CWP-813

3D seismic image processing for unconformities

Xinming Wu & Dave HaleCenter for Wave Phenomena, Colorado School of Mines, Golden, CO 80401, USA

b)a)

Figure 1. Unconformity likelihood (a) computed from a seismic image highlights unconformities, and can be used as constraintsto more accurately estimate seismic normal vectors and better flatten (b) the seismic image.

ABSTRACTWe propose a 3D seismic unconformity attribute to detect complete unconfor-

mities, highlighting both their termination areas and correlative conformities.

These detected unconformities are further used as constraints to more accu-

rately estimate seismic normal vectors at unconformities. Then, using seismic

normal vectors and detected unconformities as constraints, we can better flatten

seismic images containing unconformities.

Key words: unconformity seismic normal vectors flattening

1 INTRODUCTION

An unconformity is a non-depositional or erosional sur-face separating older strata below from younger strataabove, and thus represents a significant gap in geologictime (Vail et al., 1977). In seismic images, an unconfor-mity can be first identified by seismic reflector termi-nations (i.e., truncation, toplap, onlap or downlap) andthen be traced to its corresponding correlative confor-mity.

Unconformity detection is a significant aspect ofseismic stratigraphic interpretation, because unconfor-mities represent discontinuities in otherwise continuousdepositions and hence serve as boundaries when inter-preting seismic sequences that represent successively de-posited layers.

Moreover, unconformities pose challenges for otherautomatic techniques used in seismic interpretation.First, it is di�cult to accurately estimate normal vec-tors or slopes of seismic reflectors at an unconformitywith multiply-oriented structures due to seismic reflec-

tor terminations. Second, automatic seismic flatteningmethods cannot correctly flatten reflectors at unconfor-mities with geologic time gaps. To address these chal-lenges, we first detect unconformities and then use themas constraints for seismic normal vector estimation andimage flattening.

1.1 Unconformity detection

Seismic coherence (Bahorich and Farmer, 1995), whichhighlights reflector discontinuities, is used as a seismicattribute to detect faults and termination areas of un-conformities. However, this coherence attribute is bettersuited for detecting faults than unconformities, becausereflector discontinuities across a fault are usually moreapparent than discontinuities across an unconformity.Hoek et al. (2010) propose a better unconformity at-tribute that measures the degree of seismic reflector con-vergence (or divergence), and thereby highlights the ter-mination areas of an unconformity. Both of these meth-ods process a seismic image locally (Ringdal, 2012) to

264 X.Wu & D.Hale

compute unconformity attributes that can highlight anunconformity within its termination area, but cannotdetect its correlative conformity.

Ringdal (2012) proposes a global method that firstextracts a 2D flow field that is everywhere tangent toreflectors in a 2D seismic image. Then the flow fieldis used to compute an unconformity probability imageby repeating the following processing for each sample:four seeds are first placed at the four neighbors of thesample in the 2D flow field; the four seeds then movealong the flow field to produce trajectories; the sepa-ration rate of the trajectories is finally calculated andused as unconformity probability for that sample. Theadvantage of this method is that it can use long tra-jectories to detect the correlative conformity of an un-conformity. The disadvantage is that, to detect such acorrelative conformity, the trajectories are required tostart from the parallel area (correlative conformity) andend in the non-parallel area (termination). For 3D seis-mic images, this method processes inline and crosslineslices separately throughout the volume to compute anunconformity probability volume.

1.2 Seismic normal vector estimation

Orientation vector fields, such as vectors normal to orslopes of seismic reflectors, are useful for seismic inter-pretation. For example, estimated orientation informa-tion is used to control slope-based (Fomel, 2002) andstructure-oriented (Fehmers and Höcker, 2003; Hale,2009) filters so that they smooth along reflectors toenhance their coherencies. Seismic normal vectors orslopes are also used to flatten (Lomask et al., 2006;Parks, 2010) or unfold (Luo and Hale, 2013) seismicimages, or to generate horizon volumes (Wu and Hale,2014).

Structure tensors (van Vliet and Verbeek, 1995;Fehmers and Höcker, 2003) or plane-wave destructionfilter (Fomel, 2002) have been proposed to estimateseismic normal vectors or slopes. These methods canaccurately estimate orientation vectors for structureswith only one locally dominant orientation. This meansthat they can correctly estimate the normal vectors (orslopes) of the reflectors in conformable areas of a seis-mic image, but for an angular unconformity where twodi↵erent structures meet, these methods yield smoothedvectors that represent averages of orientations across theunconformity.

