# Aberrancy

Aberrancy is a geometric seismic attribute which measures the lateral change or gradient of curvature along a surface. This attribute is complementary to curvature and coherence as it is designed to map lateral changes in surfaces, which include faults, folds, and flexures in structural interpretation. Aberrancy delineates structural discontinuity features whose throw falls below the seismic resolution or is distributed over a zone of smaller conjugate faults[1]. This makes aberrancy a useful attribute in highlighting fractures and flexures associated with faults.

## What is aberrancy?

Figure 1. Seismic attributes highlight a variety of features of a fault system. The yellow line indicates the types of faults that coherency would delineate and where on the fault it would appear. The red and blue line indicate the types of faults and the location of the curvature attribute on these faults. Positive curvature is represented in red by k1 and negative curvature in blue by k2. The image on the bottom right displays a fault with little to no offset that falls below the seismic resolution. Aberrrancy can delineate these types of faults and is located at the main fault location where the curvature changes most rapidly[2].

Aberrancy, sometimes called flexure, measures the deformation of a surface. It is described as a vector by its magnitude and azimuth. The magnitude describes the intensity of the surface deformation and the azimuth defines the direction in which curvature decreases in signed value. Therefore, the azimuth is indicated by the direction in which curvature changes from positive to negative, consistent with that of fault plane azimuths[1].

Complementary to coherency and curvature, the aberrancy attribute will follow a similar path to coherence along the lineation of a fault which falls between the most-positive curvature anomaly, defining the footwall, and the most-negative curvature anomaly, defining the hanging wall. The coherency attribute indicates a lateral change in waveform or amplitude, while aberrancy measures the lateral changes in curvature providing a measure for the strength of the faulting as well as the direction of the downthrown side[1]. Curvature is defined as the rate of change of the direction of a curve for a particular point[3] so the third-order aberrancy attribute is generated by taking the rate of change, or derivative, of this curvature attribute. Overall, the aberrancy anomaly aligns with the fault providing a quantitative measure of magnitude and direction that can be mapped and/or correlated with image logs, production logs, and other measures of fractures[1].

Figures 1[2] and 2[1] schematically show how and where the aberrancy attribute takes effect on a fault. Figure 1[2] shows how different seismic attributes highlight various features of the fault system depending on the type and offset of the fault. The aberrancy attribute is located at the main fault location where the curvature changes most rapidly, and is present on faults with reflector offset below the seismic resolution.

Figure 2[1] illustrates two fault models with the model on the left showing a fault with finite offset revealed by a coherence anomaly.The image on the right is an unresolved fault with offset distributed over a zone of conjugate faults allowing no continuous reflector offset to give rise to a coherence anomaly. In this image, aberrancy is expressed by the lateral change of curvature pointing down and to the right revealing a shaded red aberrancy anomaly indicating that the azimuth of the aberrancy is to the east.

Figure 2. The image on the left indicates a fault with finite offset indicated by a coherence anomaly. The image on the right displays an unresolved fault image showing positive curvature defined as a red circle on the footwall and negative curvature defined as a blue circle on the hanging wall. Aberrancy is shaded in red indicating the lateral change in curvature across this fault[1].

## Theory

Figure 3. Workflow for aberrancy computation[1].

The calculation of aberrancy is complex due to the nature of this attribute as it expresses third-order surface behavior. The challenges of this computation include the robustness of high order derivatives and computational efficiency. Qi and Marfurt, 2017[1], created a workflow for calculating volumetric aberrancy based on Di and Gao's, 2014, horizon-based aberrancy calculation that computed the 3rd derivatives at equal azimuths and then searched these values for extrema[4]. They later showed how rotating the coordinate system can simplify the equations in Di and Gao, 2016[5].

The volumetric aberrancy computation described by Qi and Marfurt includes first computing the 2nd derivative of the seismic data vector dip in the x, y, and z directions[1]. Then, rotating the calculations about the local vector dip in order to simplify the computation based on Di and Gao’s, 2016[5], approach. Aberrancy is finally computed and the vectors are rotated back to the original coordinate system. The results are three roots, or extrema, to the 3rd order differential equation: maximum aberrancy, minimum aberrancy, and intermediate aberrancy vectors expressed by its magnitude and azimuth. The total vector aberrancy is the sum of the three aberrancy vectors and provides a single vector volume appropriate for structural interpretation[1].

Figure 3 displays the workflow for the volumetric aberrancy calculation[1]. The first steps include computing the inline and crossline dip derived from the seismic amplitude data. This will include the first and second derivatives of the inline and crossline dip. The next step is to rotate the original coordinate system x0 – y0 – z0 to the new coordinate system x1 – y1 – z1 from the calculations for the dip magnitude and dip azimuth. After the coordinate rotation, the extrema of third derivatives are found which compute the azimuth and magnitude of aberrancy. The final step includes rotating the new coordinate system back to the original coordinate system producing a final value for the magnitude and azimuth of aberrancy[1].

## Examples

Aberrancy can be useful for interpretation when extracted both along horizon/time slices and in cross-sections. Figure 4[1] shows horizon slices through the maximum, intermediate, minimum, and total aberrancy vectors. The total aberrancy vector image on the bottom right is the vector sum of the maximum, minimum, and total aberrancy vector images. The total aberrancy vector volume is appropriate for structural interpretation. The total aberrancy image in Figure 4 highlights linear features representing faults or flexures, while collapse features are indicated by flexures that cycle the color wheel[1]. The total aberrancy vector is most commonly used in structural interpretation, but the minimum and intermediate aberrancy vectors could indicate zone of conflicting flexure that may potentially show areas of more intense natural fracturing.[1]

Figure 5[1] is a vertical cross-section comparing how the discontinuity features are mapped by the aberrancy anomaly as compared to coherence. Aberrancy is indicated by the cyclical rainbow color bar in the top image and coherency is portrayed in yellow in the bottom image. In comparison, aberrancy highlights the faults as thin, continuous, vertical lines color-coded by the azimuthal direction of the downdip of the fault, while coherence represents faulting by a less continuous, single-colored, blocky pattern[1]

Additional examples of subsurface features examined through aberrancy and a more advanced discussion behind the theory and value of this attribute are displayed and discussed in Aberrancy Paper by Xuan Qi and Kurt J. Marfurt, 2017[1].

Figure 4. Aberrancy vectors extracted along horizon slices[1].
Figure 5. Cross-section comparison of aberrancy and coherency attributes of a faulted region[1].

## References

1. http://mcee.ou.edu/aaspi/submitted/2017/XuanQi-Aberrancy.pdf Qi, Xuan and Marfurt, Kurt J.. Volumetric Aberrancy to Map Subtle Faults and Flexures. ConocoPhillips School of Geology and Geophysics, University of Oklahoma, 2017.
2. Attribute Expression of Tectonic Deformation. Kurt J. Marfurt. The University of Oklahoma. Lecture, 2017
3. https://petraftp.ihsenergy.com/curvature.pdf Roberts, A., 2001, Curvature attributes and their application to 3D interpreted horizons: First Break, 19, 85–100.
4. https://www.sciencedirect.com/science/article/pii/S0098300414001058?via%3Dihub Di, Haibin and Gao, Dengliang. A new algorithm for evaluating 3D curvature and curvature gradient for improved fracture detection. Computer & Geosciences, 2014.
5. https://pubs.geoscienceworld.org/geophysics/article/81/2/IM13/294611/improved-estimates-of-seismic-curvature-and Di, Haibin and Gao, Dengliang. Improved estimates of seismic curvature and flexure based on 3D surface rotation in the presence of structure dip. Geophysics, 2016.