October 4, 2022

# What is Kriging? And how do I use it in Global Mapper Pro?

For many years, Global Mapper has offered an Elevation Grid Creation tool that allows elevation, or other values, to be gridded into a continuous surface. While the Elevation grid creation tool is powerful, it is basic, using either a TIN or binning grid creation method. In version 24 of Global Mapper Pro, a Variography and Kriging tool was introduced as an advanced way to create a continuous data surface from a set of lidar or vector points. While Elevation Grid Creation and Kriging both produce a surface and rely on the premise that close things are more related than far things, the difference between the tools lies in the advanced geostatistical analysis of kriging. Kriging generates not only a continuous data field from discrete points, but also calculates an uncertainty value describing the confidence in the predicted value at any given location.

Selecting the Variography and Kriging tool from the Analysis Menu in Global Mapper Pro opens two windows to start, an Empirical Variogram and the Variography and Kriging options.

## What is a Variogram?

Simply put, a variogram plots the variability (Y-axis of the below graph) of sampled values in relation to distance (X-axis of the below graph). The variance is the average of the squared deviation for all values in the set. As a formula this looks like 2γ(h) = var( Z(s+h) – Z(h) )​, where Z(s) is the value at position s and Z(s+h) the value a distance h from s. When this concept is expanded and applied to a full dataset, a plot emerges summarizing the difference in point values as it relates to distance for the dataset. The sill value is shown plotted on the empirical variogram. Moving the cursor to a data point on the plot will show the Variance(h) and h values at that location.

The empirical variogram, created from the input data, groups data into a user-specified number of bins. The average from each bin is taken from all pairs of points that are a distance h + \delta h apart, where delta h is the bin size. Each value for a bin is plotted as a point on the empirical variogram, and the error bars extending from each point show the standard error estimates calculated as two times the standard deviation over the square root of the number of samples. Looking at the variogram, the trend in the variance of the data can be explored. Additionally, the sill and range for the plotted variogram are calculated.

The sill value for a variogram is the variance at which data begins to be uncorrelated. When shown on the variogram, the sill occurs where the plot starts to flatten as it reaches the maximum variance within the dataset. The range value for the variogram is the distance at which the sill, maximum variance, occurs. Basically, keeping in mind that close things are more closely related than far things, the sill and range represent the upper limits of the meaningful variogram that will be evaluated.

### What about a Variogram Map?

Another tool to explore the variance throughout the dataset is the variogram map, which describes the directionality of the variance throughout the dataset. Shown as a shaded circular map, distance range bins are drawn as concentric circles around the origin. Direction is then represented by angle bins comprising the circular shape of the map. Remember that distance and direction are not measured from a physical location but are interpreted from the vector between sample points in the Variance(h) calculation. The shading applied to the variogram map describes the average variance for each of these distance and direction bins. The variogram map opens as a floating pane window and can be used to explore the directionality of variance throughout a dataset.

Typically, a variogram map will show lower variance toward the center, over shorter distances. However, this can be gleaned from the plotted empirical variogram. What the variogram map considers that the plotted variogram does not is the direction of the distance between samples, so it will show any directionality of variance for the data. This variation in data correlation in relation to direction is shown as a range ellipse and described as an anisotropy ratio.

## How are Kriged Estimates determined?

After plotting the empirical variogram, a theoretical variogram data model is fitted and used to generate the kriged prediction layer. Global Mapper Pro’s Variography and Kriging tool provides a few options for theoretical models and allows them to be plotted on the empirical variogram. The parameter values used in the different models are calculated by the kriging tool when plotting each model on the variogram. Additional values and statistical measures for each model and model fit are generated and displayed in the Variography and Kriging settings window. Theoretical models and sill values are plotted on the empirical variogram. The residual values for the two plotted theoretical models are shown as well.

The appropriate theoretical data model is then selected by assessing the visual fit to the plotted empirical variogram alongside an evaluation of generated model values. All models display the residual values as a histogram of delta gamma, the difference between the theoretical variogram model and empirical variogram. Additionally, a reduced chi-squared score is a good indicator of the overall fit of the model to the data to be used in the selection of the theoretical model for kriging.

### Generating Kriged Estimates in Global Mapper Pro

The selected theoretical variogram model is then used to create Krige Estimate Layers in Global Mapper Pro. Taking into account the distribution of sample points and calculated autocorrelation, weights derived from the variogram model are applied to the known point samples to generate interpolated values at non-sampled locations. Alongside kriged estimates, uncertainty values are calculated from the weights applied and used for interpolation in the kriging process.

The spacing or resolution for the kriged estimate layers is set by the user. Other parameters specific to kriging may also be set to guide the process, although Global Mapper Pro does provide default values to complete the analysis. Gridded kriged estimate layers, one containing estimate values and the other the uncertainty values, are generated with the use of the checkbox in the Variography and Kriging dialog to Publish Results of Elevation Grid. Without this option enabled, only a kriged estimate layer as an array of points will be generated, with each point holding a kriged estimate value and an uncertainty value. As another option, 3D mesh can be created as outputs from the kriging process. Finally, with the chosen data model and output options selected, kriged estimates are generated with a single click.  While the gridded and point layers generated from the kriging process appear as elevation layers in Global Mapper Pro, the values estimated by the process are created from the selected input attribute in the Variography and Kriging dialog which may not be elevation. The new layers generated from Variography and Kriging in Global Mapper Pro can then be exported and further analyzed through gridded layer shaders, raster reclassification, and any other tools in the program geared toward gridded data layer processing.