1.3 Seismic image flattening

Seismic image flattening (Lomask et al., 2006; Parks,2010; Wu and Hale, 2014) or unfolding (Luo and Hale,2013) methods are applied to a seismic image to obtaina flattened image in which all seismic reflectors are hor-izontal. From such a flattened seismic image, all seismichorizons can be identified by simply slicing horizontally.

Extracting horizons terminated by faults or uncon-formities is generally di�cult for these methods. Luoand Hale (2013) extract horizons across faults by firstunfaulting a seismic image; Wu and Hale (2014) dothe same by placing control points on opposite sidesof faults. However, none of these methods correctly flat-tens a seismic image with unconformities, which shouldproduce gaps in the flattened image (e.g., Figure 1b).

1.4 This paper

In this paper, we first propose a method to automati-cally detect an unconformity, complete with its termi-nation area and correlative conformity, as shown in Fig-ure 1a. We then describe how to more accurately esti-mate seismic normal vectors at unconformities by usingthe detected unconformities as constraints. Finally, wediscuss how to better flatten (Figure 1b) seismic imagescontaining unconformities by using these estimated seis-mic normal vectors and again using constraints derivedfrom detected unconformities.

2 UNCONFORMITY DETECTION

In manual 3D seismic stratigraphic interpretations, anunconformity is first recognized as a border at whichseismic reflectors terminate (i.e., truncation, toplap, on-lap or downlap), and then is traced to its correlativeconformities where reflectors are parallel. Therefore, toobtain a complete unconformity, an automatic methodshould be able to detect both the termination areas(green ellipse in Figure 2a) and correlative conformities(dashed blue ellipse in Figure 2a) within the unconfor-mity.

We propose an unconformity attribute that mea-sures di↵erences between two seismic normal vectorfields computed from two structure-tensor fields, one iscomputed using a vertically causal smoothing filter, andthe other using a vertically anti-causal smoothing filter.This attribute can highlight both the termination areasand correlative conformities of an unconformity.

2.1 Structure tensor

The structure tensor (van Vliet and Verbeek, 1995;Fehmers and Höcker, 2003) can be used to estimate seis-mic normal vectors that are perpendicular to seismicreflectors. For a 2D image, the structure tensor T foreach sample is a 2 ⇥ 2 symmetric positive-semidefinitematrix constructed as the smoothed outer product ofimage gradients:

T = hgg>ih,v

=

hg1g1ih,v hg1g2ih,vhg1g2ih,v hg2g2ih,v

�, (1)

where g = [g1 g2]> represents the image gradient vec-

tor computed for each image sample; h·ih,v

represents

Image processing for unconformities 265

correlative conformity

termination

a)

b)

Figure 2. A 2D synthetic seismic image (a) with an un-conformity (red curve) that is manually extended from itstermination area to its correlative unconformity. The esti-mated seismic normal vectors (magenta segments in (b) )are smoothed within the termination area, and therefore areincorrect, compared to the true seismic normal vectors (cyansegments in (b)) that are discontinuous within that area.

smoothing in both horizontal (subscript h) and verti-cal (subscript v) directions. These horizontal and ver-tical smoothing filters are commonly implemented withGaussian filters with corresponding half-widths �

h

and�

v

.As shown by Fehmers and Höcker (2003), the seis-

mic normal vector for each image sample can be es-timated from the eigen-decomposition of the structuretensor T:

T = �u

uu> + �v

vv>, (2)

where u and v are unit eigenvectors corresponding toeigenvalues �

u

and �v

of T.We choose �

u

� �v

, so that the eigenvector u,which corresponds to the largest eigenvalue �

u

, indicatesthe direction of highest change in image amplitude, andtherefore is perpendicular to locally linear features inan image, while the orthogonal eigenvector v indicatesthe direction that is parallel to such features. In otherwords, the eigenvector u is the seismic normal vectorthat is perpendicular to seismic reflectors in a seismicimage, and the eigenvector v is parallel to the reflectors.

