Structural Curvature (Fundamentals)
Curvature is a measurement of how sharply bent a curve is at a particular point. In geological terms, curvature is how fast dip changes. Sharp edges have high curvature, and lower curvature implies a smoother, softer edge.
Curvature is dependent on scale. A highresolution, picked horizon will have lots of curvature variation. An interpolated horizon will have very little variation in curvature.
Insight can calculate curvature for horizon surfaces (see Using Horizon Properties), or for a volume using a process (see Curvature Process).
Background
Curvature is mathematically defined as the inverse of the radius of curvature (K = 1/r).
On a two dimensional curve, the radius of curvature takes a single value. On a surface the radius of curvature depends on the direction and consequently, so does curvature.
On a surface, the curvature through a point can be calculated in any direction.
 For a domelike shape, the curvature will be positive, regardless of the direction.
 For a valley, the curvature will be very different across the valley as compared to along the valley floor.
 For complicated saddle structures the curvature is positive in one direction and negative in another.
When analysing curvature, we are most often interested in the maximum curvature  the largest magnitude curvature for a point, regardless of direction. This is not necessarily the most positive value  a sharp valley will have negative curvature for the maximum attribute. For flat regions where dip is not changing, maximum curvature will be closer to zero.
At each point, the minimum curvature is the curvature measured in the direction perpendicular to the maximum. In the case of a valley, the maximum curvature will be measured across the valley, while the minimum curvature will be measured along the valley floor.
The maximum curvature highlights the biggest changes in dip. This is often a good fault indicator. When looking at horizon curvature at a fault boundary, there will be a strong positive curvature on the upthrown side, and a strong negative curvature on the downthrown side. Displaying the most positive (or most negative) curvature will show only one side of the fault, which can provide a clearer view for fault discrimination. The gap between the maximum positive and maximum negative curvature is an indication of the fault heave.
Maximum and minimum curvature together can describe the shape of a feature. Dome features have positive maximum and minimum curvature, and bowl features are described by negative values in all directions.
The shape index uses the following rules to classify features by curvature shape.

+1 : Dome
 Maximum curvature: positive
 Minimum curvature: positive

+0.5 : Ridge
 Maximum curvature: positive
 Minimum curvature: near zero

0: Flat plane or saddle:
 Maximum curvature: near zero
 Minimum curvature: near zero, OR
 Max and min curvature have different signs

0.5: Valley or channel
 Maximum curvature: negative
 Minimum curvature: near zero

1 : Bowl:
 Maximum curvature: negative
 Minimum curvature: negative
Curvature shape does not take into account the amount of curvature. Use in combination with maximum curvature to identify the scale of the feature.
Curvedness or alternatively the "degree of flatness" can be calculated from the maximum and minimum curvature (the RMS average of the max and min curvature). It tends to be similar to the maximum curvature, with ridges and channels having relatively lower values than domes, bowls and saddles.
REFERENCES:
_{AlDosarry, Marfurt, 2006  3D volumetric multispectral estimates of reflector curvature and rotation. Geophysics, 2006, 71, 4151. }
_{Roberts, A., 2001  Curvature attributes and their application to 3D interpreted. Horizons: First Break, 2001, 19, 85–99.}
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