Joint PP and PS Pre-stack Seismic Inversion for Stratified Models Based on the Propagator Matrix Forward Engine

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.

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

$$\frac{{\partial {\mathbf{T}}\left( 0\right)}}{{\partial m^{ * } }} = i\omega \left[ {\begin{array}{*{20}c} {\frac{{\partial \xi_{\text{P}} }}{{\partial m^{*} }}} & {\frac{{\partial \xi_{\text{S}} }}{{\partial m^{*} }}} & {\frac{{\partial \xi_{\text{P}} }}{{\partial m^{*} }}} & {\frac{{\partial \xi_{\text{S}} }}{{\partial m^{*} }}} \\ { - \frac{{\partial \gamma_{\text{P}} }}{{\partial m^{*} }}} & { - \frac{{\partial \gamma_{\text{S}} }}{{\partial m^{*} }}} & {\frac{{\partial \gamma_{\text{P}} }}{{\partial m^{*} }}} & {\frac{{\partial \gamma_{\text{S}} }}{{\partial m^{*} }}} \\ { - \frac{{\partial X_{\text{P}} }}{{\partial m^{*} }}} & { - \frac{{\partial X_{\text{S}} }}{{\partial m^{*} }}} & { - \frac{{\partial X_{\text{P}} }}{{\partial m^{*} }}} & { - \frac{{\partial X_{\text{S}} }}{{\partial m^{*} }}} \\ {\frac{{\partial W_{\text{P}} }}{{\partial m^{*} }}} & {\frac{{\partial W_{\text{S}} }}{{\partial m^{*} }}} & { - \frac{{\partial W_{\text{P}} }}{{\partial m^{*} }}} & { - \frac{{\partial W_{\text{S}} }}{{\partial m^{*} }}} \\ \end{array} } \right] ,$$

(40)

and

$$\begin{aligned} \frac{{\partial {\mathbf{T}}\left( h \right)}}{{\partial m^{ * } }} = & \left[ {\begin{array}{*{20}c} {i\omega s_{\text{P}} h\xi_{\text{P}} + \frac{{\partial \xi_{\text{P}} }}{{\partial m^{*} }}} & {i\omega s_{\text{S}} h\xi_{\text{S}} + \frac{{\partial \xi_{\text{S}} }}{{\partial m^{*} }}} & { - i\omega s_{\text{P}} h\xi_{\text{P}} { + }\frac{{\partial \xi_{\text{P}} }}{{\partial m^{*} }}} & { - i\omega s_{\text{S}} h\xi_{\text{S}} { + }\frac{{\partial \xi_{\text{S}} }}{{\partial m^{*} }}} \\ { - i\omega s_{\text{P}} h\gamma_{\text{P}} - \frac{{\partial \gamma_{\text{P}} }}{{\partial m^{*} }}} & { - i\omega s_{\text{S}} h\gamma_{\text{S}} - \frac{{\partial \gamma_{\text{S}} }}{{\partial m^{*} }}} & { - i\omega s_{\text{P}} h\xi_{\text{P}} + \frac{{\partial \gamma_{\text{P}} }}{{\partial m^{*} }}} & { - i\omega s_{\text{S}} h\xi_{\text{S}} + \frac{{\partial \gamma_{\text{S}} }}{{\partial m^{*} }}} \\ { - i\omega s_{\text{P}} hX_{\text{P}} - \frac{{\partial X_{\text{P}} }}{{\partial m^{*} }}} & { - i\omega s_{\text{S}} hX_{\text{S}} - \frac{{\partial X_{\text{S}} }}{{\partial m^{*} }}} & { \, i\omega s_{\text{P}} hX_{\text{P}} - \frac{{\partial X_{\text{P}} }}{{\partial m^{*} }}} & {i\omega s_{\text{S}} hX_{\text{S}} - \frac{{\partial X_{\text{S}} }}{{\partial m^{*} }}} \\ {i\omega s_{\text{P}} hW_{\text{P}} + \frac{{\partial W_{\text{P}} }}{{\partial m^{*} }}} & {i\omega s_{\text{S}} hW_{\text{S}} + \frac{{\partial W_{\text{S}} }}{{\partial m^{*} }}} & { \, i\omega s_{\text{P}} hW_{\text{P}} - \frac{{\partial W_{\text{P}} }}{{\partial m^{*} }}} & {i\omega s_{\text{S}} hW_{\text{S}} - \frac{{\partial W_{\text{S}} }}{{\partial m^{*} }}} \\ \end{array} } \right] \\ & \quad \cdot \left[ {\begin{array}{*{20}c} {i\omega e^{{i\omega s_{\text{P}} h}} } & {} & {} & {} \\ {} & {i\omega e^{{i\omega s_{\text{S}} h}} } & {} & {} \\ {} & {} & {i\omega e^{{ - i\omega s_{\text{P}} h}} } & {} \\ {} & {} & {} & {i\omega e^{{ - i\omega s_{\text{S}} h}} } \\ \end{array} } \right] \\ \end{aligned} .$$

