greenspline(1) GMT greenspline(1)

## NAME

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

## SYNOPSIS

greenspline[table] [-Agradfile+f1|2|3|4|5] [-C[n|r|v]value[+ffile] ] [-Dmode] [-E[misfitfile] ] [-Ggrdfile] [-Ixinc[/yinc[/zinc]] ] [-L] [-Nnodefile] [-Qaz|x/y/z] [-Rwest/east/south/north[/zmin/zmax][+r] ] [-Sc|t|l|r|p|q[pars] ] [-Tmaskgrid] [-V[level] ] [-W[w]] [-bbinary ] [-dnodata ] [-eregexp ] [-fflags ] [-hheaders ] [-oflags ] [-x[[-]n] ] [-:[i|o] ]Note:No space is allowed between the option flag and the associated arguments.

## DESCRIPTION

greensplineuses the Greenas function G(x;xa) for the chosen spline and geometry to interpolate data at regular [or arbitrary] output loca- tions. Mathematically, the solution is composed asw(x) = sum {c(i) G(xa;x(i))}, fori= 1,n, the number of data points {x(i),w(i)}. Once thencoefficientsc(i) have been found the sum can be evaluated at any output pointx. 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 coefficientsc(i) is found by solving thenbynlinear systemw(j) = sum-over-i{c(i) G(x(j);x(i))}, forj= 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

tableThe name of one or more ASCII [or binary, see-bi] files holding thex,wdata points. If no file is given then we read standard input instead.-Agradfile+f1|2|3|4|5The solution will partly be constrained by surface gradientsv=v*n, wherevis the gradient magnitude andnits unit vector direction. The gradient direction may be specified either by Cartesian components (either unit vectornand magnitudevsepa- rately or gradient componentsvdirectly) or angles w.r.t. the coordinate axes. Append name of ASCII file with the surface gra- dients. Use+fto select one of five input formats:0: For 1-D data there is no direction, just gradient magnitude (slope) so the input format isx,gradient. Options 1-2 are for 2-D data sets:1: records containx,y,azimuth,gradient(azimuthin degrees is measured clockwise from the vertical (north) [Default]).2: records containx,y,gradient,azimuth(azimuthin degrees is measured clockwise from the vertical (north)). Options 3-5 are for either 2-D or 3-D data:3: records containx,direction(s),v(direction(s)in degrees are measured counter-clockwise from the horizontal (and for 3-D the vertical axis).4: records containx,v.5: records containx,n,v.-C[n|r|v]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 thanvalue[Default uses Gauss-Jordan elimination to solve the linear system and fit the data exactly]. Optionally, append+ffileto save the eigenvalue ratios to the specified file for further analysis. Finally, if a negativevalueis given then+ffileis required and execution will stop after saving the ei- genvalues, i.e., no surface output is produced. Specify-Cvto use the largest eigenvalues needed to explain approximatelyvalue% of the data variance. Specify-Crto use the largest eigenvalues needed to leave approximatelyvalueas the model misfit. Ifvalueis not given then-Wis required and we com- putevalueas the rms of the data uncertainties. Alternatively, use-Cnto select thevaluelargest eigenvalues. If afileis given with-Cvthen we save the eigenvalues instead of the ratios.-DmodeSets the distance flag that determines how we calculate dis- tances between data points. Selectmode0 for Cartesian 1-D spline interpolation:-D0 means (x) in user units, Cartesian distances, Selectmode1-3 for Cartesian 2-D surface spline interpolation:-D1 means (x,y) in user units, Cartesian dis- tances,-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, oth- erwise geodesics. Optionmode= 4 applies to spherical surface spline interpolation only:-D4 for (x,y) in degrees, use cosine of great circle (or geodesic) arcs. Selectmode5 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, aug- mented by two extra columns holding the spline estimate and the mis- fit.-GgrdfileName of resulting output file. (1) If options-R,-I, and possi- bly-rare set we produce an equidistant output table. This will be written to stdout unless-Gis specified. Note: for 2-D grids the-Goption is required. (2) If option-Tis selected then-Gis required and the output file is a 2-D binary grid file. Applies to 2-D interpolation only. (3) If-Nis selected then the output is an ASCII (or binary; see-bo) table; if-Gis not given then this table is written to standard output. Ignored if-Cor-C0 is given.-Ixinc[/yinc[/zinc]] Specify equidistant sampling intervals, on for each dimension, separated by slashes.-LDonotremove a linear (1-D) or planer (2-D) trend when-Dselects 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).-NnodefileASCII file with coordinates of desired output locationsxin the first column(s). The resultingwvalues are appended to each record and written to the file given in-G[or stdout if not specified]; see-bofor binary output instead. This option elim- inates the need to specify options-R,-I, and-r.-Qaz|x/y/zRather than evaluate the surface, take the directional deriva- tive in theazazimuth and return the magnitude of this deriva- tive instead. For 3-D interpolation, specify the three compo- nents of the desired vector direction (the vector will be nor- malized before use).-Rxmin/xmax[/ymin/ymax[/zmin/zmax]] Specify the domain for an equidistant lattice where output pre- dictions are required. Requires-Iand optionally-r.1-D:Givexmin/xmax, the minimum and maximumxcoordinates.2-D:Givexmin/xmax/ymin/ymax, the minimum and maximumxandycoordinates. These may be Cartesian or geographical. If geo- graphical, thenwest,east,south, andnorthspecify the Region of interest, and you may specify them in decimal degrees or in [A+-]dd:mm[:ss.xxx][W|E|S|N] format. The two shorthands-Rgand-Rdstand for global domain (0/360 and -180/+180 in longitude respectively, with -90/+90 in latitude).3-D:Givexmin/xmax/ymin/ymax/zmin/zmax, the minimum and maximumx,yandzcoordinates. See the 2-D section if your horizontal coordinates are geographical; note the shorthands-Rgand-Rdcannot 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-Dfor 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 [WesselandBercovici, 1998]; appendtension[/scale] withtensionin the 0-1 range and optionally supply a length scale [Default is the aver- age 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 [MitasovaandMitas, 1993]; again, appendtensionand optionalscale. The last two are spherical surface splines and both imply-D4: (p) Minimum curvature spline [Parker, 1994], (q) Continuous curvature spline in tension [WesselandBecker, 2008]; appendtension. The G(xa;xa) 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+elimitat the expense of longer run-time.-TmaskgridFor 2-D interpolation only. Only evaluate the solution at the nodes in themaskgridthat are not equal to NaN. This option eliminates the need to specify options-R,-I, and-r.-V[level] (morea|) 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. Appendwif weights are given instead of uncer- tainties. This results in a weighted least squares fit. Note that this only has an effect if-Cis used. [Default uses no weights or uncertainties].-bi[ncols][t] (morea|) Select native binary input. [Default is 2-4 input columns (x,w); the number depends on the chosen dimension].-bo[ncols][type] (morea|) Select native binary output.-d[i|o]nodata(morea|) Replace input columns that equalnodatawith NaN and do the reverse on output.-e[~]^<i>apattern^<i>a|-e[~]/regexp/[i] (morea|) Only accept data records that match the given pattern.-f[i|o]colinfo(morea|) Specify data types of input and/or output columns.-h[i|o][n][+c][+d][+rremark][+rtitle] (morea|) Skip or produce header record(s).-icols[+l][+sscale][+ooffset][,^<i>a|] (morea|) Select input columns and transformations (0 is first column).-ocols[,a|] (morea|) Select output columns (0 is first column).-r(morea|) Set pixel node registration [gridline].-x[[-]n] (morea|) 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 explana- tion 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.