2.2 Smoothing

The structure tensor T given in equation 1 can be usedto accurately estimate the local orientation of struc-tures in an image where there is only one locally domi-nant orientation present. However, for multiply-orientedstructures such as an unconformity where seismic re-flectors terminate (dashed green ellipse in Figure 2a),this structure tensor provides a local average of theorientations of structures. The seismic normal vectors(magenta segments in Figure 2b) estimated from T aresmoothed near the termination area, whereas the true

normal vectors (cyan segments in Figure 2b) are discon-tinuous across the unconformity.

At an unconformity where seismic reflectors termi-nate (dashed green ellipse in Figure 2a), structures of re-flectors above the unconformity are di↵erent from thoseof reflectors below. Therefore, if we compute structuretensors using vertically causal smoothing filters, whichaverage structures from above, we will obtain normalvectors at the unconformity that are di↵erent from thoseobtained using vertical anti-causal filters, which averagestructures from below.

With a vertically causal filter, the structure tensorcomputed for each sample represents structures aver-aged using only samples above. We define such a struc-ture tensor as

Tc

=

hg1g1ih,vc hg1g2ih,vchg1g2ih,vc hg2g2ih,vc

�, (3)

where h·ih,vc

represents horizontal Gaussian (subscripth) and vertically causal (subscript vc) smoothing filters.

With a vertically anti-causal filter, the structuretensor computed for each sample represents structuresaveraged using only samples below. We define such astructure tensor as

Ta

=

hg1g1ih,va hg1g2ih,vahg1g2ih,va hg2g2ih,va

�, (4)

where the subscript va denotes a vertically anti-causalsmoothing filter.

2.2.1 Vertical smoothing

To compute two structure-tensor fields that di↵er signif-icantly at an unconformity, the causal smoothing filterthat averages from above should smooth along the direc-tion perpendicular to the structures above the uncon-formity, while the anti-causal filter should smooth alongthe direction perpendicular to the structures below theunconformity. Here, we simply use vertically causal andanti-causal filters because unconformities are usuallytend to be horizontal in seismic images. We implementthese two filters with one-sided exponential smoothingfilters, which are e�cient and trivial to implement.

A one-sided causal exponential filter for input andoutput sequences x[i] and y[i] with lengths n can beimplemented in C++ (or Java) as follows:

float b = 1.0f-a;

float yi = y[0] = x[0];

for (int i=1; i=0; --i)

y[i] = yi = a*yi+b*x[i];

The parameter a in these two one-sided exponential fil-

266 X.Wu & D.Hale

b)

a)

Figure 3. Two di↵erent seismic normal vector fields es-timated using structure tensors computed with verticallycausal (yellow segments) and anti-causal (green segments)smoothing filters. In (a), the vector fields di↵er only withinthe termination area of the unconformity; in (b), these vectordi↵erences are extended to the correlative conformity.

ters controls the extent of smoothing.From structure-tensor fields T

c

and Ta

computedfor the same seismic image using vertically causal andanti-causal smoothing filters, respectively, we estimatetwo seismic normal vector fields u

c

and ua

. As shownin Figure 3a, the two seismic normal vector fields u

c

(green segments in Figure 3a) and ua

(yellow segmentsin Figure 3a) are identical in conformable areas withparallel seismic reflectors, because orientation of struc-tures locally averaged from above (used to compute T

c

)are identical to orientation of structures averaged frombelow (used to compute T

a

). However, at the termina-tion area of an unconformity, the two vector fields aredi↵erent, because the structure tensors T

c

computedwith structures locally averaged from above, should bedi↵erent from T

a

computed with structures locally av-eraged from below.

Therefore, as shown in Figure 3a, the di↵erence be-tween estimated normal vector fields u

c

and ua

pro-vides a good indication of the termination area of anunconformity. However, a complete unconformity, thatis, a curve (in 2D) or surface (in 3D) with geologic timegaps, extends from its termination area its correlativeconformity with. Thus we should extend normal vec-tor di↵erences from the termination area, where thesedi↵erences originate, to the correlative conformity.

2.2.2 Structure-oriented smoothing

To detect a correlative conformity, we extend vector dif-ferences (between u

c

and ua

) at an unconformity fromits termination area to its correlative conformity, byreplacing the horizontal Gaussian smoothing filter inequations 3 and 4 with a structure-oriented smoothingfilter (Hale, 2009) when computing structure tensors.

