**-c**- Find the best Tykhonov regularizing parameter using a "leave-one-out" cross validation method
**-e**- Estimate point density and distance
- Estimate point density and distance in map units for the input vector points within the current region extents and quit
**--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 point 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* **column**=*name*- Name of the attribute column with values to be used for approximation
- If not given and input is 3D vector map then z-coordinates are used.
**sparse_input**=*name*- Name of input vector map with sparse points
- Or data source for direct OGR access
**output**=*name*- Name for output vector map
**raster_output**=*name*- Name for output raster map
**mask**=*name*- Raster map to use for masking (applies to raster output only)
- Only cells that are not NULL and not zero are interpolated
**ew_step**=*float*- Length of each spline step in the east-west direction
- Default: 4 * east-west resolution
**ns_step**=*float*- Length of each spline step in the north-south direction
- Default: 4 * north-south resolution
**method**=*string*- Spline interpolation algorithm
- Options:
*bilinear, bicubic* - Default:
*bilinear* **bilinear**: Bilinear interpolation**bicubic**: Bicubic interpolation**lambda_i**=*float*- Tykhonov regularization parameter (affects smoothing)
- Default:
*0.01* **solver**=*name*- The type of solver which should solve the symmetric linear equation system
- Options:
*cholesky, cg* - Default:
*cholesky* **maxit**=*integer*- Maximum number of iteration used to solve the linear equation system
- Default:
*10000* **error**=*float*- Error break criteria for iterative solver
- Default:
*0.000001* **memory**=*memory in MB*- Maximum memory to be used (in MB)
- Cache size for raster rows
- Default:
*300*

From a theoretical perspective, the interpolating procedure takes
place in two parts: the first is an estimate of the linear coefficients
of a spline function is derived from the observation points using a
least squares regression; the second is the computation of the
interpolated surface (or interpolated vector points). As used here, the
splines are 2D piece-wise non-zero polynomial functions calculated
within a limited, 2D area. The length (in mapping units) of each spline
step is defined by **ew_step** for the east-west direction and
**ns_step** for the north-south direction. For optimal performance,
the length of spline step should be no less than the distance between
observation points. Each vector point observation is modeled as a
linear function of the non-zero splines in the area around the
observation. The least squares regression predicts the the coefficients
of these linear functions. Regularization, avoids the need to have one
observation and one coefficient for each spline (in order to avoid
instability).

With regularly distributed data points, a spline step corresponding to the maximum distance between two points in both the east and north directions is sufficient. But often data points are not regularly distributed and require statistial regularization or estimation. In such cases, v.surf.bspline will attempt to minimize the gradient of bilinear splines or the curvature of bicubic splines in areas lacking point observations. As a general rule, spline step length should be greater than the mean distance between observation points (twice the distance between points is a good starting point). Separate east-west and north-south spline step length arguments allows the user to account for some degree of anisotropy in the distribution of observation points. Short spline step lengths - especially spline step lengths that are less than the distance between observation points - can greatly increase the processing time.

Moreover, the maximum number of splines for each direction at each time is fixed, regardless of the spline step length. As the total number of splines used increases (i.e., with small spline step lengths), the region is automatically split into subregions for interpolation. Each subregion can contain no more than 150x150 splines. To avoid subregion boundary problems, subregions are created to partially overlap each other. A weighted mean of observations, based on point locations, is calculated within each subregion.

The Tykhonov regularization parameter (**lambda_i**) acts to
smooth the interpolation. With a small **lambda_i**, the
interpolated surface closely follows observation points; a larger
value will produce a smoother interpolation.

The input can be a 2D or 3D vector points map. If input is 3D
and **column** is not given than z-coordinates are used for
interpolation. Parameter **column** is required when input is 2D
vector map.

*v.surf.bspline* can produce a **raster_output** OR
a **output** (but NOT simultaneously). Note that topology is not
build for output vector point map. The topology can be built if
required by *v.build*.

If output is a vector points map and a **sparse** vector points
map is not specified, the output vector map will contain points at the
same locations as observation points in the input map, but the values
of the output points are interpolated values. If instead
a **sparse** vector points map is specified, the output vector map
will contain points at the same locations as the sparse vector map
points, and values will be those of the interpolated raster surface at
those points.

A cross validation "leave-one-out" analysis is available to help to
determine the optimal **lambda_i** value that produces an
interpolation that best fits the original observation data. The more
points used for cross-validation, the longer the time needed for
computation. Empirical testing indicates a threshold of a maximum of
100 points is recommended. Note that cross validation can run very
slowly if more than 100 observations are used. The cross-validation
output reports *mean* and *rms* of the residuals from the
true point value and the estimated from the interpolation for a fixed
series of **lambda_i** values. No vector nor raster output will be
created when cross-validation is selected.

v.surf.bspline input=point_vector output=interpolate_surface method=bicubic

v.surf.bspline input=point_vector raster=interpolate_surface ew_step=25 ns_step=25

v.surf.bspline -c input=point_vector

v.surf.bspline input=point_vector sparse=sparse_points output=interpolate_surface

v.surf.bspline input=point_vector raster=interpolate_surface layer=1 \ column=attrib_column

g.region region=rural_1m res=2 -p v.surf.bspline input=elev_lid792_bepts raster=elev_lid792_rast \ ew_step=5 ns_step=5 method=bicubic lambda_i=0.1

In order to avoid RAM memory problems, an auxiliary table is needed
for recording some intermediate calculations. This requires
the *GROUP BY* SQL function is used, which is not supported by
the DBF driver. For this reason, vector map output
(**output**) is not permitted with the DBF driver. There are
no problems with the raster map output from the DBF driver.

- Brovelli M. A., Cannata M., and Longoni U.M., 2004, LIDAR Data Filtering and DTM Interpolation Within GRASS, Transactions in GIS, April 2004, vol. 8, iss. 2, pp. 155-174(20), Blackwell Publishing Ltd
- Brovelli M. A. and Cannata M., 2004, Digital Terrain model reconstruction in urban areas from airborne laser scanning data: the method and an example for Pavia (Northern Italy). Computers and Geosciences 30, pp.325-331
- Brovelli M. A e Longoni U.M., 2003, Software per il filtraggio di dati LIDAR, Rivista dell'Agenzia del Territorio, n. 3-2003, pp. 11-22 (ISSN 1593-2192)
- Antolin R. and Brovelli M.A., 2007, LiDAR data Filtering with GRASS GIS for the Determination of Digital Terrain Models. Proceedings of Jornadas de SIG Libre, Girona, España. CD ISBN: 978-84-690-3886-9

Overview: Interpolation and Resampling in GRASS GIS

Update for GRASS 6 and improvements: Roberto Antolin

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

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