


To instead write the results as a series of 2-D layer grids called layer_*z*.grd, try:

gmt greenspline @Table_5_23.txt -R5/40/-5/10/5/16 -I0.25 -Sr0.85 -V -Z5 -G3D_UO2_%g.grd


Finally, to write the result to a 3-D netCDF grid, try:

gmt greenspline @Table_5_23.txt -R5/40/-5/10/5/16 -I0.25 -Sr0.85 -V -Z5 -G3D_UO2.nc


## 2-d Spherical Surface Examples¶

To recreate Parker’s [1994] example on a global 1x1 degree grid, assuming the data are in the remote file mag_obs_1990.txt, try:

gmt greenspline -V -Rg -Sp -Z3 -I1 -GP1994.nc @mag_obs_1990.txt


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

gmt greenspline -V -Rg -Sq0.85 -Z3 -I1 -GWB2008.nc @mag_obs_1990.txt


## 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. For spherical data you may consider decimating using gmtspatial 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 consider sample1d instead. It will offer traditional splines with 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.