A new look at seismic attributes

Brian Russell

Introduction

Seismic attributes have evolved into a set of seemingly

unrelated 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.

2 November, 2012

The Boonsville dataset

3

The Boonsville gas

field 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 Texas

Bureau of Economic

Geology.

The study area is

shown on the left. Hardage et al., 1996

November, 2012

4

Boonsville Geology

In the Boonsville gas field, production is from the Bend

conglomerate, a middle Pennsylvanian clastic deposited in a

fluvio-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 features

which extend up to 760 m from basement through the Bend

conglomerate, shown here in schematic:

Lucia (1995) November, 2012

A seismic volume

The 3D seismic

volume is shown here

in grey level variable

density format.

It consists of 97

inlines and 133

crosslines, each with

200 samples (800–

1200 ms).

The karst features are

illustrated by the red

ellipses.

5 November, 2012

Crossline or

y direction Inline or

x direction

Time or t

direction

Slices through the volume

Inline seismic section 146

6 November, 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.

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.

7 November, 2012

The Fourier transform

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

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

8 November, 2012

We can also perform the inverse transform from frequency

back to time.

In using the FT for attribute analysis, note that it operates

on 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.

.)( of spectrum, phase )( and1 spectrum, amplitude)(

),( of ransform Fourier t the)( frequency,,2

where,)()()( )(

tsiS

tsSff

eSSts i

ww

ww

ww w

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. 9 November, 2012

frequency

y

x

time

x y

Fourier

Transform

Spectral slices

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

Data slice 5 Hz 10 Hz 15 Hz 20 Hz

25 Hz 30 Hz 35 Hz 40 Hz 45 Hz

Spectral decomposition limitations

As seen in the previous slide, each frequency slice in a

spectral decomposition reveals a different underlying channel

structure in the original seismic amplitude slice.

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

don’t 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.

11 November, 2012

The complex time signal

Instantaneous attributes are based on the concept of the

complex 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:

12 November, 2012

.)( of ransform Hilbert t the)( and

,1 signal,complex the)(

: where)()()(

tsth

itz

tihtstz

The Hilbert transform

The Hilbert transform is a filter which applies a 90o phase

shift 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 amplitude A(t) and instantaneous phase F(t):

13 November, 2012

f(t)

time

h(t)

A(t)

s(t)

.)(

)(tan)(,)()()(

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

)()()()(

122

)(

F

FF

F

ts

thtthtstA

ttiAttA

etAtihtstz ti

The complex trace was introduced into

geophysics by Taner et al. (1979), who

also discussed its digital implementation.

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: “Théorie et applications

de la notion de signal analytique”.

The instantaneous frequency is the time derivative of the

instantaneous phase:

14 November, 2012

( )2

1

)(

)()(

)()(

)(/)(tan)()(

tA

dt

tdsth

dt

tdhts

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 90o phase

shift, but it also applies a high frequency ramp.

Instantaneous attributes

This figure shows

the instantaneous

attributes

associated with

the seismic

amplitude slice at

1000 ms.

15 November, 2012

Data slice -1.0

+1.0

Inst. Amp.

+1.0

0.0

Inst. Phase -180

o

+180o

Inst. Freq.

0 Hz

120 Hz

New attributes

Attribute analysis until the mid-90s was thus based on the

three 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.

16 November, 2012

The coherency method

The first coherency method involved finding the maximum

correlation coefficients between adjacent traces in the x

and y directions, and taking their harmonic average.

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

semblance of all combinations of J traces in a window.

17 November, 2012

This involves searching over

all x dips p and y dips q, over

a 2M + 1 sample window:

Marfurt et al. (1998)

x

y

p dip q dip

2M+1 samples

Coherency mathematics

If the covariance matrix between all locations i and j is:

18 November, 2012

),()( ,),(

1

111

jjjii

tM

tMt

iij

JJJ

J

qypxtsqypxtsc

cc

cc

qpC

.),( of diagonal main of sum ),( and 1,,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:

).,( of eigenvaluefirst ,

),(max 1

13 qpC

qpCTrcoh

A third measure, called eigen-coherence (Gursztenkorn and

Marfurt, 1999), is:

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.

19 November, 2012

Data slice -1.0

+1.0

0.7

1.1

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 the x and y dips, called p and q.

The paper by Barnes (1996) introduced two new spatial

instantaneous attributes and showed how these attributes

could 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 and

phase 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. 20 November, 2012

New instantaneous attributes

Re-visiting our earlier

3D 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 compute

spatial frequencies in

the inline (x) and

crossline (y) directions.

21 November, 2012

Crossline or

y direction Inline or

x direction

Time or t

direction

Instantaneous wavenumber

Analogous to instantaneous frequency, Barnes (1996)

therefore defined the instantaneous wavenumbers kx and

ky (using the Marfurt (2006) notation):

22 November, 2012

and ,),,(

2A

x

sh

x

hs

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

yxtk y

F

Dip and azimuth attributes

The instantaneous time dips in the x and y direction, p

and q, are given as:

23 November, 2012

.h wavelengt and period:where

, and

ww

T

Tkq

Tkp

y

y

x

x

Using the dips p and q, the azimuth f and true time dip

q are then given by:

. and

,)/(tan)/(tan

22

11

qp

kkqp yx

q

f

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. 24 November, 2012

T = 50 ms

= 36 traces

dip = 1.4

ms/trace

Dip and azimuth attributes

Next, let’s 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 and

azimuths are related

to the instantaneous

frequency and

wavenumbers. 25 November, 2012

kx

w

ky

p q

kx

w

ky

q

kx

ky f

Examples

Next, we will illustrate these instantaneous spatial

attributes using the Boonsville dataset.