Then, the structure tensors Ts,c

and Ts,a

, com-

correlative conformity

termination

b)

a)

Figure 4. Unconformity likelihoods, an attribute that evalu-ates di↵erences between two estimated seismic normal vectorfields (yellow and green segments in Figure 3b), before (a)and after (b) thinning highlight both the termination areaand correlative conformity of the unconformity.

puted with a laterally structure-oriented filter and ver-tically causal and anti-causal filters, are defined by

Ts,c

=

hg1g1is,vc hg1g2is,vchg1g2is,vc hg2g2is,vc

�, (5)

and

Ts,a

=

hg1g1is,va hg1g2is,vahg1g2is,va hg2g2is,va

�, (6)

where the subscript s represents a structure-orientedfilter that smoothes along reflectors in a seismic im-age. Note that the structure-oriented smoothing is gen-erally more expensive than the vertically causal andanti-causal smoothing. We therefore first apply thestructure-oriented smoothing filter to each element ofgg> to obtain T

s

= hgg>is

, which then is smoothedseparately by vertically causal and anti-causal filters toobtain T

s,c

and Ts,a

, respectively. By doing this, we ap-ply the relatively expensive structure-oriented smooth-ing only once. However, if we first apply the verticallycausal and anti-causal smoothing to compute two dif-ferently smoothed outer products hggi

c

and hgg>ia

,we then need to apply the structure-oriented smooth-ing twice to obtain two structure-tensor fields T

s,c

andT

s,a

.As discussed by Hale (2009, 2011), to obtain a

smoothed output image q(x) from an input p(x), thestructure-oriented smoothing method solves a finite-di↵erence approximation to the following partial di↵er-ential equation:

q(x)� �2

2r·D(x) ·r q(x) = p(x), (7)

where D(x) is a di↵usion-tensor field that shares theeigenvectors of the structure tensor computed from animage, and therefore orients the smoothing along imagestructures. Similar to the half-width � in a Gaussiansmoothing filter, the parameter � controls the extent of

Image processing for unconformities 267

(a) (b)

Figure 5. Applying our method to the synthetic image cut from Hoek et al. (2010), we obtain unconformity likelihoods before(a) and after (b) thinning.

a) b)

Figure 6. Unconformity likelihoods before (a) and after (b) thinning.

smoothing.In 2D, we use the eigenvectors u(x) and v(x), esti-

mated using the structure tensors shown in equation 1,to construct our di↵usion-tensor field

D(x) = �u

(x)u(x)u>(x) + �v

(x)v(x)v>(x). (8)

Then, because eigenvectors u(x) and v(x) are perpen-dicular and parallel to seismic reflectors, respectively,we can control the structure-orient filter to smoothalong reflectors by setting the corresponding eigenval-ues �

u

(x) = 0 and �v

(x) = 1 for all tensors in D(x).As indicated by the seismic normal vectors shown

in Figure 2b, normal vectors (magenta segments)estimated using the structure tensors computed inequation 1 are inaccurate at unconformities. However,they are accurate in conformable areas, includingthe area near correlative conformity of the unconfor-mity. Thus the structure tensors in equation 1 areadequate for constructing di↵usion-tensors D(x) forstructure-oriented smoothing along seismic reflectors,including those near the correlative conformity of anunconformity. By applying such a structure-orientedfilter to the elements of the structure tensors T

s,c

and Ts,a

, we extend structural di↵erences, originatingwithin the termination area of an unconformity, to thecorresponding correlative conformity.

As shown in Figure 3, using structure tensors

Tc

and Ta

computed with a horizontal Gaussianfilter and vertically causal and anti-causal filters, theestimated seismic normal vectors u

c

(green segments inFigure 3a) and u

a

(yellow segments in Figure 3a) di↵eronly within the termination area of the unconformity.Using structure tensors T

s,c

and Ts,a

computed witha structure-oriented smoothing filter instead of ahorizontal Gaussian filter, the di↵erences between theestimated seismic normal vectors u

s,c

(green segmentsin Figure 3b) and u

s,a

(yellow segments in Figure 3b)are extended from the termination area to the correla-tive area.

In summary, by first applying a structure-orientedfilter to each structure-tensor element of gg>, weextend any structure di↵erences, which originate withinthe termination area of an unconformity, to its correl-ative conformity. Then, applying vertically causal andanti-causal filters for each structure-tensor element,we compute two di↵erent structure-tensor fields T

s,c

and Ts,a

with seismic normal vector fields us,c

andus,a

that di↵er within both the termination area andcorrelative conformity of an unconformity. Finally, thedi↵erences between the two estimated vector fields u

s,c

and us,a

can be used as an unconformity attribute thathighlights the complete unconformity.

