1

CHAPTER ONE

STATEMENT OF THE PROBLEM

Natural fractures play an important role in petroleum exploration.

Fractures are found in nearly every reservoir, rock type and depth. Petroleum

explorationists pay a great deal of attention to locating these fractures in order to

build better reservoir models. Fractures can advance or hinder the effort in

understanding reservoir character. Fractures can be found in the source rocks,

reservoir rocks and cap rocks. Locating these fractures and identifying their

orientations can help the explorationists deal with them and benefit from their

presence or avoid their annoyances.

Surface geology, subsurface geology (i.e. core), VSP, production

performance, well logs, surface seismic and more recently direct mapping through

induced micro earthquakes, are tools that explorationists use to detect fractures.

All of these tools work, but to varying degrees. For example, well logs can detect

fractures but their area of investigation is limited laterally to a few meters. Cores

are most reliable but can be expensive and difficult to handle. Surface seismic

have been the most successful because of its greater area of coverage and because

fractures introduce azimuthal anisotropy that can be detected using surface

seismic.

Anisotropy is a general term denoting variations of a physical property

depending on the direction in which it is measured. Rocks that are fractured

exhibit this property and produce an effect known as azimuthal anisotropy where

2

P-wave velocity parallel to open fractures is always larger than the velocity

perpendicular to the fractures.

Azimuthal velocity analysis and azimuthal amplitude versus offset

analysis are the main techniques to investigate fractures (e.g. Paul, 1993; Corrigan

et al., 1996; Lynn et al., 1996). Shear-wave splitting (Lynn and Thompson, 1990;

Lynn et. al. 1995; Gaiser et al., 2002) is another technique that has been used

when 3D S-wave or P-to-S converted waves (PS waves) have been acquired. New

techniques have come out recently which are based on the seismic coherence

attribute (Ortman and Wood 1995; Skirius et al. 1998). Fractures detection using

coherence has been applied, mostly, to post stack data, and only very recently has

it been applied to prestack volumes (Chopra et al., 2000). The technology of

applying seismic coherence prestack is somewhat less mature than the other

methods of detecting fractures.

FRACTURE A fracture is a plane surface that has experienced a loss of cohesion

(Nelson, 1985). Simply put, it is a break or crack in the rock matrix. A fracture is

known as a fault when the rock shows relative displacement, and as a joint when

no such displacement exists (Nelson, 1985).

The nature of the pore structure is affected by diagenetic events and

tectonic activities during or subsequent to sediment burial. Porosity and

permeability can be substantially increased due to the dissolution of the less

chemically stable minerals in sandstones and carbonates, for example. In the case

of carbonates, dolomitization increases porosity by converting limestone to

3

dolostone. Large secondary pores formed by dissolving cement and grains can

enhance the quality of a reservoir rock. In the most prolific hydrocarbon-

producing carbonate reservoirs, the formation transmissibility is improved by a

combination of fracture –induced permeability and vugular channels( Nelson,

1985).

Natural fractures are believed to represent the local stress at the time of

fracturing. Both the maximum and minimum horizontal stress components

increase due to burial. Open fractures require stress relief in at least one direction.

Some of the geologic processes that can lead to stress relief and the formation of

open fractures include differential compaction, thrust faulting, growth faulting,

diapirism and folding.

GENERIC CLASSIFICATION OF FRACTURES All brittle fracture in rock must conform to one of the three basic fracture

types observed to form at consistent and predictable angles during laboratory

compression, extension and tensile tests (Nelson, 1985). These three fracture

types are: shear, extension and tension fractures (Nelson, 1985).

SHEAR FRACTURES Shear fractures seem displaced as they form parallel to the fracture plane.

They form at an acute angle to the maximum compressive principal stress

direction (σ 1) and at an obtuse angle to the minimum compressive stress

direction (σ 3) within the rock sample (Nelson, 1985). In every laboratory

experiment, two shear fractures can develop at an equal angle from either side of

4

σ 1 (Nelson, 1985). In such experiments, these fractures form parallel to σ 2 and at

an obtuse angle to σ 3. When all three principal stresses are compressive, shear

fractures form. The acute angle located between the shear fractures is called the

conjugate angle and depends primarily on the mechanical properties of the

material, the absolute magnitude of the minimum principal stress (σ 3), and the

magnitude of the intermediate principal stress relative to both the maximum (σ 1)

and minimum (σ 3) principal stresses (Nelson, 1985).

EXTENSION FRACTURES Extension fractures manifest perpendicular and away from the fracture

plane. They form parallel to σ 1 and σ 2 and perpendicular to σ 3 and, like shear

fractures, form when all three principal stresses are compressive (positive). In

laboratory experiments, extension fractures often form synchronously with shear

fractures (Nelson, 1985).

TENSION FRACTURES Tension fractures also have a sense of displacement perpendicular to and

away from the fracture plane and form parallel to σ 1 and σ 2. When considering

placement and orientation in terms of σ 1, tension fractures resemble extension

fractures; however, to form a tension fracture, at least one principal stress should

be negative (tensile) (Nelson, 1985).

The quality of fractured reservoirs greatly depends both on the intensity

and volume of the fracture and also, the degree of fracture network

5

communication. Fractured reservoirs are found largely in areas with significant

tectonic histories.

Nelson (1985) divides naturally fractured reservoirs into four general

types from the standpoint of productivity.

Type 1: In type 1 reservoirs, fractures account for the majority of porosity

and permeability. They contain high fracture density, may exhibit sharp

production decline, and -in the case of pressure depletion- can develop early water

or gas coning. Examples of type 1 reservoirs include the Ellenburger Fields in

Texas, La Laz-Mara in Venezuela and Amal in Libya.

Type 2: In type 2 reservoirs, the fractures provide the essential reservoir

conductivity. The nature of interporosity flow must be identified for infill drilling

or implementation of improved recovery processes. Examples include Agha Jari

and Haft Kel in Iran and the Rangely Field in Colorado.

Type 3: In type 3 reservoirs, there is adequate permeability to flow and

fractures that enhance overall permeability. In most cases, the evidence of fracture

is not clear in the early life of the field and unusual responses during pressure

support by gas or water can be observed due to permeability trends.

Type 4: In type 4 reservoirs, fractures provide no noticeable contribution

to porosity or permeability, but create significant anisotropy.

The recognition of the aforementioned categories is significant because it

can help in planning well locations, establishing a maximum efficient rate of

production and selecting an appropriate, improved oil recovery process.

6

STUDY AREAS

I used conventional p-wave seismic data volumes from two areas: the Fort

Worth Basin of North Texas and the Vinton Dome of southwestern Louisiana.

The study area falls in the Bend Arch-Fort Worth Basin in North-central Texas

(Figure1) and the southwestern corner of Oklahoma (Thompson, 1982; Ball et al.,

1996).

The Fort Worth Basin is a late Plaeozoic foreland basin, that formed along

the advancing border of the Ouachita fold and thrust belt (Walper, 1982) (Figure

2). Its evolution is attributed to the lithospheric plate convergence of the North

American and South American plates in the late Paleozoic. The basin initiated as

a rifted margin during the early Paleozoic, and was submerged by ancient

Paleozoic seas. These seas deposited the carbonate ramp deposits of the Cambro-

Ordovician age -including the Ellenburger and Viola formations (Figure 3).

Silurian and Devonian sediments were either not deposited in the Fort Worth

Basin, or were eroded during the late Paleozoic tectonic uplift (Walper, 1982).

According to Walper (1982), the Barnett shale was deposited in a deep-water

foreland basin setting during the late Mississippian. Shallow water siliciclastics

and carbonate formations such as the Marble Falls and Caddo were deposited as

the basin filled during the Early Pennsylvanian time (Walper, 1982) (Figure 2).

TARGET HORIZONS

The brittle Ordovician through Pennsylvanian carbonates (Ellenburget,

Viola, Marble Falls and Caddo, respectively) and the Barnett Shale, a

7

Mississippian unconventional shale gas reservoir, are the major targets that I used

to test my algorithms.

The Barnett shale in question originated as a normal marine shelf deposit

and rests between Pennsylvanian-age Marble Falls Limestone and the Ordovician-

age Viola Limestone/ Ellenburger group. The formation, which probably

originated due to the Middle or Late Mississippian collision of the North

American and South American and/or North African plates is about 500 feet thick

in the principal area studied. The features of the formation can be summarized as

black, organic-rich shale made up of fine-grained, non-siliciclastic rocks with

particularly low permeability. As a result of its low permeability, hydraulic

fracture treatments are needed to produce gas in commercial quantities.

VINTON DOME

Description of the geology in the Vinton Dome area of southwestern

Louisiana can be broken down into three parts, stratigraphy, structure, and salt.

Understanding of the geology requires addressing each part individually and

addressing the interrelation between each. Stratigraphy is that of a Tertiary age

prograding shelf margin with sediments in the study area ranging in age from

Oligocene to Miocene. Structure is primarily a function of the progradation of the

shelf margin resulting in growth faulting, with synchronous salt movements into

an antithetic fault. A likely scenario for the formation of the Vinton Dome is that

progradation caused growth faulting, producing normal listric faults and antithetic

faults. As the sediments moved basin-ward they moved the tabular Jurassic

8

Louann Salt along the sole of the listric fault plane. As the salt body was

displaced a portion detached and used the antithetic fault as a conduit through

which it moved. The dynamics involved in this process impacted the statigraphy

by curling the horizontal sediments close to the salt body, and creating radial

faulting on the footwall side of the antithetic fault block. The 3-D survey is

unusual in that it was shot in sources in concentric circles around the apex of the

salt dome, and the receivers in spokes radiating out from the apex.

.

FIG. 1-1. Location of the Fort Worth Basin and the Vinton Dome.

Vinton Dome

Fort Worth Basin

9

FIG. 1-2. Map of the Fort Worth Basin and the Ouachita thrust and fold belt (after Walper, 1982)

10

FIG.1-3. General stratigraphy of the Fort Worth basin (after Thompson, 1982)

11

12

12

REFERENCES

Ball, M. M. and Perry, W. J., 1996, Bend Arch-Fort Worth Basin: USG

Report for Province 45.

Chopra, S., Sudhakar, V., Larsen, G., and Leong, H., 2002, Azimuth-

based coherence for detecting faults and fractures: WorldOil

Magazine, 21.

Corrigan, D.,Withers, R., Darnall, J., and Skopinski, T., 1996, Fracture

mapping from azimuthal velocity analyses using 3-D surface

seismic data: 66th Ann. Internat. Mtg., Soc. Expl. Geophys.,

Expanded Abstracts, 1834–1837.

Lynn, H. B., Simon, K. M., and Bates, C. R., 1996, Correlation between P-

wave AVOA and S-wave traveltime anisotropy in a naturally

fractured gas reservoir: The Leading Edge, 15, 931–935.

Lynn, H.B., Simon, K.M., Layman, M., Schneider, R., Bates, C.R., Jones,

M ., 1995, Use of anisotropy in P-wave and S-wave data for

fracture characterization in a naturally fractured gas reservoir: The

leading Edge, 14, 887-893.

Lynn, H.B. and Thomson, L., 1990, Reflection shear wave data collected

near the principal axes of azimuthal anisotropy: Geophysics, 55,

147-156.

Nelson, R. A., 1985, Geological Analysis of naturally fractured reservoirs:

Gulf publishing company.

13

13

Ortmann, K. A. and Wood, L. J., 1995, Successful application of 3-D

seismic coherency models to predict stratigraphy, offshore eastern

Trinidad: 65th Annual Internat. Mtg., Soc. Expl. Geophys.,

Expanded Abstracts, 101-103.

Skirius, C., Nissen, S., Haskell, N., Marfurt, K. J., Hadley, S., Ternes, D.,

Michel, K., Reglar, I., D'Amico, D., Deliencourt, F., Romero, T,.

Romero, R., and Brown, B., 1999, 3-D seismic attributes applied to

carbonates: The Leading Edge, 18,384-389.

Thompson, D. M., 1982, Atoka Group (Lower to Middle Pennsylvanian),

Northern Fort Worth Basin, Texas: terrigenous deposiional

systems, diagenesis, and reservoir distribution and quality: The

University of Texas at Austin, Bureau of Economic Geology

Report of Investigations No. 125, 62 p.

Walper, J. L., 1982, Plate Tectonic Evolution of the Fort Woth Basin, in

Martin, C. A., ed., Petroluem Geology of the Fort Worth Basin and

Bend Arch Area: Dallas Geological Society, 237-251.

14

14

CHAPTER 2

REVIEW OF EDGE PRESERVING SMOOTHING

(To be a chapter in an SEG publication on Geometric attributes, Saleh Al-Dossary and Kurt J. Marfurt, Allied Geophysical Laboratories, University of

Houston)

INTRODUCTION

The suppression of random noise is of great importance prior to the

application of any detection algorithm, whether the algorithm is sensitive to

changes in amplitude (Luo et al., 1996), waveform as measured by coherence

(Bahovich and Farmer, 1995), or vector dip (Luo et al., 1996; Al-Dossary and

Marfurt, 2004). Coherent noise such as mismigrated fault plane reflections and

backscattered ground roll often give false edges. The majority of edge- detection

algorithms that aim to pinpoint local rapid change in seismic data are sensitive to

noise. Unfortunately smoothing algorithms such as the running average will

reduce the noise but will also smear the edges. For this reason, new algorithms

have been developed that can suppress the noise while also preserving the edges.

Two algorithms successfully applied to seismic data include

Edge-preserving smoothing (EPS) (Luo et al., 2002) and

Structure-oriented filtering (SOF) (Hoecker and Fehmers, 2002)

Both of these two algorithms have shown that noise in seismic data can be

removed along reflectors while preserving major structural and stratigraphic

discontinuities.

15

15

In EPS the smoothing process tries to select the most homogenous

window from a suite of candidate windows containing the analysis point. In

contrast, SOF damps the smoothing operation if a discontinuity threshold is

detected.

EDGE PRESERVING SMOOTHING (EPS)

Through a straightforward adjustment of the running-average smoothing

method, EPS aims to resolve the conflict between noise reduction and edge

degradation. EPS, as presented by Luo et al. (2002), avoids smearing major

discontinuities by using multiple overlapping windows. A statistic such as the

variance of the data is evaluated in each of the overlapping windows. That

window that has the best statistic (e.g. the minimum variance) is then subjected to

smoothing by using a mean, median, α-trimmed mean, or other filter. In general,

the chosen window will not span a major discontinuity and thereby not smooth

across it.

