# Curvature

Curvature is one of several geometric seismic attributes which measures how bent a curve is at a particular point on a two-dimensional or three-dimensional surface. Understanding and improving the imaging of the shape of curves in seismic data allows improved understanding and interpretation of features in subsurface data. This attribute is designed to emphasize discontinuous events, like faults and fractures in structural interpretation. In map view - or time slices and horizon slices- it can also be used to aid in the stratigraphic interpretation of features such as channel edges, reefs, compaction features, and debris flows.

## What is curvature?

Geometrically, curvature (k) is defined as the radius of a circle that is tangent to a curve. Mathematically it can be represented as k= 1/r, where k is the curvature, and r is the radius of the circle that is tangent to a curve. The smaller the radius of curvature is, the more bent the curve is. And, if the radius of curvature is infinite, then the curvature (k) would be very small having essentially zero curvature and approximating a straight line.

Curvature can be calculated in both 2D and 3D, where in 3D it attempts to fit circles, and not just a curve to the surface. You can imagine that a 3D surface can have be sliced in an infinite numbers of ways, all of which can have a calculated curvature. It has been found that for subsurface imagine, the most useful curvatures are those which are defined by planes which are orthogonal to the surface. These are called normal curvatures.

Two of the principal curvatures are the *maximum* and *minimum curvature*. A circle with a minimal radius where the curvature is very small and tight would be defined as the maximum curvature (kmax), while a second circle, perpendicular to the first, would have a greater radius than the maximum curvature, and would be defined as the minimum curvature (kmin) for that quadratic, curved surface.

The *most positive* and *most negative curvatures* are obtained by searching through all normal curvatures for the most positive and most negative values.

Two other types of curvature that are commonly used are the *mean curvature* and the *Gaussian curvature*. The use of these two types of curvature to define geometric features is discussed in detail in Roberts, 2001 ^{[2]}.

*Dip curvature* is a measure of the rate of change of the dip in the maximum dip direction. This curvature method tends to exaggerate any local relief and can help image compacted features which as channelized sand bodies and debris flows.

*Strike curvature* is calculated by extracting the curvature in a direction which is at right angles to the dip curvature. It can often be used to identify valley and ridges.

## What does curvature reveal about the subsurface?

Curvature is an extremely useful seismic attribute, as it improves the imaging of both large and small scale geometric features in the subsurface. For instance, anticlinal features would have a positive curvature, while synclinal features would have a negative curvature. Flat features, or those with a constant dipping place would exhibit a curvature close to zero. Once these shape of features is determined, geologic features can be extracted from the data.

## Subsurface features identified with the curvature attribute

Many subsurface features can be imaged in 3D seismic with curvature attributes. As with all attributes, the ability to identify features in the subsurface is dependent on the quality of the seismic data. Some of the more common features for which various curvature attributes are commonly used are Fault edges, grabens, fractures, channel edges, and compaction features.

Similarly to faults, curvature will image the curving slope of the horizon at the fault edges that define the graben. An example of this is shown in the figure.

Excellent examples subsurface features and a variety of various curvature attributes are displayed and discussed in the curvature summary paper by Andy Roberts (2001)^{[2]}.

## References

- ↑
^{1.0}^{1.1}^{1.2}Chopra, Satinder, and Kurt J. Marfurt. Seismic Attributes for Prospect Identification and Reservoir Characterization. Society of Exploration Geophysicists, 2007 - ↑
^{2.0}^{2.1}Andy Roberts. Curvature attributes and their application to 3D interpreted horizons. First Break, Vol. 19, Issue 2, Pgs 85-100, 2001

## See also

- Seismic attributes
- Semblance, coherence, and other discontinuity attributes
- Stratigraphic interpretation
- Structural interpretation
- Time slices

## External references

Curvature Attributes and their Application to 3D Interpreted Horizons