# 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.

**Example:**

**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

Where he further calculates the most-positive principal curvature

And most-negative principal curvature as:

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

and

*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

and shape index, s

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

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

and

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 *kpos*, *k2* = *kneg*. For the gently dipping environment, results of *k1/k2* are similar to *kpos/kneg.*

**Reference**

- Roberts, A., 2001, Curvature attributes and their application to 3D interpreted horizons: First Break, 19, 85-99.
- Mai, H. T., 2010, Seismic attribute analysis and its application to mapping folds and fractures: Ph.D. Dissertation, The University of Oklahoma.
- AASPI online documents. http://mcee.ou.edu/aaspi/documentation/Volumetric_Attributes-curvature3d.pdf