v.vol.rst interpolates values to a 3-dimensional grid from 3-dimensional point data (e.g. temperature, rainfall data from climatic stations, concentrations from drill holes etc.) given in a 3-D vector point file named input. The size of the output 3-D grid g3d file elev is given by the current 3D region. Sometimes, the user may want to get a 2-D map showing a modelled phenomenon at a crossection surface. In that case, cellinp and cellout options must be specified and then the output 2D grid file cellout contains crossection of interpolated volume with surface defined by cellinp 2D grid input file. As an option, simultaneously with interpolation, geometric parameters of the interpolated phenomenon can be computed (magnitude of gradient, direction of gradient defined by horizontal and vertical angles), change of gradient, Gauss-Kronecker curvature, or mean curvature). These geometric parameteres are saved as g3d files gradient, aspect1, aspect2, ncurv, gcurv, mcurv, respectively.
At first, data points are checked for identical points and points that are closer to each other than given dmin are removed. Parameters wmult and zmult allow user to re-scale the w-values and z-coordinates of the point data (useful e.g. for transformation of elevations given in feet to meters, so that the proper values of gradient and curvatures can be computed).
Regularized spline with tension method is used in the interpolation. The tension parameter tunes the character of the resulting volume from thin plate to membrane. Higher values of tension parameter reduce the overshoots that can appear in volumes with rapid change of gradient. For noisy data, it is possible to define a global smoothing parameter, smooth. With the smoothing parameter set to zero (smooth=0) the resulting volume passes exactly through the data points. Also, the user can use a spatially variable smoothing using smatt option by setting the parameter smatt to the value j for the j-th floating point attribute in the input vector point file, representing the smoothing parameter for each point. When smoothing is used, it is possible to output vector map devi containing deviations of the resulting volume from the given data.
User can define a 2D raster file named maskmap, which will be used as a mask. The interpolation is skipped for 3-dimensional cells whose 2-dimensional projection has zero value in mask. Zero values will be assigned to these cells in all output g3d files.
If the number of given points is greater than 700, segmented processing is used. The region is split into 3-dimensional "box" segments, each having less than segmax points and interpolation is performed on each segment of the region. To ensure the smooth connection of segments the interpolation function for each segment is computed using the points in given segment and the points in its neighborhood. The minimum number of points taken for interpolation is controlled by npmin , the value of which must be larger than segmax and less than 700. This limit of 700 was selected to ensure the numerical stability and efficiency of the algorithm.
v.info -c precip3d v.vol.rst -c input=precip3d wcolumn=precip segmax=700 cvdev=cvdevmap tension=10 v.db.select cvdevmap v.univar cvdevmap col=flt1 type=point
The best approach is to start with tension, smooth and zmult with rough steps, or to set zmult to a constant somewhere between 30-60. This helps to find minimal RMSE values while then finer steps can be used in all parameters. The reasonable range is tension=10...100, smooth=0.1...1.0, zmult=10...100.
In v.vol.rst the tension parameter is much more sensitive to changes than in v.surf.rst. Usually tension=10...20 provide best results. But the user should always check the result by visual inspection, sometimes CV does not provide the best results, especially when the density of data are insufficient. Then the optimal result found by CV is an oversmoothed surface.
v.vol.rst uses regularized spline with tension for interpolation from point data (as described in Mitasova and Mitas, 1993). The implementation has an improved segmentation procedure based on Oct-trees which enhances the efficiency for large data sets.
Geometric parameters - magnitude of gradient (gradient), horizontal (aspect1) and vertical (aspect2) aspects, change of gradient (ncurv), Gauss-Kronecker (gcurv) and mean curvatures (mcurv) are computed directly from the interpolation function so that the important relationships between these parameters are preserved. More information on these parameters can be found in Mitasova et al., 1995 or Thorpe, 1979.
The program gives warning when significant overshoots appear and higher tension should be used. However, with tension too high the resulting volume changes its behavior to membrane( rubber sheet stretched over the data points resulting in a peak in each given point and everywhere else the volume goes rapidly to trend). With smoothing parameter greater than zero the volume will not pass through the data points and the higher the parameter the closer the volume will be to the trend. For theory on smoothing with splines see Talmi and Gilat, 1977 or Wahba, 1990.
If a visible connection of segments appears, the program should be rerun with higher npmin to get more points from the neighborhood of given segment.
If the number of points in a vector map is less then 400, segmax should be set to 400 so that segmentation is not performed when it is not necessary.
The program gives warning when user wants to interpolate outside the "box" given by minimum and maximum coordinates in vector map, zoom into the area where the points are is suggested in this case.
For large data sets (thousands of data points) it is suggested to zoom into a smaller representative area and test whether the parameters chosen (e.g. defaults) are appropriate.
The user must run g.region before the program to set the 3D region for interpolation.
Original version of program (in FORTRAN) and GRASS enhancements:
Lubos Mitas, NCSA, University of Illinois at Urbana-Champaign, Illinois, USA,firstname.lastname@example.org
Helena Mitasova, Department of Marine, Earth and Atmospheric Sciences, North Carolina State University, Raleigh, USA, email@example.com
Modified program (translated to C, adapted for GRASS, new
Irina Kosinovsky, US Army CERL, Champaign, Illinois, USA
Dave Gerdes, US Army CERL, Champaign, Illinois, USA
Modifications for g3d library, geometric parameters,
Jaro Hofierka, Department of Geography and Regional Development, University of Presov, Presov, Slovakia, firstname.lastname@example.org, http://www.geomodel.sk
Hofierka J., Parajka J., Mitasova H., Mitas L., 2002, Multivariate Interpolation of Precipitation Using Regularized Spline with Tension. Transactions in GIS 6, pp. 135-150.
Mitas, L., Mitasova, H., 1999, Spatial Interpolation. In: P.Longley, M.F. Goodchild, D.J. Maguire, D.W.Rhind (Eds.), Geographical Information Systems: Principles, Techniques, Management and Applications, Wiley, pp.481-492
Mitas L., Brown W. M., Mitasova H., 1997, Role of dynamic cartography in simulations of landscape processes based on multi-variate fields. Computers and Geosciences, Vol. 23, No. 4, pp. 437-446 (includes CDROM and WWW: www.elsevier.nl/locate/cgvis)
Mitasova H., Mitas L., Brown W.M., D.P. Gerdes, I. Kosinovsky, Baker, T.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., Mitas, L., Brown, B., Kosinovsky, I., Baker, T., Gerdes, D. (1994): Multidimensional interpolation and visualization in GRASS GIS
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.
Mitasova, H., 1992 : New capabilities for interpolation and topographic analysis in GRASS, GRASSclippings 6, No.2 (summer), p.13.
Wahba, G., 1990 : Spline Models for Observational Data, CNMS-NSF Regional Conference series in applied mathematics, 59, SIAM, Philadelphia, Pennsylvania.
Mitas, L., Mitasova H., 1988 : General variational approach to the interpolation problem, Computers and Mathematics with Applications 16, p. 983
Talmi, A. and Gilat, G., 1977 : Method for Smooth Approximation of Data, Journal of Computational Physics, 23, p.93-123.
Thorpe, J. A. (1979): Elementary Topics in Differential Geometry. Springer-Verlag, New York, pp. 6-94.
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