## DESCRIPTION

r.resamp.bspline performs a bilinear/bicubic spline interpolation with Tykhonov regularization. The input is a raster surface map, e.g. elevation, temperature, precipitation etc. Output is a raster map. Optionally, only input NULL cells are interpolated, useful to fill NULL cells, an alternative to r.fillnulls. Using the -n flag to only interpolate NULL cells will considerably speed up the module.

The input raster map is read at its native resolution, the output raster map will be produced for the current computational region set with g.region. Any MASK will be respected, masked values will be treated as NULL cells in both the input and the output map.

Spline step values ew_step for the east-west direction and ns_step for the north-south direction should not be smaller than the east-west and north-south resolutions of the input map. For a raster map without NULL cells, 1 * resolution can be used, but check for undershoots and overshoots. For very large areas with missing values (NULL cells), larger spline step values may be required, but most of the time the defaults (1.5 x resolution) should be fine.

The Tykhonov regularization parameter (lambda) acts to smooth the interpolation. With a small lambda, the interpolated surface closely follows observation points; a larger value will produce a smoother interpolation. Reasonable values are 0.0001, 0.001, 0.005, 0.01, 0.02, 0.05, 0.1 (needs more testing). For seamless NULL cell interpolation, a small value is required. The default lambda value is set to 0.01.

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; these are 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 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 spline step values should be no less than the east-west and north-south resolutions of the input map. Each non-NULL cell 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 one observation and one coefficient for each spline (in order to avoid instability).

A cross validation "leave-one-out" analysis is available to help to determine the optimal lambda 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 values. No vector nor raster output will be created when cross-validation is selected.

## EXAMPLES

### Basic interpolation

    r.resamp.bspline input=raster_surface output=interpolated_surface method=bicubic

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

### Interpolation of NULL cells and patching

General procedure:

    # set region to area with NULL cells, align region to input map
    g.region n=north s=south e=east w=west align=input -p
    # interpolate NULL cells
    r.resamp.bspline -n input=input_raster output=interpolated_nulls method=bicubic
    # set region to area with NULL cells, align region to input map
    g.region raster=input -p
    # patch original map and interpolated NULLs
    r.patch input=input_raster,interpolated_nulls output=input_raster_gapfilled

### Interpolation of NULL cells and patching (NC data)

In this example, the SRTM elevation map in the North Carolina sample dataset is filtered for outlier elevation values; missing pixels are then re-interpolated to obtain a complete elevation map:

    g.region raster=elev_srtm_30m -p
    d.mon wx0
    d.histogram elev_srtm_30m

    r.univar -e elev_srtm_30m

    # remove too low elevations (esp. lakes)
    # Threshold: thresh = Q1 - 1.5 * (Q3 - Q1)
    r.mapcalc "elev_srtm_30m_filt = if(elev_srtm_30m < 50.0, null(), elev_srtm_30m)"

    # verify
    d.histogram elev_srtm_30m_filt
    d.erase
    d.rast elev_srtm_30m_filt

    r.resamp.bspline -n input=elev_srtm_30m_filt output=elev_srtm_30m_complete \
      method=bicubic

    d.histogram elev_srtm_30m_complete
    d.rast elev_srtm_30m_complete

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

A random sample of points should be generated first with r.random, and the current region should not include more than 100 non-NULL random cells.

    r.resamp.bspline -c input=input_raster

## 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

r.fillnulls, r.resamp.rst, r.resamp.interp, v.surf.bspline

Overview: Interpolation and Resampling in GRASS GIS

## AUTHORS

Markus Metz
based on v.surf.bspline by
Maria Antonia Brovelli, Massimiliano Cannata, Ulisse Longoni, Mirko Reguzzoni, Roberto Antolin