HRS a New Look at Seismic Attributes

7/26/2019 HRS a New Look at Seismic Attributes

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A new look at

seismic attributesBrian Russell


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Introduction

Seismic attributes have evolved into a set of seeminglyunrelated methods:

Instantaneous attributes,

Windowed frequency attributes,

Coherency and semblance attributes,

Curvature attributes,

Phase congruency, etc.

In this talk, I want to make sense of all these attributes by

showing how they are inter-related.

In particular, I will close the loop between instantaneous

attributes and trace-by-trace correlation attributes.

To illustrate the various methods, I will use the Boonsville

dataset, a karst limestone example from Texas.

2November, 2012


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The Boonsville dataset

3

The Boonsville gasfield in north Texas

(Hardage et al., 1996)

is controlled by karst

collapse features,

making it ideal for

studying attributes.

A dataset over the

field was made

available by the TexasBureau of Economic

Geology.

The study area is

shown on the left.Hardage et al., 1996November, 2012


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4

Boonsville Geology

In the Boonsville gas field, production is from the Bend

conglomerate, a middle Pennsylvanian clastic deposited in afluvio-deltaic environment.

The Bend formation is underlain by Paleozoic carbonates,

the deepest being the Ellenburger Group of Ordovician age.

The Ellenburger contains numerous karst collapse featureswhich extend up to 760 m from basement through the Bend

conglomerate, shown here in schematic:

Lucia (1995)November, 2012


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A seismic volume

The 3D seismicvolume is shown here

in grey level variable

density format.

It consists of 97inlines and 133

crosslines, each with

200 samples (800

1200 ms).

The karst features are

illustrated by the red

ellipses.

5November, 2012

Crossline orydirection Inline or

xdirection

Time or t

direction


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Slices through the volume

Inline seismic section 146

6November, 2012

Time slice at 1000 ms

1000

ms

Inline

146

Here are vertical and horizontal slices through

the seismic volume:

Note the karst feature at the east end of the line.


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A brief history of attributes

The first attributes were single trace instantaneous

attributes, introduced by Taner et al. (1979).

Barnes (1996) extended these attributes to 2D and 3D.

From 1995-1999, Amoco researchers introduced two new

types of spatial attributes:

Spectral decomposition, based on the Fourier transform.

Coherency, based on trace-to-trace correlation.

Roberts (2001) introduced curvature attributes.

Marfurt et al. (2006) then showed the relationship between

instantaneous attributes and dip and curvature.

We shall also discuss the phase congruency attribute

(Kovesi, 1996) which was developed in image analysis.

A good place to start is with the Fourier transform.7November, 2012


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The Fourier transform

The Fourier transform (FT) takes a real time signals(t) and

transforms it to a complex frequency signal S(w):

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We can also perform the inverse transform from frequency

back to time.

In using the FT for attribute analysis, note that it operateson the complete seismic trace, so cannot be localized at a

time sample to give an instantaneous attribute value.

Partyka et al. (1999) used the FT over small seismic

windows to compute the spectral decomposition attribute.

.)(ofspectrum,phase)(andspectrum,amplitude)(

),(ofransformFourier tthe)(frequency,,2

where,)()()( )(

tsS

tsSff

eSSts

ww

ww

ww w


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Spectral decomposition

As shown here, spectral decomposition involves taking a small

window around a zone of interest (here: 1000 ms +/- 50 ms)

and transforming each trace window to the frequency domain. 9November, 2012

frequency

y

x

time

xy

Fourier

Transform


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Spectral slices

Spectral slices at various frequencies, compared with seismic amplitude.10November, 2012

Data slice 5 Hz 10 Hz 15 Hz 20 Hz

25 Hz 30 Hz 35 Hz 40 Hz 45 Hz


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Spectral decomposition limitations

As seen in the previous slide, each frequency slice in aspectral decomposition reveals a different underlying channel

structure in the original seismic amplitude slice.

However, this can be seen as a limitation since we really

dont know which slice to use.

Also, as the size of the window decreases, the frequency

resolution also decreases.

This resolution was improved by Sinha et al. (2005) using the

continuous wavelet transform.

However, the fact that each frequency slice gives a different

result still remained.

This leads us to the concept of instantaneous attributes,

including instantaneous frequency.

11November, 2012


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The complex time signal

Instantaneous attributes are based on the concept of thecomplex time signal, invented by Gabor in 1947 in a classic

paper called Theory of communication.