1-D EPS

A vertical synthetic seismic section demonstrating the amplitude across an

idealized fault is shown on Figure 1. The amplitudes on the marked time slice,

which form a step function are displayed in Figure 2a.

Figure 2b shows the same step function, but with added noise. The

application of a 21- point running-average smoothing filter to the noisy function

(Figure 2b) yields the result displayed in Figure 2c in which we see a reduction in

the random noise. This result, however, is paired with a severe alteration in the

16

16

sharp step. Figure 2d is the result of applying a 21- point EPS operator to Figure

2b. The sharp edge/step is preserved and the noise is effectively reduced.

To fully understand how the EPS method works, we will take a 3-point

EPS operator instead of 21 points as an example. We must first calculate the

standard deviations for the following five windows for any given output at index

i.

window 1: ( di-2, di-1, di+0 ),

window 2: (di-1, di+0, di+1),

window 3: (di+0, di+1, di+2),

di signifies the amplitude of the ith sample in the input data. We then select the

window with the minimum standard deviation, determine the average over the

selected window and assign the average as the output at the ith output location.

Repetition of this process will yield the result shown in Figure 2D (assuming a

21-point window is used). For an N point value, we will obtain N candidate

average values for each output location. That window with minimum standard

deviation will be chosen as the ‘best’ window, and its average will be the output

filtered value.

17

17

2-D EPS

A generalization of EPS can be applied to 2-D and 3-D cases. Figure 3

shows that for any output location (x, y) in the 2-D case, we can divide its vicinity

into small fragments and calculate the standard deviation of the input data for

each fragment. The fragment corresponding to the window having the smallest

standard deviation will be selected and its average value used as the output for the

location (x,y).

Several numerical approaches can be adapted to apply the concept shown

in Figure 3. Figures 4 and 5 show a discrete implementation of a 3x3 window and

a 5x5 window respectively. Larger windows are used for nosier input data and the

vicinity of a output point can be divided into more than nine pieces.

3-D EPS

3D volume seismic data are often used in interpretation and manifests the

structures in the subsurface in 3 dimensions. To apply the method above to 3-D

seismic data, we must extend EPS from 2D to 3D. The concept of 3-D EPS is

shown in Figure 7.

Figure 6 shows the surrounding area of an outpoint point divided into a

number of wedge-shaped pieces. An average and a deviation can be calculated for

each wedge, and the average value generated in the wedge with minimum

standard deviation will be assigned as the output at the analysis point. This

implementation for 3-D EPS is similar to 2-D; however, each sample in 2-D EPS

now includes a few samples of a seismic trace.

18

18

STRUCTURE –ORIENTATION FILTERING

In order to successfully apply oriented smoothing to seismic data, three

steps must be taken. Foremost, the local orientation or “orientation analysis” of

the reflections, followed by the possible reflection terminations or “edge

detection,” must be determined. Additionally, the data must be smoothed in the

direction of the local orientation without filtering across detected edges; this

process is known as “smoothing with edge preservation.”

Each of these steps can be executed through a mass of approaches and has

been studied in academia. Studies have, however, been carried out with the

optimization for noise suppression in seismic data.

Although many image processing methods have been published,

orientated smoothing has only been documented since the 1990s, when Welkert

established anisotropic diffusion.

APPLICATION

Hoeker and Fehmens (2002) implemented edge-preserving orientated

smoothing in two manners. Their first first algorithm determines a 2-D platelet of

seismic amplitudes from 3-D seismic data, following the local structure. The

result is then written back into a 3-D output cube using EPS. Edge- preservation

tests have found simple median filters inadequate when the filter size is increased.

Kuwahara-type methods (Kuwahara, 1976) have established better edge

preservations. The statistics are computed over a set of sub-regions to determine a

19

19

possible edge. Sub-regions that exhibit deviating statistics are expected to contain

edges and are therefore assigned smaller weights in filtering. After its 1997

development, the filter became a workhorse in many Shell operating companies

(Hoecker and Fehmars, 2002).

In 1999, the “van Gogh” filter was developed (Hoecker and Fehmars,

2002). The filter is based on a 3-D implementation of the anisotropic diffusion

technique and introduced a new generation of edge- preserving oriented

smoothing. This process allows the filtering to be carried much further than SOF-

EPS filtering allows. Figure 7 illustrates this nicely. One can see that incoherent

noise and small stratigraphic features are suppressed by this method and, in

addition, both the continuity of faults is improved and the acuity of faults is

salvaged and sometimes enhanced. This “van Gogh” filter simplifies the structural

image by straightening undulating functions and eliminating minor fault-like

features, and ultimately simplifying the structure to its most basic form.

CONCLUSION

Edge-preserving smoothing algorithms have shown that noise in seismic data can

be removed along reflectors while preserving major structural and stratigraphic

discontinuities. We have reviewed the two most recently published algorithms

namely: edge-preserving smoothing (Luo et al., 2002)and structure-oriented

filtering (Hoecker and Fehmars, 2002). Both of these algorithms have improved

the edge detection capabilities of the coherence calculations and therefore have

helped the interpretation tasks to the study areas where they were applied.

Amplitude on t he marked time slice

FIG. 2-1. An idealized fault on a vertical seismic section. The amplitude extraction along the time is shown above.

(c) (d)

(a) (b)

FIG. 2-2. Concepts of EPS. (a)Input step function; (b)noise-added step function ; (c)result after regular smoothing; and (d)result after application of EPS (after Luo et al., 2002).

20

20

FIG. 2-3. Concept of 2-D EPS.

21

21

FIG. 2-4. Discrete realization of 2-D EPS operator in a 3x3 window. (a). One triangle and one square operator. There should be four triangle and four square operators in total (not depicted) (b) Central operator.

FIG. 2-5. Discrete realization of 2-D EPS operator in a 5x5 window. (a). One pentagonal and one hexagonal operator. There should be four pentagonal and four hexagonal operators in total (not depicted) (b) Central operator.

22

22

FIG. 2-6. Concept of 3-D EPS

23

23

t

(a) (b) FIG. 2-7. (a) Vertical seismic section before and (b) after applying structure oriented filtering (after Hoecker & Fehmers, 2002).

24

24

REFERENCES

Al-Dossary, S., and Marfurt, K. J., 2004, 3-D volumetric multispectral estimates

of reflector curvature and rotation: Submitted to Geophysics.

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

stratigraphic features: The coherence cube, The Leading Edge, 14,1053-

1058.

Hocker, C, and Fehmers, G., 2002, Fast structural interpretation with stucture-

oriented filtering: The Leading Edge, 21, 238-243.

Kuwahara, Hachimura, and Kinoshita., 1976, Digital Processing of Biomedical

Images., Plenum Press, 187-203.

Luo, Y., Higgs, W. G. and Kowalik, W. S., 1996, Edge detection and stratigraphic

analysis using 3-D seismic data, 66th Ann. Inter. Mtg. Soc. Expl.

Geophys. Expanded Abstract, 324-327.

Luo, Y., Marhoon, M., al-Dossary, S., and Alfaraj, M., 2002, Edge-preserving

smoothing and applications: The Leading Edge, 21, 136-158.

Nago, M. and T. Matsuyama, 1980, Complex Aerial Photographs, Kyoto

University.

25

25

CHAPTER 3

AN IMPROVED 3-D EDGE DETECTION ALGORITHM

(Presented at the 72nd annual International Meeting: Society of Exploration Geophysicists, Saleh Al-Dossary, Kurt J. Marfurt Allied Geophysical

Laboratories, University of Houston and Yi Luo Saudi Aramco)

ABSTRACT

We have developed a new algorithm that detects discontinuities or “edges”

in seismic data to reveal seismic discontinuities such as faults and channels. The

new algorithm is based on an edge-preserving smoothing method. Edge detection

based on edge-preserving smoothing (EPS-edge) is a robust algorithm that

combines the strength of the edge preserving smoothing method to suppress noise,

and the differencing method to detect faults and channels. We have successfully

applied the new method to real and synthetic data and we find that it can detect

edges while it also can suppress noise.

INTRODUCTION

The seismic expression of structural and stratigraphic discontinuities such

as faults and channels may include lateral variation in waveform, lateral variation

in dip, and lateral variation in amplitude. Estimates of seismic coherence (e.g.

Bahorich and Farmer, 1995; Marfurt et al., 1998, Gertzenkorn and Marfurt, 1999;

Marfurt and Kirlin, 2000) provide a quantitative measurement of the changes in

waveform across a discontinuity. Estimates of apparent dip ( e.g. Dalley et al.,

26

26

1989; Luo et al., 1996; Marfurt et al., 1998: Marfurt and Kirlin, 2000; Luo et al.,

2001) provide a measure of change in reflector dip/azimuth across a discontinuity.

Estimates of amplitude gradients (e.g. Marfurt and Kirlin, 2000) provide a

measure of changes in reflectivity across a discontinuity. All three of these spatial

seismic attributes are coupled with the geology, such that they give

mathematically independent, but geologically complimentary, images of lateral

changes. Most of these algorithms work well when the data quality has a signal-

to-noise ratio greater than 1:1. Unfortunately, seismic data quality is sensitive to

errors in migration velocity estimation, and often deteriorates near faults,

fractures, and channel edges where the velocity may vary rapidly due to

differential sedimentary deposition, pressure compartmentalization, diagenesis,

and in the case of fractures - anisotropy. Such local defocusing caused by under or

overmigration, provides blurred coherence images and smoothed estimates of

dip/azimuth and amplitude variations. Preprocessing such as f-xy deconvolution

can help sharpen such images. More effective are the recently derived algorithms

that smooth along estimates of instantaneous dip, including edge-preserving

smoothing (EPS) (Luo et al., 2002) and structure-oriented filtering (Hocker and

Fehmers, 2002). The EPS method works by resolving the conflict between the

noise reduction and edge degradation. It can suppress noise while keeping sharp

edges intact.

In this paper, we present a new algorithm, which detects edges in seismic

data by expanding the application of the EPS method and using it as differencing

operator. In this way, we have achieved two objectives in one step. The first

27

27

objective is to suppress random noise and the second objective is to estimate

amplitude gradients.

EDGE DETECTION BASICS

Luo and Higgs (1996) developed the first amplitude-sensitive edge detector using

simple mathematical derivative operations applied to a seismic time slice or

horizon. The first derivative, ∂/∂x will be maximum at an edge while the second

derivative, ∂²/ ∂x², will be zero at the edge where the input has its steepest

gradient.

Seismic time slice and horizon image are of discrete sample values which

may be defined by the function f (x,y). The partial derivative of a variable f(x,y)

can be defined as:

∂f(x,y)/ ∂x ≈ [f(x - ∆x,y) – f(x+∆x,y)]/2∆x , (3-1)

where ∆x is the CDP separation.

The formula above displays symmetric behavior and can be approximated by the

filter

Dx = ½[1 0 –1] , (3-2)

applied to 3 corrective samples, and is equivalent to calculating the difference

between the two neighboring and dividing by two. Similarly, in the y direction,

we can construct the filter:

Dy =1/2[1 0 –1], (3-3)

28

28

which apply to 3 adjacent samples in the crossline direction. These one

dimensional operators will detect edges perpendicular to their respective operator.

Luo and Higgs (1996) next define a vector operator that detects edges

independent of orientation:

E2=(Dx*Dx+Dy*Dy)1/2. (3-4)

An alternative edge operator can be defines as:

E1 =|Dx|+|Dy|. (3-5)

ALGORITHM DESCRIPTION

One of the basic tenets of EPS is to remove noise without blurring sharp

edges and the main idea of the difference method is to detect edges or

discontinuities in seismic data. We draw from both of these ideas and combine

them together to image discontinuities such as faults. The EPS method finds the

most homogenous neighborhood around each output point in a 3D seismic cube

and replaces the output point with the average value of the most homogenous

neighborhood.

The difference method for edge detection subtracts seismic signal on

adjacent traces. Now, instead of subtracting adjacent traces, we subtract the most

homogenous traces.

29

29

A very powerful analogy to our new method is to think of a sloping

geological bed that has several dips; however, when you pour water on this

sloping bed the water is going to follow the steepest line or the maximum slope.

Likewise, we start by calculating the most homogenous neighborhood around the

target sample and then filter the value by calculating the amplitude gradients in a

window surrounding the target sample along the instantaneous dip/azimuth.

EXAMPLES

We have applied our algorithm to synthetic and real data. The step

function is used in Figure 1 to illustrate the concept and benefits of EPS-based

edge detection. Figure 1a displays a noise-free step function while Figure 1b the

same function after adding random noise. Applying our edge detection algorithm

to Figure 1a and 1b yields the result in Figure 1c and 1d. We were able to

successfully detect the edge/step and reduce the noise. The synthetic in Figure 2a

is a noise-free “block” image. Figure 2b is identical to Figure 2a but with random

noise added. Figures 2c and 2d show the edges calculate from our new EPS-based

algorithm. We notice how our new algorithm can delineate the boundaries of the

“Block” image in spite of the low signal-to-noise ratio. In Figure 3a we show a

vertical time slice before and after EPS-edge through data collected over Green

Canyon, Gulf of Mexico, USA. In Figure 3b, we show a time slice through the

amplitude gradient attribute cube generated by our algorithm EPS-edge. We note

that the edges are extremely well detected.

30

30

CONCLUSIONS

We have combined two methods (Edge preserving smoothing and

amplitude gradient calculations) to come up with a simple but a powerful

algorithm for edge detection. Our algorithm images discontinuities in seismic data

in spite of noise by finding the most homogenous difference. More importantly,

our results prove that by combining the gradient estimation with edge-preserving

smoothing, our algorithm is very effective in detecting faults and channels even

when our data is of lesser quality.

(c) (d)

(a) (b)

FIG 3-1. Concept of EPS-edge detection. Input step function (a). Noise-added step function (b). (c) and (d) are the results after applying our EPS-Edge detection algorithm.

31

31

a)

b)

c)

d) FIG 3-2. (a) Synthetic 2-D “Block” images without noise. (b) The “Block” image with random noise added. (c) The edges of the “Block” image after applying EPS-edge. (d) The edges of the noisy “Block” after applying EPS-edge.

32

32

(a)

(b)

FIG 3-3. (a) Input amplitude time slice (b) time slice produced by applying EPS-edge detection to input data in (a). Channels are clearly revealed.

33

33

34

34

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

stratigraphic features: The coherence cube, The Leading Edge, 14,1053-

1058.

