# seismic velocity analysis _ aghazade

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Velocity analysis in the

Velocity analysis in the presence of amplitude variation

By :Debashish Sarkar,Bob Baumel and Ken [email protected]@yahoo.com

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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 its 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:

Due to assumption that amplitude is independent of offset the measure is degrade in the presence of AVO*** polarity reversal

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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.

b) Can also tackle multiple conflicting events,4

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.

Generalized traditional semblancec)To improve the estimation of velocity5

AVO sensitive semblance formalismTo incorporate amplitude variation with offset into velocity analysis ,we define the Generalized semblance as:

(2)

minimizing results inObtaining the model parameters.

:Suitably parameterized model of the trace amplitude in the move-out data

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Steps in computation generalized semblance:Defining the model function M as a linear combination of basic functions that describe the AVO.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. 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.

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Deriving the traditional semblance by generalized semblanceStarting with offset-independent model,Substituting in equation (2) :

Taking derivative respect to ,we have:

Finally:

(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.

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

(5)

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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

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

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.

Best resoloution because allows for the fewest degrees of freedom

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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

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Case 2:No noise ;window off center

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

Event 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

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 noisenegative AVO gradient with noise

Case 4 : High noisy data

Traditional semblanceAB semblance

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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