Since the Fourier transform of a real signal has a symmetric

shape on both the positive and negative frequency side,

Gabor proposed suppressing the amplitudes belonging to

negative frequencies, and multiplying the amplitudes of the

positive frequencies by two.

An inverse transformation to time gives the following

complex signal:

12November, 2012 .)(ofransformHilbert tthe)(and

,1signal,complexthe)(

:where)()()(

tsth

itz

tihtstz


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The Hilbert transform

The Hilbert transform is a filter which applies a 90

o

phaseshift to every sinusoidal component of a signal.

Notice that the complex trace can be transformed from

rectangular to polar coordinates, as shown below, to give the

instantaneous amplitudeA(t)and instantaneous phase F(t):

13November, 2012

f(t)

time

h(t)

A(t)

s(t)

.)(

)(

tan)(,)()()(

:where),(sin)()(cos)(

)()()()(

122

)(

F

FF

F

ts

th

tthtstA

ttiAttA

etAtihtstz ti

The complex trace was introduced into

geophysics by Taner et al. (1979), who

also discussed its digital implementation.


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Instantaneous frequency

Taner et al. (1979) also introduced the instantaneous

frequency of the seismic trace, which was initially derived

by J. Ville in a 1948 paper entitled: Thorie et applications

de la notion de signal analytique.

The instantaneous frequency is the time derivative of the

instantaneous phase:

14November, 2012

( )2

1

)(

)()(

)()(

)(/)(tan)()(

tA

dt

tsth

dt

tts

dt

tsthd

dt

tdt

F

w

Note that to compute w(t)we need to differentiate both the

seismic trace and its Hilbert transform.

Like the Hilbert transform, the derivative applies a 90ophase

shift, but it also applies a high frequency ramp.


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Instantaneous attributes

This figure showsthe instantaneous

attributes

associated with

the seismic

amplitude slice at1000 ms.

15November, 2012

Data slice -1.0

+1.0

Inst. Amp.

+1.0

0.0

Inst. Phase-180

o

+180o

Inst. Freq.

0 Hz

120 Hz


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New attributes

Attribute analysis until the mid-90s was thus based on thethree basic attributes introduced by Gabor and Ville in the

1940s: instantaneous amplitude, phase, and frequency.

Then, in the space of two years, papers on two new

approaches to attribute analysis appeared: The coherency method (Bahorich and Farmer, 1995)

2-D (and 3-D) complex trace analysis (Barnes, 1996)

The coherency method was a new approach which utilized

cross-correlations between traces. The paper by Barnes actually showed how instantaneous

attributes and aspects of coherency were related.

However, let us first discuss the coherency method.

16November, 2012


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The coherency method

The first coherency method involved finding the maximum

correlation coefficients between adjacent traces in thex

andydirections, and taking their harmonic average.

Marfurt et al. (1998) extended this by computing the

semblance of all combinations ofJtraces in a window.

17November, 2012

This involves searching over

allxdipspandydips q, over

a 2M+ 1 sample window:

Marfurt et al. (1998)

x

y

p dipq dip

2M+1samples


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Coherency mathematics

If the covariance matrix between all locations iandjis:

18November, 2012

),()(,),(

1

111

jjjii

tM

tMt

iij

JJJ

J

qypxtsqypxtsc

cc

cc

qpC

.),(ofdiagonalmainofsum),(and1,,1where

,),(

),(maxand

)()(max 2

2/1

2/1

3311

13

2/1

2211

121

qpCqpCTra

qpCTr

aqpCacoh

cc

c

cc

ccoh

T

T

then the two coherency measures are as follows:

).,(ofeigenvaluefirst,),(max 11

3 qpCqpCTrcoh

A third measure, called eigen-coherence (Gursztenkorn and

Marfurt, 1999), is:


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Coherency slice

The x, y and z slices through a coherency

volume with the 1000 ms data and coherency

slices on the right. Note the low amplitude

discontinuities and the highlighted event.

19November, 2012Data slice -1.0

+1.0

0.7

1.1


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Dip and azimuth attributes

At first glance, there appears to be no relationship between

instantaneous attributes and coherency. However, notice that in the coherency method it is important

to find thexandydips, calledpand q.

The paper by Barnes (1996) introduced two new spatial

instantaneous attributes and showed how these attributescould be used to find both the dips and the azimuth.

These concepts are even more important when we look at the

relationship between instantaneous attributes and curvature.

