3-d seismic attributes using a semblance-based

7/27/2019 3-D Seismic Attributes Using a Semblance-based

1/16

G EOP HYSI CS, VOL . 63, NO. 4 (JULY-AU G UST 1998); P. 11501165, 21 FIG S.

3-D seismic attributes using a semblance-basedcoherency algorithm

Kurt J. Marfurt, R. Lynn Kirlin, Steven L. Farmer,and Michael S. Bahorich

ABSTRACT

Seismic coherency is a measure of lateral changes inthe seismic response caused by variation in structure,

stratigraphy, lithology, porosity, and the presence of hy -

drocarbons. U nlike shaded relief ma ps that allow 3-D

visualization of faults and channels from horizon picks,

seismic coherency operates on the seismic data itself and

is therefore unencumbered by interpreter or automatic

picker biases.

We present a more robust, multitrace, semblance-

based coherency algorithm that allows us to analyze

dat a of lesser qua lity than our original three-trace cross-

correlation-based algorithm. This second-generation,

semblance-based coherency algorithm provides im-

proved vertical resolution over our original zero mean

crosscorrelation algorithm, resulting in reduced mixingof overlying or underlying stratigra phic features. In gen-

eral, we analyze stratigraphic fea tures using as narrow

a temporal analysis window as possible, typically de-

termined by the highest usable frequency in the input

seismic data. I n the limit, one may confi dently apply our

new semblance-based algorithm to a one-sample-thickseismic volume extracted a long a conventionally picked

stratigraphic horizon corresponding to a pea k or trough

whose amplitudes lie suffi ciently above the ambient seis-

mic noise. In contrast, near-vertical structural features,

such as faults, are better enhanced when using a longer

temporal analysis window corresponding to the lowest

usable frequency in the input data.

The ca lculation of reflector dip/azimuth throughout

the data volume allows us to generalize the calculation of

conventional complex trace attributes (including enve-

lope, phase, frequency, and bandwidth) to the calculation

of complex reflector attributes generated by slant stack-

ing the input data along the reflector dip within the co-

herency a nalysiswindow. These more robust complexre-flector attribute cubes can be combined with coherency

and d ip/azimut h cubes using conventiona l geostat istical,

clustering, and segmentation a lgorithms to provide an

integrated, multiat tribute ana lysis.

INTRODUCTION

Co mplex trace seismic attributes, including measures of seis-

mic amplitude, frequency, and phase, have been used suc-

cessfully in mapping seismic lithology changes for almost t wo

decades. Multitrace relationships, including crosscorrelation

techniques, have enjoyed an equally long history in the au-

toma tic picking of static corrections and, over the past decad e,

in the auto mat ic picking of 3-D seismic horizo ns. Nevert heless,such measurements of seismictra ce coherency lay buried inside

a numerical a lgorithm. Not until B ahorich and Farmer (1995,

1996) has seismic coherency (or, perhaps more importantly,

Presented at the 65th A nnual Interna tional Meeting, Society of Exploration G eophysicists. Manuscript received by the E ditor O ctober 17, 1996;revised manuscript received November 21, 1997.Amoco E xploration and Pro duction Technology G roup, 4502 E. 41st St. Tulsa, OK 74102-3385. E -mail: [email protected]; [email protected] niversity o f Victoria, 4821 Maplegrove St., Victoria, B ritish C olumbia V 8Y3B 9. E-mail: [email protected] che C orpo rat ion, 2000 Post O ak B lvd. #100, Ho uston, TX 77056-4400. E-mail: ba [email protected] 1998 Society of Exploration G eophysicists. All rights reserved.

lack of coherency) been displayed and used as an attribute

by itself. When there is a sufficient lateral change in acous-

tic impedance, the 3-D seismic coherency cube (B ahorich a nd

Farmer, 1995, 1996) can be extremely effective in delineating

seismic faults (Figures 1 and 2). This algorithm is also quite

effective in highlighting subtle cha nges in strat igraphy, includ-

ing 3-D images of meand ering distributary channels,point ba rs,

canyons, slumps, and tidal drainage patterns. Seismiccoherencyapplied to a 3-D cube of seismic input data offers several dis-

tinct adva ntages over ho rizon dip/azimuth and shaded relief

maps, including the ab ility to

1150


2/16

3-D Seismic Coherency 1151

1) accelerate the interpretation process by beginning struc-

tural a nd stratigraphic analysis before detailed picking

begins, even on preliminary brute sta ck data cubes;

