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NAME
v.surf.rst - Spatial approximation and topographic analysis from given point or isoline data in vector
format to floating point raster format using regularized spline with tension.
KEYWORDS
vector, interpolation
SYNOPSIS
v.surf.rst
v.surf.rst help
v.surf.rst [-zctd] input=name [layer=integer] [where=sql_query] [elev=name] [slope=name]
[aspect=name] [pcurv=name] [tcurv=name] [mcurv=name] [devi=string] [cvdev=name]
[treefile=name] [overfile=name] [maskmap=name] [zcolumn=string] [tension=float] [smooth=float]
[scolumn=string] [segmax=integer] [npmin=integer] [dmin=float] [dmax=float] [zmult=float]
[theta=float] [scalex=float] [--overwrite] [--verbose] [--quiet]
Flags:
-z
Use z-coordinates (3D vector only)
-c
Perform cross-validation procedure without raster approximation
-t
Use scale dependent tension
-d
Output partial derivatives instead of topographic parameters
--overwrite
Allow output files to overwrite existing files
--verbose
Verbose module output
--quiet
Quiet module output
Parameters:
input=name
Name of input vector map
layer=integer
Layer number
If set to 0, z coordinates are used. (3D vector only)
Default: 1
where=sql_query
WHERE conditions of SQL statement without 'where' keyword
Example: income = 10000
elev=name
Output surface raster map (elevation)
slope=name
Output slope raster map
aspect=name
Output aspect raster map
pcurv=name
Output profile curvature raster map
tcurv=name
Output tangential curvature raster map
mcurv=name
Output mean curvature raster map
devi=string
Output deviations vector point file
cvdev=name
Output cross-validation errors vector point file
treefile=name
Output vector map showing quadtree segmentation
overfile=name
Output vector map showing overlapping windows
maskmap=name
Name of the raster map used as mask
zcolumn=string
Name of the attribute column with values to be used for approximation (if layer>0)
tension=float
Tension parameter
Default: 40.
smooth=float
Smoothing parameter
scolumn=string
Name of the attribute column with smoothing parameters
segmax=integer
Maximum number of points in a segment
Default: 40
npmin=integer
Minimum number of points for approximation in a segment (>segmax)
Default: 300
dmin=float
Minimum distance between points (to remove almost identical points)
Default: 0.500000
dmax=float
Maximum distance between points on isoline (to insert additional points)
Default: 2.500000
zmult=float
Conversion factor for values used for approximation
Default: 1.0
theta=float
Anisotropy angle (in degrees counterclockwise from East)
scalex=float
Anisotropy scaling factor
DESCRIPTION
v.surf.rst
This program performs spatial approximation based on z-values (-z flag or layer=0 parameter) or
attributes (zcolumn parameter) of point or isoline data given in a vector map named input to grid cells
in the output raster map elev representing a surface. As an option, simultaneously with approximation,
topographic parameters slope, aspect, profile curvature (measured in the direction of the steepest
slope), tangential curvature (measured in the direction of a tangent to contour line) or mean curvature
are computed and saved as raster maps specified by the options slope, aspect, pcurv, tcurv, mcurv
respectively. If -d flag is set, the program outputs partial derivatives fx,fy,fxx, fyy,fxy instead of
slope, aspect, profile, tangential and mean curvatures respectively.
User can define a raster map named maskmap, which will be used as a mask. The approximation is skipped
for cells which have zero or NULL value in mask. NULL values will be assigned to these cells in all
output raster maps. Data points are checked for identical points and points that are closer to each other
than the given dmin are removed. If sparsely digitized contours or isolines are used as input,
additional points are computed between each 2 points on a line if the distance between them is greater
than specified dmax. Parameter zmult allows user to rescale the values used for approximation (useful
e.g. for transformation of elevations given in feet to meters, so that the proper values of slopes and
curvatures can be computed).
Regularized spline with tension is used for the approximation. The tension parameter tunes the character
of the resulting surface from thin plate to membrane. Smoothing parameter smooth controls the deviation
between the given points and the resulting surface and it can be very effective in smoothing noisy data
while preserving the geometrical properties of the surface. With the smoothing parameter set to zero
(smooth=0) the resulting surface passes exactly through the data points (spatial interpolation is
performed). When smoothing parameter is used, it is also possible to output a vector point file devi
containing deviations of the resulting surface from the given data.
