"Turn" kriging interpolation method

Einyboy the original link kriging interpolation method

Kriging is a high-level geostatistical process that generates an estimated surface from a set of scattered points with Z-values. Unlike other interpolation methods in the interpolation toolset, it is useful to use the Kriging tool to interact with the spatial behavior of the phenomena represented by the z-values before choosing the best estimate method for generating the output surface.

What is kriging?

IDW (inverse distance weighting) and spline interpolation tools are called deterministic interpolation methods, because these methods are based directly on the surrounding measurements or on the specified mathematical formula that determines the smoothness of the resulting surface. The second type of interpolation method consists of a geostatistical method (such as kriging), which is based on a statistical model that contains autocorrelation (that is, statistical relationships between measurement points). Therefore, the Geostatistical method not only has the function of producing the prediction surface, but also can provide some measure to the certainty or accuracy of the prediction.

Kriging assumes that the distance or direction between sample points can reflect spatial correlations that can be used to illustrate surface changes. The Kriging tool fits a mathematical function with a specified number of points or all points within a specified radius to determine the output value for each position. Kriging is a multi-step process that includes exploratory statistical analysis of data, modeling of variation functions, and creation of surfaces, including the study of variance surfaces. When you understand that space-related distances or directional deviations exist in your data, kriging is considered the most appropriate method. This method is commonly used in soil science and geology.

Kriging formula

This is similar to inverse distance weighting because kriging can weigh the surrounding measurements to produce predictions for the position that is not measured. The common formulas for these two types of interpolator consist of a weighted sum of the data:

• which

  Z (si) = measured value at the first position

  λi = unknown weight of the measured value at the first i position

  s0 = Forecast location

  N = number of measured values

In inverse distance weighting method, the weight λi only depends on the distance of the predicted position. However, when you use the Kriging method, the weights depend not only on the distance between the measurement points, the forecast position, but also on the overall spatial arrangement based on the measurement points. To use spatial permutations in weights, you must quantify spatial autocorrelation. Therefore, in ordinary kriging, the weight λi depends on the fitting model of the spatial relationship between the measurement point, the distance of the predicted position and the measured value around the predicted position. The following sections discuss how to create predictive surface maps and predictive accuracy maps using common kriging formulas.

Create a predictive surface map using kriging

To make predictions using the kriging interpolation method, there are two tasks that are required:

• Find the dependency rule.
• Make predictions.

To accomplish these two tasks, Kriging takes a two-step process:

1. Create variance functions and covariance functions to estimate the statistical correlations (called spatial autocorrelation) values that depend on the autocorrelation model (fitted model).
2. Predict unknown values (make predictions).

Because the two tasks are different, you can determine that Kriging uses two of data: the first is the spatial autocorrelation of the estimated data, and the second is the prediction.

Mutation Analysis

A fitting model or spatial modeling is also known as structural analysis or mutation analysis. In the spatial modeling of the measurement point structure, a graph of the empirical semivariogram is started, and for all position pairs separated by distance H, the following equations are calculated:

Semivariogram (distance

H

) = 0.5 * average{(value

I

–value

J

}

2

]

The formula involves calculating the squared difference between the paired positions.

Shows the pairing of a point (red dot) with all other measurement locations. This procedure is performed for each measurement point.

Calculates the squared difference between paired positions

In general, the distance from each position pair is unique and there are many point pairs. Quickly drawing all the pairings becomes difficult to handle. Instead of drawing each pair, the pairing is grouped into individual step bar cells. For example, the average half-variance of all point pairs with distances greater than 40 meters but less than 50 meters are calculated. The empirical semivariogram is a graph that represents the average Semivariogram value on the y-axis and the distance or step on the x-axis (see).

Empirical Semivariogram diagram example

Spatial autocorrelation quantification is based on the following geographic principles: Things that are closer to each other are more similar to things that are farther away. Therefore, the closer the position pair is (on the leftmost side of the x-axis of the Semivariogram cloud), the more similar the values should be (lower on the y-axis of the Semivariogram cloud). The farther away the position pair is (moving to the right on the x-axis of the Semivariogram cloud), the more different the squared of the difference is (moving upward on the y-axis of the Semivariogram cloud).

Fitting model based on empirical Semivariogram

The next step is to fit the model based on the point of the empirical semivariogram. Semivariogram modeling is a key step between spatial description and spatial prediction. The main application of kriging is to predict the value of the property at the non-sampled location. The empirical Semivariogram provides information about the spatial autocorrelation of a DataSet. However, information about all possible directions and distances is not provided. Therefore, to ensure that the kriging variance of kriging predictions is positive, it is necessary to fit the model (i.e., a continuous function or a curve) based on the empirical semivariogram. This operation is similar in theory to regression analysis, where continuous lines or curves are fitted according to data points.

To fit the model based on the empirical Semivariogram, select the function that is used as the model (for example, a spherical type that rises at the beginning and renders a horizontal state after a range becomes larger than a certain extent) (see the spherical model example below). The points on the empirical semivariogram have some deviations from the model; some points above the model curve, some points below the model curve. However, if you add a corresponding distance, each point will be above the line, or if you add another corresponding distance, each point will be below the line, and the two distance values should be similar. There are several variants of the Semivariogram model to choose from.

Semi-variant function model

The Kriging tool provides the following functions from which you can select a function to model the empirical Semivariogram:

• Round
• Spherical
• Index
• Gaussian
• Linear

The selected model affects predictions for unknown values, especially when the shape of the curve approaching the origin is significantly different. The steeper the curve near the origin, the greater the impact of the nearest adjacent element on the prediction. This will make the output surface less smooth. Each model is used to more accurately fit different kinds of phenomena.

