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Structural -> Most-positive and most-negative curvatures

Attribute Description:

Most-positive (kpos) curvature is the maximum-signed curvature at any given point on a surface defined by a quadratic equation.  Most-negative (kneg) curvature is the minimum-signed curvature at any given point on a surface defined by a quadratic equation.  These attributes define the largest and smallest curvatures defining how curved a surface is and do not depend on vector dip.

Interpretation Use:

These attributes reveal faults, flexures, anticlines, and synclines.  By definition kneg<kpos.  Therefore, if kpos and kneg are less than zero, a bowl shape is defined.  If both are greater than zero there is a dome, and if both are equal to zero, there is a plane.   

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 along the zero value is marked by white or being transparent in the middle.

Figure 1: Colorbar example for curvatures

Figure 1: Colorbar example for curvatures

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Example:

Figure 2: Most-positive (upper) and most-negative (lower) curvatures

Figure 2: Most-positive (upper) and most-negative (lower) curvatures

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Computation:

In order to capture the curvature information, the mapped plane is first defined by fitting a quadratic surface using least-square based calculation.

Most Positive - 05.png

kpos and kneg are derived by searching all possible normal curvatures, in other words, only consider the most-positive and most-negative curvature values without considering the axis rotation caused by geological events. This step can be done by setting the coefficients d and e to zero.

The definitions of most-positive and most negative curvatures are defined as

and

Most Positive - 07.png

The term most-positive and most negative curvatures (kpos and kneg) are often confused with most-positive and most-negative principal curvatures (k1 and k2). The main difference is that kpos and kneg show all the lineament contained within the surface without maintaining the shape information. For a symmetrical structure, including a flat surface, kpos and kneg will be the same like k1 and k2. However, if your surface contains folding or axis rotation, k1 and k2 will offer the curvature anomalies corresponding to the folding axis.

Figure 4: Illustration of various curvature anomalies on a folded surface (after Mai, 2010).

Figure 4: Illustration of various curvature anomalies on a folded surface (after Mai, 2010).

Figure 4 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 in value, which may not be directly linked to the axis of defamation.

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