Barnes (1996) also discussed instantaneous bandwidth andphase versus group velocity based on the work of Scheuer

and Oldenburg (1988).

However, in the following slides I will only discuss his new

spatial attributes. 20November, 2012


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New instantaneous attributes

Re-visiting our earlier3D dataset, notice that

we only computed the

frequency attribute in

the time direction

Since the Hilbert

transform is actually a

function of three

coordinates (i.e. F(t,x,y))

we can also computespatial frequencies in

the inline (x) and

crossline (y) directions.

21November, 2012

Crossline orydirection Inline or

xdirection

Time or t

direction


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Instantaneous wavenumber

Analogous to instantaneous frequency, Barnes (1996)

therefore defined the instantaneous wavenumbers kxandky (using the Marfurt (2006) notation):

22November, 2012

and,),,(

2A x

sh

x

s

x

yxtkx

F

As with instantaneous frequency, the derivative operation

can be done either in the frequency domain or using

finite differencing up to a given order.

.

),,(2A

y

sh

y

hs

y

yxt

ky

F


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The instantaneous time dips in thexandydirection,p

and q, are given as:

23November, 2012

.hwavelengtandperiod:where

,and

ww

T

Tkq

Tkp

y

y

x

x

Using the dipspand q, the azimuth fand true time dip

qare then given by:

.and

,)/(tan)/(tan

22

11

qp

kkqp yx

q

f


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The dip attribute in 2D

To understand this concept, I created a seismic volume that

consisted of a dipping cosine wave. The period, wavelength

and time dip are shown on the plot. 24November, 2012

T= 50 ms

= 36 traces

dip = 1.4

ms/trace


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Next, lets look at the

dipping cosine wave

in 3D.

In this display, we

have sliced it along

the x, y and time

axes.

For this dipping

event, it is clear how

the dips andazimuths are related

to the instantaneous

frequency and

wavenumbers. 25November, 2012

kx

w

ky

pq

kx

w

ky

kx

ky


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Examples

Next, we will illustrate these instantaneous spatialattributes using the Boonsville dataset.

There are many possible displays based on the building

blocks for these attributes.

These include the derivatives of seismic amplitude, Hilberttransform and phase in the t,x, and y directions, the three

frequencies, andp, q, true dip and azimuth.

Note that the many divisions involved in the computations

make this method very sensitive to noise when comparedto correlation based methods.

In the following displays, I show the true dip and azimuth.

26November, 2012


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True dip volume

27November, 2012

Seismic with 3 x 3 alpha-trim mean True dip


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Azimuth volume

28November, 2012

Seismic with 3 x 3 alpha-trim mean Azimuth


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Curvature attributes

Roberts (2001) shows that curvature can be estimated from

a time structure map by fitting the local quadratic surfacegiven by:

29November, 2012

eydxcxybyaxyxt 22),(

This is a combination of an ellipsoid and a dipping plane.


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Roberts (2001) computes the curvature attributes by first

picking a 3D surface on the seismic data and then finding thecoefficients athroughffrom the map grid shown below:

30November, 2012

x

tttttt

x

td

2

)()( 741963

As an example, the coefficient d,

which is the linear dip in thexdirection, is computed by:

t1 t2 t3

t4 t5 t6

t7 t8 t9

x

y

Klein et al. (2008) show how to generalize this to each point

on the seismic volume by finding the optimum time shift

between pairs of traces using cross-correlation.


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The coefficients dand eare identical to dipspand qdefined

earlier, so when a= b= c= 0, we have a dipping plane andcan also define the true dip and azimuth as before.

For a curved surface, Roberts (2001) defines the following

curvature attributes (KminandKmaxare shown on the surface):

31November, 2012

,

,

2

min

2

max

ga ussmeanmean

ga us smeanmean

KKK

KKK

.

)1(

)1()1(

and,)1(

4:where

2/322

22

2/122

2

ed

cdedbeaK

ed

cabK

mean

gau ss


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Note that the inverse relationships are:

32November, 2012

.and2

maxminmaxmin KKK

KKga ussmean

Also, can compute most positive and negative curvatureK+

andK-, which are equal toKminandKmaxwith d= e=f= 0 (or,

the eigenvalues of the quadratic involving a, b and c):

.)()(and)()( 2222 cbabaKcbabaK

Finally, we haveKeuler, where dis an azimuth angle:

)90()0(),45(

sincos)(

ooo

2

max

2

min

eulereulerga us seulermean

euler

KKKKK

KKK

ddd

The next few slides show some examples from our dataset.