## 1-D EXAMPLES

To resample thex,yGaussian 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-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 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 magni- tude 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.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 ta- ble_5.23 in Davis (1986) that containsx,y,zlocations 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-D SPHERICAL SURFACE EXAMPLES

To recreate Parkeras [1994] example on a global 1x1 degree grid, assum- ing the data are in file mag_obs_1990.txt, try greenspline -V -Rg -Sp -D3 -I1 -GP1994.nc mag_obs_1990.txt To do the same problem but applying tension of 0.85, use 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 anxndouble precision matrix for the Green function coefficients, wherenis the number of data constraints. Hence, your computeras memory may place restrictions on how large data sets you can process withgreen-spline. Pre-processing your data with doc:blockmean, doc:blockme-dian, or doc:blockmodeis recommended to avoid aliasing and may also control the size ofn. For information, ifn= 1024 then only 8 Mb memory is needed, but forn= 10240 we need 800 Mb. Note thatgreen-splineis fully 64-bit compliant if compiled as such. For spherical data you may consider decimating using doc:gmtspatialnearest neigh- bor 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 associ- ated with the smallest eigenvalues (see-C). 4. The series solution implemented for-Sqwas 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 con- trast,greensplineas 1-D spline, as is explained in note 1, doesnotspecify 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 moder- ate 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 ofscalevalues; 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,StatisticsandDataAnalysisinGeology, 2nd Edi- tion, 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,GeophysicalInverseTheory, 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 ten- sion: a Greenas function approach,Math.Geol.,30, 77-93. Wessel, P., and J. M. Becker, 2008, Interpolation using a generalized Greenas function for a spherical surface spline in tension,Geophys.J.Int,174, 21-28. Wessel, P., 2009, A general-purpose Greenas function interpolator,Com-puters&Geosciences,35, 1247-1254, doi:10.1016/j.cageo.2008.08.012.

## SEE ALSO

gmt(1),gmtmath(1),nearneighbor(1),psxy(1),sample1d(1),sphtriangulate(1),surface(1),triangulate(1),xyz2grd(1)

## COPYRIGHT

2017, P. Wessel, W. H. F. Smith, R. Scharroo, J. Luis, and F. Wobbe 5.4.2 Jun 24, 2017 greenspline(1)

gmt5 5.4.2 - Generated Thu Jun 29 13:40:54 CDT 2017