Feb04 Avo Modeling

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Introduction

AVO modeling plays an active role in three areas: new

technology development, QC data processing, and

assisting data interpretation. This paper attempts to

discuss these issues, with emphasis on the applications of

AVO modeling in data p rocessing and interpr etation. Data

modeling is introduced for its theoretical background and

its applications in isotropic and anisotropic situations. In

the data processing side, we will focus on calibrations.

Finally, some discussion is given on the applications of

AVO modeling in interpretation with additional case

studies.

AVO D ata Modeling

In the pre-stack processing stage, the cmp gathers that are

c on s id e red having app ropriate amp litude recovery or

having gone through true amplitude processing are

mod eled using AVO equations to solve AVO attr ibutes. This

can be called AVO data m odeling. Using the AVO equations

introduced in Part I of this article, data modeling is imple-

mented on the amp litud es at a given two-way time of a cmp

gather. This is often implemented using angles of incidence

for a linear fitting, or, a surface fitting for cases where two

variables, offset and azimu th for H TI media, are involved.

AVO data m odeling is usually cond ucted using least squares

method (L2 norm) or other robust m ethods such as L1 norm.

The difference between L2 and L1 norms is that L2 mini-

mizes squared deviation and L1 minimizes the absolute

deviation of the d ata from a m odel. L2 norm an d L1 norm

have the form of

Where is the single data point residu al and

d(xi) is the data and m(xi) represents an AVO equation fitted

at given offset or angle of incidence xi. Since number of d ata

points n or offsets is always greater than the n um ber of vari-

ables or AVO attributes to be solved, this is an over-deter-

mined problem. Minimization of the norms results in the

solution. For L2 norm, the m atrix form of the m inimization

is a = (ATA)-1ATd, where a is AVO attribute vector, A is

angle-dependent coefficient matrix formed by an AVO

equation corresponding to angles of incidence, and d is the

vector of d ata (amp litud es). For L1 norm , median of coeffi-

cient of an AVO equation may be used in minimization

(Press et a l., 1989). To stab ilize the solu tion, constra ints from

rock p hysical relationships m ay be brou ght in. The solution

often consists of two AVO attributes. Shueys equation, for

example, yields intercept and gradient, and Fattis equation

yields P- an d S- reflectivities.

In the amplitude fitting, L2 norm is particular appropriate

when the data contain random noise. L1 norm is consid-ered robust when a small number of data points have

deviant amplitudes, such as a multiple cutting across a

primary reflection. To demonstrate L1 norm and L2 norm

in AVO attribute extraction, a synthetic data set w ith strong

coherent interfering noise was u sed and is shown in Figure

1. Fattis equation was used for amplitude fitting in this

case. We can see that the L1 norm operation results in a

better solution wh ile the L2 norm solution is comprom ised

by the spurious data p oints.

To further demonstrate this, a synthetic gather with

primary reflections and mu ltiples was p rocessed through

L1 norm solution of Fattis equation. Figure 2a to 2d are

prim ary-only gather, inpu t cmp gather, re c o n s t ru c te dgather from th e extracted P- and S-reflectivities, and differ-

ence between Figur e 2b and Figure 2c. As expected, The L1

norm solution does a good job in rejecting the large

moveout multiples (the large moveout multiple is essen-

tially gone from the reconstru cted gather Figure 2c). Some

energy from th e small moveout m ultiples labeled 1 and 2 is

still present on the reconstru cted gather. Ideally, the recon-

structed gather should contain only primary signal (Figure

2c should look like 2a). Hence, mu ltiple attenuation m ay be

required before AVO attribute extraction.

A real data examp le of AVO data mod eling is shown in

Figure 3. We can see that the inpu t cmp gath er (Figure 3a)

contains random noise, linear coherent noise, and multi-ples. The input cmp gather (Figure 3a) is modeled by

Fattis equation with L1 norm operation. The P- and S-

reflectivities were solved and used in constructing Figure

3b. As ind icated, the Class III AVO is successfully m odeled.

