A significant number of mathematical problems associated with finding distributed unevenly in the space of information. We are talking about information systems of a geographic orientation, since it is in them that it is possible to measure the necessary values 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: the method of inverse weighted distances, the trend surface 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 in comparison with those located further. When using such an interpolation method, it is necessary to select from a certain topography in a certain neighborhood a specific point that has the greatest influence on it. This selects the maximum search radius or the number of points that are located close to a specific point. Next, the 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 closest points to the interpolated height be achieved when compared with points remote at a greater distance from a given one.
The second interpolation method is used when researchers have an interest in general surface trends. Similarly to the first method for a trend, points that are within a given surface can be used. Many best approximations are built here based on mathematical equations (splines or polynomials). Basically, the least squares technique is used, based on equations with nonlinear dependencies. The methodology is based on replacing curves and other forms of
numerical type
sequences with simple ones. In order to build a trend, each value on this surface should be substituted into the equation. The result is the only value assigned to the interpolated solution (point). For all other points, the process continues.
Another method of interpolation mentioned above, kriging, involves the optimization of the interpolation procedure, taking as a basis 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 replace some function on a certain interval with a quadratic parabola. Moreover, its extremum is calculated analytically. After its approximate location (minimum or maximum), it is necessary to set a certain range of values, after which the search for a solution to continue. Performing this procedure again, it is possible, using the iterative procedure, to refine the value of this equation to the result with the accuracy specified in the statement of the problem.