1. Testing old and new AVO methods Chuck Ursenbach CREWES Sponsors Meeting November 21, 2003

2. I.Testing pseudo-linear Zoeppritz approximations: P-wave AVO inversion II.Testing pseudo-linear Zoeppritz approximations: Multicomponent and joint AVO inversion III.Testing pseudo-linear Zoeppritz approximations: Analytical error expressions IV.Using the exact Zoeppritz equations in pseudo-linear form: Isolating the effects of input errors V.Using the exact Zoeppritz equations in pseudo-linear form: Inversion for density CREWES 2003 Research Reports

3. Outline New Inversion Methods Testing with error-free data Analytical error expressions Testing on input with errors Density inversion I, II III V IV

4. Aki-Richards Approximation Depends on  /  Snell’s Law:

5. Pseudo-Linear expression

6. Pseudo-quadratic expression

7. Accuracy depends on  / 

8. Impedance I P =  I S =   I P /I P   /  +  /   I S /I S   /  +  / 

9. P-impedance contrast is predicted accurately

10. Comparison of  I S /I S predictions

11. Comparison of R PS inversion for  I S /I S A-R P-LP-Q R PP 8.6 13 2.0 R PS 8.2 3.2.22 joint 7.2 3.6.50 Average %-errors

12. Section Summary Accurate Zoeppritz approximations can be cast into an Aki-Richards form for convenient use in AVO Errors in predicted contrasts are strongly correlated with  /  Strong cancellation of error for  /  +  /  Strong cancellation of error for  /  +  /  in Pseudo-quadratic method Pseudo-linear and Pseudo-quadratic methods give superior values of  I S /I S for R PS and joint inversion

13. Analytical Inversion Observation: Inversion of 3 points of noise-free data, (  = 0 , 15 , 30  ) gives very similar results to densely sampled data Conjecture: Inversion should be semi- analytically tractable (with aid of symbolic computation software [Maple]) Remark: For inversion of PS data only two points should be required (  = 15 , 30  )

14. Leave  / ,  / ,  / ,  /  as variables Assume their value in coefficients is exact Evaluate necessary functions at :  = 0 , 15 , 30  where sin(  ) = 0,, ½ Carry out inversion using Cramer’s rule Expand contrast estimates up to cubic order in exact contrasts, and up to first order in (  /  - ½) Method

15. S-Impedance contrast error

17. P-impedance contrast error

18. Section Summary Analytical inversion is tractable Cubic order formulae give reasonable representation of error Potential use in correcting inversion results Rigorous illustration of the superiority of P-wave impedance estimates

19. Sources of AVO error Assumptions of the Zoeppritz equations Approximations to the Zoeppritz equations Limited range of discrete offsets represented Errors in input – R (noise, processing), background parameters (velocity model, empirical relations, etc.), angles (velocity model)

20. Exact Zoeppritz in Pseudo-Linear form

21.  /  = (  /  ) exact + 0.2

22. Gaussian noise on R: magnitude 0.01

23. Section Summary AVO inversions can be carried out with the pseudo-linear form of the exact Zoeppritz equations Provides a means of examining the effect of individual input errors Provides a guide to uncertainty propagation Provides a guide to assessing the significance of approximation errors

24. An exact expression quadratic in  / 

25. Least-squares determination of  /  a, b, c are functions of ,  / , , R (  )

26.  /  = (  /  ) exact + 0.2

27. Gaussian noise on R: magnitude 0.01

28. Section Summary The exact Zoeppritz equation can be formulated to allow least-squares extraction of  /  by solution of a cubic polynomial The  /  errors from this method are distinctly different from those of 3- parameter inversion Random input errors seem to be controlled very effectively in this method