NAME
fitcircle - find mean position and pole of best-fit great
[or small] circle to points on a sphere.
SYNOPSIS
fitcircle [ xyfile ] -Lnorm [ -H ] [ -S ] [ -V ] [ -: ]
DESCRIPTION
fitcircle reads ascii lon,lat [or lat,lon] values from the
first two columns on standard input [or xyfile]. These are
converted to cartesian three-vectors on the unit sphere.
Then two locations are found: the mean of the input posi-
tions, and the pole to the great circle which best fits the
input positions. The user may choose one or both of two
possible solutions to this problem. The first is called -L1
and the second is called -L2. When the data are closely
grouped along a great circle both solutions are similar. If
the data have large dispersion, the pole to the great circle
will be less well determined than the mean. Compare both
solutions as a qualitative check.
The -L1 solution is so called because it approximates the
minimization of the sum of absolute values of cosines of
angular distances. This solution finds the mean position as
the Fisher average of the data, and the pole position as the
Fisher average of the cross-products between the mean and
the data. Averaging cross-products gives weight to points
in proportion to their distance from the mean, analogous to
the "leverage" of distant points in linear regression in the
plane.
The -L2 solution is so called because it approximates the
minimization of the sum of squares of cosines of angular
distances. It creates a 3 by 3 matrix of sums of squares of
components of the data vectors. The eigenvectors of this
matrix give the mean and pole locations. This method may be
more subject to roundoff errors when there are thousands of
data. The pole is given by the eigenvector corresponding to
the smallest eigenvalue; it is the least-well represented
factor in the data and is not easily estimated by either
method.
-L Specify the desired norm as 1 or 2, or use -L or - L3
to see both solutions.
OPTIONS
xyfile
ASCII file containing lon,lat [lat,lon] values in the
first 2 columns. If no file is specified, fitcircle
will read from standard input.
-H Input file(s) has Header record(s). Number of header
records can be changed by editing your .gmtdefaults
file. If used, GMT default is 1 header record.
-S Attempt to fit a small circle instead of a great cir-
cle. The pole will be constrained to lie on the great
circle connecting the pole of the best-fit great circle
and the mean location of the data.
-V Selects verbose mode, which will send progress reports
to stderr [Default runs "silently"].
- : Toggles between (longitude,latitude) and
(latitude,longitude) input/output. [Default is
(longitude,latitude)]
EXAMPLES
Suppose you have lon,lat,grav data along a somewhat twisty
ship track in the file ship.xyg. You want to project this
data onto a great circle and resample it in distance, in
order to filter it or check its spectrum. Try this:
fitcircle ship.xyg -L2
project ship.xyg -Oox/oy -Ppx/py -S -pz | sample1d -S-100 -
I1 > output.pg
In this example, ox/oy is the lon/lat of the mean from
fitcircle, and px/py is the lon/lat of the pole. The file
output.pg has distance, gravity data sampled every 1 km
along the great circle which best fits ship.xyg
SEE ALSO
gmt, project, sample1d
REFERENCES
Wessel, P., and W. H. F. Smith, 1995, The Generic Mapping
Tools (GMT) version 3.0 Technical Reference & Cookbook,
SOEST/NOAA.
Wessel, P., and W. H. F. Smith, 1995, New Version of the
Generic Mapping Tools Released, EOS Trans. AGU, 76, p. 329.
Wessel, P., and W. H. F. Smith, 1995, New Version of the
Generic Mapping Tools Released,
http://www.agu.org/eos_elec/95154e.html, Copyright 1995 by
the American Geophysical Union.
Wessel, P., and W. H. F. Smith, 1991, Free Software Helps
Map and Display Data, EOS Trans. AGU, 72, p. 441.