NAME¶
greenspline - Interpolate 1-D, 2-D, 3-D Cartesian or spherical surface data
using Green's function splines.
SYNOPSIS¶
greenspline [
datafile(s) ] [
-A[
1|
2|
3|
4|
5,]
gradfile ] [
-Ccut[/
file] ] [
-Dmode ] [
-F ] [
-Ggrdfile ] [
-H[
i][
nrec] ] [
-Ixinc[
yinc[
zinc]] ] [
-L ] [
-Nnodefile ] [
-Qaz|
x/y/z ] [
-Rxmin/
xmax[/
ymin/
ymax[/
zminzmax]] ] [
-Sc|t|g|p|q[
pars] ] [
-Tmaskgrid ] [
-V ] [
-:[
i|
o] ] [
-bi[
s|
S|
d|
D[
ncol]|
c[
var1/...]]
] [
-f[
i|
o]
colinfo ] [
-bo[
s|
S|
d|
D[
ncol]|
c[
var1 /...]] ]
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 then the sum can be evaluated at any
output point
x. Choose between ten 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.
OPTIONS¶
- datafile(s)
- 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.
- -A
- 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. Specify 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. Append name of ASCII file with the surface
gradients (following a comma if a format is specified).
- -C
- 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 cut [Default
uses Gauss-Jordan elimination to solve the linear system and fit the data
exactly]. Optionally, append / file to save the eigenvalue ratios
to the specified file for further analysis. Finally, if a negative
cut is given then / file is required and execution will stop
after saving the eigenvalues, i.e., no surface output is produced.
- -D
- Sets the distance flag that determines how we calculate distances between
data points. Select mode 0 for Cartesian 1-D spline interpolation:
-D 0 means (x) in user units, Cartesian distances, Select
mode 1-3 for Cartesian 2-D surface spline interpolation: -D
1 means ( x,y) in user units, Cartesian distances, -D 2 for
( x,y) in degrees, flat Earth distances, and -D 3 for
(x,y) in degrees, spherical distances in km. Then, if
ELLIPSOID is spherical, we compute great circle arcs, otherwise
geodesics. Option mode = 4 applies to spherical surface spline
interpolation only: -D 4 for (x,y) in degrees, use cosine of
great circle (or geodesic) arcs. Select mode 5 for Cartesian 3-D
surface spline interpolation: -D 5 means (x,y,z) in user
units, Cartesian distances.
- -F
- Force pixel registration. [Default is gridline registration].
- -G
- Name of resulting output file. (1) If options -R, -I, and
possibly -F 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 -C 0 is given.
- -H
- Input file(s) has header record(s). If used, the default number of header
records is N_HEADER_RECS. Use -Hi if only input data should
have header records [Default will write out header records if the input
data have them]. Blank lines and lines starting with # are always
skipped.
- -I
- 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).
- -N
- 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 -F.
- -Q
- 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).
- -R
- Specify the domain for an equidistant lattice where output predictions are
required. Requires -I and optionally -F.
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
- Select one of five 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 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 -D 4 and geographic data:
( 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; by specifying -SQ you can speed up calculations
by first pre-calculating G( x; x') for a dense set of
x values (e.g., 100,001 nodes between -1 to +1) and store them in
look-up tables. Optionally append / N (an odd integer) to specify
how many points in the spline to set [100001]
- -T
- 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 -F.
- -V
- Selects verbose mode, which will send progress reports to stderr [Default
runs "silently"].
- -bi
- Selects binary input. Append s for single precision [Default is
d (double)]. Uppercase S or D will force
byte-swapping. Optionally, append ncol, the number of columns in
your binary input file if it exceeds the columns needed by the program. Or
append c if the input file is netCDF. Optionally, append
var1 /var2/... to specify the variables
to be read. [Default is 2-4 input columns ( x,w); the number
depends on the chosen dimension].
- -f
- Special formatting of input and/or output columns (time or geographical
data). Specify i or o to make this apply only to input or
output [Default applies to both]. Give one or more columns (or column
ranges) separated by commas. Append T (absolute calendar time),
t (relative time in chosen TIME_UNIT since
TIME_EPOCH), x (longitude), y (latitude), or f
(floating point) to each column or column range item. Shorthand
-f[i|o]g means
-f[i|o]0x,1y (geographic
coordinates).
- -bo
- Selects binary output. Append s for single precision [Default is
d (double)]. Uppercase S or D will force
byte-swapping. Optionally, append ncol, the number of desired
columns in your binary output file.
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
gmtmath -T 0/10/1 0 1
NRAND = 1D.txt
psxy -R0/10/-5/5
-JX 6i/3i
-B 2f1/1
-Sc 0.1
-G black 1D.txt
-K > 1D.ps
greenspline 1D.txt
-R 0/10
-I 0.1
-Sc -V |
psxy -R -J -O -W thin >> 1D.ps
To apply a spline in tension instead, using a tension of 0.7, try
psxy -R0/10/-5/5
-JX 6i/3i
-B 2f1/1
-Sc 0.1
-G black 1D.txt
-K > 1Dt.ps
greenspline 1D.txt
-R 0/10
-I 0.1
-St 0.7
-V
|
psxy -R -J -O -W thin >> 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 Cookbook example 16, try
greenspline table_5.11
-R 0/6.5/-0.2/6.5
-I 0.1
-Sc
-V -D 1
-G S1987.grd
psxy -R0/6.5/-0.2/6.5
-JX 6i
-B 2f1
-Sc 0.1
-G black table_5.11
-K > 2D.ps
grdcontour -JX6i
-B 2f1
-O -C 25
-A 50
S1987.grd >> 2D.ps
To use Cartesian splines in tension but only evaluate the solution where the
input mask grid is not NaN, try
greenspline table_5.11
-T mask.grd
-St 0.5
-V
-D 1
-G WB1998.grd
To use Cartesian generalized splines in tension and return the magnitude of the
surface slope in the NW direction, try
greenspline table_5.11
-R 0/6.5/-0.2/6.5
-I 0.1
-Sr
0.95
-V -D 1
-Q-45
-G slopes.grd Finally, to use
Cartesian minimum curvature splines in recovering a surface where the input
data is a single surface value (pt.d) and the remaining constraints specify
only the surface slope and direction (slopes.d), use
greenspline pt.d
-R-3.2/3.2/-3.2/3.2
-I 0.1
-Sc
-V -D 1
-A 1,slopes.d
-G slopes.grd
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
greenspline table_5.23
-R 5/40/-5/10/5/16
-I 0.25
-Sr 0.85
-V -D 5
-G 3D_UO2.txt
2-D SPHERICAL SURFACE EXAMPLES¶
To recreate Parker's [1994] example on a global 1x1 degree grid, assuming the
data are in file mag_obs_1990.d, try
greenspline
-V -Rg -Sp -D 3
-I 1
-G
P1994.grd mag_obs_1990.d
To do the same problem but applying tension and use pre-calculated Green
functions, use
greenspline
-V -Rg -SQ 0.85
-D 3
-I 1
-G WB2008.grd mag_obs_1990.d
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.
(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).
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.
SEE ALSO¶
GMT(1),
gmtmath(1),
nearneighbor(1),
psxy(1),
surface(1),
triangulate(1),
xyz2grd(1)