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NAME   -  Spatial approximation and topographic analysis from given point or isoline
       data in vector format to floating  point  raster  format  using  regularized  spline  with


       vector, interpolation

SYNOPSIS help   [-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]

           Use z-coordinates (3D vector only)

           Perform cross-validation procedure without raster approximation

           Use scale dependent tension

           Output partial derivatives instead of topographic parameters

           Allow output files to overwrite existing files

           Verbose module output

           Quiet module output

           Name of input vector map

           Layer number
           If set to 0, z coordinates are used. (3D vector only)
           Default: 1

           WHERE conditions of SQL statement without 'where' keyword
           Example: income = 10000

           Output surface raster map (elevation)

           Output slope raster map

           Output aspect raster map

           Output profile curvature raster map

           Output tangential curvature raster map

           Output mean curvature raster map

           Output deviations vector point file

           Output cross-validation errors vector point file

           Output vector map showing quadtree segmentation

           Output vector map showing overlapping windows

           Name of the raster map used as mask

           Name of the attribute column with values to be used for approximation (if layer>0)

           Tension parameter
           Default: 40.

           Smoothing parameter

           Name of the attribute column with smoothing parameters

           Maximum number of points in a segment
           Default: 40

           Minimum number of points for approximation in a segment (>segmax)
           Default: 300

           Minimum distance between points (to remove almost identical points)
           Default: 0.500000

           Maximum distance between points on isoline (to insert additional points)
           Default: 2.500000

           Conversion factor for values used for approximation
           Default: 1.0

           Anisotropy angle (in degrees counterclockwise from East)

           Anisotropy scaling factor

       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 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

       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

       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 -c elevrand elevrand
       # interpolation based on all points 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) 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

       For  examples  of  applications  see  GRASS4  implementation   and   GRASS5   and   GRASS6

       The  user  must  run  g.region  before  the  program  to set the region and resolution for




       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


       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-

       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.

       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,

       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,

       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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