The rand om noise, linear coherent noise, and mu ltiples are

rejected and shown in Figure 3c. In p ractice, the difference

gathers is used to examine if any reflections have been

rejected due to poor NMO or inappropriate processing.

AVO Modeling in Seismic Processingand InterpretationPart III. ApplicationsYongyi Li, Jonathan Downton, and Y ong X u, Core Laboratoires Reservoir Technologies Division,

Calgary, Canada


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A second examp le is from a 3-D data set for stud ying AVO in a

fractured reservoir. The fractured reservoir is considered as

horizontal transverse isotropic (HTI) medium (Figure 4). For

this case, L2 norm data modeling was performed based on

Rgers equation (1996). To enh ance the r esolution and increasethe stability of the data modeling, a surface fitting approach

was taken to includ e all traces in the calculation. By examining

Figure 4c, we see th at the r eflection events have been success-

fully modeled since little primary energy leakage can be seen.

Figure 4d is a calculated theoretical amplitude surface that

illustra tes the amplitud e variation with offset and azimu th. The

resulting attr ibutes in this AVOZ analysis are zero-offset reflec-

tivity, fracture orientations, and gradients parallel and perpen-

dicular to the fracture orientations. The fracture density is

calculated based on the gra dients. Figur e 4e gives the estimates

of fracture orientation and fracture density at a carbonate

formation.

Data Calibrations

Calibration on cmp gathers and output AVO attributes can be

conducted for optimizing AVO processing. It helps to answer

the questions such as: whether the cmp gathers are properly

processed with an amplitude friendly processing flow andparameters; whether phase, tuning, signal-to-noise ratio and

other factors are influencing the solutions; and whether the

correct impedance background is used in elastic rock property

inversion. Calibration is often imp lemented by using w ell logs,

synthetic gathers, walkway VSP, and/ or know n relationships

between AVO attribu tes or between rock pro p e r t i e s .

Calibration can be perform ed locally at a cmp location, or glob-

ally on a data set.

Using synth etic cmp gather(s) to tie seismic often gives a quick

insight to the data. AVO type and its variation are often deter-

mined in this stage. Further, since AVO m odeling links seismic

responses directly to rock properties, it helps to confirm or

define reservoir condition. One may perturb the well logs torepresent the possible reservoir conditions. For example, gas

substitution m ay be perform ed on a w et well or vice verse. The

other parameters that are often changed are porosity, reservoir

thickness and lithology. Figure 5 shows an exam ple in which a

synthetic cmp gather ties to a re c o rded cmp gather.

In the zone of interest, the AVO expression has similar

c h a r a c t e r. We m ay there f o re consider that the dat a has

approp riate amp litude recovery.

Calibration at specific reflections may be performed. The

amplitudes for a given event from both the actual seismic and

the synthetic can be extracted and comp ared . Figure 6 shows an

example in which the seismic amplitudes from the top of a gas

reservoir are compared with the synthetic gather. The Class I

AVO anomaly with polarity reversal at the far offsets is

confirm ed as th e AVO expression for this reservoir.

AVO Mod eling in Seismic Pr ocessing an d Inter pr etat ionContinued from Page 42

Figure 1. Curve fitting using L1 and L2 norm.

Figure 2. Data modeling using L1 norm, a) primary-only gather, b) input gather for AV O extraction; c) reconstructedgather using P- and S-reflectivity; and d) difference between b) and c).


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Figure 4. Data modeling using Rgers equation and L2 norm for a fractured reservoir: a) input cmp gather; b) reconstructed cmpgather using intercepts and gradients; c) difference between a) and b); d) a theoretical amplitude surface; and e) fracture orientationsand fracture density. N ote that these gathers are sorted by offset, not by azimuth. Vertical discontinuities of the amplitudes in b)occur where there is a jump in azimuth within the gather.

Figure 3. Data modeling using Fattis equation and L1 norm on a cmp gather: a) input gather; b) reconstructed gather using P- and

S-reflectivity; and c) difference between a) and b).