If the number of given points is greater than segmax, segmented processing is used . The region is split
into quadtree-based rectangular segments, each having less than segmax points and approximation is
performed on each segment of the region. To ensure smooth connection of segments the approximation
function for each segment is computed using the points in the given segment and the points in its
neighborhood which are in the rectangular window surrounding the given segment. The number of points
taken for approximation is controlled by npmin, the value of which must be larger than segmax. User can
choose to output vector maps treefile and overfile which represent the quad tree used for segmentation
and overlapping neighborhoods from which additional points for approximation on each segment were taken.
Predictive error of surface approximation for given parameters can be computed using the -c flag. A
crossvalidation procedure is then performed using the data given in the vector map input and the
estimated predictive errors are stored in the vector point file cvdev. When using this flag, no raster
output files are computed. Anisotropic surfaces can be interpolated by setting anisotropy angle theta
and scaling factor scalex. The program writes values of selected input and internally computed
parameters to the history file of raster map elev.
NOTES
v.surf.rst uses regularized spline with tension for approximation from vector data. The module does not
require input data with topology, therefore both level1 (no topology) and level2 (with topology) vector
point data are supported. Additional points are used for approximation between each 2 points on a line
if the distance between them is greater than specified dmax. If dmax is small (less than cell size) the
number of added data points can be vary large and slow down approximation significantly. The
implementation has a segmentation procedure based on quadtrees which enhances the efficiency for large
data sets. Special color tables are created by the program for output raster maps.
Topographic parameters are computed directly from the approximation function so that the important
relationships between these parameters are preserved. The equations for computation of these parameters
and their interpretation is described in Mitasova and Hofierka, 1993 or Neteler and Mitasova, 2004).
Slopes and aspect are computed in degrees (0-90 and 1-360 respectively). The aspect raster map has value
0 assigned to flat areas (with slope less than 0.1%) and to singular points with undefined aspect. Aspect
points downslope and is 90 to the North, 180 to the West, 270 to the South and 360 to the East, the
values increase counterclockwise. Curvatures are positive for convex and negative for concave areas.
Singular points with undefined curvatures have assigned zero values.
Tension and smoothing allow user to tune the surface character. For most landscape scale applications
the default values should provide adequate results. The program gives warning when significant
overshoots appear in the resulting surface and higher tension or smoothing should be used. To select
parameters that will produce a surface with desired properties, it is useful to know that the method is
scale dependent and the tension works as a rescaling parameter (high tension "increases the distances
between the points" and reduces the range of impact of each point, low tension "decreases the distance"
and the points influence each other over longer range). Surface with tension set too high behaves like a
membrane (rubber sheet stretched over the data points) with peak or pit ("crater") in each given point
and everywhere else the surface goes rapidly to trend. If digitized contours are used as input data, high
tension can cause artificial waves along contours. Lower tension and higher smoothing is suggested for
such a case.
Surface with tension set too low behaves like a stiff steel plate and overshoots can appear in areas with
rapid change of gradient and segmentation can be visible. Increase in tension should solve the problems.
There are two options how tension can be applied in relation to dnorm (dnorm rescales the coordinates
depending on the average data density so that the size of segments with segmax=40 points is around 1 -
this ensures the numerical stability of the computation):
1. Default: the given tension is applied to normalized data (x/dnorm..), that means that the distances
are multiplied (rescaled) by tension/dnorm. If density of points is changed, e.g., by using higher dmin,
the dnorm changes and tension needs to be changed too to get the same result. Because the tension is
applied to normalized data its suitable value is usually within the 10-100 range and does not depend on
the actual scale (distances) of the original data (which can be km for regional applications or cm for
field experiments).
