Avo Primer

AVO Amplitude Versus Offset Analysis: A Primer

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Pre-processing prior to AVO Analysis • Data quality: reflection amplitudes should represent reflection coefficients. • Reflections must be correctly placed in the subsurface (Migration).

• Pre-processing should be able to preserve or restore relative trace amplitude in the gathers data.

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Preserving relative amplitudes Factor that change the seismic amplitudes can be grouped into three categories: the earth effects, acquisition-related effects, and noise (Dey-Sarkar et al., 1986). Earth effects: spherical divergence, absorption, transmission losses, interbed multiples, converted phases, tuning, anisotropy, and structure. Acquisition-related effects: source and receiver coupling variations, lateral changes in weathered layer properties, source and receiver arrays, and receiver sensitivity. Noise: ambient or source-generated, coherent or random.

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P and S-Waves

(a)

(b)

(c)

The above diagram shows a schematic diagram of (a) P, or compressional, waves, (b) SH, or horizontal shear-waves, and (c) SV, or vertical shear-waves, where the S-waves have been generated using a shear wave source. (Ensley, 1984)

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P and S-Waves to AVO In the previous slide, the P and SH-waves were generated at the surface by P and S-wave sources. We could use the differences between the recorded P and S reflections to discriminate gas-filled sands from wet sands, using the properties discussed in the last section. Unfortunately, most seismic surveys record P-wave data only, and S-wave data is not available. However, as shown in the next slide, if we record P-wave data at various offsets (as we always do), mode-conversion from P to SV always occurs. This means that AVO data can be used as a replacement for S-wave data.

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Mode Conversion of an Incident P-wave Incident P-wave If  > 0°, an incident P-wave will produce both P and SV reflected and transmitted waves. This is called mode conversion.

Reflected SV-wave Reflected P-wave = RP

r i

r

VP1 , VS1 , r1 VP2 , VS2 , r2

t t

Transmitted P-wave Transmitted SV-wave

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Utilizing Mode Conversion But how do we utilize mode conversion?

There are actually two ways: 1) Record the converted S-waves using two- (or three-) component receivers (in the X or Y and Z direction). 2) Interpret the amplitudes of the P-waves as a function of offset, or angle, which contain implied information about the S-waves. This is called the AVO (Amplitude versus Offset) method. When we record the converted waves, we need to be very careful in their processing and interpretation. In the AVO method, we can make use of the Zoeppritz equations, or some approximation to these equations, to extract S-wave type information from P-wave reflections at different offsets.

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Utilizing Mode Conversion But how do we utilize mode conversion?

There are actually two ways: 1) Record the converted S-waves using two- (or three-) component receivers (in the X or Y and Z direction). 2) Interpret the amplitudes of the P-waves as a function of offset, or angle, which contain implied information about the S-waves. This is called the AVO (Amplitude versus Offset) method. When we record the converted waves, we need to be very careful in their processing and interpretation. In the AVO method, we can make use of the Zoeppritz equations, or some approximation to these equations, to extract S-wave type information from P-wave reflections at different offsets.

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The Zoeppritz Equations Zoeppritz (1911) derived the amplitudes of the reflected and transmitted waves using the conservation of stress and displacement across the layer boundary, which gives four equations with four unknowns. Inverting the matrix form of the Zoeppritz equations gives us the exact amplitudes as a function of angle:

  sin 1 RP   cos  1 R    S    sin 2 1 TP      TS   cos 21 

 cos 1

sin  2

 sin 1

cos  2 r2VS22VP1 cos 21 r1VS12VP 2 r2VP 2 cos 22 r1VP1

VP1 cos 21 VS1 VS1 sin 21 VP1

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cos 2

   sin 2  r2VS 2VP1  cos 2  2 r1VS12  r2VS 2  sin 22   r1VP1

1

 sin 1   cos   1    sin 21    cos 2  1 

The Aki-Richards Equation The Aki-Richards equation is a linearized approximation to the Zoeppritz equations. The initial form (Richards and Frasier, 1976) separated the velocity and density terms:

R(  )  a where:

1 , 2 2 cos    V 2  2 S b  0.5  2  sin   ,   VP   a

2

 VS  c  4  sin 2  ,  VP 

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r

 VP VP

r 2  r1 2

b

 VS r c r VS

, r  r 2  r1 ,

VP 2  VP 1 VP  , VP  VP 2  VP 1 , 2 V  VS 1 VS  S 2 , VS  VS 2  VS 1 , 2 i  t and   . 2

Wiggins’ Version of the Aki-Richards Equation A more intuitive, but totally equivalent, form was derived by Wiggins. He separated the equation into three reflection terms, each weaker than the previous term:

