Horizontally-Biased Kriging Solid Modeling

This "Kriging - HB" solid modeling method uses the kriging method of estimation to create a model of your data. Kriging differs from some of the other modeling methods by bringing out directional influences in your data. Kriging is based on the following assumptions:

• The value for an unknown point can be estimated from neighboring points, but that the unknown point is not necessarily completely dependent upon the values of the known points.
• Variability in the G-values of a data set is a function of two factors: distance and direction. In general, points close together tend to show less variability than points far apart, and in many cases, points along certain bearings will show less variability than equidistant points along a different bearing.
  
  ! Unlike true 3D Kriging, this method does not take inclination into account. Thus, this Horizontally Biased Kriging algorithm is targeted at applications in which the geology is roughly sub-horizontal. For example, contaminants migrating though buried paleochannels would be a good application for this new algorithm. Conversely, modeling a complex, hydrothermally altered and fractured precious metal deposit would be better modeled with a higher-end product that provides for variography and Kriging in all directions and inclinations as well as performing separate geostatistics within polyhedral regions (e.g. fault blocks). 

This relationship of variability versus distance can be displayed graphically using a "variogram," which plots the variability of the G values for point pairs as a function of the 3D distance between the points.  Variograms generated for point pairs in different directions show different trends of distance versus variance.  RockWorks creates observed variograms of your data, and then finds the variogram model that offers the best fit - thus defining the distance and directional relationships in your data - and uses that equation to interpolate the solid model. 

Kriging is one of the most complex modeling methods. To keep your life simple, RockWorks can perform this analysis in an automatic way, finding the optimal point samplings and the best variogram model to use. Or, if you prefer, you can establish the variables manually and generate detailed variograms and reports; these are discussed below.   

Advantages: Kriging is a good solid modeling method for honoring directionality with data. It can prevent the bull's-eye pattern of Inverse-Distance-based algorithms.  
Disadvantages:  It can be slow. This horizontally-biased method does not take inclination into account.

We encourage you to explore the web for kriging reference information (equations, variograms, etc.).

Menu Options
Step-by-Step Summary

Menu Options

• Variogram Options:  The first decision to make when using Kriging is whether you want the program to do most of the heavy-lifting ("Automatic") or whether you want full control ("Manual").    Many users find it helpful to run the kriging automatically, with a variety of reports, and then refine the modeling manually.
  • Automatic: Click in this radio button if you want the program to set the kriging variables automatically for you (probably the best place to start). When set to Automatic, the program will determine the variogram type to use, based on the type with the highest correlation with your data.  It will search the data using a variety of spoke directions and variogram types.
  • Manual: Click this button if you prefer to set the kriging variables manually. 

    
     

    • Spoke Spacing: Since variograms represent point-to-point variability, the program creates a set of point pairs to work with.  Rather than analyzing all possible pairings between all data points (a huge number), the program will sample point pairs along specific directional bearings.  These bearing lines can be conceptualized as bi-directional "spokes" running through the data points; you define the spacing in degrees between sampling spokes.  For example, a 45 degree spoke spacing will generate 4 bi-directional spokes (180 / 45 = 4), and a 30-degree spoke spacing will generate 6 spokes. 
    • Spoke Tolerance: Since it is unlikely that many data points will lie along the narrow directional lines established above, you need to establish a "tolerance" or zone alongside each bearing line to include in the data point search. The default is 22.5 degrees.
    • Distance Increments: In addition to the program searching for data points in a directional manner, it also searches in an orderly, step-wise manner, along each bearing line at certain distance increments. These are expressed in map units. The default is 250.
    • Distance Tolerance: It is unlikely that data points lying within the directional search zone will fall right on one of the distance increments, so it is necessary to establish a tolerance zone around each increment to widen the search area. This is also expressed in map units. The default is 125.
    • Maximum Distance: This variable establishes the maximum distance from each data point that the data search will be conducted, in map units.
    • Variogram:  Use this drop-down menu to select the variogram model to apply to the modeling process. 

      

      How to know?  Here are some suggestions.

      • Click in the Edit / Examine Variogram check-box (below) to bring up an interactive window where you can actually fit different variograms to your data.
      • Activate the 2D Variogram Matrix option (discussed below) to view a series of variograms and look for the best fit.
      • Keep it simple:  You can start with a Linear no Nugget model. This indicates that there is a linear relationship between distance and point variability, and that there is not a great sampling error. 
      • Try them out:  Try several variogram models and compare the resulting maps.  Which looks best to you?  For example, if you create a triangulation-based, and inverse distance-based, and linear kriging-based contour map of the same data, you can get an idea of the study area (and the variability within it!). You can create maps using other kriging-based variogram models and compare them. 
         
  • Pre-Model Points for Variogram: This setting tells RockWorks to first create a regular block model of control points (using Inverse-Distance squared) which will then be used for generating the variogram. Keeping this turned on can create better variograms for small data sets. Turning this off can help create a better match between Automatic and Manual settings.
    
    
  • Edit / Examine Variogram: Check this box to see the interactive Variogram Editor prior to modeling.
     
• Reporting Options:  The second decision to make regarding kriging is what kind of reporting options you wish to use for generating reports and graphs of your data. 
  • Textual Report:  Check this box to generate a textual report that list the various kriging parameters that were used to create the solid model.
  • 2D Variogram Matrix:  Check this box to generate a detailed diagram that depicts all of the variograms, and a large number of other statistics.  (More.)
    • Items: Use these check-boxes to select which variogram models you wish to include in the matrix diagram.
    • Variograms per Row: Defines the maximum number of variograms to be plotted per row, for each activated variogram type.  Default = 10. 
      (Typically you won't have that many.  For a 90 degree spoke spacing, you'd have two variograms per model.  For a 45 degree spoke spacing, you'd have 4.)
       
• Kriging Options: Unlike 2D (grid) Kriging, the solid model kriging is based on the closest eight points within each octant relative to a block model node. This means that a maximum of 64 points will be used for each interpolation. There are no settings here for you to adjust.

   

Step-by-Step Summary

The steps that you and the program will follow to create the solid model will depend on the settings you've established, above, but here's a general scenario of what happens when you click the Process button in the solid modeling window:

• If you've selected Automatic kriging, the program will compute the average minimum and maximum 3D spacing of your control points, to suggest default sampling distance increments and total distance.  It will search for point pairs at 90 degree spoke increments using these lag bins, and then at successively smaller spoke increments, pitching bins without a minimum number of samples.  It will  compute the observed variograms for all spoke samples, and will determine the variogram model that has the best correlation. 
• If you've selected Manual kriging, the program will search for point pairs along the spoke and distance increments you've specified, out to the maximum distance, and will compute the observed variograms for these bearings.  It will fit the selected variogram model to the data. 
• If you have requested to Edit / Examine Variogram, the best-fit variogram model will be displayed, along with a reference range plot.  Adjust this as you wish, and click OK.  The program will create the solid model using the selected variogram model and settings. 
• If you've requested the text report or 2D variogram matix, they will be displayed along with the completed model diagram.
  
  

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