2. Flag -t : The given tension is applied to un-normalized data (rescaled tension = tension*dnorm/1000 is
applied to normalized data (x/dnorm) and therefore dnorm cancels out) so here tension truly works as a
rescaling parameter. For regional applications with distances between points in km. the suitable tension
can be 500 or higher, for detailed field scale analysis it can be 0.1. To help select how much the data
need to be rescaled the program writes dnorm and rescaled tension fi=tension*dnorm/1000 at the beginning
of the program run. This rescaled tension should be around 20-30. If it is lower or higher, the given
tension parameter should be changed accordingly.
The default is a recommended choice, however for the applications where the user needs to change density
of data and preserve the approximation character the -t flag can be helpful.
Anisotropic data (e.g. geologic phenomena) can be interpolated using theta and scalex defining
orientation and ratio of the perpendicular axes put on the longest/shortest side of the feature,
respectively. Theta is measured in degrees from East, counterclockwise. Scalex is a ratio of axes sizes.
Setting scalex in the range 0-1, results in a pattern prolonged in the direction defined by theta. Scalex
value 0.5 means that modeled feature is approximately 2 times longer in the direction of theta than in
the perpendicular direction. Scalex value 2 means that axes ratio is reverse resulting in a pattern
perpendicular to the previous example. Please note that anisotropy option has not been extensively tested
and may include bugs (for example , topographic parameters may not be computed correctly) - if there are
problems, please report to GRASS bugtracker (accessible from http://grass.osgeo.org/).
For data with values changing over several magnitudes (sometimes the concentration or density data) it is
suggested to interpolate the log of the values rather than the original ones.
The program checks the numerical stability of the algorithm by computing the values in given points, and
prints the root mean square deviation (rms) found into the history file of raster map elev. For
computation with smoothing set to 0. rms should be 0. Significant increase in tension is suggested if the
rms is unexpectedly high for this case. With smoothing parameter greater than zero the surface will not
pass exactly through the data points and the higher the parameter the closer the surface will be to the
trend. The rms then represents a measure of smoothing effect on data. More detailed analysis of smoothing
effects can be performed using the output deviations option.
SQL support
Using the where parameter, the interpolation can be limited to use only a subset of the input vectors.
Spearfish example (we simulate randomly distributed elevation measures):
g.region rast=elevation.10m -p
# random elevation extraction
r.random elevation.10m vector_output=elevrand n=200
v.info -c elevrand
v.db.select elevrand
# interpolation based on all points
v.surf.rst elevrand zcol=value elev=elev_full
r.colors elev_full rast=elevation.10m
d.rast elev_full
d.vect elevrand
# interpolation based on subset of points (only those over 1300m/asl)
v.surf.rst elevrand zcol=value elev=elev_partial where="value > 1300"
r.colors elev_partial rast=elevation.10m
d.rast elev_partial
d.vect elevrand where="value > 1300"
Cross validation procedure
The "optimal" approximation parameters for given data can be found using a cross-validation (CV)
procedure (-c flag). The CV procedure is based on removing one input data point at a time, performing
the approximation for the location of the removed point using the remaining data points and calculating
the difference between the actual and approximated value for the removed data point. The procedure is
repeated until every data point has been, in turn, removed. This form of CV is also known as the "leave-
one-out" or "jack-knife" method (Hofierka et al., 2002; Hofierka, 2005). The differences (residuals) are
then stored in the cvdev output vector map. Please note that during the CV procedure no other output
files can be set, the approximation is performed only for locations defined by input data. To find
"optimal parameters", the CV procedure must be iteratively performed for all reasonable combinations of
the approximation parameters with small incremental steps (e.g. tension, smoothing) in order to find a
combination with minimal statistical error (also called predictive error) defined by root mean squared
error (RMSE), mean absolute error (MAE) or other error characteristics. A script with loops for tested
RST parameters can do the job, necessary statistics can be calculated using e.g. v.univar. It should be
noted that crossvalidation is a time-consuming procedure, usually reasonable for up to several thousands
of points. For larger data sets, CV should be applied to a representative subset of the data. The cross-
validation procedure works well only for well-sampled phenomena and when minimizing the predictive error
is the goal. The parameters found by minimizing the predictive (CV) error may not not be the best for
for poorly sampled phenomena (result could be strongly smoothed with lost details and fluctuations) or
when significant noise is present that needs to be smoothed out.