Dalley, R M., Gevers, E. E. A., Stampli, G. M., Davies, D. J., Gastaldi, C. N.,

Ruijetnberg, P. R., and Vermeer, G. J. D., 1989, Dip and azimuth displays

for 3-D seismic interpretation: First Break, 7, 86-95.

Gersztenkorn, A., and Marfurt, K. J., 1999, Eigenstructure based coherence

computations: Geophysics, 64, 1468–1479.

Hocker, C, and Fehmers, G., 2002, Fast structural interpretation with stucture-

oriented filtering: The Leading Edge, 21, 238-243.

Luo, Y., Higgs, W. G. and Kowalik, W. S., 1996, Edge detection and stratigraphic

analysis using 3-D seismic data, 66th Ann. Inter. Mtg. Soc. Expl.

Geophys. Expanded Abstract, 324-327.

Luo, Y., Al-Dossary, S. and Marhoon, M., 2001 Generalized Hilbert transform

and its application in Geophysics, 71th Ann. Mtg. Soc. Expl. Geophys.

Expanded Abstract, 430-434.

Luo, Y., al-Dossary, S., Marhoon M., and Alfaraj, M., 2002, Edge-preserving

smoothing and applications: The Leading Edge, 21, 136-158.

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

attributes using a semblance-based coherency algorithm: Geophysics, 63,

1150-1165.

35

35

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

and amplitude: Geophysics, 65, 304-320

36

36

CHAPTER 4

LINEAMENT PRESERVING SMOOTHING

(Submitted for publication in Geophysics, Saleh Al-Dossary and Kurt J. Marfurt, Allied Geophysical Laboratories, University of Houston)

ABSTRACT

Recently developed seismic attributes such as volumetric curvature and

energy gradients enhance our ability to detect lineaments. However, since they are

based on derivatives of either dip/azimuth or the seismic data themselves, they

can also enhance high frequency noise. Recently published edge-preserving

smoothing algorithms have shown that noise in seismic data can be removed

along reflectors while preserving major structural and stratigraphic

discontinuities. In one implementation, the smoothing process tries to select the

most homogenous window from a suite of candidate windows containing the

analysis point. A second implementation damps the smoothing operation if a

discontinuity is detected. Unfortunately, neither of these algorithms preserves thin

or small lineaments that are only one voxel in width. To overcome this defect, we

have found two algorithms developed in the image-processing and synthetic

aperture radar (SAR) world that work well with seismic data and seismic

attributes: (1) the multistage median modified trimmed mean (MSMTM), and (2)

the lower upper middle (LUM) median filters. We have applied these new

algorithms to both synthetic and real 3-D attributes of fractured geology from the

Forth Worth Basin, USA. The proposed algorithms clearly show that thin and

small features are well preserved while suppressing random noise. The two

37

37

algorithms perform well in preserving linear discontinuities of only one trace in

width as well as the edges separating larger geologic blocks.

INTRODUCTION

Lineaments are found in nearly every reservoir, rock type, and depth.

Petroleum explorationists pay a great deal of attention to locate these lineaments

which are related to fractures in order to understand their reservoirs. Fractures can

advance or hinder our efforts in producing a reservoir. Fractures can be found in

source rocks, reservoir rocks, and cap rocks. Locating these fractures and

identifying their orientations can help the explorationists deal and benefit from

them or avoid the problems they pose. Geometric attributes are particularly

effective in delineating lineaments which may be related to fracture zones or

subseismic faults (Blumentritt et al., 2003; Sullivan et al., 2003; Al Dossary and

Marfurt, 2004).

On seismic time slices, lineaments are often seen as small and thin linear

features. Possible causes for the seismic contrast that causes fractures to be visible

include gas charge, porosity preservation, stress release, diagenetic alteration, and

crack fill.

Seismic attributes such as coherence (e.g. Bahorich and Farmer, 1995;

Marfurt et al., 1998, Gertzenkorn and Marfurt, 1999; Marfurt and Kirlin, 2000)

provide a quantitative measure of the changes in waveform across a discontinuity.

Estimates of apparent dip (e.g. Dalley et al., 1989; Marfurt et al., 1998; Marfurt

and Kirlin, 2000; Luo et al., 2003) provide a measure of change in reflector

38

38

dip/azimuth. Estimates of amplitude gradients (e.g. Luo et al., 1996; Marfurt and

Kirlin, 2000; Marfurt, 2004) provide a measure of changes in reflectivity.

Skirius et al. (1999) used seismic coherence in carbonates in North

America and the Arabian Gulf to detect fault and lineaments. Luo et al. (2002)

showed some examples from Saudi Arabian carbonate fields where coherence

helped delineate lineaments.

While coherence can often detect lineaments, reflector curvature is more

directly linked to fracture distribution (Lisle, 1994; Roberts, 2001; Bergbauer et

al., 2003). Hart et al. (2002) have used horizon attributes, including various

curvature attributes, to identify structural features that may be associated with

lineament-swarm sweet spots. Stewart and Wynn (2000) pointed out that it might

be necessary to examine curvature at various scales in order to account for

different wavelengths.

Roberts (2001) and Al-Dossary and Marfurt (2004) stated that volumetric

estimation of reflector curvature should be possible. Al-Dossary and Marfurt

(2004) found the most positive and negative curvatures, kpos and kneg , to be the

most useful for delineating faults, lineaments, flexures, and folds. Blumentritt et

al. (2004) used volumetric curvature attributes to determine stress regime and the

most likely direction of open fractures on a field –wide basis.

All these attributes can be contaminated by seismic noise. Noise filtering

can generally enhance the behavior of coherence, energy gradients, curvature, and

other edge detection algorithms applied to seismic data. The quality of such edge

detectors, and the reliability of the interpretation are directly related to the

39

39

effectiveness of the noise reduction filters applied prior to the calculation.

Historically, linear mean and non-linear median filters have been widely used to

improve the interpretability of the seismic data. Unfortunately, the mean filter can

severely blur coherence and other edge sensitive attributes. The edge-preserving

and impulse-removing properties are the most desirable features of the median

filter (Schulze and Pearce, 1991). While median and related alpha-trim mean

filters can preserve edges by separating fault blocks and stratigraphic features that

are several traces in width, they will, in general, obliterate narrow curvilinear

features associated with joints and fractures that are only a single trace wide.

More recently, Luo et al. (2002) have proposed a new Edge-Preserving

Smoothing algorithm (EPS). EPS attempts to resolve the conflict between noise

reduction and edge degradation via a simple modification of the running-average

smoothing method. In principle, EPS looks for the most homogeneous window

around each sample in an input data set and assigns the average value of the

selected window to that sample (Luo et al. 2002). EPS has been successfully

applied to different data sets from Saudi Arabia and other parts of the world.

However, Luo et al. (2002) admit that their proposed algorithm is inadequate in

preserving small features that are less than three voxels wide, and stated that

genuine geologic features (e.g., channels) would be suppressed if their width were

smaller than the window size. If such small features are the desired output after

running edge-detection, one should design an EPS window that is smaller than the

characteristic width of the expected features, or simply drop EPS from the

processing sequence.

40

40

In order to develop a method they call structurally oriented filtering,

Hoecker and Fehmers (2002) have introduced a forth means of random noise

suppression based on the geometric structure tensor (GST). In their approach,

they first estimate the reflector dip and azimuth by use of the eigenvalues of the

GST. If the coherence, or other measure of similarity is high, they apply a mean

filter to the data along the reflector dip and azimuth. If the reflector coherence is

low, they apply proportionately less smoothing, with no smoothing being applied

for large discontinuities. The smoothing and coherence calculation is applied

recursively, thereby simulating an annealing process. We have found this

approach to be quite robust, but it exhibits the same limitations as the first three

methods when applied to lineaments of only one trace in width.

Our work will differ from the previous literature in two ways. First, our

major focus will be on preserving small lineaments rather than large

discontinuities. Second, we will initially apply lineament-preserving smoothing to

our volumetric estimates of the components of the reflector dip vector rather than

amplitude. Improvements in our vector dip estimate will not only improve our

subsequent edge preserving smoothing, but also our estimates of reflector

curvature (Al-Dossary and Marfurt, 2004), energy gradients, and coherence

(Marfurt, 2004). With this objective in mind, we have evaluated current

techniques used in the image processing and Synthetic Aperture Radar (SAR)

world and adapted those filters that can reduce noise, preserve edges, and preserve

thin crack features often seen on seismic data.

We begin the next section with a summary of alternative filtering

techniques that have been used to smooth seismic data before applying edge

detection or coherence computations. We then briefly describe some of the more

relevant image processing and SAR algorithms, and compare and contrast the

nature of noise in SAR and image processing data and seismic data.

Finally, we apply the two most effective algorithms, the multistage median

modified trimmed mean (MSMTM) filter, and the lower upper middle median

(LUM) filter to a synthetic and to a survey over a fractured and karsted carbonate

terrain from the Fort Worth Basin, USA.

THE MEAN FILTER

The mean filter is the most well-known and simplest random noise

suppression filter. The mean filter is a low-pass filter that is typically

implemented as a running window average filter. The output data value is simply

the average of all the samples that fall within a centered analysis window. The

window size is usually an odd number, such as 3 by 3 or 5 by 5, and may be either

rectangular or elliptical. The definition of the mean filter at time t is:

∑=

=J

jjmean tdtd

1)()( , (4-1)

where dj(t) denotes the jth of J traces falling within the analysis window. In Figure 1a, we show an idealized time slice through two flat lying

reflectors that have different amplitudes and are separated by a NW-SE fault.

Both reflectors are further cut by a system of narrow NE-SW trending

41

41

‘lineaments” of only one trace wide. In Figure 1b, we show the same synthetic

contaminated with random noise. In Figure 2a, we show the effect of applying a 3

by 3 running average mean filter to the image shown in Figure 1b. We note that

the overall signal to noise ratio is improved, but that the fault edge is blurred and

the fractures diminished.

MEDIAN FILTER

The median filter is one of the most widely used nonlinear techniques in

signal and image processing (Schulze and Pearce, 1991). In the seismic world, the

median filter is routinely used in velocity filtering of VSP data to distinguish

between down- and up-going events using the differences in their apparent

velocities. The median filter works by replacing each sample in a window of a

seismic trace by the median of the samples falling within the analysis window.

The window size is typically an odd number (e.g. 3 by 3 or 5 by 5). One way to

calculate the median is simply to order all of the J samples in the analysis window

using an ordering index, k:

)()...()(...)()( )()1()()2()1( tdtdtdtdtd Jjkjkjjj ≤≤≤≤≤ + . (4-2)

The median is then given by:

)()( )2/]1[( tdtd Jkjmedian +== . (4-3)

The α-trimmed mean is given by:

∑−

+=

=J

Jkkj tdtd

)1(

1)( )()(

α

αα , (4-4)

42

42

43

43

where the value of α varies between 0 and 0.5. If α=0.5, we obtain the median

filter. If α=0.0, we obtain the conventional mean filter.

The median filter is well known for preserving sharp discontinuities and

removing impulse noise in the signal. The median and α-trimmed mean filters

performs better than the mean filter in suppressing noise and preserving details, as

seen by the improved fault edge shown in Figure 2b and 2c. However, we note

that neither of these two filters is capable of preserving the thin cracks

EDGE PRESERVING SMOOTHING FILTER Luo et al.’s (2002) edge-preserving smoothing algorithm avoids

smearing major discontinuities by using multiple overlapping windows. A statistic

such as the variance of the data is evaluated in each of the overlapping windows.

That window that has the best statistic (e.g. the minimum variance) is then

subjected to smoothing by using a mean, median, α-trimmed mean, or other filter.

In general, the chosen window will not span a major discontinuity and thereby not

smooth across it. We show the effect of EPS using five overlapping 3x3 windows

on the synthetic shown in Figure 1b, with a mean (Figure 2d), median (Figure 2e)

and α-trimmed mean (Figure 2f) applied to the window having the lowest

variance. While the main fault is enhanced, the narrow lineaments are only

partially illuminated.

44

44

NOISE

Seismic data, SAR, and digital photographic images are rarely noise-free.

Noise can corrupt the image and make the tasks of feature detection and

interpretation difficult. Seismic data can be contaminated by both random and

coherent noise, including electronic and cultural noise, backscattered surface

waves, and variation in surface conditions. In general, we define noise as any part

of the observed waveform that we do not wish to treat as signal. Noise

suppression techniques that range from stacking to f-k filtering to wavelet

transforms are routinely used to improve the signal to noise ratio. The coherent

patterns (broadly described by the term ‘acquisition footprint’ (e.g. Marfurt et al.,

1998b) have the strongest, negative impact on geometric attributes, which

overprint desired fault and crack lineaments with undesired lineaments that

correspond to the source/receiver acquisition program. In general, we expect that

any algorithm that preserves cracks will also preserve acquisition footprint. If at

all possible, we recommend that acquisition footprint be addressed in acquisition

design and prestack data processing stages. Deeper in the seismic section, noise

that leaks through the stack array, processing, and migration, appears to be

random, giving rise to a ‘salt and pepper’ appearance on our geometric attribute

volumes. In contrast, SAR and photographic images contaminated by electronic

noise or ambient noise such as rain or dust are referred to as ‘speckles’.

Thus, the noise model for seismic images, and SAR and digital

photographic images are naturally different. In seismic data, the noise can be

45

45

expressed using an additive model; however, in SAR and photographic image

processing, noise is better expressed using a multiplicative model. Both models,

additive (salt and pepper) and multiplicative (speckles), cause an image to look

uneven, bumpy, or jagged. In general, noise-free images will appear to be piece-

wise smooth. The calculation of such smoothed images is an important step in

edge detection and coherence calculation because it allows us to increase our

ability to resolve fine details within the image and make the image more

interpretable.

ADDITIVE NOISE MODEL

Additive noise is defined as the sum of the intensity of the pixels plus the

noise. Additive noise can be modeled as:

d(t) = s(t) + n(t) (4-5)

where d(t) is the measured data, s(t) is the signal, and n(t) is the additive noise.

Mean, median, and f-x deconvolution have been widely used to smooth the

data with additive noise, especially before seismic edge detection or seismic

coherence. Luo et al. (2002) introduced a particularly effective technique for

smoothing salt and pepper noise before edge detection using their edge-preserving

smoothing (EPS) technique.

46

46

MULTIPLICATIVE NOISE MODEL

Multiplicative noise is defined as the product of the intensity of the pixel

times the noise. Multiplicative noise can be modeled as:

d(t) = s(t)n(t) (4-6)

where s(t) is the signal and n(t) is the multiplicative noise.