R(  )  A  B sin 2   C tan2  sin 2  where:

1   VP r  A    2  V p r  2

2

VS   VS VS  r 1  VP B  4   2  2 Vp VP  VS VP  r 1  VP C 2 Vp

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Aki-Richards Equation The first term, A, is a linearized version of the zero offset reflection coefficient and is thus a function of only density and P-wave velocity. The second term, B, is a gradient multiplied by sin2, and has the biggest effect on amplitude change as a function of offset. It is dependent on changes in P-wave velocity, S-wave velocity, and density. The third term, C, is called the curvature term and is dependent on changes in Pwave velocity only. It is multiplied by tan2*sin2 and thus contributes very little to the amplitude effects below angles of 30 degrees. (Note: Prove to yourself that tan2*sin2 = tan2 - sin2, since the equation is often written in this form.)

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Ostrander’s Paper

Ostrander (1984) was one of the first to write about AVO effects in gas sands and proposed a simple two-layer model which encased a low impedance, low Poisson’s ratio sand, between two higher impedance, higher Poisson’s ratio shales. This model is shown in the next slide. Ostrander’s model worked well in the Sacramento valley gas fields. However, it represents only one type of AVO anomaly (Class 3) and the others will be discussed in the next section.

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Ostrander’s Model

the model consists of a low acoustic impedance and Poisson’s ratio gas sand encased between two shales.

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Synthetic from Ostrander’s Model

(a) Well log responses for the model.

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(b) Synthetic seismic.

AVO Curves from Ostrander’s Model (a) Response from top of model to 45o. Note that the transmitted Pwave amplitude is shifted.

(b) Response from base of model to 45o. Note that the transmitted Pwave amplitude is shifted.

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Rutherford/Williams Classification Rutherford and Williams (1989) derived the following classification scheme for AVO anomalies, with further modifications by Ross and Kinman (1995) and Castagna (1997):

Class 1: High acoustic impedance contrast Class 2: Near-zero impedance contrast Class 2p: Same as 2, with polarity change Class 3: Low impedance contrast Class 4: Very low impedance contrast

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Class 4 The Rutherford and Williams classification scheme as modified by Ross and Kinman (1995) and Castagna (1997).

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Wet and Gas Models Let us now see how to get from the geology to the seismic. We will do this by using the two models shown below. Model A consists of a wet sand, and Model B consists of a gas-saturated sand.

(a) Wet model

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(b) Gas model

AVO Models In the next two slides, we are going to compute the top and base event responses from Models A and B, using the following values, where the Wet and Gas cases were computed using the Biot-Gassmann equations:

Wet: VP= 2500 m/s, VS= 1250 m/s, r = 2.11 g/cc, s = 0.33 Gas: VP= 2000 m/s, VS= 1310 m/s, r = 1.95 g/cc, s = 0.12 Shale: VP= 2250 m/s, VS= 1125 m/s, r = 2.0 g/cc, s = 0.33 We will consider the AVO effects with and without the third term in the AkiRichards equation.

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AVO Wet Model AVO - Wet Sand (Model A) Base

0.100

0.000

0.080

-0.020

Amplitude

Amplitude

AVO - Wet Sand (Model A) Top

0.060 0.040 0.020 0.000

-0.040 -0.060 -0.080 -0.100

0

5

10

15

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25

30

35

40

45

Angle (degrees) R (All three terms)

(a)

R (First two terms)

0

5

10

15

20

25

30

35

40

Angle (degrees) R (All three terms)

R (First two terms)

(b)

The above figures show the AVO responses from the (a) top and (b) base of the wet sand. Notice the decrease of amplitude, and also the fact that the two-term approximation is only valid out to 30 degrees.

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45

AVO Gas Model AVO - Gas Sand (Model B) Base

0.000

0.250

-0.050

0.200

Amplitude

Amplitude

AVO - Gas Sand (Model B) Top

-0.100 -0.150 -0.200

0.150 0.100 0.050

-0.250

0.000

0

5

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35

40

45

Angle (degrees) R (All three terms)

(a)

R (First two terms)

0

5

10

15

20

25

30

35

40

45

Angle (degrees) R (All three terms)

R (First two terms)

(b)

The above figures show the AVO responses from the (a) top and (b) base of the gas sand. Notice the increase of amplitude, and again the fact that the two-term approximation is only valid out to 30 degrees.

