Factorial Kriging

By Jeffrey Yarus, QGSI, Houston, USA

Factorial what? Richard Chambers and I presented an oral paper at the AAPG in Salt Lake City, UT in May 2003 on using variograms to identify subtle stratigraphic and facies trends. We avoided using the name, “Factorial Kriging” for fear it would frighten people away, but instead introduced the concept as a form feature extraction, a more comfortable term to most of us geologists. We were overwhelmed by the positive response we had, and thought a summary of the major idea would be valuable to our readers.

Factorial Kriging is a geostatistical approach to isolating and extracting features related to the modeled variogram. Commonly, the advantage to modeling a variogram with “nested” structures is to provide a more accurate depiction of the spatial components that make up the variogram (nugget, scale1, sill1, scale 2, sill2, scale x sill x), and to thereby provide an improved interpolation. Isatis provides an additional advantage in allowing the users to “decompose” the variogram into its component parts (factors), and visualize their separate contributions to a Kriged 3D volume or 2D map. The practical aspect to this is simply to extract and verify the appropriateness of the individual factors with regard to the conceptual model, and to potentially use each structure as a template for further analysis or model building. There are numerous examples of what can be done with the extracted features, or factor maps. For instance, if the variographic structures are related to separate facies, the factor maps can be used as templates to restrict the conditional simulation of petrophysical properties (Figure 3, a-c). Alternatively, a variographic structure could be related to an artifact, like an acquisition footprint in seismic data. In this case, the footprint can be isolated and removed very simply.

Figure 1 is the base image and can be the result from Kriging or CoKriging a variable(s), or any produced grid (2D or 3D), like from a seismic attribute. The image shows multiple features which can be related to the input variogram or from a variogram derived from the grid itself. The major features that can be observed in this example include a drift (approximately ENE-WSW), a NW - SE intermediate direction of continuity, and an ENE-WSW small scale continuity.

Figure 2 shows the base variogram model with the 3 major structures. 2a shows the traditional variogram model and 2b shows the variogram map.

Figure 3 is composed of the four factor maps. Figure 3a shows the isolated drift term achieved by removing the effect of short and intermediate scale variographic structures. 3b depicts the Kriged map without the drift term. 3c is the factor map showing only the intermediate variographic structure, and 3d is the isolated contribution of the short scale structure.

If you wish to try your hand a Factorial Kriging, select from the main menu, select Interpolate. Then select Estimation and (CO)Kriging. In the (Co) Kriging menu open Spatial Model Options (Figure 4). It is simple to use and very powerful.