Filtering of multiplicative noise has been well investigated in the SAR and

image-processing world. Simple mean and median filters have been supplanted by

more advanced Lee and Frost filters (Lee, 1986; Frost et al., 1982). The more

recently published speckle reducing anisotropic diffusion filters (Yu and Acton,

2002) also are effective in reducing speckle noise. Although these filters were

developed to smooth images with speckle noise, some of them have been

modified to smooth images with additive noise. Others have been developed to

take care of both kinds of noise.

MULTISTAGE MEDIAN BASED MTM FILTER (MSMTM)

Wu and Kundu (1991) proposed the multistage median-based MTM filter

(MSMTM) as a modification and improvement of the modified trimmed mean

(MTM) filter. The MTM filter --an enhancement of the α-trimmed mean filter--

was designed by Lee and Kassam in 1985 to lessen the edge blurring typical of

the standard mean filter. The modified trimmed mean filter is also known as the

range trimmed mean filter. The MTM filter works sufficiently for some images;

however, since it is not a detail-preserving filter, it cannot collect all of the image

details. As a remedy, the MSMTM filter was developed and can equally be used

as a noise filter.

Like the α-trimmed mean filter, the modified trimmed mean filter is a

running window estimator that selects only a subset of the samples inside the

window to calculate an average. In this section, we will simplify our notation by

omitting the argument t (the indication of the time sample), with the

understanding that the analysis window is either along a time or a horizon

(interpreted reflector) slice including the analysis point. The samples, dj, within

the analysis window are selected if they fall within the following range:

qddqd medianjmedian +≤≤− (4-7)

where dmedian is given by equation (3), and q is a pre-selected threshold value

between edge-preservation and smoothing efficiency. Unlike the α-trimmed

mean, the ordered samples are generally selected in a non-symmetric manner,

with the number of data dependent selected samples in any given estimate.

The result of the filter is the average of the selected samples:

, (4-8) ∑=

=J

jjjmedianMTM tddqdbd

1)(),,(

where b(dmedain,q,dj) is the ‘box-car’ function defined as: 47

47

⎭⎬⎫

⎩⎨⎧

=01

),,( jmedian dqdbotherwise

qddqd medianjmedian +≤≤−. (4-9)

q is an important parameter in selection of the samples. If q has a value of zero,

the resulting filter reduces to the median filter. As q increases all of the samples of

the window will eventually be included, such that the filter becomes the mean

filter. Unfortunately, while the MTM filter is good for edge preservation, it is still

based on the median and mean filters, and thus it cannot preserve internal details

such as lineaments.

Wu and Kundu (1991) combined the MTM filter with a detail-preserving filter,

the multi stage median filter, and dubbed the new filter the multistage median

modified trimmed mean (MSMTM). The MSMTM filter is an MTM filter based

on a multistage median (MSM) filter. A data sample is selected if its value falls

into the range of [m – q, m + q] where m is a value-calculated form the data

samples (Wu and Kundu, 1991). Since the MSM filter is a detail-preserving filter,

the MSMTM filter will be able to preserve cracks. The MSMTM filter is efficient,

smoothes noise, and preserves both edges and lineaments. Like all our filters

discussed in this paper, the MSMTM filter is implemented as a running window

estimator. Like the α-trimmed mean and MTM filters, the MSMTM filter selects

a subset of samples inside a window and calculates an average. Like the MTM

algorithm, the samples are selected if they are in the range:

qddqd MSMjMSM +≤≤− . (4-10)

48

48

Now that we define our range with reference to the result of the application of the

multistage median (MSM) filter, dMSM, rather than with reference to the median,

dmedian, as in equation (6), we calculate the multistage median, dMSM, using the

following four steps:

1) Define four one-dimensional linear sub windows Wk align in the N-S, E-W,

and NE-SW and NW-SE of the larger 2-dimensional area (2N+1) by (2N+1)

centered about the trace at (m,n):

W1 = { d(m+i,n) , -N ≤ i ≤ N }, W2 = { d(m+i,n+i) , -N ≤ i ≤ N }, W3 = { d(m,n+i) , -N ≤ i ≤ N }, and W4 = { d(m+i,n-i) , -N ≤ i ≤ N }. (4-11)

2 ) Calculate the median, Z(Wj) of each of the four subwindows :

Z(Wj) = median [ djk ε Wj]. (4-12)

3) Calculate the second stage medians defined as:

M13 = median [Z(W1) ,Z(W3) , dmn], M24 = median [Z(W2) ,Z(W4) , dmn], (4-13)

where dmn is the data value at the center of the analysis window. 4) Finally we calculate the final stage median and obtain the multistage median, dMSM:

dMSM = median [M13 , M24 , dmn]. (4-14)

The result of the filter is the average of the selected samples:

(4-15) ∑=

=J

jjjMSMMSMTM tddqdbd

1

)(),,(

For clarity, we provide an example of 3x3 analysis in Figure 3.

49

49

50

50

We show the effect of MSMTM using 3x3 window on the synthetic shown in

Figure 1b, in Figure 2g.While the noise has been suppressed, the narrow cracks

are well preserved.

LOWER-UPPER-MIDDLE (LUM) FILTER

Boncelet et al. (1991) designed the LUM filter for smoothing and sharpening.

Typically, they use a running 3x3 and 5x5 square window centered about each

analysis point The LUM filter calculates the median by the following four steps:

1) Sort the samples in the window as given by equation (2).

As with the α trimmed mean, the user defines lower and upper order statistics;

only these values will be used in subsequent analysis.

2) The value of the center sample, dC, of the window is compared with these

two order statistics:

For smoothing, the output is taken to be the median of the lower order d(k),

the upper order d(N-k+1) statistics and the center sample dC of the window

will be:

dLUM(k) = med[d(k) , dC, d(N-k+1)] (4-16)

51

51

Thus, the output will be dC if the center value falls within the range of the

“normal” values [ d(k), d(N-k+1) ]. If this is not the case, the output will be the

value of the two order statistics d(k), d(N-k+1) that is closer to dc. These

“extreme” center values are brought in toward the “normal” values.

If k=1, the output is always the same center value, dC. If k=(N+1)/2, the

output is always d(N+1)/2, the median of the window. Therefore, the parameter k

adjusts smoothing from none, (k=1), to that of a median, (k=(N+1)/2).

For sharpening, the output is dC if dC is outside of the range of the

“normal” values [ d(k), d(N-k+1) ] . If not, the output is the closest of the two

order statistics, d(k) or d(N-k+1)).

dLUM(k) = median (d(k) , dC, d(N-k+1)) = dC , if ( dC < d(k) ) or ( dC > d(N-k+1) );

d(k), if ( |dC - d(k)| ) < ( |dC - d(N-k+1)| );

d(N-k+1) if ( |dC – dk)| ) > ( |dC - d(N-k+1)| ).

(4-17)

So, if the center sample is “extreme,” it remains unchanged. If it is “normal,”

the filter will bring it outward to the closer of the two “normal” values.

52

52

If k=(N+1)/2, the output is always the same center value, dC . So, the

parameter k adjusts sharpening from none, (k=(N+1)/2) to the maximum,

(k=1).

For clarity, we provide an example of 3x3 analysis in Figure 4:

We show the effect of MSMTM using 3x3 window on the synthetic shown in

Figure 1b, (Figure 2h). Again while the noise has been suppressed, the narrow

cracks are well preserved.

REAL DATA

In this section we apply our new filters to a data set from the Fort Worth

Basin that is faulted, fractured, and karsted. The faults and fractures (or

lineaments) have little or no displacement or rotation about them. We speculate

that their illumination by curvature attributes is related to velocity changes due to

lateral changes in porosity, diagenetic alternation, gas charge, or crack

cementation. We note in particular that the waveform across these cracks is nearly

unchanged, as quantified by the coherence horizon extraction shown in Figure 5a

and time slice at 0.8 s in Figure 5b.

In Figures 6a and 6b, we show a time slice at 0.8 s through the inline and

crossline dip cubes. Short wavelength curvature estimates (Al-Dossary and

Marfurt, 2004) based on such dip volumes are particularly sensitive to short

wavelength noise. In Figure 6c we see how the quality of our curvature estimates

of crack lineaments deteriorates in those areas covered by the two older surveys.

53

53

In Figure 7a, we show the impact on curvature after applying one pass of edge

preserving smoothing (Luo et al. 2002) to the inline and crossline components of

dip using five overlapping 3x3 windows We notice that while the random noise

has been attenuated, some of the small features indicated by arrows have been

blurred. In Figures 7b and 7c we show the same image after application of the

MSMTM and LUM filters. We notice that our most negative curvature time slices

have been improved and the cracks are well preserved. In particular, note that we

are able to follow lineations (indicated by arrows) through the two areas covered

by the lower quality older surveys. To demonstrate the effectiveness of MSMTM

and LUM filters, we will display the difference between the filtered data and the

unfiltered data. In Figure 8a we show the difference between the unfiltered

negative curvature and the curvature filtered with EPS. We notice that the

difference, not only contains some of the random noise, but also some of the

signal, especially the signal of the linear cracks (circled). Now on Figure 8b we

show the difference between the unfiltered data and data filtered with MSMTM.

We notice that the figure contains less of the signal of the linear features. Finally,

we show the difference between the unfiltered data and the filtered data with

LUM. Again we notice here that our signal has not been filtered out.

CONCLUSIONS

Seismic attributes are sensitive to subtle changes in signal and noise.

Popular random noise suppression algorithms can blur small linear features such

as fractures. To overcome the defect of current noise smoothing and edge

54

54

preserving algorithms, we have found two algorithms developed in the image

processing and SAR world. MSMTM and LUM filters allow us to preserve the

edges as well as preserve the thin and linear features. Both of the algorithms that

we applied produced good results when used in conjunction with coherence, edge,

and curvature detection. The imaging processing world also offered many

algorithms we found to be less applicable to seismic attribute enhancement.

Anisotropic diffusion, speckle reducing, and phase preserving denoising

algorithms produced results we judged to be inferior to MSMTM and LUM for

our application. The algorithms we applied, MSMTM and LUM, are simple to

implement, simple to use, and produce very good results in terms of preserving

details such as thin lineaments and in terms of smoothing noise.

(a)

(b) FIG. 4-1. (a)An idealized time slice through two flat lying reflectors that have

different amplitudes and are separated by a NW-SE fault. Both reflectors are

further cut by a system of narrow NE-SW trending ‘lineaments’ of only one trace

wide. (b) The idealized time slice corrupted by noise.

55

55

(a)

FIG. 4-2. The image in Figure 1b after applying a 3x3 running window (a) mean

filter. (b)median filter, and (c)α-trimmed mean filter.(d) The image in Figure 1b

after applying 3x3 EPS multi window mean (e) median (f) α-trimmed filters (g)

the image in figure 1b after applying the MSMTM and (h) LUM filters. Note that

while the 3x3 mean, median and α-trimmed filters preserve the main fault, the

thick fractures are unacceptably attenuated. In contrast, the MSMTM filter

preserves these smaller features. The more aggressive LUM filter results in holes

in our fractures.

56

56

Figure 4-2 (continued)

(b)

(c)

57

57

Figure 4-2 (continued)

(d)

(e)

58

58

Figure 4-2 (continued)

(f)

(g)

59

59

Figure 4-2 (continued)

(h)

60

60

61

61

Let us define a window (3x3) centered about the 5th sample, d5 :

After sorting, the nine samples we obtain the results in Figure 3b :

d9 d1 d2 d8 d3 d7 d5 d6 d4

1 2 4 7 8 10 11 14 15

Then: W1 = { d2, d5, d8 } = { 4,11,7} with Z(W1) = median(W1) = 7, W2 = { d1, d5C, d9 } = { 2,11,1} with Z(W2) = median(W2) = 2, W3 = { d4, d5, d6 } = {15,11,14} with Z3 = median(W3) = 14, W4 = { d3, d5, d7 } = { 8,11,10} with Z4 = median(W4) = 10, M13 = median [Z(W1) ,Z(W3) , dC] = 11, M24 = median [ZW2 ,ZW4 , dC] = 10, and dMSM = median [M13 , M24 , dC] = 11.

For q = 3, the selected samples are:

d3 d7 d5 d6 9 9 11 14 , such that

d

MSMTM = mean ( d3 , d7 , d5 , d6 ) = 10.75.

d1=2 d2=4 d3=8

d4=15 d5=11 d6=14

d7=10 d8=7 d9=1

FIG. 4-3. Example of MSMTM applied to a 3x3 window of samples.

62

62

- Sort the samples: d9 , d1 , d2 , d8 , d3 , d7 , d5 , d6 , d4 I = { 1 2 4 7 8 10 11 14 15 } - For parameter k = 4: [ d(k) , d(N-k+1) ] = [ d(4) , d(6) ] = [ 7 , 10 ]

- For smoothing: dLUM(k=4) = median(d4 , dC, d6) = median( 7, 11, 10) = 10

- For sharpening: dLUM(k=4) = median(d4 , dC, d6) = median( 7, 11, 10) =

11 FIG. 4-4. Example of LUM applied to a 3x3 window of samples.

(a)

(b)

FIG. 4-5. (a) Principal component coherence along Caddo horizon and (b) on a

time slice, at t=0.800 s (approximately Caddo/Atoka time) through a survey from

the Fort Worth Basin, TX, USA.

63

63

(a)

(b)

FIG.4-6. (a) inline dip (b) crossline dip (c) most negative curvature , kneg, for

spectral components defined by α=1.5 without smoothing.

64

64

Figure.4-6 (continued)

(c)

65

65

(a)

(b)

FIG. 4-7. most negative curvature, kneg, (a) with EPS filter smoothing, (b) with

MSMTM filter smoothing, and (c) most negative curvature, kneg, with LUM filter

smoothing .

66

66

Figure 4-7 (continued)

(c)

67

67

(a)

(b)

(c)

68

68

69

69

FIG. 4-8. The difference between the unfiltered image of kneg shown in Figure

6c and the three smoothed images shown in Figure 7a-c. Arrows in (a) indicate

lineaments that have been attenuated by conventional EPS filter. Note these

lineaments are either smaller in amplitude or non existent in (b)a and (c).

indicating that they were preserved.

70

70

REFERENCES

Al-Dossary, S., and Marfurt, K. J., 2004, 3-D volumetric multispectral estimates

of reflector curvature and rotation: Submitted to Geophysics.

