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.
Overview: Interpolation and Resampling in GRASS GIS
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