(41)

Differentiating Eq. (4) with respect to model parameter $m^{{ * { = }\alpha ,\beta ,\rho }}$, we obtain

$$\begin{aligned} \frac{\partial \xi }{{\partial m^{ * } }} = & \frac{1}{\xi }\left\{ {\frac{{\left( {\alpha s^{2} \frac{\partial \alpha }{{\partial m^{*} }} + \alpha^{2} s\frac{\partial s}{{\partial m^{*} }} + \beta p^{2} \frac{\partial \beta }{{\partial m^{*} }} + \beta^{2} p\frac{\partial p}{{\partial m^{*} }}} \right) \cdot \left[ {\left( {\alpha^{2} + \beta^{2} } \right)\left( {p^{2} + s^{2} } \right) - 2} \right]}}{{\left[ {\left( {\alpha^{2} + \beta^{2} } \right)\left( {p^{2} + s^{ 2} } \right) - 2} \right]^{2} }}} \right. \\ & \left. {\quad \frac{{\left. { - \left( {\alpha^{2} s_{{}}^{2} + \beta^{2} p^{2} - 1} \right) \cdot \left[ {\left( {p^{2} + s_{{}}^{2} } \right)\left( {\alpha \frac{\partial \alpha }{{\partial m^{*} }} + \beta \frac{\partial \beta }{{\partial m^{*} }}} \right) + \left( {\alpha^{2} + \beta^{2} } \right)\left( {p\frac{\partial p}{{\partial m^{*} }} + s\frac{\partial s}{{\partial m^{*} }}} \right)} \right.} \right]}}{{\left[ {\left( {\alpha^{2} + \beta^{2} } \right)\left( {p^{2} + s^{ 2} } \right) - 2} \right]^{2} }}} \right\} \\ \end{aligned} ,$$

(42)

and

$$\begin{aligned} \frac{\partial \gamma }{{\partial m^{ * } }} = & \frac{1}{\gamma }\left\{ {\frac{{\left( {\alpha p^{2} \frac{\partial \alpha }{{\partial m^{*} }} + \alpha^{2} p\frac{\partial s}{{\partial m^{*} }} + \beta s^{2} \frac{\partial \beta }{{\partial m^{*} }} + \beta^{2} s\frac{\partial s}{{\partial m^{*} }}} \right) \cdot \left[ {\left( {\alpha^{2} + \beta^{2} } \right)\left( {p^{2} + s^{2} } \right) - 2} \right]}}{{\left[ {\left( {\alpha^{2} + \beta^{2} } \right)\left( {p^{2} + s^{2} } \right) - 2} \right]^{2} }}} \right. \\ & \quad \left. {\frac{{\left. { - \left( {\alpha^{2} p^{2} + \beta^{2} s^{2} - 1} \right) \cdot \left[ {\left( {p^{2} + s^{2} } \right)\left( {\alpha \frac{\partial \alpha }{{\partial m^{*} }} + \beta \frac{\partial \beta }{{\partial m^{*} }}} \right) + \left( {\alpha^{2} + \beta^{2} } \right)\left( {p\frac{\partial p}{{\partial m^{*} }} + s\frac{\partial s}{{\partial m^{*} }}} \right)} \right.} \right]}}{{\left[ {\left( {\alpha^{2} + \beta^{2} } \right)\left( {p^{2} + s^{ 2} } \right) - 2} \right]^{2} }}} \right\} \\ \end{aligned} .$$

(43)