268 X.Wu & D.Hale

a)

b)

Figure 7. Unconformity likelihoods before (a) and after (b) thinning. Thinned unconformity likelihoods form unconformitysurfaces (b).

Image processing for unconformities 269

d)c)

u1

u2

a) b)

c) d)

Figure 8. Vertical (u1) and horizontal (u2) components of seismic normal vectors estimated using structure tensors computedwith (b, d) and without (a, c) unconformity constraints.

2.3 Unconformity likelihood

As shown in Figure 3b, the vectors us,c

(green segments)and u

s,a

(yellow segments) are identical everywhere ex-cept at the unconformity, including its termination areaand correlative conformity. Therefore, we define an un-conformity likelihood attribute g, that evaluates the dif-ferences between u

s,c

and us,a

, to highlight unconfor-mities:

g ⌘ 1� (us,c

· us,a

)p. (9)

A large power p (p � 1) increases the contrast betweensamples with low and high unconformity likelihoods. Forthe example shown in Figure 4a, the unconformity like-lihoods are computed with p = 200.

Using a process similar to that used by Hale (2012)for extracting ridges of fault likelihoods, we extractridges of unconformity likelihood by simply scanningeach vertical column of the unconformity likelihood im-age (Figure 4a), preserving only local maxima, and set-ting unconformity likelihoods elsewhere to zero. Fig-ure 4b shows that ridges of unconformity likelihood co-incide with the unconformity that appears in the syn-thetic seismic image.

Figure 5 shows a more complicated 2D syntheticimage used by Hoek et al. (2010). The geometric at-tributes they compute highlight only the termination ar-eas of unconformities apparent in this synthetic image.In comparison, unconformity likelihoods before (Fig-ure 5a) and after (Figure 5b) thinning, computed us-ing our method, highlight the complete unconformities,including their termination areas as well as correlativeconformities.

Figure 6 shows an example of a real 2D seismicimage, in which generated unconformity likelihoods be-fore (Figure 6a) and after (Figure 6b) thinning correctly

highlight two unconformities apparent in the seismic im-age.

For a 3D seismic image, following the same processas above, we compute an unconformity-likelihood vol-ume as shown in Figure 7, which correctly highlightstwo apparent unconformities. In the time slices of un-conformity likelihoods before and after thinning, we ob-serve that samples in the lower-left and upper-right ar-eas, separated by high unconformity likelihoods, suggestdi↵erent seismic facies. This indicates that they belongto two di↵erent depositional sequences that have di↵er-ent geologic times.

From ridges of unconformity likelihoods (Fig-ure 7b), we connect adjacent samples with high un-conformity likelihoods to form unconformity surfaces asshown in upper-right panel of Figure 7b.

3 APPLICATIONS

We first use unconformity likelihoods as constraints tomore accurately estimate seismic normal vectors at un-conformities. Then, using more accurate normal vectorsand unconformity likelihoods as constraints in our seis-mic image flattening method, we are able to better flat-ten an image containing unconformitities.

3.1 Estimation of seismic normal vectors atunconformities

Using structure tensors computed with horizontal andvertical Gaussian filters as shown in equation 1, we findsmoothed seismic normal vectors (magenta segments inFigure 2b) in the termination area, because discontin-uous structures across the unconformity are smoothed

270 X.Wu & D.Hale

a) b)

c) d)

RGT

Figure 9. RGT (a) and flattened (c) images generated with inaccurate seismic normal vectors (Figures 8a and 8c) and withoutunconformity constraints. Improved RGT (b) and flattened (d) images with more accurate seismic normal vectors (Figures 8band 8d) and constraints from unconformity likelihoods (Figure 6).

by symmetric Gaussian filters. Therefore, to obtain cor-rect normal vectors (cyan segments in Figure 2b) thatare discontinuous in the termination area, we must usemore appropriate filters to compute structure tensors.

To preserve structure discontinuities, we computethe structure tensors using horizontal and vertical fil-ters that do not smooth across unconformities:

T =

hg1g1ish,sv hg1g2ish,svhg1g2ish,sv hg2g2ish,sv

�, (10)

where the h·ish,sv

represent horizontal (subscript sh)and vertical (subscript sv) filters that vary spatially,and for which the extent of smoothing is controlled bythe thinned unconformity likelihoods.

