Variogram Range
(Weight Envelope)
Shown in 2D rather than 3D for simplicity
It turns out that one of the main differences between many of the different Kriging algorithms – Ordinary Kriging, Simple Kriging, Kriging with an External Drift, Co-Kriging, etc., is how “extrapolation” occurs – i.e., how grid values are computed in the cases of little or no data.
For example, if Simple Kriging were being used in our example, then when no data is available with which to use the Krig weight function, the node value will be set equal to the average value of all the points (Global Mean).
If the chosen algorithm had been Ordinary Kriging, then the node value would be some local average of a subset of the total data. For each type of Kriging we discuss, you will learn it’s alternate method of computation in these
“extrapolated” (little or no data) areas.
Here is a graphic example of a data set where some areas contain dense data and some contain sparse data. [q 3]
Data in this area are further away than the variogram range from the grid node being computed.
What happens here depends on which Kriging algorithm you use
Data in this area are as close as the variogram range to the grid node being computed.
Here, data is weighted according to the variogram.
SPARSE DATA
DENSE DATA
Review Questions
1. When there is no data in the Weight Envelope, Kriging changes its behavior and makes use of an __________ algorithm for computing the node value. (alternate)
2. If there is no data in the weight envelope, but data in within the search range for a node, then the kriging weight function is applied to that data instead.(T/F). (F)
3. “Extrapolation” is a term used for the method of computing a grid node value when _____________________. (there is little or no data).
4. One of the main differences between the many Kriging algorithms is the way in which they handle _____________ (extrapolation, or sparse data)
Subject: Part 3- Modeling
Section: Gridding Methods Chapter: Anisotropy
Page Title: What is Anisotropy?
First we’ll define anisotropy as a characteristic of data sets. It measures the direction and the degree to which attribute values (porosity, rock type, whatever, …) differ. For example, if saturation values do not vary at all when looking towards the East, but vary significantly over short distances when looking towards the North, then this is a clear indication of anisotropy which gives information about how the property is distributed in a particular facies. This information is useful in the gridding step, and allows the system to give preferential weighting to data in a certain direction. Variograms are used to let the system know that a data set is anisotropic. When you create a variogram for your data, you establish the direction and degree of anisotropy using the variogram parameters.
With most geostatistical variogramming tools, anisotropy will be defined only
horizontally, by defining two horizontal variograms - one for the major direction and one for the minor direction. The ranges of each of these variograms will differ to define the degree of anisotropy. [q 1] , [q 4]
The major and minor directions of anisotropy are specified by an azimuth (in degrees), with the minor axis perpendicular to the major. A horizontal variogram for each direction is required.
The degree of anisotropy is defined by the ratio of variogram ranges or sills between the major and minor directions. Sometimes, the degree of anisotropy is also referred to as the eccentricity of the ellipse which is formed according to the major and minor ranges for the two directions (axes). When there is no anisotropy, then variograms for the two directions are the identical. We will show more about defining anisotropy with
variograms in the next section. [q 2], [q 3]
An example of anisotropy
If you were to measure particulate size in the channel complex shown below, its variability across the channels will be much higher than along the channels.
In geostatistics, we define anisotropy as a characteristic of a set of data values. If there is a clear difference in how data values change in one direction versus how they change in another direction, then the data set is said to be anisotropic.
If you suspect this kind of directional bias in your property data source, you should determine its direction and degree by using one of the two methods available:
1. Finding the direction by trial and error with the horizontal variogram tool 2. Use a variogram map to display the direction and degree
Afterwards, use the variogram you create in the modeling operation for the property.
When you do have anisotropy in your model, then your variogram will have two horizontal variograms – a variogram for the Major direction and one for the Minor direction. The Major variogram will have a larger Range than the Minor one, and thus the weight function in the x,y plane will be elliptical.
The Geostatistical 3D ellipsoidal weight function
The three variogram RANGES – Major Horizontal, Minor Horizontal, and Vertical are combined into a single weight function for Kriging operations. When there is no
anisotropy, then the two horizontal ranges are the same and data in all directions has the same weight.