Blumentritt, C.H., Marfurt K. J., and Sullivan, E. C., 2003, Limits to seismic

resolution of stratigraphic features – Applications to the Devonian

Thirtone Formation, Cenetral Basin Platform [Expanded Abstract];

Proceedings, West Texas Geological Society Annual Symposium.

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

stratigraphic features: The coherence cube: The Leading Edge, 14, 1053-

1058.

Boncelet, C., Hardie, R., and Arce, G., 1991, LUM filters for smoothing and

sharpening: Nonlinear image processing II: SPIE, 1451, 70-73.

Dalley, R M., Gevers, E. E. A., Stampli, G. M., Davies, D. J., Gastaldi, C. N.,

Ruijetnberg, P. R., and Vermeer, G. J. D., 1989, Dip and azimuth displays

for 3-D seismic interpretation: First Break, 7, 86-95.

Frost, V. S., Stiles, J. A., Shanmugan, K.S., and Hotzman, J. C., 1982, A model

for radar images and its application to adaptive digital filtering of

multiplicative noise: IEEE Transsaction on Pattern analysis and Machine

Intelligence , 4, 157-166

Gersztenkorn, A., and Marfurt, K. J., 1999, Eigenstructure based coherence

computations: Geophysics, 64, 1468–1479.

Hart, B.S., Pearson, R.A., and Rawling, G.C., 2002, 3-D Seismic horizon-based

approaches to fracture-swarm sweet spot definition in tight-gas reservoirs:

71

71

The Leading Edge, 21, 28-35.

Hocker, C, and Fehmers, G., 2002, Fast structural interpretation with structure-

oriented filtering: The Leading Edge, 21, 238-243.

Lisle, R.J., 1994, Detection of zones of abnormal strains in structures using

Gaussian curvature analysis: AAPG Bulletin, 78, 1811-1819.

Lee, J. S., 1986, Speckle suppression and analysis for synthetic aperture radar

images: Optical Engineering, 25, 636-643

Lee, Y. and Kassam, S., 1985, Generalized median filtering and related nonlinear

filtering techniques: IEEE Transactions on Acoust. Speech, Image Proc.,

33, 672-683.

Luo, Y., Al-Dossary S., Marhoon M., and Alfaraj, M., 2003 Generalized Hilbert

transform and its application in geophysics: The Leading Edge, 22, 198-

202.

Luo, Y., Higgs, W. G. and Kowalik, W. S., 1996, Edge detection and stratigraphic

analysis using 3-D seismic data, 66th Ann. Inter. Mtg. Soc. Expl.

Geophys. Expanded Abstract, 324-327.

Luo, Y., Marhoon, M., Al-Dossary, S., and Alfaraj, M., 2002, Edge-preserving

smoothing and applications: The Leading Edge, 21, 136-158.

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

attributes using a semblance-based coherency algorithm: Geophysics, 63,

1150-1165.

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

and amplitude: Geophysics, 65, 304-320.

72

72

Marfurt, K. J., 2004, Amplitude Gradient : Submitted to geophysics.

Pratt, K. W., 1978, Digital Image Processing: John Wiley & Sons, Inc.

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

horizons: First Break, 19, 85-99.

Schulze, M., and Pearce, J., 1991, Some Properties of the Two-Dimensional

Pseudomedian Filter: Nonlinear image processing II, SPIE, 1451, 48-57.

Skirius, C., Nissen, S., Haskell, N., Marfurt, K. J., Hadley, S., Ternes, D., Michel,

K., Reglar, I., D'Amico, D., Deliencourt, F., Romero, T,. Romero, R., and

Brown, B., 1999, 3-D seismic attributes applied to carbonates: The

Leading Edge, 18,384-389.

Soumekh, M., 1999, Synthetic Aperture Radar Signal Processing with Matlab

Algorithems: John Wiley & Sons, Inc.

Stewart, S.A., and Wynn, T.J., 2000, Mapping spatial variation in rock properties

in relationship to scale-dependent structure using spectral curvature.

Geology, 28, p. 691-694.

Sullivan, E. S., Marfurt, K. J., Lacazette, A., and Ammerman, M., 2003, Bottoms-

up karst: Submitted to Geophysics.

Wu, W.-R., and Kundu, A., 1991, A new type of modified trimmed mean filter:

Nonlinear image processing II: SPIE, 1451, 13-20.

Yu, Y., and Acton, S., 2002, Speckle Reducing Anisotropic Diffusion: IEEE

Transactions on Image Processing, 11, 1260-1270

73

73

CHAPTER FIVE

3-D VOLUMETRIC MULTISPECTRAL ESTIMATES OF REFLECTOR

CURVATURE AND ROTATION (Submitted for publication in Geophysics 3/5/2004, Saleh Al-Dossary and Kurt J.

Marfurt, Allied Geophysical Laboratories, University of Houston)

ABSTRACT While seismic attributes such as acoustic impedance and spectral

decomposition are directly related to porosity and reservoir thickness, geometric

attributes are only indirectly related to reservoir properties. By the use of both

geologic models and paleo and modern geologic analogues, geometric attributes

provide a means of unraveling the history of tectonic deformation and

depositional environment. This in turn allows us to infer petrophysical properties

such as: sand/shale ratios, diagenetic alteration, and the likelihood of fractures.

One of the most accepted geologic models is the relation between reflector

curvature and the presence of open and closed fractures. Such fractures, as well

as other small discontinuities, are relatively small and below the imaging range of

conventional seismic data. Depending on the tectonic regime, structural geologists

link open fractures to either Gaussian curvature or to curvature in the dip or strike

directions. Reflector curvature is fractal in nature, with different tectonic and

lithologic effects being illuminated at the 50 m and 1000 m scales.

Until now, such curvature estimates have been limited to the analysis of

picked horizons. We have developed what we feel to be the first volumetric

spectral estimates of reflector curvature. We find that the most positive and

74

74

negative (also called principal) curvatures are the most valuable in the

conventional mapping of lineament -including faults, folds, and flexures. Such

lineament is mathematically independent of, and interpretatively complementary

to, the well-established coherence geometric attribute. We find the long spectral

wavelength curvature estimates to be of particular value in extracting subtle,

broad features in the seismic data such as block faults, karst, and

compartmentalization. While single trace attributes can be calibrated by vertical

well control, we feel that the calibration of the fracture prediction capability of

geometric attributes will be best addressed using one or more of the following:

horizontal image logs, production history, pressure-transient tests, and tracer tests.

We will illustrate the value of these spectral curvature estimates and compare

them to other attributes through application to two land datasets – a salt dome

from the onshore Louisiana Gulf Coast, and a fractured/karsted data volume from

Fort Worth Basin of North Texas.

INTRODUCTION

The seismic expression of structural and stratigraphic discontinuities such

as faults and channels may include lateral variation in waveform, dip, and

amplitude. Estimates of seismic coherence (e.g. Bahorich and Farmer, 1995;

Marfurt et al., 1998, Gertzenkorn and Marfurt, 1999; Marfurt and Kirlin, 2000)

provide a quantitative measure of the changes in waveform across a discontinuity.

Estimates of apparent dip (e.g. Dalley et al., 1989; Luo et al., 1996; Marfurt et al.,

1998: Marfurt and Kirlin, 2000; Marfurt, 2003) provide a measure of change in

75

75

reflector dip magnitude and azimuth across a discontinuity. Additionally,

estimates of amplitude or energy gradients (e.g. Luo et al., 1996; Marfurt and

Kirlin, 2000; Marfurt, 2003) provide a measure of change in reflectivity

amplitude as energy across a discontinuity. Such discontinuity measures highlight

the boundaries between both: different fault blocks and stratigraphic units, and

hydrocarbon accumulation and diagenetic changes.

Although also used as edge detectors, the later two attributes highlight

subtle changes within the coherent blocks of data. These changes include:

flexures, joints and differential compaction. These geometric seismic attributes

are coupled through geology, such that they give mathematically independent -but

geologically complimentary- images of lateral changes. Each of the two gradient

measures requires a robust estimate of directional derivatives. Luo et al. (2003)

have presented a new amplitude gradient method based on the generalized Hilbert

transform that can detect abrupt and gradual amplitude changes associated with

fault and channels.

One of the major goals of exploration seismology is the delineation of

fractures. Fractures are found in nearly every reservoir, rock type, and depth; they

may also be found in source rocks, reservoir rocks and cap rocks. Petroleum

explorationists pay a great deal of attention to locating these fractures in order to

predict reservoir performance. Fractures can advance or hinder our efforts in

producing a reservoir. Locating these fractures and identifying their orientations

can help the explorationists benefit from their presence or avoid their related

inconvenience.

76

76

Using seismic coherence to detect fractures has been investigated since the

first emergence of the coherence cube as a new attribute of seismic data.

Skirius et al. (1999) used seismic coherence in carbonates in North America and

the Arabian Gulf to detect faults and fractures. Luo et al. (2002) showed some

examples from a Saudi Arabian carbonate field where amplitude gradients helped

in delineating fractures. While coherence can often detect lineaments, reflector

curvature is more directly linked to fracture distribution (Lisle, 1994; Roberts,

2001, Bergbauer et al., 2003). Hart et al. (2002) have used horizon attributes

(including various curvature attributes) to identify structural features that may be

associated with fracture-swarm sweet spots. Stewart and Wynn (2000) pointed

out that it may be necessary to examine curvature at various scales to account for

different wavelengths. Roberts (2001) stated that volumetric estimation of

reflector curvature should be possible. This paper demonstrates such examples. In

the next section, we begin with a summary of alternative estimates of derivatives

and show how we can use concepts presented by Cooper and Cowan (2003) as the

building blocks for multispectral curvature analysis -as discussed by Stewart and

Wynn. Next, we show how Robert’s (2001) measures of reflector curvature and

independent measure of reflector rotation can be calculated directly from

volumetric estimates of reflector dip (e.g. Marfurt, 2003; Barnes 2000). Finally,

we apply these new attributes to data from onshore areas of Louisiana and Texas

comparing them to state of the art coherence volumes showing how they are most

effective in delineating subtle faults, folds, fractures, other tectonic effects, infill

77

77

and differential compaction above karsts, channels and other stratigraphic

features.

EVALUATION OF ALTERNATIVE DERIVATIVE CALCULATIONS

Lateral changes in reflector amplitude can be highlighted by calculating

their derivatives (with the magnitude of the derivative vector) or gradient and

estimating the position of discontinuities. Luo et al. (1996) applied such

derivatives for seismic edge detection to an uninterpreted cube of seismic data.

For the first derivative edge detection, the amplitude gradient can be

approximated by convolving the seismic data with the vector [-1,0, +1].

Clearly, if the first derivative is a good thing to calculate, we can achieve a

more accurate approximation by replacing our 3 sample, 2nd order accurate [-1,0,

+1] operator with a longer-length, higher-order, accurate approximation of the

first derivative. We might hope that this higher-order approximation will provide

us with a more robust estimate of the gradient. Alternatively, we may obtain

better derivative-based edge detection by exploiting recent advances made in the

2-D image processing literature (Torreao and Amaral, 2002) and applying them to

3-D seismic data. A third alternative is to modify the fractional order horizontal

gradients -developed and applied to 2-D potential field data by Cooper and

Cowan (2003)- and modify them to the 3-D seismic curvature estimation. Such

fractional order horizontal gradients should allow us to analyze our data over a

range of wavelengths as to delineate different structures from the same time slice

of 3-D seismic data.

In this paper, we are primarily interested in alternative measures of

curvature, rather than edge detection. Fortunately, even when viewed on time

slices, vector dip is of relatively slow variance when compared to seismic

amplitude. In fact, the lateral variability of vector dip is closer to those seen in

photographic images and potential field data rather than seismic amplitudes. In

the following sections we summarize the theory and present the spectral response

of each of our three alternate approaches.

HIGHER ORDER APPROXIMATION OF THE FIRST DERIVATIVE

78

78

)

Higher order accuracy approximations to derivatives are routinely used in

numerical modeling of geophysical phenomena (e.g. Fornberg, 1987). What is not

intuitively obvious is that we can interpret these approximations as an exact

derivative multiplied by a low pass filter. It is this low pass filter framework of

the exact derivative that will allow us to compare and evaluate the three

seemingly disparate edge detection algorithms described in these sections. As an

illustration of the higher order approximation of the first derivative, we derive a

fourth order-accurate derivative by expanding a function (u x h± and

where h is the separation between samples in a Taylor’s series: ( 2u x h± )

2 3 42 3 45

2 3 4( ) ( ) ( ) ( ) ( ) ( ) ( )2! 3! 4!h h hd d d du x h u x h u x u x u x u x O h

dx dx dx dx+ = + + + + + (5-1)

2 3 42 3 45

2 3 4( ) ( ) ( ) ( ) ( ) ( ) ( )2! 3! 4!h h hd d d du x h u x h u x u x u x u x O h

dx dx dx dx− = − + − + + (5-2)

2 3 42 3 45

2 2 4

(2 ) (2 ) (2 )( 2 ) ( ) 2 ( ) ( ) ( ) ( ) ( )2! 3! 4!h h hd d d du x h u x h u x u x u x u x O h

dx dx dx dx+ = + + + + +

(5-3)

2 3 42 3 45

2 3 4

(2 ) (2 ) (2 )( 2 ) ( ) 2 ( ) ( ) ( ) ( ) ( )2! 3! 4!h h hd d d du x h u x h u x u x u x u x O h

dx dx dx dx− = − + − + +

(5-4)

To obtain the first derivative dudx

, we multiply equations (1), (2), (3) and (4) by

parameters a1, a2, a3, and a4, respectively, and sum them to obtain:

1 2 3 4 1 2 3 42 3 4

2 3 41 2 3 42 3 4

( ) ( ) ( 2 ) ( 2 ) ( ) ( )

( ) ( ) ( ) ( )

a u x h a u x h a u x h a u x h a a a a u x

d d d de h u x e h u x e h u x e h u xdx dx dx dx

+ + − + + + − = + + + +

+ + +

(5-5)

where e1, e2, e3 and e4 are given by:

1 1 2 3 4

2 1 2 3 4

3 1 2 3 4

4 1 2 3 4

2 2 ,1 1 2 2 ,2 21 1 4 4 2 ,6 6 3 31 1 2 2 2 .24 24 3 3

e a a a a

e a a a a

e a a a a and

e a a a a

= − + −

= + + +

= − + −

= + + +

(5-6)

79

79

To express the first derivative only in terms of ( )u x h± and ( 2 )u x h± , we need to

choose the coefficients a such that e2=e3=e4=0 and e1=1. Doing so gives

1 2 32 2, ,3 3

a a a= − = = −1

12 and 4

112

a = , which, when inserted into equation (5-5),

provides an approximation of the first derivative that is fourth order accurate:

2 2 1 1( ) ( ) ( 2 ) ( 2 )( ) 3 3 12 12u x h u x h u x h u x hdu x

dx h

+ − − − + + −= . (5-7)

Fornberg (1987) has carried these approximations out for a complete suite of

higher order accurate operators (Figure 1). In the limit, the first order derivative is

approximated by

[1 ( ( )xdu F ik F u xdx

−= ] , where F and F-1 denote the forward and reverse Fourier

transform, kx is the wavenumber, and 1i ≡ − . In this paper, we interpret Figure 1

as a suite of low pass filters applied to the exact derivative operator. We will show

that these low pass (read: less accurate) approximations provide more useful

images than the “exact” derivative.

