Variogram Analysis Simplified: Part-2: Variogram Parameters & How they affect the result

Understanding Variogram is tricky and simplifying the fundamentals of variograms can indeed be challenging. By breaking down the concepts into separate articles, I am trying to make materials more systematically accessible and easier to understand for readers. Each article will be focussed on a specific aspect of variograms, providing clear explanations, examples, and visual aids to help convey the key points effectively.

Previous articles related to Variogram as below;

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Variogram Analysis Simplified Part-1. Variogram calculation and fitting a variogram model https://www.dhirubhai.net/pulse/variogram-analysis-simplified-part-1-mahapatra-mba-pmp--nsp7f

This current part is focussing on how the variogram parameters affect the result, understanding this we can use the variogram to get our desired output. Below is a standard variogram plot with spherical model fitting shows the common parameters like Nugget, Range & Sill.

Find attached the excel sheet for clear snapshots and details as mentioned below.

Variogram Analysis Simplified: Part-2: Variogram Parameters & How they affect the result

Key Variogram Parameters:

1. Nugget: The Nugget is the value of the semi-variogram at a lag distance of zero. Think it as, if we acquired data at same place/point there should be always same value and should not be any difference/variance but geological samples always show some variance or nugget. It represents the variance that cannot be explained by the spatial model and is often associated with measurement errors or micro-scale variability. Helps distinguish between true spatial variability and noise or error in the data.

2. Sill: The Sill is the value where the variogram generally flattens off and reaches a constant value. The sill is also the value where the variogram stabilizes, indicating that the spatial correlation between data points no longer increases with distance. The maximum sill value is typically equal to the variance of the dataset. In geostatistical terms, the sill is expected to approach the variance of the data when the lag distance (the distance between data pairs) is large enough that the pairs are no longer spatially correlated. There is no absolute numerical limit to the variance or sill value—it depends on the scale and spread of your data. For example, if you have a dataset with values ranging widely, the variance (and thus the sill) will be higher.

3. Range: The Range is the distance at which the variogram reaches the Sill. It represents the effective distance over which spatial correlation or dependence is significant. Defines the distance beyond which data points are no longer spatially correlated.

There are two other parameters during variogram analysis which also affects the results,

4. Variogram Model Type : The model which we use to fit the variogram also affect our results. Choice of variogram model directly impacts the accuracy and reliability of spatial analysis, predictions. Depending on data variability and extent, variogram model type i.e. Spherical, Exponential or Gaussian provides different results.

5. Azimuth/Anisotropy in the data : Anisotropy refers to the directional dependence of spatial properties, meaning that the spatial correlation of a variable varies depending on the direction in which it is measured. In the context of variograms, anisotropy can significantly affect the results and interpretation.

Now let's consider each parameter and its effect with help of examples. For this example I have considered a random data, Each Case used same input data (as shown below), keeping same resolution and same area for analysis.

Using the data above, the Kriging interpolation method was applied to create a map under different variogram input scenarios. The table below discusses various cases that directly impact the results, with each color representing the effect of a specific parameter on the variation.

Range Effect:

Please note that as the range value increases, the continuity in the output results improves. However, beyond a certain range, the impact becomes negligible, as the maximum area of influence has already been reached. This is why Cases 3 and 4 yield the same results.

Nugget Effect:

With a higher nugget value, continuity in the output results decreases. Therefore, exercise caution when working with high-nugget data, as it indicates significant variation over short distances. In some cases, it might even be advisable to disregard such data, as it suggests a high level of small-scale variability.

Anistropy/Azimuth Effect:

If you closely examine the results, you'll notice that Cases 3, 8, and 9 all represent isotropic conditions, where the range values for the major and minor directions are the same. This means there is no variation with changes in azimuth or direction, so altering the azimuth value does not affect the results. In contrast, Cases 10, 11, 12, and 13 illustrate different scenarios of anisotropy, where changes in direction or azimuth alter the distribution angle of the results. A closer look reveals that the change in the major and minor range directly affects the width of continuity, as seen when comparing Case 11 with Case 13.

Model Type Effect:

The Spherical and Exponential models (Case 3 and Case 14) yield similar outputs with only minor visible differences. However, the Gaussian model (Case 15) produces a more noticeable change. Cases 16 and 17 clearly demonstrate the impact of directional anisotropy and the effect of using different model types in the analysis.

Sill Effect:

Selecting an appropriate sill value is crucial for accurate variogram modeling. An overestimated sill can lead to overly smoothed predictions, while an underestimated sill can result in exaggerated variability in the spatial model. If the range of data values is large, the variance and hence the sill can be large. Example: If you have porosity values ranging from 5% to 45%, the variance will be higher than if the values range from 20% to 25%. In geological or environmental datasets, typical variance (and hence sill) values could range from small fractions to several hundred units, depending on the scale of measurement (e.g., porosity percentages, permeability in millidarcies, etc.). For example, a sill value might be 0.01 for a tightly clustered porosity dataset, or it could be 100 or more for a highly variable dataset.

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Please feel free to cite this article and visualizations as:

Mahapatra, Mahabir Prasad (2024), Variogram Analysis Simplified: Part-2: Variogram Parameters & How they affect the result https://www.dhirubhai.net/pulse/variogram-analysis-simplified-part-2-parameters-how-mahabir-prasad-5w4mf/

More Literatures to read:

• The Variogram Basics: A visual introduction to one of the most useful geostatistical concepts | CSEG RECORDER

References:

• The Importance of the Model Choice for Experimental Semivariogram Modeling and Its Consequence in Evaluation Process

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