# greenspline

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

## Synopsis

gmt greenspline [ table ] -G grdfile [ -A gradfile+f|0|1|2|3|4|5 ] [ -C[[n|r|v]value[%]][+c][+f file][+i][+n] ] [ -D[+x xname][+y yname][+z zname][+v vname][+s scale][+o offset][+n invalid][+t title][+r remark] ] [ -E[misfitfile][+r reportfile] ] [ -I xinc[/yinc[/zinc]] ] [ -L[t][r] ] [ -N nodefile ] [ -Q[az|x/y/z] ] [ -R xmin/xmax[/ymin/ymax[/zmin/zmax]] ] [ -S c|l|p|q|r|t[tension[/scale]][+e limit][+n odd] ] [ -T maskgrid ] [ -V[level] ] [ -W[w]] [ -Z mode ] [ -b binary ] [ -d nodata[+c col] ] [ -e regexp ] [ -f flags ] [ -g gaps ] [ -h headers ] [ -i flags ] [ -o flags ] [ -q flags ] [ -r reg ] [ -s flags ] [ -w flags ] [ -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

\[w(\mathbf{x}) = T(\mathbf{x}) + \sum_{j=1}^{n} \alpha_j g(\mathbf{x}; \mathbf{x}_j),\]

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.

If there are also m observed constraints on the gradient s of the curve or surface (-A) then we must add additional m unknown coefficients and use the gradient of the Green’s functions to satisfy the m extra constraints:

\[s(\mathbf{x}_k) = \nabla w(\mathbf{x}_k) = \sum_{j=1}^{n} \alpha_j \nabla g(\mathbf{x}_k; \mathbf{x}_j) \mathbf{n}_k, \quad k = 1,m\]

where the gradient of the Green’s functions is dotted with the unit vector of the observed gradient.

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.

-G grdfile

  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 standard output 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 standard output. (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.

## Optional Arguments

-A gradfile+f|0|1|2|3|4|5

  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 modifier +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, \(v\) (1-D data set).

  - 1 - Records contain x, y, azimuth, \(v\) (azimuth in degrees is measured clockwise from the vertical (north) [Default] (2-D data set).

  - 2 - Records contain x, y, \(v\), azimuth (azimuth in degrees is measured clockwise from the vertical (north); 2-D data set).

  - 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 (2-D or 3-D data set).

  - 4 - Records contain x, \(\mathbf{v}\) (2-D or 3-D data set).

  - 5 - Records contain x, \(\hat{\mathbf{n}}\), \(v\) (2-D or 3-D data set).

  Note: The slope constraints must not be at the same locations as the data constraints. That scenario has not yet been implemented.

-C[[n|r|v]value[%]][+c][+f file][+i][+n]

  Find an approximate surface fit: Solve the linear system for the spline coefficients by SVD and eliminate the contribution from smaller eigenvalues [Default uses Gauss-Jordan elimination to solve the linear system and fit the data exactly (unless -W is used)]. Append a directive and value to determine which eigenvalues to keep:

  - n - Retain only the value numbers largest eigenvalues [all]. Optionally, append % to indicate value is given in percentage.

  - r - Retain those eigenvalues whose ratio to the largest eigenvalue is less than value [Default, with value = 0].

  - v - Retain the eigenvalues needed to ensure the model prediction variance fraction is at least value. Optionally, append % to indicate value is given in percentage.

  Several optional modifiers are available:

  - +c - Produce the cumulative sum of these contributions, one grid per eigenvalue (2-D only).

  - +f - Append file to save the eigenvalues to the specified file for further analysis.

  - +n - If given then +f file is required and execution will stop after saving the eigenvalues, i.e., no surface output is produced.

  - +i - Produce the incremental sum of these contributions, one grid per eigenvalue (2-D only).

  Notes: (1) Modifiers ++c and +i require a file name with a suitable extension to be given via -G (we automatically insert “_cum_###” or “_inc_###” before the extension, using a fixed integer format for the eigenvalue number, starting at 0). (2) Use both modifiers to write both types of intermediate grids.

-D[+x xname][+y yname][+z zname][+c[-|cpt]][+d dname][+s scale][+o offset][+n invalid][+t title][+r remark][+v varname]

  Control names and units of netCDF grid and cube meta-data. For dimensions with units, add the unit in square bracket (e.g., “distance [km]”). Select one or more of these modifiers:

  - +c - Append cpt to set a default CPT for this grid [turbo] or give - to remove any default CPT already set.

  - +d - Set dname, the data value name.

  - +n - Set the invalid number used to indicate a NaN or missing value.

  - +o - Set the offset to add to data after first scaling the data [0].

  - +r - Set a remark used for this grid (any sentence you prefer).

  - +s - Set the scale used fto multiply data values after they are read [1].

  - +t - Set a title used for this grid (any sentence you prefer).

  - +v - Append varname, the variable name of the data set.

  - +x - Append xname, the name of the x-coordinate (and optional unit in brackets).

  - +y - Append yname, the name of the y-coordinate (and optional unit in brackets).

  - +z - For 3-D cubes; append zname, the name of the z-coordinate (and optional unit in brackets).