2) carefully analyze structural and stratigraphic features

over the entire dat a volume,including zonestha t are shal-

lower, deeper, or adjacent to the primary zone of interest;

3) identify and interpret subtle features that a re not repre-

sentable b y picks on peaks, tro ughs, or zero crossings;

4) generate paleoenvironmental maps of channels and fa ns

corresponding to sequence versus reflector boundaries;

a nd

5) analyze features that are either internal or parallel to

pickable formation tops and bottoms.

O ur C1 coherency algorithm (Bahorich and Farmer, 1995,

1996) is based on classical normalized crosscorrelation. We fi rst

define the in-line l-lag crosscorrelation, x , at time t between

a)

b)

FIG. 1. Time slices through a seismic data volume at (a) t=1200 ms and (b) t= 1600 ms. D ata courtesy of G eco-Pra kla.

data traces u at po sitions (xi , yi ) and (xi +1, yi ) to be

x (t, ,xi , yi ) =

+w=w

u(t , xi , yi )u(t ,xi +1, yi )

+w=w

u2(t , xi , yi )+w

=w

u2(t ,xi +1, yi )

,

(1a)

where 2w is the temporal length of the correlation window.

Next, we defi ne the cross-line m-lag crosscorrelation, y , a t

time t between data traces u a t (xi , yi ) and (xi , yi +1) to be

a)

b)

FIG. 2. Time slices through the C1 coherency cube at (a) t =1200ms a nd (b) t= 1600 ms, correspond ing to the seismic datain Figure 1a nd using the three-trace crosscorrelation a lgorithmgiven by equa tions (1) and (2). Temporal a nalysis half-windoww = 32 ms.


3/16

1152 Marfurt et al.

y (t,m,xi , yi ) =

+w=w

u(t , xi , yi )u(t m,xi , yi +1)

+w

=wu2(t , xi , yi )

+w

=wu2(t m,xi , yi +1)

.

(1b)

(, p, q) =

J

j =1

u( px j q yj ,xj , yj )

2+

J

j =1

uH( px j q yj ,xi , yj )

2

J

Jj =1

[u( px j q yj ,xj , yj )]

2 +

uH( px j q yj ,xj , yj )2 , (3)

These in-line (-lag) and cross-line (m-lag) correlation coef-

ficients can then be combined to generate a 3-D estimate of

coherency, xy :

x y =

ma x

x (t, ,xi , yi )

ma x

my (t,m,xi , yi )

, (2)

w here ma x x (t, ,xi , yi ) a nd ma xm y(t,m,xi , yi ) denot e

those lags a nd m for which x a nd y are maximum. For

high-quality data, lags a nd m approximately measure the ap-

parent time dip per trace in the x- and y-directions. For dat a

contaminat ed by coherent noise, estimates of apparent dip us-

ing only two traces can be quite noisy, which is a limitation of

the crosscorrelation a lgorithm.

Crosscorrelating each trace against its neighbor for various

time lags forms a different 2 2 covariance matrix for eachlag pair (, m). Extending equation (1) beyond three traces

requires a more general analysis of higher order covariance

matrices using eigenvalue analysis (G ersztenkorn and Ma rfurt,

1996).

A second limitation of the three-point crosscorrelation al-

gorithm is the assumption of zero-mean seismic signals. This is

approximately true when the correlation window [2w in equa-

tion (1)] exceeds the length of a seismic wavelet. For seismic

data containing a 10-Hz component of energy, this req uires

a rather long 100-ms window that will mix stratigraphy asso-

ciated with both deeper and shallower times about the zone

of interest. Shortening this operator window (Figure 3) to

2w = 16 ms results in increased art ifacts att ributed to the seis-

mic wavelet. U nfortunately, a more rigorous, nonzero meanrunning window crosscorrelation algorithm is computationa lly

much more expensive.