The program writes the values of parameters used in computation into the comment part of history file
elev as well as the following values which help to evaluate the results and choose the suitable
parameters: minimum and maximum z values in the data file (zmin_data, zmax_data) and in the interpolated
raster map (zmin_int, zmax_int), rescaling parameter used for normalization (dnorm), which influences the
tension.
If visible connection of segments appears, the program should be rerun with higher npmin to get more
points from the neighborhood of given segment and/or with higher tension.
When the number of points in a vector map is not too large (less than 800), the user can skip
segmentation by setting segmax to the number of data points or segmax=700.
The program gives warning when user wants to interpolate outside the rectangle given by minimum and
maximum coordinates in the vector map, zoom into the area where the given data are is suggested in this
case.
When a mask is used, the program takes all points in the given region for approximation, including those
in the area which is masked out, to ensure proper approximation along the border of the mask. It
therefore does not mask out the data points, if this is desirable, it must be done outside v.surf.rst.
For examples of applications see GRASS4 implementation and GRASS5 and GRASS6 implementation.
The user must run g.region before the program to set the region and resolution for approximation.
SEE ALSO
v.vol.rst
AUTHORS
Original version of program (in FORTRAN) and GRASS enhancements:
Lubos Mitas, NCSA, University of Illinois at Urbana Champaign, Illinois, USA (1990-2000); Department of
Physics, North Carolina State University, Raleigh
Helena Mitasova, USA CERL, Department of Geography, University of Illinois at Urbana-Champaign, USA
(1990-2001); MEAS, North Carolina State University, Raleigh
Modified program (translated to C, adapted for GRASS, new segmentation procedure):
Irina Kosinovsky, US Army CERL, Dave Gerdes, US Army CERL
Modifications for new sites format and timestamping:
Darrel McCauley, Purdue University, Bill Brown, US Army CERL
Update for GRASS5.7, GRASS6 and addition of crossvalidation: Jaroslav Hofierka, University of Presov;
Radim Blazek, ITC-irst
REFERENCES
Mitasova, H., Mitas, L. and Harmon, R.S., 2005, Simultaneous spline approximation and topographic
analysis for lidar elevation data in open source GIS, IEEE GRSL 2 (4), 375- 379.
Hofierka, J., 2005, Interpolation of Radioactivity Data Using Regularized Spline with Tension. Applied
GIS, Vol. 1, No. 2, pp. 16-01 to 16-13. DOI: 10.2104/ag050016
Hofierka J., Parajka J., Mitasova H., Mitas L., 2002, Multivariate Interpolation of Precipitation Using
Regularized Spline with Tension. Transactions in GIS 6(2), pp. 135-150.
H. Mitasova, L. Mitas, B.M. Brown, D.P. Gerdes, I. Kosinovsky, 1995, Modeling spatially and temporally
distributed phenomena: New methods and tools for GRASS GIS. International Journal of GIS, 9 (4), special
issue on Integrating GIS and Environmental modeling, 433-446.
Mitasova, H. and Mitas, L., 1993: Interpolation by Regularized Spline with Tension: I. Theory and
Implementation, Mathematical Geology ,25, 641-655.
Mitasova, H. and Hofierka, J., 1993: Interpolation by Regularized Spline with Tension: II. Application to
Terrain Modeling and Surface Geometry Analysis, Mathematical Geology 25, 657-667.
Mitas, L., and Mitasova H., 1988, General variational approach to the approximation problem, Computers
and Mathematics with Applications, v.16, p. 983-992.
Neteler, M. and Mitasova, H., 2008, Open Source GIS: A GRASS GIS Approach, 3rd Edition, Springer, New
York, 406 pages.
Talmi, A. and Gilat, G., 1977 : Method for Smooth Approximation of Data, Journal of Computational
Physics, 23, p.93-123.
Wahba, G., 1990, : Spline Models for Observational Data, CNMS-NSF Regional Conference series in applied
mathematics, 59, SIAM, Philadelphia, Pennsylvania.
Last changed: $Date: 2011-11-08 03:29:50 -0800 (Tue, 08 Nov 2011) $
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