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 +of a spline function, which 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 +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 +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 +directions is sufficient. However, data points are often not regularly +distributed and require statistical 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 +point observations. As a general rule, the 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 - +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 +
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 +number of splines 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 +
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 +
The input can be a 2D or 3D point vector layer. 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. +vector layer. + +
v.surf.bspline can produce raster (raster_output) OR +vector output but NOT simultaneously. Note that topology is not +built for output point vector layer. If required, the topology can be built +using v.build. + +
If output is a point vector layer and sparse is not specified, +the output vector layer will contain points at the +same locations as observation points in the input layer but the values +of the output points will be interpolated values. If a sparse +point vector layer is specified, the output vector layer will contain points +at the same locations as the sparse vector layer points. The 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 @@ -96,7 +95,7 @@
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 +the GROUP BY SQL function is used. This function is not +supported by the DBF driver. For this reason, vector output (output) is not permitted with the DBF driver. There are no problems with the raster map output from the DBF driver.