Using Eq. (3), the derivatives of the intermediate variables $X$ and $Z$ are

$$\begin{aligned} \frac{\partial X}{{\partial m^{ * } }} = & 2\alpha \rho \left( {\xi p + s\gamma } \right)\frac{\partial \alpha }{{\partial m^{ * } }} - 4\beta \rho \xi p\frac{\partial \beta }{{\partial m^{ * } }} + \left( {\alpha^{2} \xi p + \alpha^{2} \gamma s - 2\beta^{2} \xi p} \right)\frac{\partial \rho }{{\partial m^{ * } }} \\ & \quad { + }\,\left( {\alpha^{2} - 2\beta^{2} } \right)\rho p\frac{\partial \xi }{{\partial m^{ * } }} + \left( {\alpha^{2} - 2\beta^{2} } \right)\rho \xi \frac{\partial p}{{\partial m^{ * } }} + \alpha^{2} \rho s\frac{\partial \gamma }{{\partial m^{ * } }} + \alpha^{2} \rho \gamma \frac{\partial s}{{\partial m^{ * } }} \\ \end{aligned} ,$$

(44)

and

$$\begin{aligned} \frac{\partial W}{{\partial m^{ * } }} = & 2\beta \rho \left( {\gamma p + \xi s} \right)\frac{\partial \beta }{{\partial m^{ * } }} + \beta^{2} \left( {\gamma p + \xi s} \right)\frac{\partial \rho }{{\partial m^{ * } }} \\ & \quad + \beta^{2} \rho \left( {p\frac{\partial \gamma }{{\partial m^{ * } }} + \gamma \frac{\partial p}{{\partial m^{ * } }} + s\frac{\partial \xi }{{\partial m^{ * } }} + \xi \frac{\partial s}{{\partial m^{ * } }}} \right) \\ \end{aligned} .$$

(45)

The vertical slowness $s$ is a function of the model parameters $m^{*}$, and the corresponding derivatives are derived according to Eq. (5):

$$\frac{\partial s}{{\partial m^{ *} }} = \pm \frac{1}{2\sqrt 2 } \cdot \frac{{\left( {\sqrt {K_{1}^{2} - 4K_{2} K_{3} } \mp K_{1} } \right)\frac{{\partial K_{1} }}{{\partial m^{ *} }} - 2\frac{{\partial K_{2} }}{{\partial m^{ *} }}K_{3} - 2K_{2} \frac{{\partial K_{3} }}{{\partial m^{ *} }}}}{{\sqrt {\left( {K_{1}^{3} - 4K_{1}^{{}} K_{2} K_{3} } \right) \mp \left( {K_{1}^{2} - 4K_{2} K_{3} } \right)^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} } }} ,$$

(46)

where

$$\frac{{\partial K_{1} }}{{\partial m^{ *} }}{ = } - \frac{ 2}{{\alpha^{ 3} }} \cdot \frac{\partial \alpha }{{\partial m^{ *} }} - \frac{ 2}{{\beta^{ 3} }} \cdot \frac{\partial \beta }{{\partial m^{ *} }} - 4p\frac{\partial p}{{\partial m^{ *} }} ,$$

(47a)

$$\frac{{\partial K_{2} }}{{\partial m^{ *} }}{ = 2}p\frac{\partial p}{{\partial m^{ *} }} + \frac{ 2}{{\alpha^{ 3} }} \cdot \frac{\partial \alpha }{{\partial m^{ *} }} ,$$

(47b)

$$\frac{{\partial K_{3} }}{{\partial m^{ *} }}{ = 2}p\frac{\partial p}{{\partial m^{ *} }} + \frac{ 2}{{\beta^{ 3} }} \cdot \frac{\partial \beta }{{\partial m^{ *} }} .$$

(47c)

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

$$\frac{\partial \alpha }{\partial \alpha } = 1,\;\frac{\partial \alpha }{\partial \beta } = 0,\;\frac{\partial \alpha }{\partial \rho } = 0,$$

(48a)

$$\frac{\partial \beta }{\partial \alpha }{ = }0,\;\frac{\partial \beta }{\partial \beta } = 1,\;\frac{\partial \beta }{\partial \rho } = 0,$$

(48b)

$$\frac{\partial \rho }{\partial \alpha } = 0,\;\frac{\partial \rho }{\partial \beta } = 0,\;\frac{\partial \rho }{\partial \rho } = 1.$$

(48c)

Based on the derivatives above, one can obtain the partial derivatives of the reflection coefficients and then the Fréchet derivatives.