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Shuey’s Equation Shuey (1985) rewrote the Aki-Richards equation using VP, r, and s. Only the gradient is different than in the Aki-Richards expression:

1  2s  s  B  A D  2( 1  D )  2  1  s ( 1  s )  

VP / VP where : D  , VP / VP  r / r s 2 s1 s 2

s  s 2  s 1

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Gas Sand Model Aki-Richards vs Shuey 0.250 0.200 0.150 0.100 Amplitude

This figure shows a comparison between the two forms of the Aki-Richards equation for the gas sand.

0.050 0.000 -0.050 -0.100 -0.150 -0.200 -0.250 0

5

10

15

20

25

30

35

40

Angle (degrees)

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A-R Top

Shuey Top

A-R Base

Shuey Base

45

Single Layer Models The previous exercise showed us that for a gas sand with a low acoustic impedance, we can expect absolute amplitude increases with offset at the both the top and bottom of the sand. For the models, we used P and S-wave velocity. Another approach is to use the Poisson’s ratio change as the key parameter.

The next figure shows four single-layer boundaries consisting of all combinations of increasing and decreasing acoustic impedance and Poisson’s ratio. Note that the sign of the gradient is generally to same as the sign of s. (This is not true in the case of a Class 4 sand, as we shall see in a later theory section.)

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Four Single Layer Models

(a) r, VP, and s all increase.

(b) r, VP increase, s decreases.

(c) r, VP decrease, s increases.

(d) r, VP, and s all decrease.

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Multi-layer AVO Modeling

Multi-layer modeling in the AVO program consists first of creating a stack of N layers, generally using well logs, and defining the thickness, P-wave velocity, Swave velocity, and density for each layer.

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Multi-layer AVO Modeling

You must then decide what effects are to be included in the model: primaries only, converted waves, multiples, or some combination of these.

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AVO Modeling Options There are three main options for the modeling process: Zoeppritz - Primaries only using the Zoeppritz equations for calculation. Aki-Richards - Primaries only using the Aki-Richards equations for calculation. Full Elastic Wave - Computation of the full elastic wave solution (with optional an elastic effects), which includes primaries, converted waves, and multiples. The following example, taken from a paper by Simmons and Backus (AVO Modeling and the locally converted shear wave, Geophysics 59, p1237, August, 1994), illustrates the effect of wave equation modeling.

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The Oil Sand Model

Simmons and Backus used the thin bed oil sand model shown above.

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The Possible Modeled Events

Simmons and Backus (1994)

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Responses to Various Algorithms (A) Primaries-only Zoeppritz, (B) + single leg shear, (C) + double-leg shear, (D) + multiples,(E) Wave equation solution, (F) Linearized approximation.

Primaries only Zoeppritz + single leg shear + double leg shear + multiples Wave equation Aki-Richards Simmons and Backus (1994)

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Logs from the Real Data Example

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Models from a Real Data Example

(a) Full elastic wave. (b) Zoeppritz eqn. (c) Aki-Richards eqn.

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Anisotropy and AVO So far, we have considered only the isotropic case, in which earth parameters such as velocity do not depend on seismic propagation angle. In the next few slides, we will discuss anisotropy, in particular the case of Transverse Isotropy with a vertical symmetry axis, or VTI.

We will then see how anisotropy affects the AVO response. Finally, we will look at this effect on our original model.

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Isotropic versus Anisotropic (VTI) Velocity As mentioned, in an isotropic earth P and S-wave velocities are independent of angle. VTI velocities depend on angle, as shown below for three different angles:

VP(90o) VP(45o) VP(0o) VTI can be extrinsic, caused by fine layering of the earth, or intrinsic, caused by particle alignment as in a shale.

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Velocities for Weak Anisotropy Although the equations for full anisotropy are quite complex, Thomsen (1986) showed that for weakly anisotropic materials the velocities can be written as follows, where e, d, and g are called Thomsen’s parameters. Note that, for AVO and converted wave studies, we are only interested in the first two velocities and constants. Note also that VSV(0o) = VSH(0o):

VP ( )  VP (0o ) 1  d sin 2  cos 2   e sin 4   2 o   V (0 ) o 2 2 P VSV ( )  VSV (0 ) 1  2 o (e  d ) sin  cos    VSV (0 ) 

VSH ( )  VSH (0o ) 1  g sin 2  

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Thomsen’s Parameters Thomsen’s parameters are simply combinations of the differences between the P and S velocities at 0, 45, and 90 degrees. The following relationships can be derived quite easily using the velocities in the previous slide:

VP ( 90 o )  VP ( 0 o ) e VP ( 0 o )

VSH ( 90 o )  VSH ( 0 o ) g VSH ( 0 o )

VP ( 45 o )  VP ( 0 o )  VP ( 45 o )  VP ( 0 o )  d  4   e  d  e  4  o o V ( 0 ) V ( 0 ) P P     In the next slide, we will look at VP and VSV as a function of angle for different values of d and e. (As mentioned, VSH will not be used in AVO).

