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] ] [ -Dmode ] [ -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]] [ -bbinary ] [ -dnodata ] [ -eregexp ] [ -fflags ] [ -hheaders ] [ -oflags ] [ -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(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.

Required Arguments

None.

Optional 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.
-Agradfile+f1|2|3|4|5
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. 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, 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.
-C[n]value[+ffile]
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.
-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.
-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) 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[/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 -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 [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.
-Tmaskgrid
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[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].
-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.
-h[i|o][n][+c][+d][+rremark][+rtitle] (more …)
Skip or produce header record(s).
-icols[+l][+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).
-r[g|p] (more …)
Set 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 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 exits.
-? or no arguments
Print a complete usage (help) message, including the explanation of all options, then exits.
--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 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 -D1 -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 -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.txt -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.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 -D1 -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 -D5 -G3D_UO2.txt

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 -D3 -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 -D3 -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 doc:blockmean, doc:blockmedian, or doc: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 doc: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.