Changes

The relationship between the reflection coefficient and the angle of incidence in write down the Zoeppritz equations by Karl Zoeppritz since the early 20th century and then this equations was developed again by some figures such as Bortfeld (1961), Aki, Richard and Frasier (1976), Hilterman (1983), and Shuey (1985).  

The relationship between the reflection coefficient and the angle of incidence in write down the Zoeppritz equations by Karl Zoeppritz since the early 20th century and then this equations was developed again by some figures such as Bortfeld (1961), Aki, Richard and Frasier (1976), Hilterman (1983), and Shuey (1985).  

== Zoeppritz Equations ==

==Zoeppritz Equations==

Zoeppritz derived the amplitudes of the reflected and transmitted waves using the conservation of stress and displacement across the layer boundary.

Zoeppritz derived the amplitudes of the reflected and transmitted waves using the conservation of stress and displacement across the layer boundary.

Because Zoeppritz equations give four equations with four unknowns, there are some of approximations in order to perform physical interpretation and visualization of the effect of some parameters to reflection coefficient. Some of the approximations are:

Because Zoeppritz equations give four equations with four unknowns, there are some of approximations in order to perform physical interpretation and visualization of the effect of some parameters to reflection coefficient. Some of the approximations are:

'''a) Bortfeld Approximation (1961)'''

===Bortfeld Approximation (1961)===

With assumptions that there is slight change in the layer properties, Bortfeld generates this approximation :  

With assumptions that there is slight change in the layer properties, Bortfeld generates this approximation :  

<math> R(\Theta _1) = \frac{1}{2}\ln \left ( \frac{V_{p2}\rho _2\cos \theta _1}{V_{p1}\rho _1\cos \theta _2} \right )+\frac{\sin^{2}\theta_1}{V_{p1}}\left ( V_{s1}^{2}-V_{s2}^{2} \right )\left [ 2+\frac{\ln\left ( \frac{\rho _2}{\rho_1} \right )}{\ln\left ( \frac{Vp_2}{Vp_1} \right )- \ln\left ( \frac{Vp_2 Vs_1}{Vp_1Vs_2} \right )} \right ] </math>

<math> R(\Theta _1) = \frac{1}{2}\ln \left ( \frac{V_{p2}\rho _2\cos \theta _1}{V_{p1}\rho _1\cos \theta _2} \right )+\frac{\sin^{2}\theta_1}{V_{p1}}\left ( V_{s1}^{2}-V_{s2}^{2} \right )\left [ 2+\frac{\ln\left ( \frac{\rho _2}{\rho_1} \right )}{\ln\left ( \frac{Vp_2}{Vp_1} \right )- \ln\left ( \frac{Vp_2 Vs_1}{Vp_1Vs_2} \right )} \right ] </math>

'''b) Aki, Richard and Frasier Approximation (1976)'''

===Aki, Richard and Frasier Approximation (1976)===

Bortfeld approximation then revised by Richard-Frasier (1976) and by Richard-Aki (1980). Approximation of Richard-Frasier is giving a simple equation because it is written in a clear three forms, namely the right-hand side includes the first P wave velocity, the second node density and the last wave velocity.

Bortfeld approximation then revised by Richard-Frasier (1976) and by Richard-Aki (1980). Approximation of Richard-Frasier is giving a simple equation because it is written in a clear three forms, namely the right-hand side includes the first P wave velocity, the second node density and the last wave velocity.

     <math> \theta = \frac{\theta_2+\theta_1}{2} </math>

     <math> \theta = \frac{\theta_2+\theta_1}{2} </math>

 

===Hilterman (1983)===

'''c) Hilterman (1983)'''

Hilterman simplified Bortfeld equations by separating the reflection coefficients into the form of acoustic and elastic :

Hilterman simplified Bortfeld equations by separating the reflection coefficients into the form of acoustic and elastic :

<math> R(\theta_1) = \frac{Vp_2\rho_2\cos\theta_1 - Vp_1\rho_1\cos\theta_2}{Vp_2\rho_2\cos\theta_1 + Vp_1\rho_1\cos\theta_2 }+ \left ( \frac{\sin\theta_1 }{Vp_1}\right )\left ( Vs_1+Vs2 \right )\left [ 3\left ( Vs_1-Vs2 \right ) + \frac{2\left ( Vs_2\rho_1 - Vs_1\rho_2 \right )}{\rho_2 + \rho_1} \right ] </math>

<math> R(\theta_1) = \frac{Vp_2\rho_2\cos\theta_1 - Vp_1\rho_1\cos\theta_2}{Vp_2\rho_2\cos\theta_1 + Vp_1\rho_1\cos\theta_2 }+ \left ( \frac{\sin\theta_1 }{Vp_1}\right )\left ( Vs_1+Vs2 \right )\left [ 3\left ( Vs_1-Vs2 \right ) + \frac{2\left ( Vs_2\rho_1 - Vs_1\rho_2 \right )}{\rho_2 + \rho_1} \right ] </math>

'''d) Shuey Approximation (1985)'''

===Shuey Approximation (1985)===

Shuey modified Aki and Richard equation by using variable of poisson ratio as follows:

Shuey modified Aki and Richard equation by using variable of poisson ratio as follows:

       <math> B = \frac{\frac{\Delta \alpha }{\alpha }}{\frac{\Delta \alpha }{\alpha }+ \frac{\Delta \rho }{\rho }}</math>

