Extending AVO Inversion Techniques Charles Ursenbach Robert Stewart

CREWES Report 2001 Ursenbach & Stewart, “Extending AVO inversion techniques” (Chapter 28) Ursenbach, “A generalized Gardner relation” (Chapter 7)

Objectives Develop improved AVO linear inversion techniques Develop an effective approach to assessment of AVO approximations

The Linear Approximation Zoeppritz Equations: Aki-Richards Approximation:

From 3 variables to 2 Aki-Richards Approximation: - Convert to impedances Use Gardner’s relation - Discard remainder Smith-Gidlow: Fatti et al:

Smith-Gidlow approximation

Fatti et al. approximation

Comparison of Smith-Gidlow and Fatti et al.

Possible improvements #1 Combine strengths of Smith-Gidlow & Fatti et al. #2 Improve on Gardner relation #3 Extract / #4 Assess approximations efficiently

#1 How to combine best features of Smith-Gidlow and Fatti et al.? The Full Offset approximation

#2 How to improve on Gardner’s relation? Lithology specific – Castagna et al., 1993 Based on Vs – CREWES, 1997, 1998, 1999 Laboratory data – Wang, 2000 -VP and -VS shale, clean sandstone, shaley sandstone, unconsolidated sandstone, limestone, dolomite

#2 How to improve on the Gardner relation? Lithology independent, -(VP, VS) The generalized Gardner relation

Density contrast approximations #3 How to estimate /? Density contrast approximations

#4 How to efficiently assess the value of an approximation? Choose lithology Generate a representative sampling of (,VP,VS) values for upper and lower layers For each sample generate synthetic offset gather for some angle range (spike wavelet) Using chosen approximation, find best fit value of VP / VP, VS / VS, etc. Average over all samples for chosen lithology Incorporate in user-friendly interactive program

Another assessment issue How does predicted result compare to exact value? How does predicted result compare to Aki-Richards? How does Aki-Richards compare to exact value?

Results for Aki-Richards Quantity (method) / (-, -) IP/ IP (IP -) / -)  IS / IS (IS -) / (-, shale/sst 1.18, 8.83 2.28 6.24, 6.87 10.25 155, 18.4 shale/ limestone 0.95, 14.4 2.33 22.9, 23.8 41.1 258, 11.2 dolostone 0.41, 22.1 0.41 14.7, 14.7 11.4 376, 13.0 anhydrite/ 0.81, 16.0 3.02 20.1, 21.1 141 14.6, 0.51 anhydrite/dolostone 1.86, 58.1 4.99 40.1, 41.4 49.8 272, 8.64 Linear inversions have high potential accuracy

Results for Smith-Gidlow and Fatti et al. %-error in contrast / (A-R, Smith-Gidlow) IP / IP (A-R, Fatti et al.) / IS / IS shale/sst 1.18, 38.8 2.28, 3.07 6.24, 35.1 10.25, 9.99 shale/limestone 0.95, 39.2 2.33, 2.46 22.9, 113 41.1, 29.8 shale/dolostone 0.41, 133 0.41, 0.31 14.7, 249 11.4, 14.5 anhydrite/ limestone 0.81, 130 3.02, 10.5 20.1, 168 141, 88.7 anhydrite/limestone (0 to 50 degrees) 0.80, 137 14.5, 82.5 20.4, 239 53.6, 143 dolostone 1.86, 39.4 4.99, 1.60 40.1, 69.4 49.8, 48.5 Fatti et al. superior; wide variance in values

Results for Full Offset method %-error of I/I IP (A-R, Fatti et al., Full Offset) IS (A-R, Fatti et al., Full Offset) shale/sst 2.28, 3.07, 2.74 10.25, 9.99, 11.1 shale/ limestone 2.33, 2.46, 2.30 41.1, 29.8, 32.8 dolostone 0.41, 0.31, 0.23 11.4, 14.5, 15.3 anhydrite/ 3.02, 10.5, 10.43 141, 88.7, 83.2 4.99, 1.60, 1.04 49.8, 48.5, 47.8 Full Offset improves on Fatti et al. for IP

Results for generalized Gardner No significant improvement in present form

Results for density inversion %-error (/) VP-  VS-  IP-  IS-  Aki-Richards - shale/sst 512 4160 233 18.4 shale/ limestone 181 1344 118 11.2 dolostone 126 1477 143 13.0 anhydrite/ 95.7 129 104 0.51 162 335 81.1 8.64 Still considerable potential for improvement

Conclusions The method of Fatti et al. is more reliable than that of Smith and Gidlow Considerable variation in accuracy is possible for a given lithology The Full Offset approximation yields a clear improvement on the method of Fatti et al. The generalized Gardner relation as presently formulated does not improve AVO The Aki-Richards approximation could provide the basis for a suitable density inversion scheme

Acknowledgments The authors wish to thank the sponsors of CREWES for financial support of this research