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Figure 5. Seismic cmp gather (left) ties to synthetic cmp gather (right).

Further, calibration can be conducted on AVO attributes or

inverted r ock prop erties. P- and S-reflectivity synthetic may tie

to stacked P- and S-reflectivity sections. It also can be

condu cted in cross-plotting spaces su ch as P-reflectivity against

S-reflectivity, and against . Figure 7 shows an example

using w ell logs to calibrate inverted elastic rock prop erties for a

gas charged dolomite reservoir. In Figures 7a and 7b, the

inverted elastic rock properties of the reservoir are high lighted

in black squares. The overlain empirical relations are shale

(solid black), water saturated sand (solid blue), limestone

(dashed black), dolomite (dashed gr een), and gas charged clean

sand (red ). We can see that the da ta points from the gas-charged

reservoir are shifted towards low values and low / ratio.

Figures 7c and 7d show the cross-plots of dipole sonic logs. The

data from a gas charged dolomite reservoir are highlighted

with red squares, and brine-saturated porous dolomite with

green squares. This comparison leads to an interpretation of a

gas-charged dolomite based on the seismically-derived elastic

rock prop erties (Figures 7a and 7b). In ad dition, the porosity of

the reservoir is similar to the porous dolomite that is high-

lighted by the green squares.

Global calibration implies a way to QC data on entire data set.

For example, the amplitude variation with offset within a time

window can be calculated and compared to that from synthetic

gathers. Consequ ently, offset-variant scaling corrections m ay be

applied to the data set. Calibration may also be conducted

based on relationships of AVO attributes such as P- and S-

reflectivities. Background constraints that are used in data

modeling and elastic rock property inversion may also be cali-

brated through this method.

Interpretation

AVO modeling assists interpretation on cmp gathers, AVO

attributes, and inverted elastic rock prop erties. It helps in vali-

dating AVO respon ses and linking seismic expression to know n

reservoir cond itions. AVO mod eling can increase confidence in

interpretation an d red uce risk in reservoir characterization as it

provides independent information. We can use synthetic to

identify AVO anomalies and determine AVO types on a cmp

gather. Also, we can use synthetic pre-stack data to determ ine

the S-wave information that often is ignored in conventional

data processing. For instance, a strong S-impedance contrast

may exist for a reservoir even though P-impedance contrast is

small. The derived AVO attributes such as fluid factor, and

inverted elastic rock prop erties such as Vp/ Vs ratio, Poissons

ratio, and / ratio can be used to infer the fluid type in a

reservoir.

Tuning may invoke or mask an AVO anomaly. The synthetics

with varied bandwidth or reservoir thickness may give

answ ers to tu ning q uestions. Special lithologies or lithological

contrasts may generate AVO anomalies and can be proved by

AVO modeling. Lithological complexity may also bring in diffi-

culties in interpretation. Tight streaks may m anifest themselvesFigure 6. Comparison of amplitudes from seismic and synthetic cmp gathers.


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as AVO anomalies and brighten in a stacked section. Coal,

carbonate, and the lithologies that do not follow water-

saturated trend of rock properties for clastics may complicate

fluid stack anomalies. The lack of understanding on some

seismic rock properties may prevent one to effectively explore

those types of reservoirs. Further, high clay content in sand

may result in low gas saturation. This type of partial gas satu-

ration may still have relative high Vp/ Vs ratio that contrad icts

traditional theories of partial gas saturation. Therefore, AVO

modeling may provide opportunities for distinguishing the

partial gas satu rated reservoirs based on the lithological effect

on rock properties.


Figure 7. a) and b): elastic rock properties from inverted seismic; and c) and d): from well logs.

Figure 8. Workflow using AV O modeling in assisting data processing and interpretation.


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Figure 9. Zoeppritz modeling and elastic modeling for a reservoir in the Wabamun Formation in the

WCSB: a) Zoeppritz modeling; b) elastic modeling with reservoir; c) elastic modeling with no reservoir;and d) difference between reservoir and non-reservoir cases.

