Logarithmic interpolation

Logarithmic interpolation is intended for building the smoothed surface, provided that the height of the surface at the location of the reference point shall be equal to the height of this point.

Logarithmic interpolation is based on modelling a local surface in a vicinity of a defined point. The local surface is calculated by the adjacent reference points taking into account their relative positioning.

Calculating the value of an element of a matrix at logarithmic interpolation is carried out by the following algorithm:

1. By sectors or irrespective of a direction the nearest points round a defined element are selected. Search of points is carried out until the number of points specified in parameters of the task will be found.

2. For all pairs of points the system of equations is made in which unknown parameters are weight of points. The coefficients of the equations are equal to product of the logarithm from distance between points onto the distance (logarithmic function). Remainders of system are values of the investigated characteristic (height) of a point.

3. Weights of points are calculated from the solution of the equations system by the Gauss method with a choice of the major element.

4. The interpolated value is obtained as the sum of products of logarithmic function of distance between defined and a reference point onto the calculated weight of a point.

Items 1-3 are concerned to the construction of a local surface, i.e. calculating the scales of points depending on their mutual positioning. In item 4 the interpolation is performed properly.

Advantages of the method:

- builds the smooth surface coinciding by height with reference points.

Disadvantages of the method:

- it is processed more slowly than interpolation by triangles;

- there is no possibility to customize modelling function, as in Kriging.