# greenspline¶

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

## Synopsis¶

**gmt greenspline** [ *table* ]
**-G***grdfile*
[ **-A***gradfile***+f****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*] ]
[ **-I***xinc*[/*yinc*[/*zinc*]] ]
[ **-L**[**t**][**r**] ]
[ **-N***nodefile* ]
[ **-Q**[*az*|*x/y/z*] ]
[ **-R***xmin*/*xmax*[/*ymin*/*ymax*[/*zmin*/*zmax*]] ]
[ **-S****c|t|l|r|p|q**[*pars*] ] [ **-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

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:

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:

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

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**-C**0 is given.

## Optional Arguments¶

**-A***gradfile***+f****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

**+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**, \(\hat{\mathbf{n}}\), \(v\).**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**will retain only the*value*largest eigenvalues [all],**r**[Default] will retain those eigenvalues whose ratio to the largest eigenvalue is less than*value*[0], while**v**will retain the eigenvalues needed to ensure the model prediction variance fraction is at least*value*. For**n**and**v**you may append % if*value*is given as a*percentage*of the total instead. Several optional modifiers are available: Append**+f***file*to save the eigenvalues to the specified file for further analysis. If**+n**is given then**+f***file*is required and execution will stop after saving the eigenvalues, i.e., no surface output is produced. The two other modifiers (**+c**and**+i**) are only available for 2-D gridding and can be used to write intermediate grids, one per added eigenvalue, and thus 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). The**+i**modifier will write the**i**ncremental contributions to the grid for each eigenvalue added, while**+c**will instead produce the**c**umulative sum of these contributions. 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*]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]”),*cpt*to set a default CPT for this grid [turbo] or give - to remove any default CPT already set,*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**+v***varname*.

**-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. 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 \(\chi^2\).

**-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.

**-S****c|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**-Z**4: (**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**+n***N*(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**+e***limit*at the expense of longer run-time.

**-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:**-Z**0 means (*x*) in user units, Cartesian distances. Select*mode*1-3 for Cartesian 2-D surface spline interpolation:**-Z**1 means (*x*,*y*) in user units, Cartesian distances,**-Z**2 for (*x*,*y*) in degrees, Flat Earth distances, and**-Z**3 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:**-Z**4 for (*x*,*y*) in degrees, use cosine of great circle (or geodesic) arcs. Select*mode*5 for Cartesian 3-D surface spline interpolation:**-Z**5 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*[,…][,**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*|*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 argumentsPrint 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 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¶

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.

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.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**).The series solution implemented for

**-Sq**was developed by Robert L. Parker, Scripps Institution of Oceanography, which we gratefully acknowledge.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.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¶

## 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