TORREAO AND AMARAL’S EDGE DETECTOR

Torreao and Amaral (2002) were not interested in curvature at all, but

rather developed a robust edge detector that has derivative-like properties. They

noticed that on many image processing applications, we wish to detect edges that

segment different regions of smoothly varying signals that are contaminated by 80

80

rapidly varying noise. They therefore chose to estimate signals that had the

behavior

2( ) ( ) ( )u x L u x o x+ = + , (5-8)

for all values of x, where u(x+L) is equal to the signal at location x+L.

Using a Taylor’s series expansion they rewrite equation (8) as:

2

2( ) ( )1! 2!L u L uu x u x

x x∂ ∂

+ + =∂ ∂

. (5-9)

The Green’s function solution corresponding to equation (9) is

2

0 0( ) 2 sin( )exp( ) 0

xG x x x x

L L L

<⎧⎪+ = ⎨

− >⎪⎩

. (5-10)

Next, they modify equation (9) to solve for the signal at x-L. If the signal is

smoothly varied, these limits should be identical. They therefore form a difference

operator,

[2 2 21( ) ( ) ( )

2D x G x G x

L= − − + ] . (5-11)

Note that in equation (-11) we use the symbol D2 rather than ux∂∂

. According to

equation (9), D2(x)≡0 for both linear and parabolic signal variation. For

information about higher order D3 operator and a hybrid D23 operator by 81

81

82

82

combining D2 and D3 operators, read Torreao and Amaral (2002) who find that

D23 is a better edge detector operator. Accordingly, D23 will be the operator that

we will use for our edge detection algorithm.

We show the frequency response of Torreao and Amaral’s (2002) D23 operator in

Figure 2 for values of L=h, 2h, 3h, 4h, and 5h, where h is the separation between

seismic traces. We note that the operator for L=h is indistinguishable from the 2nd

order finite difference operator shown in Figure 1. Increasing the value of L has

the following three effects: an increase in the number of seismic traces used in the

computation, an increase in the initial slope in the spectral response above 1.0,

and a decrease in the spectral content towards lower wavenumbers. We have

found Torreao and Amaral’s (2002) D23 operator to be the most effective of the

plethora of recently developed image processing edge detectors when applied to

seismic data. In particular, it produces more robust edges and curvature estimates

than the classic derivatives shown in Figure 1. We were, however, troubled by the

bimodal spectral response seen at values of L=4h and L=5h in Figure 2.

Furthermore, the steep low frequency slope of the spectra points bears a similarity

to the fractional derivatives presented by Cooper and Cowan (2003).

FRACTIONAL ORDER HORIZONTAL GRADIENTS

Cooper and Cowan (2003) applied different order gradients to gravity and

magnetic data thereby delineating linear features that are wavelength dependent.

To show the mathematics behind the fractional gradient, let us assume that we

83

83

have a function u(x), and first derivative ∂u/∂x . In the wavenumber domain, the

first derivative is given by

F(∂u/∂x)=–ikxF(kx). (5-12)

The fractional horizontal gradient in the wavenumber domain is thus equal to

Fα(∂u/∂x)=–i(kx)α F(kx), (5-13)

where α is fractional real number.

Cooper and Cowan (2003) do not present implementation details. While a

fractional derivative may be represented formally as [ikx]α F(kx), we have found it

to be more useful to keep the phase change at a constant value of i, or 90o. In our

implementation, we retain the amplitude spectrum weighting of the fractional

derivative, but keep the phase spectrum to be that of the conventional first

derivative. Furthermore, building on experience in analyzing the efficacy of the

Torreao and Amaral (2002) algorithm, we high cut each filter by applying a

simple raised cosine. The peak of the raised cosine is at 0.5 α kNyquist , where

kNyquist =1/(2h). For scaling purposes we find it useful to normalize the energy of

each filter to a constant that is equal to the energy of the filter associated with

α=1.0 (Figure 3). In this manner, our fractional derivative for α=1.0 is identical

to that of the 2nd order finite differences and D32 operator with L=h.

84

84

ALTERNATIVE MEASURES OF REFLECTOR SHAPE

Seismic reflectors are rarely planar, but are usually folded or even broken.

Many regions of the earth’s subsurface can best be described as chaotic. Most

published work in mapping reflector shape has been restricted to represent

interpreted horizons by their curvature (Lisle, 1994; Stewart and Wynn, 2000;

Roberts, 2001; Sigismondi and Soldo, 2003). This work in turn has been based on

a great deal of literature in mapping surface topography or terrain (e.g. Mitsova

and Hofierka, 1993; Wood, 1996). In this paper, we wish to develop an algorithm

that estimates reflector shape on a complete cube of seismic data without the need

for prior interpretation. While assigning a reflector surface to each point in a

given seismic data volume proves intractable, assigning a vector dip (or

alternative dip magnitude and dip azimuth) is not. Barnes (1996, 2000) shows

how to calculate reflector dip and azimuth using a 3-D generalization of

instantaneous frequency. Instantaneous frequency, ω, (and wavenumbers kx and

ky) estimates suffer from waveform interference, so considerable smoothing needs

to be done to stabilize the calculation. We have found that estimates of reflector

dip based on a multiwindow coherence scan (Marfurt, 2003) produce both stable

and high lateral resolution results. The examples shown in this paper will use this

latter technique as input.; however, we have found that long wavelength (low

wavenumber) estimates of reflector shape using either of these two input

algorithms are roughly equivalent.

Given a cube of estimated vector dip:

u(z,x,y)=xux(z,x,y)+yuy(z,x,z)+zuz(z,x,y) (5-14)

85

85

where ux, uy, and uz are the direction cosines defining the reflector as normal, and

where x, y, and z are unit vectors along the Cartesian axes, we can mathematically

decompose them into two parts:

kmean=0.5∇•u, and (5-15)

r=∇×u. (5-16)

Equation (15), which in Robert’s (2000) notation is called the mean curvature,

kmean, is proportional to the divergence of the vector dip; u. Equation (16), which

we will call the reflector rotation, r, is proportional to the curl of the vector dip. In

principal, any arbitrary vector field can be expressed by some linear combination

of equations (15) and (16). We have found the component of rotation about the z

axis

rz=∂ux/∂y-∂uy/∂x, (8-17)

to be of interpretational value, since it represents the rotation of a reflector across

a vertical plane. This often occurs when there is a strike/slip component of

deformation. The other components of the rotation vector correspond to

acquistion rather than (approximately) depositional axes; we have not found them

to be particularly useful. Other than some preliminary work by Marfurt and Kirlin

(2000), we have not seen any published literature using this measurement on

interpreted surfaces.

In contrast, there is a great deal of relevant literature published on the use

of curvature. Following the notation of Roberts (2000), we can represent a

reflector surface, z(x,y), by a quadratic surface:

86

86

z(x,y)=ax2+cxy+by2+dx+ey+f . (5-18)

A fixed depth (or time), slice through this surface will be:

an ellipse; if c2 – 4ab > 0, (5-19)

a hyperbola; if c2 – 4ab< 0, or (5-20)

a parabola; if c2 – 4ab= 0. (5-21)

Using our input estimates of reflector dip, ux and uy, the coefficients in equation

(18) become at x=y=0: Dx( Dxz)=2a=Dxux,

Dy( Dyz)=2b= Dyuy

Dxuy+Dyux=2c,

d=ux, and

e=uy, (5-22)

where the operators Dx and Dy can be any of the numerical approximations to the

first derivative that was discussed in the previous section. We should note that by

construction, equation (18) does not express any rotational component, since

rz=DyDxz-DxDyz=c-c=0. (5-23)

equation (18) expresses the reflector surface in the acquisition coordinate system.

When the coefficient, c, is not zero, the conic is said to be rotated with respect to

its principal axes. To find the maximum and minimum (or principal) curvatures,

kmin and kmax, we need to rotate the coordinate system to another frame. Details

can be found in Roberts (2000) as well as in advanced mathematics books on

solid geometry and 3-D computer graphics. We will use the terminology (and

equations) presented by Roberts (2000) and calculate the mean curvature, kmean:

87

87

kmean=[a(1+-e2)+b(1+-d2)-cde]/(1+d2+e2 )3/2, (5-24)

the Gaussian curvature, kgauss:

kgauss=(4ab-c2)/(1+d2+e2 )2, (5-25)

the maximum curvature, kmax:

kmax= kmean +( kmean2

- kgauss)1/2 , (5-26)

the minimum curvature, kmin,

kmin= kmean -( kmean2

- kgauss)1/2 , (5-27)

the most positive curvature, kpos:

kpos = (a+b)+[ (a-b)2+ c2]1/2 , (5-28)

the most negative curvature, kneg: (5-29)

kneg = (a+b)-[ (a-b)2+ c2]1/2 , (5-30)

the dip curvature:

kdip = 2(ad2+be2+cde)/[(d2+e2)(1+d2+e2)3/2], (5-31)

the strike curvature, kstrike:

kstrike = 2(ae2+bd2-cde)/[(d2+e2)(1+d2+e2)1/2], (5-32)

the shape index, s:

s = 2/π tan-1 [( kmax + kman)/ (kmax - kmin)] (5-33)

the curvedness, r:

r = ( k2max + k2

min)1/2 (5-34)

and finally, the azimuth of the maximum curvature, δ:

δ=tan-1[c/(a-b)], if a≠b, and

δ=π/4, if a=b. (5-35)

88

88

The principal curvatures kmax and kmin measure the maximum and minimum

bending of the surface at each point (Lisle, 1994). Once we have the principal

curvatures, we can calculate the Gaussian curvature kgauss by multiplying the two

principal curvatures:

kgauss=kminkmax (5-36)

The Gaussian curvature, kgauss, is positive for spheres, negative for hyberboloids

and zero for planes everywhere (Figure 4). The Gaussian curvature kgauss -

sometimes referred to as the total curvature- is named after Gauss and his

Theorema Egreium or “wonderful theory” (Roberts,2001). Lisle (1994) suggested

Gaussian curvature as a method of delineating faults, but Roberts (2001) found

that Gaussian curvature is not a good attribute for this type of operation. We have

tested this hypothesis and feel that Roberts’ (2001) example was either too

contaminated by noise or -more likely- simply did not have any elliptical features.

In the following section, we will show how Gaussian curvature clearly delineates

elliptically shaped, infilled karsts.

We found the most positive and most negative curvatures kpos and kneg to

be the most useful in delineating faults, fractures, flexures, and folds. We have

also found the ‘reflector rotation’ attribute, rz, to be more sensitive to acquisition

footprint than others (at least for land data where there is a strong azimuthal and

offset bias on output lines). Our initial calculation of rotation is quite simple, and

is perhaps the easy-to-calculate equivalent of the mean curvature, kmean. We

89

89

expect that there are more useful expressions of ‘tears’ in the reflector surface;

however, we leave this analysis to a future paper.

APPLICATION

To illustrate the value of our method, I have calculated curvature and

rotation attributes for two data sets – the Vinton salt dome from the onshore

Louisiana Gulf Coast, and a data volume with small scale faults, and sinkhole-like

collapse fractures from the Fort Worth Basin of North Texas. We begin by

comparing a suite of time slices through attribute cubes generated for Vinton

Dome Louisiana at 1.000 s. As a baseline we plot time slices of the principal

component of coherence and vector dip in Figure 5. Zones of low coherence

correspond to lateral change in waveform, rather than in changes of amplitude,

dip, or curvature. The vector dip serves as input to the curvature and rotation

calculations shown in Figure 6. The images in Figure 6 are complementary to

coherence and independent amplitude-sensitive ‘edge detectors’. Many of the

features are similar, due to their being expressions of the same geology. In Figure

6a we show the mean curvature, kmean, calculated using a value of α=1.00 in

equation (13). We note that the reflector rotation attribute, rz, shown in Figure 6b,

shows considerable reflector rotation along the radial faults in the Northwest part

of the timeslice. Other faults that show up on the most negative and positive

curvatures, kneg and kpos, (Figures 6c and 6d) appear to have only minimal

rotation. We feel that kneg and kpos provide cleaner, less ambiguous images of

faults than does the mean curvature.

90

90

In Figure 7, we reproduce the same attributes with a low resolution and

long wavelength implementations of Dx and Dy by using the value of α=0.25 in

equation (13). In addition to being less sensitive to noise we enhance more subtle

long features. Next, we turn our attention to a time slice through one of the Fort

Worth Basin surveys roughly at the Caddo/Atoka horizon at a depth of 0.8s. In

Figure 8 we display the coherence, North apparent dip, and East apparent dip. We

begin with a sensitivity study, and show a suite of negative curvature images, kneg,

in Figure 9 for values of α=2.00, 1.75. , 1,50, 1.25, 1.00, 0.75, 0.50, and 0.25 .

We agree with Stewart and Wynn’s (2000) observation that multispectral analysis

brings out different features – highly localized faults and fracture swarms for

larger values of shorter wavelength α, (Figure 9c) and more regional warping and

flexures of small values of α (Figure 9f). In Figure 10, we display a full suite of

long wavelength shape attributes - kmean, kgauss, kpos, kdip, kstrike, and rz, at t=0.800 s

using a value of α=0.25 . We note that the lineaments seen in kmean are mixed,

and less clear than those seen in kpos and kneg, and that kgauss reflects the elliptical

collapse chimneys described by Sullivan et al. (2003) and Jyosula (2003), and

which we will investigate in greater detail at a deeper, Ellenburger level. We plot

kdip and kstrike to complete our set of images. If in a compressional terrane, Roberts

(2001) and Hart et al. (2002) predict that large values of kstrike will be correlated to

open, vs. closed fractures.

