Velocity analysis in the presence of amplitude variation

By :Debashish Sarkar,Bob Baumel and Ken Larner

Aghazade.kamal@ut.ac.irAghazade.kamal@yahoo.com

introductionConventional semblance velocity analysis(Taner and Koehler,1969) is equivalent to modeling prestack seismic data events that have hyperbolic moveout but no amplitude variation with offset(AVO).As a result of it’s assumption that amplitude is independent of offset,this method may not perform well for event with strong AVO ,specially for events with polarity reversals at large offset.To account for AVO,the semblance method can be extended by modeling the data with events that have not only hyperbolic moveout,but also amplitude variation.

History and methods Corcoran (1989) and Sarkar,lamb and Castagna(1999) have shown that

the semblance measure of taner and Koehler(1969) is based on the implicit assumption that the wavelet dose not vary with offset.

the conventional semblance measure evaluated at zero-offset time is defined as:

Where:

𝑡 0

1

1

2

1

1 0 21

( , )( , )

( , )vt x

vt x

D t xS v t

N D t x

𝐷𝑣 (𝑡1 ,𝑥)=𝐷 [𝑡𝑣(𝑡1 ,𝑥 ) ,𝑥 ]

𝑡𝑣 (𝑡1 ,𝑥)=√𝑡12+ 𝑥2𝑣2

𝑡1 : zerooffset×within awindow centered at t 0

𝑁 :number of traces

𝐷𝑣 :𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡 h𝑡 𝑒 𝑑𝑎𝑡𝑎𝑚𝑜𝑣𝑒𝑜𝑢𝑡 𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑 h𝑤𝑖𝑡 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑣

differential semblance method(symes and kern,1994) works by taking the difference of adjacent traces.the objective function is such that it eliminates secondary maxima and produces a broad primary maximum Poor velocity resolution

due to broad semblance curves obtained through this method,it may not suitable for standard semblance panels. Eigenvalue methods(Biondi and Kostov,1989;Key and

Smithson,1990),exploit the fact that the signal covariance matrix is of low rank in the absence of noise,easily incorporate AVO.

An amplitude dependent function(e.g Shuy 1985),can incorporate in the semblance measure(Corcoran 1989).

This method successfully estimates velocities of seismic events with large AVO and polarity reversals.Increasing number of parameters result in loss of velocity

precision ,when the range of incident angle is small.Sarkar(1999),suggested solving the problem as a mixed-determined

problem(menke 1984) that incorporates the use of a regularization parameter.

AVO sensitive semblance formalism

To incorporate amplitude variation with offset into velocity analysis ,we define the “Generalized semblance” as:

(2)

minimizing results inObtaining the model parameters.

2

0 2( , ) 1 vG

v

M DS v t

D

: data after move out correction with velocity “”

: trial velocity: zero offset time at the center of the semblance window

1( , )v vD D t x

1( , )M M t x :Suitably parameterized model of the trace amplitude in the move-out data

‖𝑀−𝐷𝑣 ‖2

Steps in computation generalized semblance:

i. Defining the model function M as a linear combination of basic functions that describe the AVO.

ii. Determining the coefficient of the model M, by setting the derivatives of equation (2) with respect to the model coefficients equal to zero and solving.

iii. substituting M back into equation (2) to obtain generalized semblance for each zero offset time and trial stacking velocity.

the trial velocity that maximize the semblance measure is taken as the

stacking velocity.