Shows two common models and determines the difference between functions:

Spherical Model Example

The model shows the gradual reduction of spatial autocorrelation (equivalent to the increase of half variance) to the zero-dependent autocorrelation after a certain distance. The spherical model is one of the most commonly used models.

Spherical Model Example

Example of an exponential model

The model is applied when the spatial autocorrelation decreases exponentially with the increase of distance. Here, self-correlation disappears completely at infinity. Exponential models are also common models. To choose which model to use based on the spatial autocorrelation of the data and the prior knowledge of the data phenomenon.

Example of an exponential model

For more information on mathematical models, see below.

Understanding Semivariogram-Variations, abutment, and Nugget

As mentioned earlier, the Semivariogram shows the spatial autocorrelation of the measured sample points. Because of the basic principles of geography (the closer things are, the more similar), usually, the difference of the square of the close measurement points is less than the square of the difference between the measured points far away. The positions are adjusted to be drawn, and then the model is fitted according to these positions. These models are typically described using range, abutment, and Nugget.

Variable range and abutment

When you look at the model of a semivariogram, you'll notice that the model presents a horizontal state at a certain distance. The distance the model first renders a horizontal state is called a range. The spatial autocorrelation of the sample location separated from the distance of the range is not related to the spatial autocorrelation of the sample location far from the range.

Illustrations of range, abutment and Nugget

The value (the value on the y-axis) obtained by the Semivariogram model at the variable range is called the abutment. The offset abutment is equal to the base station minus the block gold. The nugget is described in the following sections.

Block Gold

Theoretically, at 0 spacing (for example, step = 0), the Semivariogram value is 0. However, at infinitely small distances, the semivariogram usually shows the nugget effect, that is, the value is greater than 0. If the Semivariogram model has a intercept of 2 on the y-axis, the block gold is 2.

The nugget effect can be attributed to a measurement error or to a spatial variation source (or both) that is less than the sampling interval distance. Due to the inherent error in the measuring device, the measurement error is present. The natural phenomenon can change with the scale range and produce the spatial change. Micro-scale changes that are smaller than the sample distance will appear as part of the nugget effect. Before collecting data, it is important to understand the proportion of spatial changes that you are concerned about.

Make predictions

Find correlations or autocorrelation in your data (see the Variance analysis section above) and complete the first Data application (that is, use spatial information in the data to calculate distances and perform spatial autocorrelation modeling), you can use the fitted model for predictions. Thereafter, the empirical Semivariogram is set aside.

You can now use this data to make predictions. Similar to inverse distance weighted interpolation, Kriging generates weights from surrounding measurements to predict non-measured locations. The same as the inverse distance weighting method, the most recent measurement value is most affected by the distance from the measured position. However, the kriging weights of the surrounding measurement points are more complex than the inverse distance weighting method weights. The inverse distance weighting method uses a simple distance-based algorithm, but the kriging weights are derived from the Semivariogram developed by viewing the spatial characteristics of the data. To create a continuous surface of a phenomenon, predictions are made for each location or cell center in the study area, which is based on the spatial arrangement of the semivariogram and the nearby measured values.

Kriging methods

There are two methods of kriging: Ordinary kriging and Universal kriging.

Ordinary kriging is the most common and widely used kriging method, and is a default method. The method assumes a constant and unknown average. If it is not possible to refute the scientific basis, this is a reasonable assumption.

Universal Kriging assumes that there are coverage trends in the data, such as prevailing winds that can be modeled by deterministic functions (polynomial). The polynomial is deducted from the original measurement point and the autocorrelation is modeled by random errors. After the model is fitted with random errors, the polynomial is added back to the forecast to produce meaningful results before the prediction is made. You should use this method only if you understand that there is a trend in the data and that you can provide scientific judgments describing the pan-kriging method.

Semivariogram graphs

Kriging is a complex process that requires more knowledge about spatial statistics than is described in this topic. Before you use kriging, you should fully understand its fundamentals and evaluate the suitability of the data that is modeled using that technology. If you do not fully understand the process, it is highly recommended that you review some of the bibliography listed at the end of this topic.

Kriging is based on the regional variable theory, which assumes that the spatial changes in the phenomena represented by the z-values are consistent across the entire surface in terms of statistical significance (for example, the same pattern of change can be observed at all locations of the surface). The spatial consistency hypothesis is very important for the theory of regional variables.

Mathematical models

The following are common shapes and equations used to describe the mathematical model of semi-variance.

Illustration of a spherical half-variance model

Illustration of a circular half-variance model

Illustration of exponential half-variance model

Illustration of Gaussian half-variance model

Illustration of linear half-variance model

Reference books

Burrough, P. A. Principles of Geographical information Systems for land Resources assessment.new York:oxford University Pr Ess. 1986.

Heine, G. W. "A controlled Study of Some two-dimensional interpolation Methods." COGS Computer Contributions 3 (No. 2): 60–72. 1986.

McBratney, A. B., and R. Webster. "Choosing Functions for Semi-variograms of Soil Properties and Fitting them to sampling estimates." Journal of Soil Science 37:617–639. 1986.

Oliver, M. A. "Kriging:a Method of interpolation for geographical information Systems." International Journal of Geographic Information Systems 4:313–332. 1990.

Press, W. H., S. A. Teukolsky, W. T. vetterling, and B. P. Flannery. Numerical Recipes in c:the Art of scientific computing.new York:cambridge University Press. 1988.

Royle, A. G., F. Clausen, and P. Frederiksen. "Practical Universal Kriging and Automatic contouring." Geoprocessing 1:377–394. 1981

"Turn" kriging interpolation method