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Get to know eigenvectors

Terms like orientation tensor (Scheidegger 1965), Bingham distribution (Bingham 1964), eigenvector and eigenvalue, symmetric matrix, and symmetry axis are probably long forgotten by you. They might fill you with fear, or curiosity. But what makes them important for a geologist, and how are they linked to locating more oil?

To see how, let’s start with some synthetic data. An image log can help determine paleo-transport directions in a sedimentary system.

Planar features are manually interpreted and visualized on the image log as sine curves (above left). Often a 3D visualization of the planes on the image are used as guidance but this typically delivers more confusion than explanation. The identified planes are often represented as tadpoles on a depth-based plot, where the tadpoles correspond to the sine curves on the image log.

Stereographic projections of the planes are the primary tool to catch any possible symmetry of the (many) planar features, now abstracted as poles on a two dimensional plane (above middle). The open circle is the pole of theaxis of the great circle which fits the poles of the planar features, and also the eigenvector corresponding to the the third eigenvalue of the orientation tensor based on their poles.

Poles are normal vectors to planes defined by polar dip and azimuth coordinates. The challenge now is to apply spherical statistics to differentiate between a cluster or girdle distribution (Woodcock 1977), and to find a corresponding measure of statistical significance. A cluster here is a set of poles with similar dip and azimuth, while poles on a girdle can be estimated by a great circle (very much like the flight path between Paris and New York) which has a plunge and a direction.

Linear algebra to the rescue

From linear algebra we know that a 3-by-3 symmetric matrix will have three eigenvectors with three corresponding eigenvalues. The eigenvalues, being real numbers, can be sorted in descending order, allowing us to refer to eigenvectors 1 to 3. For the full story, find ‘Symmetric matrix’ on Wikipedia.

It turns out that this is very helpful to us. The best mean pole of a population of poles is defined by the eigenvector belonging to the biggest eigenvalue of the 3-by-3 orientation tensor based on the given population. Even better, the third eigenvector (the one with the smallest eigenvalue with respect to the orientation tensor) defines the pole of the axis of the best great-circle fit through the given population of poles. This is useful!

Step by step

For simplicity, I have ignored structural dip removal for my synthetic example. The method would be:

  1. Identify features on the image log, in this case an FMI image.
  2. Look at the stereographic projection (forget 3D visualization — it will makes your eyes sick).
  3. Perform spherical statistics by calculating the eigenvalues and eigenvectors of the orientation matrix based on the set of poles selected.
  4. Incorporate other sources from the area, for example suggestions from seismic or other data.
  5. Voilà! If we assumed a fluvial system or a channelized turbidite system, we have found a symmetry axis defined by the third eigenvector suggesting north–south trending channels (as shown in the figure). 

References

Bingham, C (1964). Distributions on the sphere and on the projective plane, PhD dissertation, Yale University.

Scheidegger, A (1965). On the statistics of the orientation of bedding planes, grain axes, and similar sedimentological data. US Geological Survey Prof. Paper 525C. 

Woodcock, N (1977). Specification of fabric shapes using an eigenvalue method, Geological Society of America
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