Deriving the traditional semblance by generalized semblance

Starting with offset-independent model,Substituting in equation (2) :

Taking derivative respect to ,we have:

Finally:

1 1( , ) ( )M t x A t

21 11 0 2

1

( ) ( , )( , ) 1

( , )x

Gvx

A t D t xS v t

D t x

1( )A t

1 1( ) ( , )vx

NA t D t x

211 0 2

1

( , )( , )

( , )vx

Gvx

D t xS v t

N D t x

(3)

(4)

To account for AVO we choose a model parameterization based on Shuey(1985) simplification:

Where is the angle of incidence at the reflector.

i. estimating A(t1) and B(t1) coefficient for each time t1 inside the semblance window.ii. substitution into equation(5) and (2) respectively.iii. sequence of above steps repeated for each trial velocity “v”. this approach is called “AB semblance”

1 1 1( , ) ( ) ( )sin xM t x A t B t

1( , )x x t v

(5)

AB semblance approach allows too much freedom to fit events with combinations of incorrect velocity and incorrect AVO.resulting in:“POOR VELOCITY RESOLUTION” So what is the solution to this problem?

We must reduce the degree of freedom!!!!

The amplitude at every point within the wavelet should be a scaled version of the amplitude at the peak of the wavelet.

The amplitude variation along any moveout curve with zero offset time t1 within a wavelet should be a scaled version of the amplitude variation with offset along the moveout trajectory traced out by the center of the wavelet.

Thus,the ratio of the amplitude gradient to zero-offset amplitude should be constant for all moveout curves within a wavelet.

We make the ratio constant through out each semblance window.then we have:

Refers to “AK semblance”

1

1

( )( )B tKA t

21 1( , ) ( )(1 sin )xM t x A t K

Tests with modeled data

applying the three methods(traditional semblance,AB semblance and AK semblance ) to the synthetic CMP gather.

Event A has no amplitude variation and event B exhibits polarity reversal .

Lets see the result!!!!

resultsCase1: No noise ;polarity reversal with offset

a) event A,No amplitude variation. the three methods have their maximum Semblance value near the correct stacking velocity(1685 m/s) of the three methods,traditional semblance Has the smallest width.(Good velocity resolution) In the presence of little or no AVO the AK Semblance matches the resolution and accuracy Performance of traditional semblance without Need of regularization parameter.

b) Event B,polarity reversal along a reflector

traditional semblance method fails_in terms Of velocity accuracy and resolution. AB and AK semblance peak near the correct stacking velocity(1720 m/s).

results

Case 2:No noise ;window off center

• Using zero-offset time,that differs from that of the center of the wavelet.

a) Zero offset time that is 13 ms earlier than the center of the wavelet. drifting the semblance peak for the case window is off center. AB semblance peak stayNear unity.others decrease..

Event AEvent B

b) Zero offset time that is 13 ms later than the center of the wavelet. traditional semblance yields unacceptable estimates of stacking

velocity in the presence of polarity reversal. the AB semblance seems to Perform better than AK semblance.

Event AEvent B

Case3 :noise contaminated data;Amplitude variation

a) Positive AVO gradient

reduction the peak amplitude of the traditional semblance,comparing to the case with no AVO effect

compare

b) negative AVO gradient

the peak value of the traditional semblance decreases .for moderate changes the traditional semblance serve the purpose acceptably.

but if changes include polarity reversals amplitude dependent measures perform better.

Noise contaminated data

the presence of noise has greatly reduced the peak semblance value. AB semblance method shows the poorest resolution.

Positive AVO gradient with noise negative AVO gradient with noise

Case 4 : High noisy data

Traditional semblance AB semblance

the AB and AK semblance panels show higher semblance and hence preserve the relevant peaks.

because the greater degree of freedom in amplitude Dependent measure allows for higher semblance Value.For AB semblance this benefit achieved at the Cost of loss in velocity resolution.

AK semblance

Conclusion: detection of polarity reversal is crucial because they may indicate the

presence of hydrocarbons. with the AK methods,we follow previous work based on Shuy

equation(which allows AVO ).using more fitting parameters than does traditional semblance but fewer than the number required in AB semblance.

the amplitude dependent semblance method are based on the premise that amplitude variation inside the semblance window is along a single reflection.thus this measure degrade when two or more events overlap within the window.

the computational costs of AB and AK semblance are comparable.

GOOD LUCK