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

**-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

**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

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
f_{x},f_{y},f_{xx},
f_{yy},f_{xy} 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.

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):

- 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). - 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.

*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*.

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.

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

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"

- 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.

Overview: Interpolation and Resampling in GRASS GIS

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

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

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

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