## table of contents

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

# NAME¶

**v.surf.rst** - Performs surface interpolation from
vector points map by splines.

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, surface, interpolation, splines, RST, 3D, no-data filling

# SYNOPSIS¶

**v.surf.rst**

**v.surf.rst --help**

**v.surf.rst** [-**ctd**] **input**=*name*
[**layer**=*string*] [**zcolumn**=*name*]
[**where**=*sql_query*] [**elevation**=*name*]
[**slope**=*name*] [**aspect**=*name*]
[**pcurvature**=*name*] [**tcurvature**=*name*]
[**mcurvature**=*name*] [**deviations**=*name*]
[**cvdev**=*name*] [**treeseg**=*name*]
[**overwin**=*name*] [**nprocs**=*integer*]
[**mask**=*name*] [**tension**=*float*]
[**smooth**=*float*] [**smooth_column**=*string*]
[**segmax**=*integer*] [**npmin**=*integer*]
[**dmin**=*float*] [**dmax**=*float*]
[**zscale**=*float*] [**theta**=*float*]
[**scalex**=*float*] [--**overwrite**] [--**help**]
[--**verbose**] [--**quiet**] [--**ui**]

## Flags:¶

**-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 **--help**-

Print usage summary **--verbose**-

Verbose module output **--quiet**-

Quiet module output **--ui**-

Force launching GUI dialog

## Parameters:¶

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

Name of input vector map

Or data source for direct OGR access **layer**=*string*-

Layer number or name

Vector features can have category values in different layers. This number determines which layer to use. When used with direct OGR access this is the layer name.

Default:*1* **zcolumn**=*name*-

Name of the attribute column with values to be used for approximation

If not given and input is 2D vector map then category values are used. If input is 3D vector map then z-coordinates are used. **where**=*sql_query*-

WHERE conditions of SQL statement without ’where’ keyword

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

Name for output surface elevation raster map **slope**=*name*-

Name for output slope raster map **aspect**=*name*-

Name for output aspect raster map **pcurvature**=*name*-

Name for output profile curvature raster map **tcurvature**=*name*-

Name for output tangential curvature raster map **mcurvature**=*name*-

Name for output mean curvature raster map **deviations**=*name*-

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

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

Name for output vector map showing quadtree segmentation **overwin**=*name*-

Name for output vector map showing overlapping windows **nprocs**=*integer*-

Number of threads for parallel computing

Default:*1* **mask**=*name*-

Name of raster map used as mask **tension**=*float*-

Tension parameter

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

Smoothing parameter

Smoothing is by default 0.5 unless smooth_column is specified **smooth_column**=*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) **dmax**=*float*-

Maximum distance between points on isoline (to insert additional points) **zscale**=*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* program performs spatial approximation based on
*z-values* (input vector map is 3D and **zcolumn** parameter is not
given), *categories* (input vector map is 2D and **zcolumn**
parameter is not given), or *attributes* (**zcolumn** parameter is
given) of point or isoline data given in a vector map named **input** to
grid cells in the output raster map **elevation** 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, *v.surf.rst* outputs
partial derivatives fx,fy,fxx, fyy,fxy instead of slope, aspect, profile,
tangential and mean curvatures respectively. If the input vector map have
time stamp, the program creates time stamp for all output maps.

User can either use *r.mask* to set a mask or specify a
raster map in **mask** option, 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 map **deviations**
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 **treeseg** and **overwin** 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 map **cvdev**.
When using this flag, no raster output maps 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
**elevation**.

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

# 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 https://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.

*v.surf.rst* 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 **elevation**.
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.

*v.surf.rst* also writes the values of parameters used in
computation into the comment part of history file **elevation** 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**.

*v.surf.rst* 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*.

## 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 maps 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.

# EXAMPLE¶

## Setting for lidar point cloud¶

Lidar point clouds as well as UAS SfM-based (phodar) point clouds
tend to be dense in relation to the desired raster resolution and thus a
different set of parameters is more advantageous, e.g. in comparison to a
typical temperature data interpolation.

v.surf.rst input=points elevation=elevation npmin=100

## Usage of the where parameter¶

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

North Carolina example (we simulate randomly distributed elevation
measures which we interpolate to a gap-free elevation surface):

g.region raster=elevation -p # random elevation extraction of 500 samplings r.random elevation vector_output=elevrand n=500 v.info -c elevrand v.db.select elevrand # interpolation based on all points v.surf.rst elevrand zcol=value elevation=elev_full # apply the color table of the original raster map r.colors elev_full raster=elevation d.rast elev_full d.vect elevrand # interpolation based on subset of points (only those over 1300m/asl) v.surf.rst elevrand zcol=value elevation=elev_partial where="value > 1300" r.colors elev_partial raster=elevation d.rast elev_partial d.vect elevrand where="value > 1300"

# 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.

# SEE ALSO¶

*v.vol.rst,* *v.surf.idw,*
*v.surf.bspline,* *r.fillnulls,* *g.region*

Overview: Interpolation and Resampling in GRASS GIS

For examples of applications see GRASS4 implementation and GRASS5 and GRASS6 implementation.

# 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

*Parallelization using OpenMP:*

Stanislav Zubal, Czech Technical University in Prague

Michal Lacko, Pavol Jozef Safarik University in Kosice

# SOURCE CODE¶

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

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