Finally in Figure 11, we display the coherence (North apparent dip) and

East apparent dip at t=1.200 s that cuts the highly irregular Ellenberger horizon

91

91

and the overlying Barnett Shale. In Figure 11, we display a suite of short

wavelength, α=1.00, curvature and rotation time slices; in Figure 13, we display

the corresponding long-wavelength, α=0.25, time slices through cubes of

reflector rotation and curvature. The geologic interpretation of these images is

addressed by Sullivan et al. (2003). For the purposes of this paper we note that the

strong, negative curvature (concave up towards negative time) corresponds to

sinkhole-like feature and associated infill. These circular to rhomboid features can

be seen to be aligned along major NE-SW and NW-SE faults or lineaments. In the

long wavelength negative curvature image (Figure 13c), we can see details of

lineaments surrounding these circular sinkhole-like features. Some of these

features are elongated (indicated by arrows) along the fault, and may be related to

collapse associated with small pull-apart structures (Lacazette et al., 2004). The

overall long wavelength pattern is surprisingly periodic in both the NE-SW and

the NW-SE directions. The Gaussian curvature image, Figures 12e and 13e,

highlight these elliptical collapse features at the expense of detailed definition of

lineaments.

CONCLUSIONS

My geometric attribute work has supplemented and enhanced the suite of

powerful, new AGL-developed multi-trace volumetric seismic attributes are

applied to entire uninterpreted cubes of seismic data. These attributes, which

include measures of reflector rotation and curvature, are independent of, and

complementary to, the popular measures of seismic coherence. In particular,

92

92

reflector rotation is found to be a good indicator of scissor movement along a

fault. The negative and positive curvatures appear to be the most unambiguous of

the curvature images in highlighting and illuminating lineaments. The Gaussian

curvature shows what it was designed to show - surfaces that exhibit an elliptical

bowl or dome component- and appears to be a good indicator of small collapse

structures. In addition to highlighting lineaments, structural geologists have

theoretical and empirical evidence linking Gaussian and strike curvatures to

fracture density. We are now able to make such predictions through a complete,

uninterpreted data cube.

Stewart and Wynn (2000) found that measurements of reflector shape,

such as curvature and rotation, are fractal in nature, rendering them amenable to

multispectral analysis. The application of my research to Permian Basin and Fort

Worth Basin data demonstrates that the low wave number time slices are

particularly functional in extracting structural and stratigraphic information that

was previously difficult or impossible to see (Blumentritt et al., 2003; Serrano et

al., 2003; Blumentritt et al., 2004) .

Curvature and rotation (and other possible measures of reflector shape) are

mathematically independent of coherence and amplitude. While we expect the

impact of 3-D volumetric estimates of reflector shape to be every bit as big as the

impact of coherence on seismic interpretation, we also anticipate a good workflow

to include all of these geometric attribute tools. What we find most encouraging is

that there is a firm basis in using these attributes on maps by the structural

geology community.

93

93

This work is in its infancy. Thus, we see a need to quantitatively calibrate

the features seen in curvature and rotation to direct measures of fractures through

both horizontal image logs and microseismicity epicenter location during

hydraulic fracture well tests. We also see a need to qualitatively calibrate these

curvature and rotation features directly to core, log and production data (eg. Fu et

al., 2003).

If there is a major limitation to this technology, it is in the calculation

itself. The vector dip is a true 3-D calculation that follows the best reflector,

which includes the analysis point. However, since we wish to calculate curvature

even when there are only piecewise continuous reflectors available, we calculate

the derivatives of dip on time slices. For steeply dipping horizons and low

wavenumber estimates, we expect these measures will undesirably mix geology

of different formations. While we anticipate near term improvements in our

estimates to better follow the local dip, we know such improvements will not be

trivial to implement.

FIG. 5-1. Spectral response of finite difference approximations to the first derivative (after Fornberg, 1987).

FIG. 5-2. Spectral response of Torreao and Amaral’s (2002) D32 edge detector for values of L=h, 2h, 3h, 4h, and 5h.

94

94

FIG. 5-3. Spectral response of our high cut filtered fractional derivative operator, Dα, given by equation (13) for values of α=2.00, 1.75, 1.50, 1.25, 1.00, 0.75, 0.50, and 0.25.

FIG. 5-4. The classification of points on folded surface based on signs of mean curvatures and the Gaussian curvature (after Bergbauer et al., 2003).

95

95

5a) 5b

5c) FIG. 5-5. (a) Principal component coherence, (b ) North apparent dip, and (c) East apparent dip, at t=1.000 s through a survey at Vinton Dome, LA, USA.

96

96

6a)

6b)

6c)

FIG. 5-6. High resolution (α=1.00) estimates of (a) mean curvature, kmean, (b), reflector rotation, rz, (c) most negative curvature kneg, (d) most postive curvature, kpos, (e) shape index, s, (f) curvedness, r, and (g) shape index modulated by curvedness corresponding to the same time slice shown in Figure 4.

97

97

Figure 5-6 (continued)

6d)

6e)

6f)

6g)

98

98

7a)

7b)

7c) FIG. 5-7. Low resolution (α=0.25) estimates of (a) mean curvature, kmean, (b), reflector rotation, rz, (c) most negative curvature kneg (d) most postive curvature, kpos, (e) shape index, s, (f) curvedness, r, and (g) shape index modulated by curvedness corresponding to the same time slice shown in Figures 4 and 5.

99

99

Figure 5-7 (continued)

7d)

7e)

7f)

7g)

100

100

8a)

8b)

8c)

FIG. 5-8. (a) Principal component coherence, (b ) North apparent dip, and (c) East apparent dip, at t=0.800 s (approximately Caddo/Atoka time) through a survey from the Fort Worth Basin, TX, USA.

101

101

9a)

9b)

9c)

FIG. 5-9. Time slice at t=0.800 s from a survey in the Fort Worth Basin showing most negative curvature, kneg, for spectral components, defined by α=2.00, 1.75, 1.50, 1.25, 1.00, 0.75, 0.50, and 0.25.

102

102

Figure 5-9 (continued)

9d)

9e)

9f)

9g)

103

103

Figure 5-9 (continued)

9h)

104

104

10a)

10b)

10c)

FIG. 5-10. Full suite of curvature values for the same time slice shown in Figure 8, for spectral components defined by α=0.25: (a) mean curvature, kmean, (b), reflector rotation, rz, (c) most negative curvature kneg, (d) most positive curvature, kpos, (e) Gaussian curvature, kgauss, (f) dip curvature, kdip, (g) strike curvature, kstrike, (h) shape index, s, (i) curvedness, r, and (j) shape index modulated by curvedness.

105

105

Figure 5-10 (continued)

10d)

10e)

10f)

10g)

106

106

Figure 5-10 (continued)

10h)

10i)

10j)

107

107

11a)

11b

11c) FIG. 5-11. (a) Principal component coherence, (b ) North apparent dip, and (c) East apparent dip, at t=1.200 s (approximately Ellenburger/Bartnett time) through a survey from the Fort Worth Basin, TX, USA.

108

108

12a)

12b)

12c)

FIG. 5-12. Full suite of curvature values for the same time slice shown in Figure 11, for spectral components defined by α=1.00: (a) mean curvature, kmean, (b), reflector rotation, rz, (c) most negative curvature kneg, (d) most positive curvature, kpos, (e) Gaussian curvature, kgauss, (f) dip curvature, kdip, and (g) strike curvature, kstrike, (h) shape index, s, (i) curvedness, r, and (j) shape index modulated by curvedness.

109

109

Figure 5-12 (continued)

12d)

12e) 12f)

12g)

110

110

Figure 5-12 (continued)

12h)

12i)

12j)

111

111

13a)

13b)

13c)

FIG. 5-13. Full suite of curvature values for the same time slice shown in Figure 10, for spectral components defined by α=0.25: (a) mean curvature, kmean, (b), reflector rotation, rz, (c) most negative curvature kneg, (d) most positive curvature, kpos, (e) Gaussian curvature, kgauss, (f) dip curvature, kdip, and (g) strike curvature, kstrike, (h) shape index, s, (i) curvedness, r, (j) shape index modulated by curvedness.

112

112

Figure 5-12 (continued)

13d)

13e)

13f)

13g)

113

113

Figure 5-12 (continued)

13h)

13i)

13j)

114

114

115

115

REFERENCES

Bahorich, M.S., and Farmer, S.L., 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.

Barnes, A. E., 2000, Weighted average seismic attributes: Geophysics, 65, 275-

285.

Bergbauer, S., Mukerji, T., and Hennings, P., 2003, Improving curvature analyses

of deformed horizons using scale-dependent filtering techniques: AAPG

Bulletin, 87, 1255-1272.

Blumentritt, C.H., Marfurt K.J., and Sullivan, E.C., 2003, New attributes

illuminate old structures on the Central Basin Platform: [Extended

Abstract]: Proceedings, West Texas Geologic Society Annual Symposium.

Cooper, G. R., and Cowan, D. R., 2003, Sunshading geophysical data using

fractional order horizontal gradients: The Leading Edge, 22, 204-205.

Dalley, R M., Gevers, E. E. A., Stampli, G. M., Davies, D. J., Gastaldi, C. N.,

Ruijetnberg, P. R., and Vermeer, G. J. D., 1989, Dip and azimuth displays

for 3-D seismic interpretation: First Break, 7, 86-95.

Fornberg, B., 1987,The pseudospectral method: Comparisons with finite

differences for the elastic wave equation: Geophysics, 52, 483-501.

116

116

Serrano, I.C., Blumentritt, C.H., Sullivan, C., Marfurt, K.J., and M. Murphy.,

Seismic attribute delineation of lineaments and reservoir

compartmentalization: An example from the Devonian Dollarhide Field,

Central Basin Platform, West Texas [Extended Abstract]: Proceedings,

Society for Exploration Geophysicists 73rd Annual Symposium.

Fu, D., Sondhi, A., Sullivan, C., and Marfurt, K., 2003, Petrophysical analysis and

seismic response of carbonates and chert in the Thirtyone Formation, West

Texas [Extended Abstract]: Proceedings, West Texas Geologic Society

Annual Symposium.

Gersztenkorn, A., and Marfurt, K. J., 1999, Eigenstructure based coherence

computations: Geophysics, 64, 1468–1479.

Hocker, C., and Fehmers, G., 2002, Fast structural interpretation with structure-

oriented filtering: The Leading Edge, 21, 238-243.

Hart, B.S., Pearson, R.A., and Rawling, G.C., 2002, 3-D Seismic horizon-based

approaches to fracture-swarm sweet spot Definition in Tight-Gas

Reservoirs. The Leading Edge, 21, 28-35.

Lisle, R.J., 1994, Detection of zones of abnormal strains in structures using

Gaussian curvature analysis. AAPG Bulletin, 78, 1811-1819.

Luo, Y., Higgs, W. G. and Kowalik, W. S., 1996, Edge detection and stratigraphic

analysis using 3-D seismic data, 66th Annual International. Meeting

Society of Exploration Geophysicists,Expanded Abstracts, 324-327.

117

117

Luo, Y., al-Dossary S., Marhoon M., and Alfaraj, M., 2003, Generalized Hilbert

transform and its application in Geophysics: The Leading Edge, 22, 198-

202

Luo, Y., Marhoon M., al-Dossary, S., and Alfaraj, M., 2002, Edge-preserving

smoothing and applications: The Leading Edge, 21, 136-158.

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

attributes using a semblance-based coherency algorithm: Geophysics, 63,

1150-1165.

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

and amplitude: Geophysics, 65, 304-320.

Marfurt, K. J., 2003, Robust estimates of 3-D reflector dip: Submitted to

Geophysics.

Mitasova, H. and Hofierka, J., 1993, Interpolation by regionalized spline with

tension: II. Application to terrain modeling and surface geometry analysis:

Mathematical Geology, 25, 657-669.

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

horizons. First Break, 19, 85-99.

Skirius, C., Nissen, S., Haskell, N., Marfurt, K. J., Hadley, S., Ternes, D.,

Michel, K., Reglar, I., D'Amico, D., Deliencourt, F., Romero, T,.

Romero, R., and Brown, B., 1999, 3-D seismic attributes applied to

carbonates: The Leading Edge, 18, 384-389.

118

118

Sigismondi, E.M., and Soldo, C.J., 2003, Curvature attributes and seismic

interpretation: Case studies from Argentina basins: The Leading Edge, 22,

1122-1126

Stewart, S.A., and Wynn, T.J., 2000, Mapping spatial variation in rock properties

in relationship to scale-dependent structure using spectral curvature.

Geology, 28, 691-694.

Sullivan, E. S., Marfurt, K. J., Lacazette, A., and Ammerman, M., 2003, Bottoms-

up karst: Submitted to Geophysics.

Torreao, J.R.A. and Amaral, M.S., 2002, Signal differentiation through a Green’s

function approach. Pattern Recognition Letters, 23, 1755-1759.

Wood, J.D., The geomorphological characterization of digital elevation models.

PhD Thesis, University of Leicester, UK.

119

119

CHAPTER 6

INTER AZIMUTH COHERENCE ATTRIBUTE FOR FRACTURE DETECTION

(Submitted to the 74th annual International Meeting, Society of Exploration

Geophysicists, Saleh Al-Dossary, Yves Simon and Kurt Marfurt, Allied Geophysical Laboratories, University of Houston)

SUMMARY

Fractures occur on many scales in the earth. Fractures on a sub-seismic scale

of less than tens of meters are of great interest in a reservoir context. Locating

areas of greatest fracture density and determining the orientation of these fractures

within a reservoir represents a significant technical challenge for geophysicists. 3-

D surface seismic data can image fractures and faults more effectively if it is

sorted into common azimuth bins and analyzed separately for each azimuth bin.

Based on this fact, we have developed a new algorithm to detect sub-seismic

faults and fractures by calculating coherence cubes between prestack limited

azimuth seismic data. The new algorithm will calculate coherence in prestack

azimuth-sorted space, rather than poststack full azimuth space. We have applied

our algorithm to a survey over a fractured reservoir in Texas, and had interesting

results.

INTRODUCTION

Natural fractures play an important role in petroleum exploration. Fractures

are found in nearly every reservoir, rock type, and depth. Petroleum

120

120

explorationists pay a great deal of attention to locating these fractures in order to

understand the reservoirs. Fractures can advance or hinder the effort in

understanding reservoir character. Fractures can be found in source rocks,

reservoir rocks and cap rocks. Locating these fractures and identifying their

orientation can help the explorationists to deal with them and benefit from their

presence or avoid their annoyances.