Figure 10. AV O modeling and interpretation for a gas charged dolomite reservoir, a) synthetic cmp gather, and b) stack section and cmp gathers at well locations.

Figure 8 shows an ideal workflow in using AVO modeling to

assist data p rocessing an d inter pretation. We can see that AVO

modeling workflow is the same as that of data processing.

Therefore, seismic data and synthetics can be compared in the

stages of cmp gath ers, AVO attributes and inverted elastic rock

properties. In interpretation, the information from all three

branches can be integrated. The risk in reservoir characteriza-

tion may thus be reduced since the interpretation is broadly

based, involving und erstand ing of seismic, rock physical prop-

erties and geology.

Several AVO modeling examples for AVO interpretation have

been given in Part 1 and Part 2 of this article (Li et al., 2003;

Li et al. 2004). Figure 9 shows an example using AVO modeling

to understand interference of multiples and converted energy

at the Wabamu n dolomite poro s i t y. The mod eling was

conducted using both 1) Zoeppritz modeling with ray-tracing,

and 2) full wave elastic wave equation. For the elastic

modeling, two cases were modeled: reservoir case (with

porosity) and n on-reservoir case (withou t porosity). The obser-

vations that can be m ade for this typical study includ e: a) Class

a) b) c) d)


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III AVO anom aly (a trough brightens w ith offset) at the top of

the reservoir presents in the Zoeppritz mod eling but it does not

show in th e elastic modeling; b) the elastic mod eling show s that

mu ltiples and converted energy (w hich are not accounted for in

a Zoeppritz model) interfere with the reflections from both the

top and the base of the reservoir; and c) the difference between

the reservoir case (Figur e 9b) and the n on-reservoir case (Figure

9c) indicates that it could be enough for differentiating the

porosity case from non-porosity case. Further, we may notice

the inter-bedded mu ltiples and converted energy generated by

the reservoir (Figure 9d). This study provides the information

of wave prop agation and interference. We ma y use it as a guide

for attenuating the coherent noise and performing amp litude

recovery.

The second example is from a study on carbonate reservoirs

(Li et al., 2003). In Figure 10a, AVO modeling shows that a gas

charged dolom ite reservoir prod uces a Class III AVO anomaly.

This is consistent with the AVO response in the cmp gather at

the location of the p rodu cing w ell. We can see that at the tight

well locations, a completely different AVO response presents.

The information provided by AVO modeling validates the

information from seism ic.

Conclusions

This paper, the third part of AVO modeling in seismic

processing and interpretation, demonstrates some applications

of AVO mod eling involving da ta processing and interpr etation.

The discussion of data modeling provides an insight in AVO

attribute extraction. Calibration using AVO modeling in data

processing sheds som e light on how to optimize AVO solution.

Combined with rock physical property analysis, petrophysical

analysis, and geological information, AVO modeling provides

useful information in interpretation and thus incre a s e s

certainty in reservoir characterization.

AcknowledgementsThe authors thank Core Laboratories Reservoir Technologies Division for

supporting this work.

ReferencesLi, Y., Down ton , J., and G ood wa y, B., 2003, Recent ap plication of AVO to

carbonate reservoirs in the Western Canadian Sedimentary Basin, The Leading

Edge, 22, 671-674.

Li, Y., Down ton , J., and Xu, Y., 2003, AVO modelin g in seism ic processing and

interpretation, P art 1: fund amentals, Recorder, 28 December, 43-52.

Li, Y., Down ton , J., and Xu, Y., 2004, AVO modelin g in seism ic processing and

interpretation, P art 2: method ologies, Recorder, 29 January, 36-42.

Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, W .T., 1989, Num ericalRecipes, The Art of Scientific Comp uting, Cam bridge U niversity Press.

R g e r, A ., 1996, Reflection Coefficients and Azimu thal AVO Ana lysis in

Anisotrop ic Media, Ph.D. Thesis, Colorado School of Mines.

This is the third and final part of the series.

Article ContdAVO Mod eling in Seismic Pr ocessing an d Inter pr etat ionContinued from Page 47

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