There are many possible displays based on the “building

blocks” for these attributes.

These include the derivatives of seismic amplitude, Hilbert

transform and phase in the t, x, and y directions, the three

frequencies, and p, q, true dip and azimuth.

Note that the many divisions involved in the computations

make this method very sensitive to noise when compared

to correlation based methods.

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

26 November, 2012

True dip volume

27 November, 2012

Seismic with 3 x 3 alpha-trim mean True dip

Azimuth volume

28 November, 2012

Seismic with 3 x 3 alpha-trim mean Azimuth

Curvature attributes

Roberts (2001) shows that curvature can be estimated from

a time structure map by fitting the local quadratic surface

given by:

29 November, 2012

feydxcxybyaxyxt 22),(

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

Curvature attributes

Roberts (2001) computes the curvature attributes by first

picking a 3D surface on the seismic data and then finding the

coefficients a through f from the map grid shown below:

30 November, 2012

x

tttttt

x

td

2

)()( 741963

As an example, the coefficient d,

which is the linear dip in the x

direction, 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.

Curvature attributes

The coefficients d and e are identical to dips p and q defined

earlier, so when a = b = c = 0, we have a dipping plane and

can also define the true dip and azimuth as before.

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

curvature attributes (Kmin and Kmax are shown on the surface):

31 November, 2012

,

,

2

min

2

max

gaussmeanmean

gaussmeanmean

KKKK

KKKK

.)1(

)1()1(

and,)1(

4 :where

2/322

22

2/122

2

ed

cdedbeaK

ed

cabK

mean

gauss

Curvature attributes

Note that the inverse relationships are:

32 November, 2012

. and 2

maxminmaxmin KKK

KKK gaussmean

Also, can compute most positive and negative curvature K+

and K-, which are equal to Kmin and Kmax with d = e = f = 0 (or,

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

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

Finally, we have Keuler, where d is an azimuth angle:

)90()0(),45(

sincos)(

ooo

2

max

2

min

eulereulergausseulermean

euler

KKKKK

KKK

ddd

The next few slides show some examples from our dataset.

Minimum Curvature

33 November, 2012

Seismic with 3 x 3 alpha-trim mean Minimum curvature

Maximum Curvature

34 November, 2012

Seismic with 3 x 3 alpha-trim mean Maximum curvature

Azimuth from Curvature

35 November, 2012

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

Azimuth comparison

36 November, 2012

Instantaneous Azimuth Azimuth from curvature by correlation

Curvature and instantaneous attributes

37 November, 2012

Differentiating the Roberts quadratic, we find that at x = y = 0

we get the following relationships for the coefficients:

fqypxxyx

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 from

the instantaneous dip attributes described earlier, using

a second differentiation.

The next figure shows a comparison between maximum

curvature derived the two different ways.

Max Curvature Comparison

38 November, 2012

Instantaneous maximum

curvature Correlation maximum

curvature

High

Low

Data slice

Notice that the curvature features are similar, but that

instantaneous curvature shows higher frequency events.

Possible instability

39 November, 2012

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

differentiation.

To see this, let’s expand the p term:

volume.ansformHilbert tr and volumeseismic

:where,)/(

hs

x

t

sh

t

hs

x

sh

x

hs

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.

Amplitude curvature

40 November, 2012

2

2

2

2

,,,y

s

x

s

y

s

x

s

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, a represents the amplitude

curvature at an azimuth angle and emean is the mean

amplitude curvature.

Examples are shown in the next slides.

.2

1 and sincos

2

2

2

2

y

s

x

se

y

s

x

ssa mean

X derivative

41 November, 2012

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

Y derivative

42 November, 2012

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

45o azimuth amp curvature

43 November, 2012

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

44

The phase congruency method

This concludes our discussion of the most common

attributes 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 to

coherency.

The following few slides give a brief overview of the theory

and then an application to our dataset.

November, 2012

45

The phase congruency method

The concept behind the phase congruency algorithm in

the 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

November, 2012

Phase congruency flow chart

Input data slice

in x-y space

Data slice in

kx-ky space

2D FFT

Create N radial

filters in kx-ky space

Create M angular

filters in kx-ky space

Multiply to create

N*M filters

Inverse 2D FFT

Apply filters

Apply moment

analysis

Normalize and sum radial

terms for each angle

Maximum moment

= edges

Minimum moment

= corners

Seismic Volume Time slices

FFT

Frequency partitioning Phase spectrum at Freq

Partitions Map the Phase

Phase Congruency on 3D Seismic

The phase congruency algorithm is applied to seismic slices,

as shown above. November, 2012 47

Seismic vs Phase Congruency

48

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

Seismic vs Coherency

49

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

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.

50 November, 2012

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.

51 November, 2012

Acknowledgements

I first met instantaneous attributes while working at Chevron

Geophysical in Houston in 1979 and am indebted to Tury

Taner, 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 did

not 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. 52 November, 2012

References

53 November, 2012

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

reflector curvature and rotation: Geophysics, 71, 41–51.

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

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

1053–1058.

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

Geophysics, 61, 264–272.

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 coherence

computations as an aid to 3D structural and stratigraphic mapping:

Geophysics, 64, 1468–1479.

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

approach for seismic interpretation: First Break, 26, 105–111.

References

54 November, 2012

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, 304–320.

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attributes using a running window semblance-based algorithm:

Geophysics, 63, 1150–1165.

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1999, Coherency calculations in the presence of structural dip:

Geophysics 64, 104–111.

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

horizons: First Break, 19, 85–100.

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Geophysics, 44, 1041–1063.