## table of contents

v.vol.rst(1grass) | GRASS GIS User's Manual | v.vol.rst(1grass) |

# NAME¶

**v.vol.rst** - Interpolates point data to a 3D raster
map using regularized spline with tension (RST) algorithm.

# KEYWORDS¶

vector, voxel, surface, interpolation, RST, 3D, no-data filling

# SYNOPSIS¶

**v.vol.rst**

**v.vol.rst --help**

**v.vol.rst** [-**c**] **input**=*name*
[**cross_input**=*name*] [**wcolumn**=*name*]
[**tension**=*float*] [**smooth**=*float*]
[**smooth_column**=*name*] [**where**=*sql_query*]
[**deviations**=*name*] [**cvdev**=*name*]
[**maskmap**=*name*] [**segmax**=*integer*]
[**npmin**=*integer*] [**npmax**=*integer*]
[**dmin**=*float*] [**wscale**=*float*]
[**zscale**=*float*] [**cross_output**=*name*]
[**elevation**=*name*] [**gradient**=*name*]
[**aspect_horizontal**=*name*] [**aspect_vertical**=*name*]
[**ncurvature**=*name*] [**gcurvature**=*name*]
[**mcurvature**=*name*] [--**overwrite**] [--**help**]
[--**verbose**] [--**quiet**] [--**ui**]

## Flags:¶

## Parameters:¶

**input**=*name***[required]**-

Name of input 3D vector points map **cross_input**=*name*-

Name of input surface raster map for cross-section **wcolumn**=*name*-

Name of column containing w-values attribute to interpolate **tension**=*float*-

Tension parameter

Default:*40.* **smooth**=*float*-

Smoothing parameter

Default:*0.1* **smooth_column**=*name*-

Name of column with smoothing parameters **where**=*sql_query*-

WHERE conditions of SQL statement without ’where’ keyword

Example: income < 1000 and population >= 10000 **deviations**=*name*-

Name for output deviations vector point map **cvdev**=*name*-

Name for output cross-validation errors vector point map **maskmap**=*name*-

Name of input raster map used as mask **segmax**=*integer*-

Maximum number of points in a segment

Default:*50* **npmin**=*integer*-

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

Default:*200* **npmax**=*integer*-

Maximum number of points for approximation in a segment (>npmin)

Default:*700* **dmin**=*float*-

Minimum distance between points (to remove almost identical points) **wscale**=*float*-

Conversion factor for w-values used for interpolation

Default:*1.0* **zscale**=*float*-

Conversion factor for z-values

Default:*1.0* **cross_output**=*name*-

Name for output cross-section raster map **elevation**=*name*-

Name for output elevation 3D raster map **gradient**=*name*-

Name for output gradient magnitude 3D raster map **aspect_horizontal**=*name*-

Name for output gradient horizontal angle 3D raster map **aspect_vertical**=*name*-

Name for output gradient vertical angle 3D raster map **ncurvature**=*name*-

Name for output change of gradient 3D raster map **gcurvature**=*name*-

Name for output Gaussian curvature 3D raster map **mcurvature**=*name*-

Name for output mean curvature 3D raster map

# DESCRIPTION¶

*v.vol.rst* interpolates values to a 3-dimensional raster map
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 3D raster map
**elevation** 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, **cross_input** and **cross_output** options
must be specified, with the output 2D raster map **cross_output**
containing the crossection of the interpolated volume with a surface defined
by **cross_input** 2D raster map. 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 3D raster maps
**gradient, aspect_horizontal, aspect_vertical, ncurvature, gcurvature,
mcurvature**, respectively. Maps **aspect_horizontal** and
**aspect_vertical** are in degrees.

At first, data points are checked for identical positions and
points that are closer to each other than given **dmin** are removed.
Parameters **wscale** and **zscale** allow the 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). Rescaling of z-coordinates (**zscale**)
is also needed when the distances in vertical direction are much smaller
than the horizontal distances; if that is the case, the value of
**zscale** should be selected so that the vertical and horizontal
distances have about the same magnitude.

Regularized spline with tension method is used in the
interpolation. The **tension** parameter controls the distance over which
each given point influences the resulting volume (with very high tension,
each point influences only its close neighborhood and the volume goes
rapidly to trend between the points). 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. When smoothing is used, it is possible to output a vector map
**deviations** containing deviations of the resulting volume from the
given data.

The user can define a 2D raster map named **maskmap**, which
will be used as a mask. The interpolation is skipped for 3-dimensional cells
whose 2-dimensional projection has a zero value in the mask. Zero values
will be assigned to these cells in all output 3D raster maps.

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 the 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.

## SQL support¶

Using the **where** parameter, the interpolation can be limited
to use only a subset of the input vectors.

# preparation as in above example v.vol.rst elevrand_3d wcol=soilrange elevation=soilrange zscale=100 where="soilrange > 3"

## Cross validation procedure¶

Sometimes it can be difficult to figure out the proper values of
interpolation parameters. In this case, the user can use a crossvalidation
procedure using **-c** flag (a.k.a. "jack-knife" method) to
find optimal parameters for given data. In this method, every point in the
input point file is temporarily excluded from the computation and
interpolation error for this point location is computed. During this
procedure no output grid files can be simultanuously computed. The procedure
for larger datasets may take a very long time, so it might be worth to use
just a sample data representing the whole dataset.

