January 25, 2017
### Amplitude Variations with Offset

Seismic reflection amplitude changes with distance between shotpoint and receiver,It happens due to energy partitioning.When seismic wave hit a boundary,some of the energy gets reflected and some gets refracted.The reflected and refracted energy amplitude depends upon physical properties contrast across the boundary.If incident waves is not normally incident,incident P waves get partitioned into reflected and refracted P & S waves.Reflected P& S wave amplitude depends upon reflection coeffecient with respect to angle of incident of seismic wave as explained by Zoepprittz equation,Reflection coefficient for normally incident wave is a function of compressional wave velocity,shear wave velocity and density. AVO analysis helps geophysicist determine differences in lithology and fluid content above and below the reflector.

**Figure:**** Reflected and Refracted P&S waves due to contrast in physical properties and angle of incidence on both sides of reflector **

** **AVO technique is widely used by geophysicist to identify fluid type for example in case of gas bearing sandstone amplitude increases with offset on the other hand in case of coal amplitude decreases with offset .In order to do proper AVO analysis,its important to model rock property for known fluid content,with this information we can model other fluid contents also.Bright spots are one of the most important indicators of gas zones but bright spots can also be due to tight rock so in order to identify hydrocarbon special handling of data for AVO purpose is needed so that high amplitude bright spot from gas can be distinguished from tight sand.

AVO behavior is sometimes observed on pre-stack data,stacking the data enhances the signal to noise ratio but at the same time destroys AVO information so AVO analysis requires processed gathers that are not stacked.

The equation for reflection of compressional and shear wave was derived by Karl Zoeppritz.For normally incident waves reflection coefficient is function of acoustic impedance contrast between two media:-

Reflection coefficient= ( V_{2}r_{2}-V_{1}r_{1}) / ( V_{2}r_{2}+V_{1}r_{1})

Z_{2}= V_{2}r_{2}

Z_{1=}V_{1}r_{1}

Where

V_{2}=Seismic Velocity of lower layer

V_{1}=Seismic Velocity of upper layer

r_{1}=Density of upper layer

r_{2}=Density of lower layer

Z_{1}= acoustic impedance of upper layer

Z_{2}= acoustic impedance of lower layer

AVO utilizes the fact that reflection coefficient varies with changing offsets,and this behavior can be observed in pre-stack gather.As discussed earlier an incident compressional wave falling at the reflector gets partitioned into reflected P wave ,reflected S wave,refracted P wave and Refracted S waves,as defined by Zoeppritz’s equation reflection coefficient of these waves depends on acoustic impedance contrast between two media and angle of incidence.* Acoustic impedance of material which is a function of density and velocity in turn *depends on the physical parameters such as lithology, porosity, pore fluid and pressure. Zoeppritz equation for incident wave is given as below:-

Where

*R*_{P},*R*_{S},*T*_{P}, and*T*_{S}, are the reflected P, reflected S, refracted P, and refracted S-wave amplitude coefficients f1,f2,q1and q2 are as per

*R*_{P}, *R*_{S}, *T*_{P}, and *T*_{S}, can be calculated from matrix form of Zoeppritz equation by inverting the matrix.

Zoeppritz equation provides exact estimation of *R*_{P}, *R*_{S}, *T*_{P}, and *T*_{S}_{ }but this equation is an complex matrix equation,modelling can be done on Zoeppritz equations but realtime seismic data are estimated on the basis of linearized approximation of Zoeppritz equations.These equations are as below:-

- Aki–Richards equation
- Shue equation
- Fatti equation
- Bortfeld equation
- Hilterman approximation
- Verm & Hiilterman approximation

Another way to look at AVO analysis is in terms of VP/Vs ratio vs Poisson’s ratio.Poisson’s ratio is ratio of transverse strain to longitudinal strain,Poisson’s ratio can also be defined in terms of VP/Vs ratio as below:-

PR={(Vp/Vs)^{2}-2}/{2(Vp/Vs)^{2}-2}

Where

PR = Poisson’s ratio

Vp=Compressional wave velocity

Vs=Shear wave velocity

For ease of understanding lets look at relationship between poisson’s ratio and Vp/Vs ratio graphically.

**Figure:- Plot of Poisson’s ratio versus ***Vp/Vs***. **

As observed in above figure poisson’s ratio decreases with decreasing Vp/Vs.Poisson’s ratio decreases rapidly with decreasing Vp/Vs ratio in the range of Vp/Vs= 1.4 to 3,which is the region where most of the Vp/Vs ratio falls in subsurface conditions.Although Low Vp/Vs ratio is associated with gas but Vp/Vs ratio also decreases due to changes in lithology,pressure,pore fluid volume and porosity.

** **

**Fig****ure:- a) Plot showing gas saturation vs Vp ,b) Plot showing gas saturation vs Vs **** Plot showing gas saturation vs Poisson****’s Ratio and ****d) Plot showing gas saturation vs Vp/Vs;reference:-****Fluid and Frequency Effects on Sonic Velocity”, R. A. Anderson, SPWLA 1984**

** **As shown in figure above Vp decreases rapidly with very small increase in gas saturation at low gas saturation , compared to the water filled case. Shear velocity keeps more or less linear relationship with changing gas saturation. Poisson’s Ratio and Vp/Vs also decrease with small gas saturation of upto 20% gas saturation and after that does not changes much with changing gas saturation.Above figure implies that Poisson’s ratio or Vp/Vs is good to identify presence of gas but once gas saturation is higher than 20% both the ratio’s remain almost constant.

As discussed above presence of gas increase reflection coefficient with increasing offset/incidence angle and when gas is not present absolute reflection coefficient decreases with increasing offset/incidence angle.This theory was first applied successfully by Ostrander for seismic data interpretation,and his work started the application of AVO in petroleum industry.

Rutherford and Williams(1989) in Geophysics volume 54 classified AVO anomalies on the basis of acoustic impedance as below:-

- Class1 -High impedance sands:- Generally found in hard-rock,onshore area,exhibits higher impedance than the encasing shale with relativity large positive values for normally incident reflection coefficient.
- Class 2 -Near-zero impedance contrast sands :- These sands are found on both onshore and offshore area,they have nearly the same impedance as the encasing shale and are characterized by near zero value of normally incident reflection coefficient.
- Class 3 -Low-impedance sands:- Mainly found in offshore environment,they exhibit lower impedance than the encasing shale with negative, large magnitude values for normally incident reflection coefficient.

**Figure:- Zoeppritz P-wave reflection coefficients for a shale-gas sand interface for a range of R _{0} values. The Poisson’s ratio and density of the shale were assumed to be 0.38 and 2.4gm/cc respectively. The Poisson’s ratio and density of the gas sand were assumed to be 0.15 and 2.0 gm/cc respectively.Here Ro is normally incident reflection coefficient;reference:-**