# Kriging Interpolation in GIS Raster Data: Semivariogram and Universal Kriging (Especially for GATE-Geospatial 2022)

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Kriging regression is a method of interpolation for which the interpolated values are modeled by a Gaussian process governed by prior covariances. Under suitable assumptions on the priors, kriging gives the best linear unbiased prediction of the intermediate values. Interpolating methods based on other criteria such as smoothness (e. g. , smoothing spline) may not yield the most likely intermediate values. The method is widely used in the domain of spatial analysis and computer experiments. Kriging is used to interpolate the values from the control points in the GIS raster data set.

## Kriging Interpolation

Kriging is like IDW (Inverse Distance Weighted) in that it weights the surrounding measured values to derive a prediction for an unmeasured location. The general formula for both interpolators is formed as a weighted sum of the data:

Where:

• the measured value at the location
• an unknown weight for the measured value at the location
• the prediction location
• the number of measured values

In IDW, the weight , depends on the distance to the prediction location. To use the spatial arrangement in the weights, the spatial autocorrelation must be quantified.

## Semivariogram

Kriging uses the Semivariogram to measure the spatially correlated component, a component that is also called spatial dependence or spatial autocorrelation. As a measure of spatial autocorrelation, a semi-variance overlaps with Moran՚s I and G-statistic. The semivariance is computed by:

Where is the semivariance between known points, and separated by the distance h; and z is the attribute value.

## Universal Kriging

Universal Kriging assumes that the spatial variation in z values has a drift or a trend in addition to the spatial correlation between the sample points. Typically, universal Kriging incorporates a first order or a second order polynomial in the Kriging process. A first order polynomial is: .

Where M is the drift, and are the x-, y-coordinates of sampled point I, and and are the drift coefficients. A second-order polynomial is: .

## Limitations of Higher-Order Polynomials

Higher-order polynomials are usually not recommended for two reasons.

• Kriging is performed on the residuals after the trend is removed. A higher-order polynomial will leave little variation in the residuals for assessing uncertainty.
• A higher-order polynomial means a larger number of the coefficients, which must be estimated along with the weights and a larger set of simultaneous equations to be solved.

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