In t his paper, we present our second-generation, o r C2, algo-

rithm, which estimates coherency using a semblance analysis

over an arbitrary number of traces. In addition to more robust

measures of coherency, dip, and azimuth in noisy data envi-

ronments, the vertical analysis window can be limited to only

a f ew time samples, allowing us to accurately ma p thin, subtle,

stratigraphic fea tures.

ALGORITHM DESCRIPTION

Just as crosscorrelation forms the basis of automatic first

break and reflector picking, multitrace semblance estimates

form the basis of both conventional and tomographic seismic

velocity ana lysis (Taner and Koe hler, 1969; La nda et a l., 1993).

Likewise, semblance ca lculations a re intrinsic to robust (, p)

ana lysis algorithms (Stoffa et al., 1981; Yilmaz and Taner, 1994)

a nd f-x deconvolution a lgorithms.

We begin by defining an elliptical or rectangular analysis

window containing J traces centered about the analysis point

(Figure 4). If w e center the loca l (x , y) axis about this analysis

point, we define the semblance, (, p, q), to be

where the triple (, p, q) defines a local planar event at time ,

where p a nd q are the apparent dips in the x a nd y directions,

a)

b)

FIG. 3. Time slices thro ugh the C1 coherency cube, correspond-ing to those shown in Figure 2 but with a temporal analysishalf-window of w = 8 ms.


4/16

3-D Seismic Coherency 1153

measured in milliseconds per meter (Figure 5), and where t he

superscript H denotes the Hilbert transform or quadrature

component of the real seismic trace, u. C alculating the sem-

blance of the analytic trace will allow us to obta in robust es-

timates of coherency even abo ut the zero crossings of seismic

reflection events. The numerator of equation (3) is the 3-D

(, p, q) transform, U(, p, q), of the data u(t,x, y) and is re-

lated closely to the least-squares R ado n transform for 3-D dip

filtering and trace interpolation:

U(, p, q) =J

j =1

u[ (px j + q yj ),xj , yj ]. (4)

c(, p, q) =

+Kk=K

J

j =1

u(+ kt px j q yj ,xj , yj )

2+

J

j =1

uH(+ kt px j q yj ,xj , yj )

2

J

+Kk=K

Jj =1

[u(+ kt px j q yj ,xj , yj )]

2 +

uH(+ kt px j q yj ,xj , yj )2 , (5)

FIG. 4. (a) E lliptical and (b) rectangular analysis window s cen-tered about a n analysis point defined by length of major axisa, length of minor axis b, and azimuth of major axis a .

Indeed, we exploited such 3-D R adon transform dip fi lter-

ing (Marf urt et a l., 1998) to minimize the a cquisition fo otprint

because of cable feathering and dip movement artifa cts as part

of the data processing leading up to Figure 1. The semblance

estimate given by eq uation (3) will be unstable for some small

but coherent seismic events such as might occur if we were to

sum along the zero crossings of a planar coherent event. We

therefore exploit the same trickused in semblance-based veloc-

ity a nalysis; explicitly, we will calculate an average semblance

over a vertical analysis window of height 2w ms or of half-

height K =w/ samples. We define this average semblance

to be our coherency estimate, c:

where t is the temporal sample increment. Since our ana lysiswindow is always centered a bout (x = 0, y = 0), the intercept

time is identical to t.

In general, we do not know, but wish to estimate, the value

of p, q associated with the local dip and azimuth of a hypo-

thetical planar 3-D reflection event. Horizon-based estimates

of dip a nd azimuth have proven to be a n extremely powerful

interpretation t ool (D alley et al., 1989; Mond t, 1990; R ijks and

Jauffred, 1991). In this paper, we will estimate (p, q) through

a straightforward search over a user-defined range of discrete

apparent d ips. We assume the interpreter is able to estimate t he

maximum true d ip, dma x, mea sured in milliseconds per met er,

from conventional in-line and cross-line seismic displays of the

dat a (Figure 6), thereby limiting the apparent dips top2 + q2 +dma x. (6)

I f a a nd b are the half-widths of the major and minor axes

of our analysis window (Figure 4), and if fma x is the highest

temporal frequency component contained in the seismic data,

then the Nyq uist criterion of sampling the data at two points

per period restricts the apparent dip increments, p a nd q ,

to

p 1

2a fma x(7a)

FIG. 5. Ca lculation of coherency over an elliptical analysis win-dow with apparent dips (p, q) = (0.1 m s/m, 0.1 m s/m) .


