# Structural -> Maximum and Minimum Curvature

**Attribute Description:**

Maximum (*kmax*) and minimum (*kmin*) curvatures define the curvature anomalies in magnitude yet, still, captures the folding axis that contains geological information. This is done by fitting two circles tangent to a surface. The circle with the minimum radius is the maximum curvature (*kmax*) and the circle which is perpendicular to the first circle is the minimum curvature (*kmin*).

**Interpretation Use:**

These attributes reveal faults, flexures, anticlines, and synclines. The maximum curvature has the larger absolute value and the minimum the smaller absolute value. For this reason, the minimum curvature can define the elevation change along the long axis of a valley or ridge which is identified by maximum curvature. These attributes are also very good at defining upthrown and downthrown portions of fault systems.

**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.

**Example:**

**Computation:**

At any point on the mapped surface, two radiuses are derived (Figure 4) by fitting two circles to the perpendicular planes. The reciprocal of the radius of each circle is called apparent curvature. Within the infinite plane combinations, the circles with the two planes that have the minimum and maximum radiuses, then produce the maximum curvature (*kmax*) and minimum curvature (*kmin*).

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

Gaussian (*kGauss*) and mean (*kmean*) curvatures are then derived from the quadratic surface. After that, *KGauss *and *kmean* can further formulate the most-positive and most-negative principal curvatures (*k1* and *k2*).

For more details on how *kGauss* and *kmean*, *k1* and *k2* were derived, please refers to principal curvatures computation section.

After *k1* and *k2* are derived, the maximum and minimum curvatures (*kmax* and *kmin*) are defined by the largest and smallest principal curvatures in magnitude, regardless of the curving direction. The derivation of *kmax* and *kmin* is conceptually equivalent to the step described above that looks for the most representative circles on the two planes.

and

**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