*Example* *(based on Slovakia3d dataset):*

v.info -c precip3d g.region n=5530000 s=5275000 w=4186000 e=4631000 res=500 -p v.vol.rst -c input=precip3d wcolumn=precip zscale=50 segmax=700 cvdev=cvdevmap tension=10 v.db.select cvdevmap v.univar cvdevmap col=flt1 type=pointBased on these results, the parameters will have to be optimized. It is recommended to plot the CV error as curve while modifying the parameters.

The best approach is to start with **tension**, **smooth**
and **zscale** with rough steps, or to set **zscale** 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,
**zscale**=10...100.

In *v.vol.rst* the tension parameter is much more sensitive
to changes than in *v.surf.rst*, therefore the user should always check
the result by visual inspection. Minimizing CV does not always provide the
best result, especially when the density of data are insufficient. Then the
optimal result found by CV is an oversmoothed surface.

# NOTES¶

The vector points map must be a 3D vector map (x, y, z as geometry). The module v.in.db can be used to generate a 3D vector map from a table containing x,y,z columns. Also, the input data should be in a projected coordinate system, such as Universal Transverse Mercator. The module does not appear to have support for geographic (Lat/Long) coordinates as of May 2009.

*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 (**aspect_horizontal**) and vertical
(**aspect_vertical)**aspects, change of gradient (**ncurvature**),
Gauss-Kronecker (**gcurvature**) and mean curvatures (**mcurvature**)
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 will have local maximum in each given point and everywhere else the volume goes rapidly to trend. With a 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 than 400,
**segmax** should be set to 400 so that segmentation is not performed
when it is not necessary.

The program gives a warning when the user wants to interpolate outside the "box" given by minimum and maximum coordinates in the input vector map. To remedy this, zoom into the area encompassing the input vector data points.

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.

# EXAMPLES¶

Spearfish example (we first simulate 3D soil range data):

g.region -dp # define volume g.region res=100 tbres=100 res3=100 b=0 t=1500 -ap3 ### First part: generate synthetic 3D data (true 3D soil data preferred) # generate random positions from elevation map (2D) r.random elevation.10m vector_output=elevrand n=200 # generate synthetic values v.db.addcolumn elevrand col="x double precision, y double precision" v.to.db elevrand option=coor col=x,y v.db.select elevrand # create new 3D map v.in.db elevrand out=elevrand_3d x=x y=y z=value key=cat v.info -c elevrand_3d v.info -t elevrand_3d # remove the now superfluous ’x’, ’y’ and ’value’ (z) columns v.db.dropcolumn elevrand_3d col=x v.db.dropcolumn elevrand_3d col=y v.db.dropcolumn elevrand_3d col=value # add attribute to have data available for 3D interpolation # (Soil range types taken from the USDA Soil Survey) d.mon wx0 d.rast soils.range d.vect elevrand_3d v.db.addcolumn elevrand_3d col="soilrange integer" v.what.rast elevrand_3d col=soilrange rast=soils.range # fix 0 (no data in raster map) to NULL: v.db.update elevrand_3d col=soilrange value=NULL where="soilrange=0" v.db.select elevrand_3d # optionally: check 3D points in Paraview v.out.vtk input=elevrand_3d output=elevrand_3d.vtk type=point dp=2 paraview --data=elevrand_3d.vtk ### Second part: 3D interpolation from 3D point data # interpolate volume to "soilrange" voxel map v.vol.rst input=elevrand_3d wcol=soilrange elevation=soilrange zscale=100 # visualize I: in GRASS GIS wxGUI g.gui # load: 2D raster map: elevation.10m # 3D raster map: soilrange # visualize II: export to Paraview r.mapcalc "bottom = 0.0" r3.out.vtk -s input=soilrange top=elevation.10m bottom=bottom dp=2 output=volume.vtk paraview --data=volume.vtk

# KNOWN ISSUES¶

**deviations** file is written as 2D and deviations are not
written as attributes.

# REFERENCES¶

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.

# SEE ALSO¶

*g.region,* *v.in.ascii,* *r3.mask,*
*v.in.db,* *v.surf.rst,* *v.univar*

# AUTHOR¶

Original version of program (in FORTRAN) and GRASS enhancements:

Lubos Mitas, NCSA, University of Illinois at Urbana-Champaign, Illinois, USA,
since 2000 at Department of Physics, North Carolina State University,
Raleigh, USA lubos_mitas@ncsu.edu

Helena Mitasova, Department of Marine, Earth and Atmospheric Sciences, North
Carolina State University, Raleigh, USA, hmitaso@unity.ncsu.edu

Modified program (translated to C, adapted for GRASS, new
segmentation procedure):

Irina Kosinovsky, US Army CERL, Champaign, Illinois, USA

Dave Gerdes, US Army CERL, Champaign, Illinois, USA

Modifications for g3d library, geometric parameters,
cross-validation, deviations:

Jaro Hofierka, Department of Geography and Regional Development, University of
Presov, Presov, Slovakia, hofierka@fhpv.unipo.sk, http://www.geomodel.sk

# SOURCE CODE¶

Available at: v.vol.rst source code (history)

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