The horizontal and vertical filters are similar tothe edge-preserving smoothing filter discussed in Hale(2011):

q(x)� �2

2r· c2(x) ·r q(x) = p(x). (11)

We compute c(x) = 1 � gt

(x) to prevent this fil-ter from smoothing across unconformities. g

t

(x) is athinned unconformity likelihood image as shown in Fig-ure 6b, which has large values (close to 1) only at un-conformities and zeros elsewhere.

Figure 8 shows seismic normal vectors estimated forthe image with two unconformities shown in Figure 6.Both the vertical (Figure 8a) and horizontal (Figure 8c)components of seismic normal vectors, estimated fromstructure tensors computed as in equation 1, are smoothat the unconformities; those estimated from structuretensors computed as in equation 10 preserve disconti-nuities at unconformities (Figures 8b and 8d).

3.2 Seismic image flattening at unconformities

Seismic normal vectors or dips can be used to flatten(Lomask et al., 2006; Parks, 2010) or unfold (Luo andHale, 2013) a seismic image to generate a horizon vol-ume (Wu and Hale, 2014), that allows for the extractionof all seismic horizons in the image. Faults and uncon-formities, which represent discontinuities of reflectors ina seismic image, present challenges for these methods.Luo and Hale (2013) and Wu and Hale (2014) have ex-tended their methods to handle faults by first unfaultingthe seismic image or by placing control points on oppo-site sides of a fault. However, neither of these methodscorrectly handles seismic images with unconformities,because estimated seismic normal vectors or dips areinaccurate at unconformities, and because unconformi-ties are not automatically detected and then used asconstraints in these methods.

In this paper we have proposed methods to auto-matically detect unconformities and more accurately es-timate seismic normal vectors at unconformities. There-fore, we can easily extend the flattening method de-scribed in Wu and Hale (2014), to better flatten aseismic image at unconformities, by using seismic nor-mal vectors estimated from structure tensors computedwith equation 10, and by incorporating constraints de-rived from unconformity likelihoods into the flatten-ing method. We incorporate unconformity constraintsin our flattening method by weighting the equations forflattening using unconformity likelihoods, and then us-ing the unconformity likelihoods to construct precondi-tioner in the conjugate gradient method used to solvethose equations.

Image processing for unconformities 271

a)

b)

Figure 10. Generated RGT volume (a) and flattened (b) 3D seismic image. Discontinuities in the RGT volume correspond tovertical gaps in the flattened image at unconformities.

272 X.Wu & D.Hale

3.2.1 Weighting

To generate a horizon volume or to flatten a seismicimage, we first solve for vertical shifts s(x, y, z) as dis-cussed in Wu and Hale (2014):

2

66664

w(� @s@x

� p @s@z

)

w(� @s@y

� p @s@z

)

✏

@s

@z

3

77775⇡

2

66664

wp

wq

0

3

77775, (12)

where p(x, y, z) and q(x, y, z) are inline and crosslinereflector slopes computed from seismic normal vectors;w(x, y, z) represent weights for the equations; and thethird equation ✏ @s

@z

⇡ 0, scaled by a small constant ✏, isused to reduce rapid vertical variations in the shifts.

For a seismic image with unconformities, we in-corporate constraints derived from unconformity like-lihoods into the equations 12 by setting w(x, y, z) =1 � g

t

(x, y, z) and we use a spatially variant ✏(x, y, z)instead of a constant value:

✏(x, y, z) = ✏0(1� gt(x, y, z)), (13)

where ✏0 is a small constant number (we use ✏0 = 0.01for all examples in this paper), and g

t

(x, y, z) denotesthe thinned unconformity likelihoods, such as thoseshown in Figure 7b.

This spatially variant ✏(x, y, z), with smaller values(nearly 0) at unconformities, helps to generate more rea-sonable shifts with gradual variations everywhere exceptat unconformities.

