Volumetric aberrancy: a complement to coherence and curvature Xuan Qi, Ph.D. candidate 2016 AASPI Consortium Meeting 11/23/2018 2016 AASPI Consortium Meeting

2016 AASPI Consortium Meeting Outline Background Introduction Motivation Method Results Conclusion 11/23/2018 2016 AASPI Consortium Meeting

Background: curvature Anticline Syncline Dipping Plane Flat Plane k>0 k=0 k<0 R x z Anticline: k > 0 Plane: k = 0 Syncline: k < 0 11/23/2018 2016 AASPI Consortium Meeting

Introduction: aberrancy – curvature gradient Depth Dip Aberrancy measures the gradient change along curvature. For example, in a two-dimensional perfect circle, the curvature does not change at every single sample point, thus the aberrancy is zero at every single sample points. Using the perfect circle analogy, we would observe large aberrancy in terms of magnitude for an anticlinal geometry, which is a common geometry for folds or faults structure. Curvature 11/23/2018 2016 AASPI Consortium Meeting

Introduction: aberrancy – curvature gradient Depth Depth Curvature Aberrancy measures the gradient change along curvature. For example, in a two-dimensional perfect circle, the curvature does not change at every single sample point, thus the aberrancy is zero at every single sample points. Using the perfect circle analogy, we would observe large aberrancy in terms of magnitude for an anticlinal geometry, which is a common geometry for folds or faults structure. Aberrancy 11/23/2018 2016 AASPI Consortium Meeting

Motivation: Accurately map sub-seismic resolution faults Coherence Aberrancy Aberrancy measures the gradient change along curvature. For example, in a two-dimensional perfect circle, the curvature does not change at every single sample point, thus the aberrancy is zero at every single sample points. Using the perfect circle analogy, we would observe large aberrancy in terms of magnitude for an anticlinal geometry, which is a common geometry for folds or faults structure. 11/23/2018 2016 AASPI Consortium Meeting

Internal Steps Step 1: Compute Derivatives Inline dip, p crossline dip, q Rotate to prime system (p’=0, q’=0) Find extrema of third derivatives as function of azimuth, φ Rotate back to original system Azimuth and magnitude, amax Azimuth and magnitude, amin Internal Steps Compute Derivatives Step 1: Due to the nature of aberrancy, it characterizes the third-order surface behavior. Computation of aberrancy involves two main challenges, high order equations and high dimensional calculation. The original aberrancy equation proposed by Di & Gao, (2014) is absolutely complicated, it is almost impossible to derive an analytical solution from it. That is why we have to rotate the coordinate system to simplify the equation (Di and Gao, 2016). After rotation the coordinate system, the super difficult equation simplified into a third order polynomial equation. 11/23/2018 2016 AASPI Consortium Meeting

Rotate Coordinate system Inline dip, p crossline dip, q Rotate to prime system (p’=0, q’=0) Find extrema of third derivatives as function of azimuth, φ Rotate back to original system Azimuth and magnitude, amax Azimuth and magnitude, amin Internal Steps Compute Derivatives Step 1: Rotate Coordinate system Step 2: Due to the nature of aberrancy, it characterizes the third-order surface behavior. Computation of aberrancy involves two main challenges, high order equations and high dimensional calculation. The original aberrancy equation proposed by Di & Gao, (2014) is absolutely complicated, it is almost impossible to derive an analytical solution from it. That is why we have to rotate the coordinate system to simplify the equation (Di and Gao, 2016). After rotation the coordinate system, the super difficult equation simplified into a third order polynomial equation. 11/23/2018 2016 AASPI Consortium Meeting

Rotate Coordinate system Compute the extreme aberrancy Inline dip, p crossline dip, q Rotate to prime system (p’=0, q’=0) Find extrema of third derivatives as function of azimuth, φ Rotate back to original system Azimuth and magnitude, amax Azimuth and magnitude, amin Internal Steps Compute Derivatives Step 1: Rotate Coordinate system Step 2: Due to the nature of aberrancy, it characterizes the third-order surface behavior. Computation of aberrancy involves two main challenges, high order equations and high dimensional calculation. The original aberrancy equation proposed by Di & Gao, (2014) is absolutely complicated, it is almost impossible to derive an analytical solution from it. That is why we have to rotate the coordinate system to simplify the equation (Di and Gao, 2016). After rotation the coordinate system, the super difficult equation simplified into a third order polynomial equation. Compute the extreme aberrancy Step 3: 11/23/2018 2016 AASPI Consortium Meeting