       <math> B = \frac{\frac{\Delta \alpha }{\alpha }}{\frac{\Delta \alpha }{\alpha }+ \frac{\Delta \rho }{\rho }}</math>

 

==AVO Classification==

== AVO Classification ==

In 1989 Rutherford and Williams introduced a three-fold classification of AVO (amplitude versus offset) characteristics for seismic reflections from the interface between shales and underlying gas sands. The classification scheme they proposed is explicitly defined for gas sands and has become the industry standard; it has proven its validity and usefulness in countless exploration efforts. In 1997 Castagna and Swan proposed AVO crossplotting wherein an estimate of the normal-incidence reflectivity is plotted against a measure of the offset dependent reflectivity. Using this approach Castagna and Swan graphically illustrated the continuum between the classes and defined the characteristics of the classes using what they termed AVO Intercept and AVO Gradient. They also added a class 4 <ref> Young, Roger A. and LoPiccolo, Robert D. (2004). Conforming and Non-conforming Sands – An Organizing Framework for Seismic Rock Properties. </ref>.

In 1989 Rutherford and Williams introduced a three-fold classification of AVO (amplitude versus offset) characteristics for seismic reflections from the interface between shales and underlying gas sands. The classification scheme they proposed is explicitly defined for gas sands and has become the industry standard; it has proven its validity and usefulness in countless exploration efforts. In 1997 Castagna and Swan proposed AVO crossplotting wherein an estimate of the normal-incidence reflectivity is plotted against a measure of the offset dependent reflectivity. Using this approach Castagna and Swan graphically illustrated the continuum between the classes and defined the characteristics of the classes using what they termed AVO Intercept and AVO Gradient. They also added a class 4 <ref> Young, Roger A. and LoPiccolo, Robert D. (2004). Conforming and Non-conforming Sands – An Organizing Framework for Seismic Rock Properties. </ref>.

[[File:Fig.4.classification avo.jpg|framed|center|Fig.4. Classifications of AVO <ref>Ahmed,Haseb. Institute of Geologi,University of the punjab </ref>]]

[[File:Fig.4.classification avo.jpg|framed|center|Fig.4. Classifications of AVO <ref>Ahmed,Haseb. Institute of Geologi,University of the punjab </ref>]]

'''Class 1 : High impedance Gas-Sandstone'''

===Class 1 : High impedance Gas-Sandstone===

Class 1 sandstone has higher impedance than its cover (shale). Interface between shale and this kind of sandstone will generate a high reflection coefficient and a positive zero offset, but has amplitude magnitude decreasing in order to offset. Class 1 has greater gradient than class 2 and class 3. Sandstone at class 1 is having a change in polarity in certain angle, then the amplitude will be increasing proportionally to the offset.

Class 1 sandstone has higher impedance than its cover (shale). Interface between shale and this kind of sandstone will generate a high reflection coefficient and a positive zero offset, but has amplitude magnitude decreasing in order to offset. Class 1 has greater gradient than class 2 and class 3. Sandstone at class 1 is having a change in polarity in certain angle, then the amplitude will be increasing proportionally to the offset.

'''Class 2 : Near zero impedance contrast gas sandstone'''

===Class 2 : Near zero impedance contrast gas sandstone===

Class 2 sandstone has almost equally acoustic impedance with its cover (seal rock) and the amplitude which is increasing proportionally to the offset. Class 2 sandstone divided into class 2 and class 2p. class 2 sandstone has negative reflection coefficient at zero offset while class 2p has positive at zero offset.

Class 2 sandstone has almost equally acoustic impedance with its cover (seal rock) and the amplitude which is increasing proportionally to the offset. Class 2 sandstone divided into class 2 and class 2p. class 2 sandstone has negative reflection coefficient at zero offset while class 2p has positive at zero offset.

'''Class 3 : Low impedance gas sandstone'''

===Class 3 : Low impedance gas sandstone===

Class 3 has lower acoustic impedance than its cover.

Class 3 has lower acoustic impedance than its cover.

'''Class 4 '''

===Class 4===

Class 4 has negative reflection coefficient at zero offset and lower impedance with amplitude that is decreasing against the offset. There is a change in polarity at a certain angle and then amplitude will increasing proportionally to the offset.  

Class 4 has negative reflection coefficient at zero offset and lower impedance with amplitude that is decreasing against the offset. There is a change in polarity at a certain angle and then amplitude will increasing proportionally to the offset.  

|}  

|}  

= Reference =

==References==

<references />

<references />

*Rutherford,S.R. and Williams R.H. (1989). Amplitude-versus-offset Variations in Gas Sands.

*Rutherford,S.R. and Williams R.H. (1989). Amplitude-versus-offset Variations in Gas Sands.

*C. Ecker, D. Lumley, S. Levin, T. Rekdal, A. Berlioux, R. Clapp, Y. Wang & J. Ji. (2001). An AVO Analysis Project.

*C. Ecker, D. Lumley, S. Levin, T. Rekdal, A. Berlioux, R. Clapp, Y. Wang & J. Ji. (2001). An AVO Analysis Project.

*http://www.ipt.ntnu.no/pyrex/stash/avo_theory.pdf , accessed on June 2015.

*http://www.ipt.ntnu.no/pyrex/stash/avo_theory.pdf , accessed on June 2015.