A significant number of mathematical problems is associated with finding information distributed unevenly in space. We are talking about information systems of a geographical orientation, since it is in them that it is possible to measure the necessary quantities at certain points. To solve these problems, one or another interpolation method is often used.
Definition
Interpolation is a way of calculating intermediate values of quantities from an available discrete set of values. The most common interpolation methods are: inverse distance weighting, trend surfaces and kriging.
Basic interpolation methods
So, let's take a closer look at the first method, its essence lies in the influence of points that are closer to the estimated ones in comparison with those located farther. When using such an interpolation method, it involves choosing from some topography in a certain neighborhood a specific point that has the greatest influence on it. This is how the maximum search radius or the number of points thatlocated close to a certain point. Next, a weight is set for the height at each specific point, calculated depending on the distance from this point. Only in this way can a greater contribution of the nearest points to the interpolated height be achieved when compared with points further away from the given one.
The second interpolation method is used when researchers have an interest in general surface trends. Similarly to the first method, points that are within a given surface can be used for the trend. Here, a best fit set is built based on mathematical equations (splines or polynomials). Basically, the least squares technique is used, which is based on equations with non-linear dependencies. The technique is based on the replacement of curves and other forms of sequences of numerical type with simple ones. In order to build a trend, each value on a given surface must be substituted into the equation. The result is a single value assigned to the interpolated solution (point). For all other points, the process continues.
Another interpolation method mentioned above, kriging, optimizes the interpolation procedure based on the statistical nature of the surface.
Using quadratic interpolation
There is another tool for determining specific points - the method of quadratic interpolation, the essence of which is to replacesome function on a certain interval by a quadratic parabola. At the same time, its extremum is calculated analytically. After its approximate finding (minimum or maximum), it is necessary to set a certain interval of values, after which the search for finding a solution should be continued. By repeating this procedure, it is possible, using an iterative procedure, to refine the value of this equation to the result with the accuracy specified in the problem statement.
