Dr. Tom Smith presenting on Machine Learning at the 3D Seismic Symposium on March 6th in Denver
What is the "holy grail" of Machine Learning in seismic interpretation? by Dr. Tom Smith, GSH Luncheon 2018
Using Attributes to Interpret the Environment of Deposition - A Video Course. Taught by Kurt Marfurt, Rocky Roden, and ChingWen Chen
Dr. Kurt Marfurt and Dr. Tom Smith featured in the July edition of AOGR on Machine Learning and Multi-Attribute Analysis
Rocky Roden and Ching Wen Chen in May edition of First Break - Interpretation of DHI Characteristics using Machine Learning

Structural -> Principal Component Curvature k1 and k2

Attribute Description:

k1 and k2 measure the most positive and most negative curvature on the direction of first and second principal component. 

Interpretation Use:

Principal curvatures identify the structure corresponding to the axis of the fold, rather than local folding that associates with horizontal maximum and minimum. k1 and k2 along with other curvature volumes are often seen as regional stress and strain indicators.

Recommended Color Palette:

Colorbar for curvatures often includes two color themes such as blue and red or black and red to mark the anomalies. The less anomalous value around zero is marked by white or light color.

  Figure 2: Colorbar example for curvatures

Figure 2: Colorbar example for curvatures

Example:

  Figure 1: k1(left) and k2 (right)

Figure 1: k1(left) and k2 (right)

Computation:

(This section contains descriptions for k1, k2, positive, negative, maximum, minimum, mean and Gaussian curvatures; shape index and curvedness.)

Roberts (2001) describes the curvature definition by first fitting a quadratic surface for any point on a mapped surface, using the surrounding grid values in the least squares sense.

The mapped surface has been defined by inline and crossline dip component from prior reflector dip analysis.

Based on the fitted quadratic, the author defines the mean curvature

and the Gaussian curvature

PCC - 07.png

Where he further calculates the most-positive principal curvature

PCC - 08.png

And most-negative principal curvature as:

PCC - 09.png

The maximum and minimum curvatures here are defined by comparing the absolute value of k1 and k2, i.e.

PCC - 10.png

and

PCC - 11.png

kmax and kmin are designed to highlight the largest eigenvalue in magnitude, regardless of the shape of the structure where the anomalies happen.
 
An alternative way to reveal and define structural deformation is to view curvedness

PCC - 12.png

and shape index, s

PCC - 13.png

where the value of shape index ranges from -1 to 1. Figure 4 illustrates the relation of shape index and the associated quadratic shapes.

  Figure 4: Structure shapes and their relationship with shape index and Curvedness. [ AASPI online document ]

Figure 4: Structure shapes and their relationship with shape index and Curvedness. [AASPI online document]

Roberts (2001) also defines the most-positive and most-negative curvatures

PCC - 15.png

and

  Figure 5: Illustration of various curvature anomalies on a folded surface

Figure 5: Illustration of various curvature anomalies on a folded surface

    Figure 5 shows the lateral displacement associated with different curvature anomalies (Mai, 2010). Most-positive and most-negative principal curvature anomalies correspond to the fold axes, whereas most-positive and most-negative curvatures are associated with the local maximum and minimum, which may not be directly linked to the axis of defamation.

    When the axis of deformation is symmetric, i.e. k1 will equal to kposk2 = kneg. For the gently dipping environment, results of k1/k2 are similar to kpos/kneg.

    Reference