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Group Angle versus Phase Angle For anisotropic velocities, it is important to note the difference between the phase angle , which is computed normal to the seismic wavefront, and the group or ray angle , along which energy propagates. This is illustrated below.



x



Ray Wavefront Normal

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Wavefront

z

Anisotropic P and SV VTI velocities

(a) VTI medium with d = 0.2 and e = 0.2.

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(b) VTI medium with d = 0.1 and e = 0.2.

(c) VTI medium with d = 0.2 and e = 0.1.

Solving for e and d using the Velocity

VP(90o)= 2600 m/s

600 m/s

VP(45o)= 2225 m/s

225 m/s VP(0o)= 2000 m/s

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Solving for e and d using the Velocity

VP ( 90 o )  VP ( 0 o ) e  0.3 o VP ( 0 )

VP ( 45 o )  VP ( 0 o )  d  4   e  0.45  0.3  0.15 o VP ( 0 )  

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AVO and Transverse Isotropy Thomsen (1993) showed that a transversely isotropic term could be added to the Aki-Richards equation using his weak anisotropic parameters d and e, where Ran( ) is the anisotropic AVO response and Ris( ) is the isotropic AVO response. Ruger (2002) gave the following corrected form of Thomsen’s original equation:

Ran (  )  Ris (  )  where :

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d 2

sin  

d  d 2  d 1 e  e 2  e 1

2

e 2

sin 2  tan2  ,

Typical Values for Delta, Epsilon and Gamma Typical values for delta, epsilon, and gamma were given by Thomsen (1986). Here are some representative values from his table: Lithology

VP(m/s)

VS(m/s)

rho(g/cc)

epsilon

delta

gamma

sandstone_1

3368

1829

2.50

0.110

-0.035

0.255

sandstone_2

4869

2911

2.50

0.033

0.040

-0.019

calcareous sandstone

5460

3219

2.69

0.000

-0.264

-0.007

immature sandstone

4099

2346

2.45

0.077

0.010

0.066

shale_1

3383

2438

2.35

0.065

0.059

0.071

shale_2

3901

2682

2.64

0.137

-0.012

0.026

mudshale

4529

2703

2.52

0.034

0.211

0.046

clayshale

3794

2074

2.56

0.189

0.204

0.175

silty limestone

4972

2899

2.63

0.056

-0.003

0.067

laminated siltstone

4449

2585

2.57

0.091

0.565

0.046

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AVO and Transverse Isotropy Blangy (1997) computed the effect of anisotropy on models of the three RutherfordWilliams type. Blangy’s models are shown below, but since he used Thomsen’s formulation for the linearized approximation, his figures have been recomputed in the next slide for the wet and gas cases using Ruger’s formulation. The slide after that shows our example.

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d = -0.15 e = -0.3

Transverse Isotropy – AVO Effects Class 1

Class 1 Class 2 Class 2

Class 3 Class 3

Isotropic (a) Gas sandstone case: Note that the effect of d and e is to --- Anisotropic increase the AVO effects.

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(b) Wet sandstone case: Note that the effect of d and e is to create apparent AVO decreases.

Anisotropy Applied to Gas Sand Example Isotropic vs Anisotropic AVO Gas Sand Top, d = -0.15, e = -0.3

Amplitude

0.000 -0.100 -0.200 -0.300 -0.400 0

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Angle (degrees) R (Isotropic)

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R (Anisotropic)

45

Anisotropic AVO Model Example

Notice that only the gas sand is isotropic.

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Anisotropic AVO Synthetics

(a) Isotropic

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(b) Anisotropic

(a) – (b)

(a) Isotropic

(b) Anisotropic

(a) – (b)

In the above display, the synthetic responses in the previous slide are shown using colour amplitude scale.

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Summary This section introduced the theory of AVO and considered a number of modeled examples. Our first modeled example looked at both a wet sand and a gas sand, which were based on typical values found in a reservoir. As we will see in the next section, this is the most common response and is called a Class 3 anomaly. We also found that modeling can be very sensitive to the type of algorithm used. For thin beds, wave equation modeling is suggested. Finally, anisotropy should also be modeled, since it can have a large effect on the AVO response.

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