3.2.2 Preconditioner

As discussed in Wu and Hale (2014), to obtain the shiftss(x, y, z) in equation 12 for a 3D seismic image with Nsamples, we solve its corresponding least-squares prob-lem expressed in a matrix form:

(WG)>WGs = (WG)>Wv, (14)

where s is an N⇥1 vector containing the unknown shiftss(x, y, z), G is a 3N ⇥ N sparse matrix representingfinite-di↵erence approximations of partial derivatives,W is a 3N ⇥ 3N diagonal matrix containing weightsw(x, y, z) and ✏(x, y, z), and v is a 3N ⇥ 1 vector with2N slopes p and q, and N zeros.

Because the matrix (WG)>WG is symmetricpositive-semidefinite, we can solve the linear system ofequation 14 using the preconditioned conjugate gradi-ent method, with a preconditioner M�1 as in Wu andHale (2014):

M�1 = Sx

Sy

Sz

S>z

S>y

S>x

, (15)

where Sx

, Sy

and Sz

are filters that smooth in the x, yand z directions, respectively.

For a seismic image with unconformities, the fil-ters S

x

, Sy

and Sz

are spatially variant filters designed

as in equation 11, to preserve discontinuities in shiftss(x, y, z) at unconformities.

3.2.3 Results

With the computed shifts s(x, y, z), we first generatea relative geologic time (RGT) volume ⌧(x, y, z) =z + s(x, y, z) (Figures 9a and 10a). We then use theRGT volume to map a seismic image f(x, y, z) (Fig-ures 6 or 7) in the depth-space domain to a flattenedimage f̃(x, y, ⌧) (Figures 9b or 10b) in the RGT-spacedomain.

From the 2D example shown in Figure 9, the RGT(Figure 9a) and flattened (Figure 9c) images, gener-ated with inaccurate seismic normal vectors (Figure 8aand 8c) and without unconformity constraints, are in-correct at unconformities, where we expect discontinu-ities in the RGT image and corresponding gaps in theflattened image. With more accurate seismic normalvectors (Figure 8b and 8d) and with constraints derivedfrom unconformity likelihoods (Figure 6), we obtain animproved RGT image (Figure 9b) with discontinuitiesat unconformities. Using this RGT image, we obtain animproved flattened image (Figure 9c), in which seismicreflectors are horizontally flattened and unconformitiesappear as vertical gaps.

Figure 10 shows a 3D example with two unconfor-mity surfaces, highlighted by unconformity likelihoodsin Figure 7. We observe obvious discontinuities in RGTat unconformities in our generated RGT volume (Fig-ure 10a). These RGT discontinuities result in verticalgaps in the corresponding flattened seismic image (Fig-ure 10b). The time slice of an RGT image shows largeRGT variations between samples in the lower-left andupper-right areas that are separated by an unconfor-mity. This indicates that the sediments, represented bythe samples in the two di↵erent areas, are deposited intwo di↵erent sedimentary sequences occurring at di↵er-ent geologic times.

4 CONCLUSION

We have proposed a method to obtain an unconfor-mity likelihood attribute from the di↵erences betweentwo seismic normal vector fields estimated from twostructure-tensor fields, one is computed using a verti-cally causal smoothing filter, and the other using a ver-tically anti-causal filter. From a seismic image, we firstcompute smoothed outer products of image gradients byapplying a structure-oriented smoothing filter to eachelement of these outer products. These smoothed outerproducts are then smoothed using vertically causal andanti-causal filters to compute two di↵erent structure-tensor fields, and their corresponding normal vectorfields.

Using structure-oriented smoothing filters for the

Image processing for unconformities 273

outer products, we extend structure variations from atermination area to the corresponding correlative con-formity. In doing this, we assume that the correlativeconformity is not dislocated by faults. If faults appearin the seismic image, we could perform unfaulting (Luoand Hale, 2013) before attempting to detect unconfor-mities.

We use separate vertically causal and anti-causalfilters to obtain structure tensors that di↵er at uncon-formities. Unconformity likelihoods might be further im-proved by instead using causal and anti-causal filtersthat smooth in directions orthogonal to unconformities.

As examples of applications, we have shown how toestimate more accurate seismic normal vectors and bet-ter flatten seismic reflectors at unconformities by usingunconformity likelihoods as constraints.

ACKNOWLEDGMENTS

This research is supported by the sponsor companiesof the Consortium Project on Seismic Inverse Methodsfor Complex Structures. All of the real seismic imagesshown in this paper were provided by the US Depart-ment of Energy and by dGB Earth Sciences. The 2Dsynthetic seismic image used in the example shown inFigure 5 is that shown in Hoek et al. (2010).

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