## DESCRIPTION

v.surf.bspline performs a bilinear/bicubic spline interpolation with Tykhonov regularization. The input is a 2D or 3D vector points map. Values to interpolate can be the z values of 3D points or the values in a user-specified attribute column in a 2D or 3D vector map. Output can be a raster (raster_output) or vector (output) map. Optionally, a "sparse point" vector map can be input which indicates the location of output vector points.

## NOTES

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.

## EXAMPLES

### Basic interpolation

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

A bicubic spline interpolation will be done and a vector points map with estimated (i.e., interpolated) values will be created.

### Basic interpolation and raster output with a longer spline step

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

A bilinear spline interpolation will be done with a spline step length of 25 map units. An interpolated raster map will be created at the current region resolution.

### Estimation of lambda_i parameter with a cross validation process

    v.surf.bspline -c input=point_vector

### Estimation on sparse points

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

An output map of vector points will be created, corresponding to the sparse vector map, with interpolated values.

### Using attribute values instead z-coordinates

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

The interpolation will be done using the values in attrib_column, in the table associated with layer 1.

### North carolina location example using z-coordinates for interpolation

    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

## KNOWN ISSUES

Known issues:

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.

## REFERENCES

- 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

## SEE ALSO

v.surf.idw, v.surf.rst

Overview: Interpolation and Resampling in GRASS GIS

## AUTHORS

Original version (s.bspline.reg) in GRASS 5.4: Maria Antonia Brovelli, Massimiliano Cannata, Ulisse Longoni, Mirko Reguzzoni
Update for GRASS 6 and improvements: Roberto Antolin

Last changed: $Date: 2017-07-24 23:59:39 +0200 (Mon, 24 Jul 2017) $