5/16

1154 Marfurt et al.

a nd

q 1

2b fma x. (7b)

Our search for a n estimate of the apparent dip ( p, q) of

the seismic reflector at time t is thus reduced to the direct

calculation o f c(t, p, qm ) over np nq discrete apparent dip

pairs (p, qm ) where np = 2dma x/p + 1 a n d nq = 2dmax /q + 1.

We declare the apparent dip pair (pL, qM) to be an esti-mate of the reflector apparent dips ( p, q) and the coherency

c(t, pL , qM) to be an estimate of the reflector coherency, c,

when

c(t, pL, qM) c(p, qm ) (8a)

for all np +np,nq m +nq . If necessary, we can

obtain a more accurate estimate of ( p, q) a nd c by 2-D inter-

polation. Here, we fi t a para boloid, g(p, q), through a subset of

the discrete measures of c(t, p, qm ) centered a bout (pL , qM).

a)

b)

c)

FIG. 6. Vertical slice through seismic dat a corresponding tolines (a) A A, (b) B B , and (c) C C, shown in Figure 1.

Then

c = max[g(p, q)] (8b)

and ( p, q) is that value of (p, q) where the maximum occurs

or, mathema tically,

( p, q) = g1(c), (8c)

where g1 denotes the inverse mapping of c to (p, q).The estimated apparent dips ( p, q) are related to the esti-

mated true dip and azimuth, ( d, ) by the simple geometric

relationships

p = d sin() (9a )

a nd

q = d cos(), (9b)

where d is measured in milliseconds per meter and the angle

is measured clockwise fro m the positive x (or north) axis. A

simple coordinate rotation by angle o is necessary when the

in-line acquisition axis, x , is not aligned with the northsouth,

or x

, axis (Figure 4).

Solid-angle discretization anddisplay

Optima l angular discretization is important fo r two rea sons.

First, we wish to minimize the computational cost, which in-

creases linearly with the number of angles searched. Second,

at the writing of this paper, the two most popular commercial

interpretive workstation software systems are limited to only

32 or 64 colors, greatly limiting the number of angles we can

display. Since seismic surveys are o ften designed to line up with

the predo minant geo logic strike a nd d ip, the simplest tessella-

tion of d ip/azimut h is tha t alon g the appa rent dip axes, p a nd q

(Figure 7a). I n practice, we use this tessellation o nly when we

wish to illuminate faults cutting perpendicular to a particularstrike and dip (Figure 7b).

Many geologic terranes are characterized by highly variable

strike and dip; these include salt diapirs, shale diapirs, pinnacle

reefs, prograd ing delta s, tilted fault blocks, a nd po p-up struc-

tures associated with w rench faulting. Even in simple geologic

terranes, rarely is a single strike and dip valid over large surveys

thousands of square kilometers in size.

Instead of a search over apparent dips (p, q), we could just

as easily have made our search over dip and azimuth (d, ).

Figure 8a shows the d iscretization of apparent dip using eq ual

incrementsp a nd q subject to equation (7), while Figure 8b

shows the discretization using equa l increments d a nd .

Clea rly, we do not w ish to sample dip d= 0 ms/m in Figu re 8b

for ten d ifferent azimuths. The Chinese checker t essellationof Figure 8c more closely represents an equal and therefore

more economic sampling of the solid angle ( d, ) surface with

a minimum number of points. For the angular discretization

shown in Figure8ca nd circular analysisra diusa = b,wechoose

the incremental dip d to be

d

3-d seismic attributes using a semblance-based

Documents