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Minimum Curvature

33November, 2012

Seismic with 3 x 3 alpha-trim mean Minimum curvature


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Maximum Curvature

34November, 2012

Seismic with 3 x 3 alpha-trim mean Maximum curvature


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Azimuth from Curvature

35November, 2012

Seismic with 3 x 3 alpha-trim mean Azimuth from curvature by correlation


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Azimuth comparison

36November, 2012

Instantaneous Azimuth Azimuth from curvature by correlation


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Curvature and instantaneous attributes

37November, 2012

Differentiating the Roberts quadratic, we find that atx = y = 0

we get the following relationships for the coefficients:

qypxxyx

q

y

py

y

qx

x

pyxt

2

1

2

1

2

1),( 22

.and,2

1,

2

1,

2

1qepd

x

q

y

pc

y

qb

x

pa

This leads to the following quadratic relationship:

Thus, all of the curvature attributes can be derived fromthe instantaneous dip attributes described earlier, using

a second differentiation.

The next figure shows a comparison between maximum

curvature derived the two different ways.


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Max Curvature Comparison

38November, 2012

Instantaneous maximum

curvatureCorrelation maximum

curvature

High

Low

Data slice

Notice that the curvature features are similar, but that

instantaneous curvature shows higher frequency events.


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Possible instability

39November, 2012

However, there is a lot of complexity hidden in the second

differentiation.

To see this, lets expand thepterm:

volume.ansformHilbert trandvolumeseismic

:where,)/(

hs

x

t

s

ht

h

sx

s

hx

h

s

x

k

x

p x w

Performing this differentiation will produce a large number of

seismic and Hilbert transform derivatives in both the

numerator and denominator, which can cause instability in

some datasets.


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Amplitude curvature

40November, 2012

2

2

2

2

,,,y

sx

sys

xs

To avoid this instability, Chopra (2012) recommends a new

approach to curvature called amplitude curvature (as

opposed to the previous structural curvature), which involves

first and second derivatives of only the seismic data:

In the above options, arepresents the amplitude

curvature at an azimuth angle and emeanis the mean

amplitude curvature.

Examples are shown in the next slides.

.2

1andsincos

2

2

2

2

y

s

x

se

y

s

x

ssa mean


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X derivative

41November, 2012

Seismic with 3 x 3 alpha-trim mean Amplitude derivative in x-direction


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Y derivative

42November, 2012

Seismic with 3 x 3 alpha-trim mean Amplitude derivative in y-direction


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45oazimuth amp curvature

43November, 2012

Seismic with 3 x 3 alpha-trim mean Amplitude curvature at 45oazimuth


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44

The phase congruency method

This concludes our discussion of the most commonattributes available to interpreters.

But I want to finish with yet another spatial attribute, which

is not as commonly used.

This attribute is called the phase congruency algorithm(Kovesi, 1996), and was developed to detect corners and

edges on 2D images.

The algorithm is applied on a slice-by-slice basis to the

seismic volume and produces results that are similar tocoherency.

The following few slides give a brief overview of the theory

and then an application to our dataset.

November, 2012


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45

The phase congruency method

The concept behind the phase congruency algorithm inthe 1D situation is shown below:

Note that the Fourier components in

the above figure are all in phase for

a step.

By lining these components up

as vectors, we can measure the

phase congruency.

Figures from Kovesi, 1996

E(x)

f(x)

Imaginary

Real

An

)(xf

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Phase congruency flow chart

Input data slice

inx-yspace

Data slice in

kx-kyspace

2D FFT

Create Nradial

filters in kx-kyspace

Create M angular

filters in kx-kyspace

Multiply to create

N*Mfilters

Inverse 2D FFT

Apply filters

Apply moment

analysis

Normalize and sum radial

terms for each angle

Maximum moment

= edges

Minimum moment

= corners


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Seismic Volume Time slices

FFT

Frequency partitioning Phase spectrum at Freq

PartitionsMap the Phase

Phase Congruency on 3D Seismic

The phase congruency algorithm is applied to seismic slices,

as shown above.November, 2012 47


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Movie of seismic slices

48

Animation of seismic amplitude

for each vertical slice in the

Boonsville dataset.

Animation of seismic amplitude

for each time slice in the

Boonsville dataset.


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Coherency and phase congruency

49

Animation of seismic

coherency for each time slice

in the Boonsville dataset.

Animation of seismic phase

congruency for each time slice

in the Boonsville dataset.