Rotate back to original coordinate system Inline dip, p crossline dip, q Rotate to prime system (p’=0, q’=0) Find extrema of third derivatives as function of azimuth, φ Rotate back to original system Azimuth and magnitude, amax Azimuth and magnitude, amin Internal Steps Due to the nature of aberrancy, it characterizes the third-order surface behavior. Computation of aberrancy involves two main challenges, high order equations and high dimensional calculation. The original aberrancy equation proposed by Di & Gao, (2014) is absolutely complicated, it is almost impossible to derive an analytical solution from it. That is why we have to rotate the coordinate system to simplify the equation (Di and Gao, 2016). After rotation the coordinate system, the super difficult equation simplified into a third order polynomial equation. Rotate back to original coordinate system Step 4: 11/23/2018 2016 AASPI Consortium Meeting

Theory: Aberrancy before rotation 𝝋: azimuth [0,360) 11/23/2018 2016 AASPI Consortium Meeting

Theory: aberrancy after rotation (Di and Gao 2015) 𝜙 1 : azimuth [0,360) 11/23/2018 2016 AASPI Consortium Meeting

Fault-controlled karst – Fort Worth Basin, Texas Results Fault-controlled karst – Fort Worth Basin, Texas 11/23/2018 2016 AASPI Consortium Meeting

Case study : Fault-controlled karst – Fort Worth Basin, Texas Genetic paleocave model for the Lower Ordovician of West Texas showing cave floor, cave roof, and collapsed breccia (modified after Kerans, 1988, 1989). The Fort Worth Basin is one of several basins that formed during the late Paleozoic Ouachita Orogeny, generated by convergence of Laurasia and Gondwana (Bruner and Smosna, 2011). The Mississippian-age organic-rich Barnett Shale gas reservoir is a major resource play in the Fort Worth Basin. It extends more than 28;000 mi2 with most production coming from a limited area, in which the shale is relatively thick and isolated between effective hydraulic fracture barriers. Conformably overlying the Barnett Shale is the Marble Falls Formation. The lower Marble Falls consists of a lower member of interbedded dark limestone and gray-black shale. Underlying the Barnett Shale are the Ordovician Viola-Simpson Formations, which dominantly consist of dense limestone, and dolomitic Lower Ordovician Ellenburger Group. (Qi et al. 2014) Qi et al,. 2014 11/23/2018 2016 AASPI Consortium Meeting

Results: variance on top of Marble Falls Fault Karst Extract the coherence and project on the top Marble Falls surface. 11/23/2018 2016 AASPI Consortium Meeting

Results: Co-rendered curvature and variance on top of Marble Falls Fault Karst How curvature 11/23/2018 2016 AASPI Consortium Meeting

2016 AASPI Consortium Meeting Results: Co-render aberrancy amax and amax azimuth on top of Marble Falls Fault Karst 11/23/2018 2016 AASPI Consortium Meeting

Results: Co-render variance and amax on top of Marble Falls Fault Karst As you can see that, aberrancy provide more detailed information where coherence is not able to pick up. 11/23/2018 2016 AASPI Consortium Meeting

Results: top of Marble Falls time structure 11/23/2018 2016 AASPI Consortium Meeting

Results: top of Marble Falls time structure 11/23/2018 2016 AASPI Consortium Meeting

Results: top of Marble Falls time structure 11/23/2018 2016 AASPI Consortium Meeting

Results: vertical section of AA’ Aberrancy Compare aberrancy with coherence. 1）some of the features we can observe from aberrancy are not able or poorly to be picked up from coherence. Variance 11/23/2018 2016 AASPI Consortium Meeting

Results: vertical section of BB’ Aberrancy Compare how aberrancy and coherence characterize faults differently. The faults were highlighted as vertical thin lines (higher resolution, continuous), while in coherence cross section, faults were observed as less continuous dots. Variance 11/23/2018 2016 AASPI Consortium Meeting

Results: vertical section of CC’ Aberrancy Compare how aberrancy and coherence characterize karst features differently. Similar to faults. Variance 11/23/2018 2016 AASPI Consortium Meeting

2016 AASPI Consortium Meeting Conclusions 11/23/2018 2016 AASPI Consortium Meeting

AASPI volumetric attributes curvature 3d 11/23/2018 2016 AASPI Consortium Meeting

2016 AASPI Consortium Meeting THANK YOU QUESTIONS? 11/23/2018 2016 AASPI Consortium Meeting

2016 AASPI Consortium Meeting Rotational matrix Rotated along z axis clock wise with phi as the rotational andl 11/23/2018 2016 AASPI Consortium Meeting

2016 AASPI Consortium Meeting Theory: aberrancy 11/23/2018 2016 AASPI Consortium Meeting

Short wavelength vs long wavelength 11/23/2018 2016 AASPI Consortium Meeting

Figuring out the meaning of the cubic roots… Three roots scenario One root scenario If the discriminator is less than 0, we will expect to have three distinct roots (a), if the discriminator is larger than 0, there will be 2 imaginary roots and one real root (b). 11/23/2018 2016 AASPI Consortium Meeting

Variance As you can see that, aberrancy provide more detailed information where coherence is not able to pick up.