Interpolation is the use of known samples to predict or estimate the values of unknown samples. Interpolation is to deduce the data of unknown points in the same area from the data of known points. Extrapolation is the extrapolation of data from known regions to other regions. Both the interpolation method and the extrapolation method are the commonly used interpolation ideas in the interpolation process. SuperMap provides three interpolation methods for simulating or creating a surface: Inverse Distance Weighting (IDW), Kriging, and Radial Basis Function (RBF). The method you choose for interpolation usually depends on the distribution of the sample data and the type of surface you want to create. No matter which interpolation method is chosen, the more data are known and the wider the distribution is, the closer the interpolation result will be to the actual situation.
Inverse Distance Weighting interpolation
Inverse Distance Weighting interpolation (Inverse Distance Weighted, IDW) is based on the principle that the sampling points are close and similar. It is assumed that the closer the distance between two samples is, the more similar their properties are, and vice versa, the farther the distance is, the less similar they are. It estimates the value of the cell by calculating the Weighted Mean value of the sample points in the nearby area, and the closer to the center of the sample point, the greater the weight value. This is a simple and effective data interpolation method, and the operation speed is relatively fast.
In addition to the weight distance, the power and the search radius are also important factors for IDW interpolation.
- Power degree: The power degree is related to the calculation of the weight distance, and the power index has a great influence on the interpolation results of IDW. The lower the power value, the smoother the interpolation result; the higher the power value, the more detailed the interpolation result. The default power is 2.
- Search radius: There are two types of search radius for Inverse Distance Weighting interpolation:
- Fixed Count: The specified number of sample points nearest to the Distance Raster unit participate in the interpolation calculation. For each grid cell, the number of sampling points participating in the interpolation operation is fixed, while the radius used for searching is variable, and the searching radius depends on the density of the sampling points around the grid cell. If the sampling point exceeds the maximum search range, the sampling point will not participate in the interpolation operation;
- Fixed Radius: All sampling points within the specified radius participate in the interpolation operation of grid cells. If the number of sampling points participating in the interpolation operation within the specified radius is less than specified minimum number, the search radius will be expanded to include more sampling points to ensure that the number of sampling points participating in the calculation reaches the specified minimum number.
The following figure shows the surface interpolation effect using the Inverse Distance Weighting interpolation method, where the interpolation field is elevation, the resolution is 100, the search radius type is Fixed Count, the number of points is all points, and the power is 1, 2, and 3.
Spline interpolation
Spline function is a mathematical method to imitate manual spline and draw a smooth curve through a series of sample points. Spline interpolation is a relatively accurate interpolation technique, which assumes that the change is smooth. It has two characteristics: (1) the surface must pass through all the sample points accurately; (2) the surface must have the minimum curvature. Spline interpolation has advantages in creating visually demanding curves and contours.
The spline interpolation method is suitable for the interpolation calculation of a large number of sample points and the requirement of obtaining a smooth surface. In the case of smooth surface changes, a more ideal result will be obtained. However, this method is not applicable when the surface value changes sharply within a short Horizontal Distance, or when the obtained sample data is not accurate enough.
There are three types of search radii for spline interpolation:
- Fixed Count: The specified number of sample points nearest to the Distance Raster unit participate in the interpolation calculation. For each grid cell, the number of sampling points participating in the interpolation operation is fixed, while the radius used for searching is variable, and the searching radius depends on the density of the sampling points around the grid cell. If the sampling point exceeds the maximum search range, this part of the sampling point will not participate in the interpolation operation.
- Fixed Radius: All sampling points within the specified radius participate in the interpolation operation of grid cells. If the number of sampling points participating in the interpolation operation within the specified radius is less than specified minimum number, the search radius will be expanded to include more sampling points to ensure that the number of sampling points participating in the calculation reaches the specified minimum number.
- Block search: Dataset is divided into blocks according to the set maximum number of points in each block, and interpolation operation is performed using the points in the block.
Krügin interpolation
Based on the Spatial Autocorrelation of data, the Klugkin interpolation method uses the variogram model to make unbiased estimation of the unknown sample points in a limited area. When there is Spatial Autocorrelation or directional trend in the sample points, Klugin is the most suitable interpolation method. The sample data in the same distribution area are interdependent, that is, Spatial Autocorrelation. And the closer the distance between the two samples, the stronger the correlation. At present, the Klugin interpolation method is widely used in soil science and geology.
- Semivariogram model
- The spherical function type shows that when the Spatial Autocorrelation relationship is gradually reduced (that is, the semivariogram value is gradually increased), until beyond a certain distance, the Spatial Autocorrelation relationship is 0. Spherical functions are more commonly used.
- Exponential is used when the Spatial Autocorrelation decreases exponentially with distance. Exponential model is widely used.
- The Gaussian type is suitable for the case where the value of the semivariance function asymptotically approaches the value of the sill.
- Parameter Description
- Autocorrelation threshold: The semivariogram value will tend to a constant value after reaching a certain distance (X axis). This distance is the range of the autocorrelation threshold. Within the range of the autocorrelation threshold, the data of sample points are correlated, while outside the autocorrelation threshold, the data of sample points are not correlated, that is, the sample points outside the autocorrelation threshold do not affect the estimation results.
- Sill value: The vertex value reached by the semivariogram (Y axis). The sill value minus the nugget effect is the partial sill value. As shown in the following figure:
- Nugget effect: At H = 0, the value of the semivariogram intersecting the Y axis, known as the "nugget effect" in geostatistics, is characterized by large spatial variability over a very short distance. It can be caused by measurement errors, or it can come from the microscopic variability of the mineralization phenomenon.
SuperMap supports spherical, Exponential and Gaussian semivariograms.
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| Figure: Schematic diagram of spherical function model |
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| Figure: Exponential model diagram |
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| Figure: Illustration of Gaussian model |
The semivariogram model characterizes the Spatial Autocorrelation of the sampling points. The semivariogram model is described by autocorrelation threshold, sill value and nugget effect.
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| Figure: Illustration of autocorrelation threshold, sill value and nugget effect |
SuperMap provides three Kriging interpolation methods, namely Ordinary Kriging, Simple Kriging, and Universal Kriging.
- Ordinary Kriging : is a linear estimate of the regionalized variable, assuming that the observed data are normally distributed and that the expected value of the regionalized variable is unknown.
- Simple Kriging : is a linear estimate of the regionalized variable, assuming that the observed data is normally distributed and that the expected value of the regionalized variable is a fixed constant.
- Universal Kriging: When there is a trend in the observed data, and the trend can be fitted by a definite function or polynomial, the Universal Kriging interpolation method can be used.
Different interpolation methods have their own applicable conditions. Therefore, when interpolating the data, it is necessary to select the appropriate interpolation method according to the characteristics of the data to ensure the best interpolation effect. The table below compares the five interpolation methods in the following four aspects:
| Method | Extrapolation capability | Degree of approximation | Computing power | Scope of application |
| Inverse Distance Weighting | When the distribution is uniform, it is better | Poor | Hurry up | Uniformly distributed |
| Spline | High | Strong | Hurry up | When the distribution is dense |
| Ordinary Kriging | High | Strong | Slower | All right |
| Simple Kriging | High | Strong | Slower | All right |
| Universal Kriging | High | Strong | Slower | All right |
Related topics
Inverse Distance Weighting interpolation
Or dinary Kriging interpolation
Universal Kriging interpolation