Haiyan Zhang and Arthur B. Weglein Direct non-linear inversion of multi-parameter 1D elastic media using the inverse scattering series Haiyan Zhang and Arthur B. Weglein M-OSRP Annual Meeting University of Houston May 10 –12, 2006

Outline Background and Motivation Theory: Inverse Scattering Series Application on three parameter 1D elastic media Inversion in PS space An approach with PP data only and its numerical tests Full non-linear elastic inversion with four components of data Conclusions and plan Acknowledgements

Background and motivation Change in earth properties Non-linear Wave response Current Inversion methods: Linear inversion or Born approximation Model matching Direct non-linear inversion method Towards fundamentally new comprehensive and realistic target identification.

Derivation of the inverse series In actual medium: In reference medium: Perturbation: L-S equation: Forward scattering Series: Inverse scattering series:

Inverse Scattering Series Linear Non-linear

Three parameter elastic inversion References: Weglein and Stolt (1992) : introduced an elastic L-S equation and provided a specific linear inverse formalism for parameter estimation. Matson (1997): pioneered the development and application of methods for attenuating ocean bottom and on-shore multiples.

Three parameter elastic inversion In PS space In actual medium: In reference medium: Perturbation: L-S equation: Linear inversion: First term beyond linear: .

Elastic inversion: linear

Elastic inversion: linear Then, in domain, we get

Elastic inversion: linear Where For 1D earth, we have

Elastic inversion: non-linear

Three parameter elastic inversion When the acoustic work is extended to elastic case, more issues would be involved. For example: Data requirements: four components of data. One more velocity, four mislocations (3 for 1D earth).

Three parameter elastic inversion For towed streamer case, only PP data is available. Can we perform non-linear elastic inversion only using PP data? A particular non-linear approach--- only use PP data as input, to do the non-linear elastic inversion.

Elastic inversion: linear Then, in domain, we get

Elastic inversion: non-linear

Non-linear elastic inversion: PP data only

Numerical test: PP data only One interface model x a z

Numerical tests: PP data only x One interface, a z Model 1: shale (0.2 porosity) over oil sand (0.1 porosity) Model 2: shale (0.2 porosity) over oil sand (0.2 porosity) Model 3: shale (0.2 porosity) over oil sand (0.3 porosity) Model 4: oil sand (0.2 porosity) over wet sand (0.2 porosity)

Summary of the numerical tests: PP data only Before correction: For one of the parameter (shear modulus) in model 3, second order approximation almost no difference, even a little bit worse than linear approximation. After correction: For all of the parameters of the four models, second order approximation provides improvements beyond the linear approximation.

Numerical tests: PP data only x Model 2 shale (0.20 porosity) a oil sand (0.20 porosity) z

Inversion of density Linear+nonlinear Linear

Inversion of P-bulk modulus Linear Linear+nonlinear

Inversion of Shear modulus Linear Linear+nonlinear

Numerical tests: PP data only x Model 4 oil sand (0.20 porosity) a wet sand (0.20 porosity) z

Inversion of density Linear+nonlinear Linear

Inversion of P-bulk modulus Linear Linear+nonlinear

Inversion of Shear modulus Linear Linear+nonlinear

Non-linear elastic inversion: all four components of data

Elastic inversion: linear Then, in domain, we get

Non-linear elastic inversion: all four components of data in the linear solutions equal, we have Given , we can get the following

Non-linear elastic inversion: all four components of data

Non-linear elastic inversion: all four components of data Through the similar way, we can get the solution for

Non-linear elastic inversion: all four components of data Based on this idea, the solution for the first equation is

Non-linear elastic inversion: all four components of data The solution for the second equation is

Non-linear elastic inversion: all four components of data Continued

Non-linear elastic inversion: all four components of data Continued

Non-linear elastic inversion: all four components of data Continued The end

Non-linear elastic inversion: all four components of data The solutions for the third and fourth equations Please see the annual report.

Three parameter elastic inversion Inversion for 1D elastic earth with three parameters. All four components of data need to be known in order to do non-linear elastic inversion. A particular non-linear approach--- only use PP data as input, is performed and encouraging results are obtained. A consistent method is provided to do full direct non-linear inversion using all four components of data.

Conclusions A framework and algorithm for more accurate target identification have been developed. The elastic non-linear inversion requires all four components of data. Elastic non-linear inversion showed benefit in all the cases with PP data only. How accurately PP data effectively synthesize PS, SP and SS data determined the degree of benefit provided by the non-linear elastic inversion. We anticipate collecting PP, PS, SP and SS would provide further benefit. We will test this hypothesis this summer.

Acknowledgements We are grateful to Robert Keys, Douglas Foster and Simon Shaw for useful comments and suggestions. The M-OSRP sponsors are thanked for supporting this research.