The use of seismic coherence to detect fractures has been investigated

since the first emergence of the coherence cube as a new attribute of seismic data.

Estimates of seismic coherence (e.g. Bahorich and Farmer, 1995; Marfurt et al.,

1998, Gertzenkorn and Marfurt, 1999; Marfurt and Kirlin, 2000) provide a

quantitative measure of the changes in waveform across a discontinuity. Estimates

of apparent dip (e.g. Dalley et al., 1989; Luo et al., 1996; Marfurt et al., 1998;

Marfurt and Kirlin, 2000; Luo et al., 2001) provide a measure of change in

reflector dip/azimuth across a discontinuity. Estimates of amplitude gradients (e.g.

Luo et al., 1996; Marfurt and Kirlin, 2000) provide a measure of changes in

reflectivity across a discontinuity. More recently Al-Dossary and Marfurt (2004)

have used spectrally limited volumetric curvature to help predict fractures. Skirius

et al. (1999) used seismic coherence in carbonates in North America and the

Arabian Gulf to detect fault and fractures. Luo et al. (2002) showed some

examples from a Saudi Arabian carbonate field where amplitude gradients have

helped in delineating fractures.

121

121

These investigations, however, have all been made using post stack data.

In their work, Chopra et al. (2000) applied a fairly simple process of sorting the

data according to source-receiver azimuth bins by migrating the partially stacked

data, and applying coherence to each volume. In general, stacking all the data into

a single volume using an inaccurate velocity smears the data, thereby increasing

the overall coherence of the image, blurring edges and other discontinuities.

While having lower fold and hence exhibiting lower signal to noise ratios, Chopra

et al.’s (2000) common azimuth images show better definition of edges inferred

to be microfaulting or fractures. We have developed a new algorithm to detect

fractures by calculating coherence between prestack data volumes. We will

calculate coherence on the traces of the same offset but of different azimuth. The

offsets we will work with are the near (0° to 20°) and the far (>20°) incident

angle. The azimuths (NE and NW) are approximately parallel and perpendicular

to expected fractures.

ALGORITHM DESCRIPTION

Azimuthal variations caused by fracture-induced anisotropy affect P-wave

attributes such as traveltime, amplitude and velocity (Lynn et al., 1996).

To implement our algorithm, we have sorted the data according to azimuth

(parallel and perpendicular) and to offsets (near and far), generating four

subvolumes:

1. Parallel azimuth and near offset (subvolume1),

122

122

2. Perpendicular azimuth and near offset (subvolume2),

3. Parallel azimuth and far offset (subvolume3), and

4. Perpendicular azimuth and far offset (subvolume4).

We then calculated coherence between traces having the same offset, but with

different azimuth.

MODEL

To illustrate our methodology, we have generated a simple model having

2-D symmetry. The model consists of three horizontal layers. The first and the

third layers are isotropic. The middle layer (fractured layer) includes azimuthally

anisotropic zones, where figure 1a depicts a source-receiver azimuth parallel to

the fracture and figure 1b depicts a source-receive azimuth perpendicular to the

fracture. In this case the thickness of our middle layer is 100 meters. Figures 2a

and 2b depict the simulated seismic responses for parallel and perpendicular

azimuth respectively. Cross correlating the two simulated seismic responses

yields the result shown in figure 3. We notice that the result of the cross

correlation detects the change of velocity induced by the fractures.

Now we reduce the thickness of our middle layer to 25 meters. Figure 4a and 4b

depicts the idealized earth model. The 25 meters thick layer is below resolution,

and only one series of peaks is visible in the simulated seismic response (figure 5a

and 5b). Figure 6 depicts the result of cross correlating the traces from the

parallel azimuth dataset against those from the perpendicular azimuth dataset. We

123

123

notice that the results for fractures inside a 25 meters thick layer, where detecting

changes by interval velocity or travel time thickness analysis becomes intractable,

are very similar to those for a 100 meters thick layer.

FIELD DATA

We now apply our algorithm to seismic data from a 3D-wide azimuth survey

over a fractured reservoir in Texas. In Figures 7a and 7b, we display time slices

through the near incident angle (0o – 20o) NE azimuth and NW azimuth seismic

data volumes respectively. Figures 8a and 8b show the same datasets but for

incident angle (20o – 50o). Figures 9a and 9b show the waveform changes

between NE and NW volumes, for the near and the far data sets.

Unlike other attributes, the cross-correlation between NE and NW volumes

shows lineaments oriented NW-SE and NE-SW in the reservoir. This is very

encouraging, as the natural stress is oriented NE-SW, and micro-fractures

detected in cores are oriented NW-SE (the stress direction has changed with

time).

We found only a small, but significant, linear inverse correlation by cross

plots with production data of 110 wells. We are investigating possible geologic

explanations of these results, and will test our new algorithm with more

sophisticated, spatial statistical tools (like co-kriging) against other seismic

attributes and physical measurements.

124

124

CONCLUSIONS

We have developed a new algorithm for direct detection of azimuthal

anisotropy that we believe will provide insight necessary to characterize fracture

systems, stress anisotropy, and potential permeability directions.

We have applied this new algorithm on modeled and field data sorted in two

azimuths at ninety degrees to each other and same limited offsets (near or

far), with promising results.

By maximizing the difference between, in addition to avoiding manual

velocity picking, we believe that our methodology can provide a sensitive tool to

identify fractures within layer thickness smaller than a seismic wavelength.

0 km 2 km4604 m/s

5457 m/s

5031 m/s

100 m

0 km 2 km0 km 2 km4604 m/s

5457 m/s

5031 m/s

4604 m/s

5457 m/s

5031 m/s

100 m

(a)

FIGsou equfrac

4604 m/s

5457 m/s5031 m/s

4756 m/s4880 m/s

4604 m/s

5457 m/s5031 m/s

4756 m/s4880 m/s

(b)

. 6-1. (a) Idealized earth model with thickness equals 100m. The model depicts a rce- receiver azimuth parallel to the fractures. (b) Idealized earth model with thicknessals 100m. The model depicts a source- receiver azimuth perpendicular to the tures.

125 125

1 1001 1001 100

(a)

1 1001 11 10000

(b) FIG. 6-2. (a) Simulated seismic response for parallel azimuth (figure 1a). (b) Simulated seismic response for perpendicular azimuth (figure 1b).

126

126

1 1.86.90

.94

.981.0

xcor

r

001 1.86.90

.94

.981.0

xcor

r

001 11 1.86.90

.94

.981.0

.86

.90

.94

.981.0

xcor

r

0000 FIG. 6-3. Cross-correlation of the traces from parallel azimuth dataset against those from perpendicular azimuth dataset.

127

127

4604 m/s

5457 m/s

5031 m/s

0 km 2 km

25 m

4604 m/s

5457 m/s

5031 m/s

0 km 2 km

25 m

4604 m/s

5457 m/s

5031 m/s

0 km 2 km0 km 2 km

25 m

(a)

4604 m/s

5457 m/s5031 m/s

4756 m/s4880 m/s

4604 m/s

5457 m/s5031 m/s

4756 m/s4880 m/s

(b) FIG. 6-4. (a) Idealized earth model with reservoir thickness equal to 25m. The velocity model depicts (a) source- receiver azimuth parallel to the fractures and (b) source- receiver azimuth perpendicular to the fractures.

128

128

1 1001 100

(a)

1 1001 100

(b) FIG. 6-5. (a) Simulated seismic response for (a) parallel (Figure4a) and (b) perpendicular azimuth (Figure 4b).

1 1.70

.80

.90

1.0

xcor

r

001 11 1.70

.80

.90

1.0

.70

.80

.90

1.0

xcor

r

0000 FIG. 6-6. Cross-correlation of the traces from parallel azimuth dataset against those from perpendicular azimuth, 25 m thickness.

129

129

0 1 0 k f t3 K m

0 1 0 k f t3 K m

0 1 0 k f t3 K m

(a)

0 1 0 k f t3 K m

0 1 0 k f t3 K m

0 1 0 k f t3 K m

(b) FIG. 6-7. Time slice at 1.236 s through the near incident angle (0o – 20o) (a) NE azimuthand (b) NW azimuth seismic data volumes.

130 130

131

131

0 1 0 k f t3 K m

0 1 0 k f t3 K m

0 1 0 k f t3 K m

(a)

0 1 0 k f t3 K m

0 1 0 k f t3 K m

0 1 0 k f t3 K m

(b) FIG. 6-8. Time slice at 1.236 s through the far incident angle (20o – 50o) (a) NE azimuth and (b) NW azimuth seismic data volumes.

132

132

. 0

0 . 5

1 . 0

0 1 0 k f t3 K m

. 0

0 . 5

1 . 0

. 0

0 . 5

1 . 0

0 1 0 k f t3 K m

(a)

. 0

0 . 5

1 . 0

0 1 0 k f t3 K m

. 0

0 . 5

1 . 0

. 0

0 . 5

1 . 0

0 1 0 k f t3 K m

(b)

FIG. 6-9. (a) Extraction along the top of the fractured reservoir of the coherence cube between mid incident angle NE and NW azimuth datasets. Extraction along the top of the fractured reservoir of the coherence cube between far incident angle NE and NW azimuth datasets. Areas of low coherence are indicative of wave form change due to either anistrophy or signal to noise ratios.

133

133

REFERENCES:

Al-Dossary, S., and Marfurt, K. J., 2004, 3-D volumetric multispectral estimates

of reflector curvature and rotation, submitted to Geophysics.

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

stratigraphic features: The coherence cube, The leading Edge, 16, 1053-

1058.

Chopra, S., Sudhakar, V., Larsen, G., and Leong, H., 2002, Azimuth-based

coherence for detecting faults and fractures: World Oil Magazine, 21.

Dalley, R M., Gevers, E. E. A., Stampli, G. M., Davies, D. J., Gastaldi, C. N.,

Ruijetnberg, P. R., and Vermeer, G. J. D., 1989, Dip and azimuth displays

for 3-D seismic interpretation: First Break, 7, 86-95.

Gersztenkorn, A., and Marfurt, K. J., 1999, Eigenstructure based coherence

computations: Geophysics, 64, 1468–1479.

Luo, Y., Higgs, W. G. and Kowalik, W. S., 1996, Edge detection and stratigraphic

analysis using 3-D seismic data, 66th Ann. Inter. Mtg. Soc. Expl.

Geophys. Expanded Abstract, 324-327.

Luo, Y., Al-Dossary, S. and Marhoon, M., 2001 Generalized Hilbert transform

and its application in Geophysics, 71th Ann. Mtg. Soc. Expl. Geophys.

Expanded Abstract, 430-434.

Luo, Y., Marhoon M., Al-Dossary S., and Alfaraj, M., 2001 Edge-Preserving

134

134

Smoothing and its application in Geophysics, 71th Ann. Mtg. Soc. Expl.

Geophys. Expanded Abstract, 370-374

Lynn, H. B., Simon, K. M., and Bates, C. R., 1996, Correlation between P-wave

AVOA and S-wave traveltime anisotropy in a naturally fractured gas

reservoir: The Leading Edge, 15, 931–935.

Marfurt, K. J., and Duncan, W., 2002 Comparison of 3-d edge detection seismic

attributes to Vinton Dome Louisiana: 72nd Ann. Internat. Mtg., Soc. Expl.

Geophys.,Expanded Abstracts, 723–730.

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

attributes using a semblance-based coherency algorithm: Geophysics, 63,

1150- 1165.

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

and amplitude: Geophysics, 65, 304-320.

Skrius, C., Nissen, S., Haskell, N., Marfurt, K. J., Hadley, S., Ternes, D., Michel,

K., Reglar, I., D'Amico, D., Deliencourt, F., Romero, T,. Romero, R., and

Brown, B., 1999, 3-D seismic attributes applied to carbonates: The Leading Edge,

18, 384-389.

135

CHAPTER 7

CONCLUSIONS

In this dissertation, I have shown through application to data from the Fort

Worth Basin, Texas and Vinton Dome, Louisiana that modern multi-trace seismic

attributes display features (some of which are subseismic) useful in resolving

those structural relationships and fault patterns that are impossible or difficult to

see on conventional seismic displays.

To help improve the fidelity of these multi-trace seismic attribute images,

noise has to be suppressed. To achieve this, I have developed and calibrated new

edge-preserving and lineament-preserving smoothing algorithms that remove

random noise along reflectors, while preserving major structure and stratigraphy

as well as preserving lineaments of only one trace in width. I have also introduced

a new edge detection algorithm that can highlight and emphasis seismic

discontinuities by converting the edge-preserving smoothing algorithm into an

edge detection algorithm that can both detect edges and reduce noise in one

operation. Using dip/ azimuth volumes generated by others at AGL, I have

generalized a suite of powerful, new seismic reflector shape attributes that were

previously limited to only interpreted horizons. These attributes, which include

measures of reflector rotation and curvature, are independent of, and

complementary to, the popular measures of seismic coherence. I found the

negative and positive curvatures to be the most unambiguous of the curvature

136

images in highlighting and illuminating lineations. The Gaussian curvature shows

what it was designed to show - surfaces that exhibit an elliptical bowl or dome

component- and appears to be a good indicator of collapse infill. In addition to

highlighting lineations, structural geologists have theoretical and empirical

evidence linking Gaussian and strike curvatures to fracture density. Interpreters

are now able to make such predictions through a complete, uninterpreted data

cube.

I agree with Stewart and Wynn (2000) that measurements of reflector

shape, such as curvature and rotation, are fractal in nature, rendering them

amenable to multispectral analysis. I have found that the low wave number

curvature time slices are particularly useful in extracting information that was

previously difficult or impossible to see.

I have developed a new algorithm for direct detection of azimuthal

anisotropy that can provide the insight necessary to characterize fracture systems,

stress anisotropy, and, potential permeability directions. By comparing the data

sorted in two azimuths at ninety degrees (orthogonal) to each other for fixed

offsets (near or far), we have produced an attribute that is sensitive to changes in

tuning thickness due to azimuthal anisotropy. In addition to avoiding manual

velocity picking, I believe that this methodology can provide sensitive tools to

identify fractures within layer thicknesses smaller than a seismic wavelength.

Although curvature and inter azimuth coherence clearly delineates

structural deformation and small scale features and enables us to predict open or

137

closed fractures, the challenge remains to find a statistical link between

hydrocarbon production and our new family of curvature attributes and azimuthal

coherence attributes. Such statistical relationship will require calibration with

image log, early production, tracer and acoustic data.