  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 rename this netCDF variable via +v.

-E[misfitfile][+r reportfile]

  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. Alternatively, if -C is used and history is computed (via one or more of modifiers +c and +i), then we will instead write a table with eigenvalue number, eigenvalue, percent of model variance explained, and rms misfit. If -W is used we also append \(\chi^2\). Optionally append this modifier:

  - +r - Write misfit and variance evaluation statistics to reportfile. Output order is Data Model Explained(%) N Mean Std.dev RMS. If -W is used we add \(\chi^2\) as a final 8th column. Note: If +r is not used then -V i will report the information via verbose messages.

-I xinc[/yinc[/zinc]]

  Specify equidistant sampling intervals, on for each dimension, separated by slashes.

-L[t][r]

  Specifically control how we detrend (i.e., adjust \(T(\mathbf{x})\)) and normalize the data (and possibly gradients) prior to determining the solution coefficients. The order of adjustments is always the same even if some steps may be deselected:

  - We always determine and then remove and restore the mean data value \(\bar{w}\).

  - We determine a linear least-squares trend (directive t) and remove this trend from residual data and slopes. Note: For spherical and 3-D interpolation the t directive is not available.

  - We determine the maximum absolute value of the minimum and maximum data residuals (directive r) and normalize the residual data and slopes by that range value.

  After evaluating the solution based on the residuals, we undo any normalization and detrending in reverse order. Use -L and specifically append one or both of these directives to override the default. If no directives are given then no detrending nor normalization will take place.

-N nodefile

  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 standard output if not specified]; see -bo for binary output instead. This option eliminates the need to specify options -R, -I, and -r.

-Q[az|x/y/z]

  Rather than evaluate the solution w(x), take the first derivative of a 1-D solution. For 2-D, select directional derivative in the az azimuth and return the magnitude of this derivative instead. For a 3-D interpolation, specify the three components of the desired vector direction (the vector will be normalized before use).

-R xmin/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|l|p|q|r|t[tension[/scale]][+e limit][+n odd]

  Select one of six different splines. Some are 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 < tension < 1. Choose among these directives:

  - c - Minimum curvature spline [Sandwell, 1987] (1-D, 2-D, or 3-D Cartesian spline).

  - l - Linear or bilinear spline; these produce output that do not exceed the range of the given data (1-D or 2-D Cartesian spline).

  - p - Minimum curvature spline [Parker, 1994] (spherical surface splines and implies -Z).

  - 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 (spherical surface spline and implies -Z).

  - r - Regularized spline in tension [Mitasova and Mitas, 1993]; again, append tension and optional scale (2-D or 3-D spline).

  - 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] (1-D, 2-D, or 3-D Cartesian spline).

  Note: Directive q may take two optional modifiers:

  - +e - The finite Legendre sum has a truncation error [1e-6]; you can lower that by appending limit at the expense of longer run-time.

  - +n - Change how many points to use in the spline setup by appending odd [10001] (must be an odd integer).

-T maskgrid

  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].

-Z mode

  Sets the distance mode 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 record[+b|l] (more …)

  Select native binary format for primary table input. [Default is 2-4 input columns (x,w); the number depends on the chosen dimension].

-bo record[+b|l] (more …)

  Select native binary format for table output.

-d[i|o][+c col]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.

-h[i|o][n][+c][+d][+m segheader][+r remark][+t title] (more …)

  Skip or produce header record(s).

-i cols[+l][+d divisor][+s scale|d|k][+o offset][,…][,t[word]] (more …)

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

-o cols[+l][+d divisor][+s scale|d|k][+o offset][,…][,t[word]] (more …)

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

-q[i|o][~]rows|limits[+c col][+a|t|s] (more …)

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

-r[g|p] (more …)

  Set node registration [gridline].

-wy|a|w|d|h|m|s|c period[/phase][+c col] (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 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 output file based on the main grid name (such as contribution_cum_033.nc), try:

    gmt greenspline @Table_5_11.txt -R0/6.5/-0.2/6.5 -I0.1 -Gcontribution.nc -Sc -Z1 -C+c

Finally, to use Cartesian minimum curvature splines in recovering a surface where the input data represent 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 preprocessing 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.

6. It may be difficult to know how many eigenvalues are needed for a suitable approximate fit. The -C modifiers allow you to explore this further by creating solutions for all cutoff selections and estimate model variance and data misfit as a function of how many eigenvalues are used. The large set of such solutions can be animated so it is easier to explore the changes between solutions and to make a good selection for the -C directive values. See the animations for one or more examples of this exploration.

## 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.

## Deprecations

- 6.3.0: Replace +m and +M modifiers for -C. #5714

- 6.3.0: Use +n instead of negative value for -C to set dry-run. #5725

## 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, https://doi.org/10.1023/A:1021713421882.

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, https://doi.org/10.1111/j.1365-246X.2008.03829.x.

Wessel, P., 2009, A general-purpose Green’s function interpolator, Computers & Geosciences, 35, 1247-1254, https://doi.org/10.1016/j.cageo.2008.08.012.

## See Also

gmt, gmtmath, nearneighbor, plot, sample1d, sphtriangulate, surface, triangulate, xyz2grd