Abstract
Pre-stack seismic inversion of the P- and S-wave velocities and bulk density is important in seismic exploration for evaluating lithological units and fluid properties. Generally, this inversion is based on ray-tracing modeling, which introduces errors and requires substantial pre-processing for stratified models due to its oversimplified single-interface assumption. To overcome those problems, we propose a pre-stack inversion method, using wave-equation-based modeling as a forward engine. Most wave-equation-based pre-stack inversions are based on the reflectivity method and adopt nonlinear optimization algorithms, although accurate, but computationally expensive. Hence, we use a fast propagator matrix (PM) method valid for layered media. To improve the stability and accuracy, the PP data inversion is extended to joint PP and PS PM-based inversion (JPMI). A linear inversion scheme is adopted to reduce the computational cost, and the Fréchet derivatives are computed accordingly. Moreover, to obtain an optimal model solution, the L-BFGS (Limited-memory Broyden–Fletcher–Goldfarb–Shanno) optimization algorithm and L-curve criterion, an adaptive regularization parameter acquisition method, are implemented. A posterior probability analysis shows that the method has a higher parameter sensitivity than the joint exact Zoeppritz-based inversion and gives better estimations than the single-data inversion. We discuss the effects of dataset weight, internal multi-reflections, time window setting, noise level and initial model by using model tests. Synthetic and real-data examples demonstrate that the algorithm is better than the single PP inversion in terms of stability and accuracy, especially for S-wave velocity and density estimations.
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Acknowledgements
The authors are grateful to the support of the China Postdoctoral Science Foundation (Grant No. 2019M661716), the Fundamental Research Funds for the Central Universities, China (Grant No. B200202135), and the National Natural Science Foundation of China (Grant No. 41974123).
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Appendix
Appendix
From Eq. (2), the partial derivatives of \({\mathbf{T}}\left( 0\right)\) and \({\mathbf{T}}\left( h \right)\) with respect to the parameters \(m^{{ * { = }\alpha ,\beta ,\rho }}\) are
and
Differentiating Eq. (4) with respect to model parameter \(m^{{ * { = }\alpha ,\beta ,\rho }}\), we obtain
and
Using Eq. (3), the derivatives of the intermediate variables \(X\) and \(Z\) are
and
The vertical slowness \(s\) is a function of the model parameters \(m^{*}\), and the corresponding derivatives are derived according to Eq. (5):
where
The partial derivatives of P-wave velocity α, S-wave velocity β and density ρ, with respect to the parameters \(m^{{ * { = }\alpha ,\beta ,\rho }}\) are
Based on the derivatives above, one can obtain the partial derivatives of the reflection coefficients and then the Fréchet derivatives.
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Luo, C., Ba, J., Carcione, J.M. et al. Joint PP and PS Pre-stack Seismic Inversion for Stratified Models Based on the Propagator Matrix Forward Engine. Surv Geophys 41, 987–1028 (2020). https://doi.org/10.1007/s10712-020-09605-5
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DOI: https://doi.org/10.1007/s10712-020-09605-5