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Seismic vs Phase Congruency

50

A vertical slice showing karst features superimposed on a

horizontal slice at 1080 ms, roughly halfway through the karst

collapse. Seismic on left and the phase congruency on right.

Note the good definition of the collapse on phase congruency.November, 2012


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Seismic vs Coherency

51

A vertical slice showing karst features superimposed on a

horizontal slice at 1080 ms, roughly halfway through the karst

collapse. Seismic on left and coherency on right.

Note the definition of the collapse on the coherency volume.November, 2012


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Conclusions

In this document, I have given an overview of most of the

commonly used attributes and how they are related.

The most basic attributes are derived from the Fourier

transform, but cannot be localized.

Instantaneous attributes involve combinations of the

derivatives of the seismic amplitude volume and its Hilbert

transform, and were initially just done in the time direction.

Correlation attributes involve computing the cross-correlation

between pairs of traces in the inline and crossline directions.

Coherency is based on the correlation amplitude and

curvature on the correlation time-shift.

Instantaneous attributes in all three seismic directions can be

combined to give dip, azimuth and curvature.

52November, 2012


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Conclusions (continued)

Finally, we discussed the phase congruency algorithm which

involved both the 2D Fourier transform and radial filters, and

looks at where the Fourier phase components line up.

I illustrated these methods using the Boonsville dataset,

which was over a gas field trapped in karst topography.

It is important to note that the seismic volume itself is

probably our best seismic attribute!

However, each of the attributes that I discussed has its own

advantages and disadvantages when used to extend the

interpretation process.

By helping to show the relationships among the various

attributes, I hope that I have shed some light on when these

attributes can be used to best advantage by interpreters.

53November, 2012


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Acknowledgements

I first met instantaneous attributes while working at Chevron

Geophysical in Houston in 1979 and am indebted to TuryTaner, Fulton Koehler and Bob Sheriff for introducing them.

I became fascinated by the coherency and spectral decomp

attributes introduced by Mike Bahorich, Kurt Marfurt, Greg

Partyka and their colleagues at Amoco in the 1990s, but didnot at first see their relationship to the earlier methods.

I also failed to see the importance of Art Barnes work on 2D

and 3D instantaneous attributes when it first appeared, but

now appreciate how important his work has been. The ongoing work by Professor Kurt Marfurt and his students

also provided the inspiration for much of this talk.

Finally, my thanks go to Satinder Chopra for both his work on

attributes and his book, co-authored with Kurt Marfurt. 54November, 2012


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References

55November, 2012

al-Dossary, S., and K. J. Marfurt, 2006, 3D volumetric multispectral estimates of

reflector curvature and rotation: Geophysics, 71, 4151.

Bahorich, M. S., and S. L. Farmer, 1995, 3-D seismic discontinuity for faults and

stratigraphic features, The coherence cube: The Leading Edge, 16,

10531058.

Barnes, A. E., 1996, Theory of two-dimensional complex seismic trace analysis:

Geophysics, 61, 264272.

Cohen, L., 1995, Time-Frequency Analysis: Prentice-Hall PTR.

Gabor, D., 1946, Theory of communication, part I: J. Int. Elect. Eng., v. 93,

part III, p. 429-441.

Gersztenkorn, A., and K. J. Marfurt, 1999, Eigenstructure based coherencecomputations as an aid to 3D structural and stratigraphic mapping:


Klein, P., L. Richard and H. James, 2008, 3D curvature attributes: a new

approach for seismic interpretation: First Break, 26, 105111.


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References

Marfurt, K. J., 2006, Robust estimates of 3D reflector dip: Geophysics, 71, 29-40

Marfurt, K. J., and R. L. Kirlin, 2000, 3D broadband estimates of reflector dip and

amplitude: Geophysics, 65, 304320.

Marfurt, K. J., R. L. Kirlin, S. H. Farmer, and M. S. Bahorich, 1998, 3D seismic

attributes using a running window semblance-based algorithm:


Marfurt, K. J., V. Sudhakar, A. Gersztenkorn, K. D. Crawford, and S. E. Nissen,

1999, Coherency calculations in the presence of structural dip:

Geophysics 64, 104111.

Roberts, A., 2001, Curvature attributes and their application to 3D interpreted

horizons: First Break, 19, 85100.

Taner, M. T., F. Koehler, and R. E. Sheriff, 1979,Complex seismic trace analysis:


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HRS a New Look at Seismic Attributes