Moving surface

Algorithm

The Moving surface operation is a point interpolation which requires a point map as input and returns a raster map as output. For each output pixel, a polynomial surface is calculated by a least square method approaching weighted point values.

First, for each output pixel, the distances of all points towards the output pixel are calculated to determine weight factors for the points:

For each output pixel, weight factors for the points are then calculated according to the weight function specified by the user. Two weight functions are available: inverse distance and linear decrease.

Inverse distance:	weight = (1 / dⁿ ) - 1
Linear decrease:	weight = 1 - dⁿ

where:

d	=	D/D₀ = relative distance of point to output pixel
D	=	Euclidean distance of point to output pixel. When the spherical distance option is used, distances (D) are calculated over the sphere using the projection of the coordinate system that is used by the georeference of the output raster map.
D₀	=	limiting distance
n	=	weight exponent

Figures 1 and 2 below show the manner in which weight values decrease with increasing distance, for different values of n. The X-axes represent d: the distance of a point towards an output pixel divided by the limiting distance. The Y-axes represent the calculated weight values.


	Fig. 1: Inverse distance weight = (1/dⁿ ) - 1 X-axis: d = D/D0 Y-axis: weight values	Fig. 2: Linear decrease weight = 1 - dⁿ X-axis: d = D/D0 Y-axis: weight values

The weight functions ensure that points close to an output pixel obtain a larger weight value than points which are farther away from an output pixel.

See that when the distance of a point towards an output pixel equals the limiting distance (value 1.0 at X-axis), or when the distance of a point towards an output pixel is larger than the limiting distance, the calculated weight value will equal 0; the weight functions are thus continuous.

Tips:

The inverse distance function can be selected when you have very accurately measured point values and when local variation, within a pixel, is small. This function ensures that the computed output values equal the input point values.
The linear decrease function can be selected for point maps in which you know there are measurement errors, and when points which are close to each other have different values. This function will decrease the overall error by correcting erroneous measurements with other close points. The consequence is that the computed output values will not necessarily coincide with the measured point values.

Then, for each output pixel, an output value is calculated by fitting a polynomial surface through all weighted point values of points that fall within the limiting distance towards this pixel. To each output pixel, the calculated surface value is assigned.

Below the functions and surface types are listed, as well as the absolute minimum number of points that are mathematically required to fit such a surface. To obtain good results, for each output pixel, you need more points within the limiting distance than this absolute mathematical minimum.

In general, the use of simple surfaces is preferred, as these will produce the least artificial extreme values.

Plane:	the surface is a plane; formula: z = a + bx + cy Minimum number of points required: 3
2nd degree Linear:	the surface is planar but tilted, i.e. first order plane; formula: z = a + bx + cy + dxy Minimum number of points required: 4
2nd degree Parabolic:	the surface is a second order polynomial surface; formula: z = a + bx + cy + ex² + fy² Minimum number of points required: 5
2nd degree:	the surface is a full second order polynomial surface; formula: z = a + bx + cy + dxy + ex² + fy² Minimum number of points required: 6
3rd degree:	the surface is a third order polynomial surface; formula: z = a + ... + gx³ + hx²y + ixy² + jy³ Minimum number of points required: 10
4th degree:	the surface is a fourth order polynomial surface; formula: z = a + ... + kx⁴ + lx³y + mx²y² + nxy³ + oy⁴ Minimum number of points required: 15
5th degree:	the surface is a fifth order polynomial surface; formula: z = a + ... +px⁵ + qx⁴y +rx³y² + ... + uy⁵ Minimum number of points required: 21
6th degree:	the surface is a sixth order polynomial surface z = a + ... + vx⁶+... Minimum number of points required: 28

Tip:

For time efficiency reasons, it is strongly advised to choose a rather large pixel size for the output raster map. Further interpolation on the raster values can be performed with the Densify operation or the Resample operation (using bicubic interpolation).