Thin interbed AVA inversion based on a fast algorithm for reflectivity

Abstract

Zoeppritz equations form the theoretical basis of most existing amplitude variation with incident angle (AVA) inversion methods. Assuming that only primary reflections exist, that is, the multiples are fully suppressed and the transmission loss and geometric spreading are completely compensated for, Zoeppritz equations can be used to solve for the elastic parameters of strata effectively. However, for thin interbeds, conventional seismic data processing technologies cannot suppress the internal multiples effectively, nor can they compensate for the transmission loss accurately. Therefore, AVA inversion methods based on Zoeppritz equations or their approximations are not applicable to thin interbeds. In this study, we propose a prestack AVA inversion method based on a fast algorithm for reflectivity. The fast reflectivity method can compute the full-wave responses, including the reflection, transmission, mode conversion, and internal multiples, which is beneficial to the seismic inversion of thin interbeds. A further advantage of the fast reflectivity method is that the partial derivatives of the reflection coefficient with respect to the elastic parameters can be expressed as analytical solutions. Based on the Gauss–Newton method, we construct the objective function and model-updating formula considering sparse constraint, where the Jacobian matrix takes the form of an analytical solution, which can significantly accelerate the inversion convergence. We validate our inversion method using numerical examples and field seismic data. The inversion results demonstrate that the fast reflectivity-based inversion method is more effective for thin interbed models in which the wave-propagation effects, such as interval multiples, are difficult to eliminate.

Appendices

Appendix 1

T_n is the 6 × 6 delta matrix for the nth layer. The elements of T_n are formed from the Dunkin matrix, which is frequency independent. The following is a list of the 16 independent elements of the matrix T_n:

$$\begin{aligned} t_{11} & = - (p^{2} + q^{p} q^{s} )/\mu = t_{16} , \\ t_{12} & = - 2pq^{p} /\mu , \\ t_{13} & = - (p^{2} { - }q^{p} q^{s} )/\mu = { - }t_{14} , \\ t_{15} & = - 2pq^{s} /\mu , \\ t_{21} & = iq^{s} /\beta^{2} = - t_{23} = - t_{24} = - t_{26} , \\ t_{31} & = - ip(\Gamma + 2q^{p} q^{s} ) = t_{36} = t_{41} = t_{46} , \\ t_{32} & = - 4ip^{2} q^{p} , \\ t_{33} & = - ip(\Gamma { - }2q^{p} q^{s} ) = t_{43} = { - }t_{34} = { - }t_{44} , \\ t_{35} & = - 2i\Gamma q^{s} , \\ t_{42} & = - 2i\Gamma q^{p} , \\ t_{45} & = - 4ip^{2} q^{s} , \\ t_{{{5}1}} & = - iq^{p} /\beta^{2} = t_{53} = t_{54} = - t_{56} , \\ t_{61} & = - \mu (\Gamma^{2} + 4p^{2} q^{p} q^{s} ) = t_{66} , \\ t_{62} & = - 4\mu \Gamma pq^{p} , \\ t_{63} & = - \mu (\Gamma^{2} { - }4p^{2} q^{p} q^{s} ) = { - }t_{64} , \\ t_{65} & = - 4\mu \Gamma pq^{{\text{S}}} ,\;{\text{and}} \\ t_{22} & = t_{25} =t_{55} =t_{52} =0, \\ \end{aligned}$$

where $\Gamma =2p^{2} - 1/\beta^{2}$, $\mu =\rho \beta^{2}$, $q^{{\text{P}}} = (\alpha^{{{ - }2}} - p^{2} )^{1/2}$, $q^{{\text{S}}} = (\beta^{{{ - }2}} - p^{2} )^{1/2}$, and $p = \sin \theta_{p} /\alpha = \sin \theta_{S} /\beta$.

${\mathbf{T}}_{n}^{ - 1}$ is the inverse matrix of T_n. The elements of ${\mathbf{T}}_{n}^{ - 1}$ are simply a rearrangement of the elements of ${\mathbf{T}}_{n}$:

$${\mathbf{T}}_{n}^{ - 1} = \left[ {\begin{array}{*{20}c} {t_{61} } & {t_{51} } & {t_{31} } & {t_{31} } & {t_{21} } & {t_{11} } \\ { - t_{65} } & 0 & { - t_{45} } & { - t_{35} } & 0 & { - t_{15} } \\ { - t_{63} } & { - t_{51} } & { - t_{33} } & { - t_{33} } & {t_{21} } & { - t_{13} } \\ {t_{63} } & { - t_{51} } & {t_{33} } & {t_{33} } & {t_{21} } & {t_{13} } \\ { - t_{62} } & 0 & { - t_{42} } & { - t_{32} } & 0 & { - t_{12} } \\ {t_{61} } & { - t_{51} } & {t_{31} } & {t_{31} } & { - t_{21} } & {t_{11} } \\ \end{array} } \right].$$

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Appendix 2

Partial derivation of E_n and T_n

In general, the incident angle of angle gather will be controlled within 90°. Therefore, the vertical slowness and E_n can be expressed as:

$$q^{{\text{P}}} = (\alpha^{{{ - }2}} - p^{2} )^{1/2} =\frac{{\cos \theta_{{\text{p}}} }}{\alpha }$$

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$$q^{{\text{S}}} = (\beta^{{{ - }2}} - p^{2} )^{1/2} =\frac{{\cos \theta_{{\text{S}}} }}{\beta }$$

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$${\mathbf{E}}_{n} = {\text{diag}}\left[ {\begin{array}{*{20}c} {{\text{e}}^{{ - i\omega d_{n} (\cos \theta_{{\text{p}}} /\alpha + \cos \theta_{{\text{S}}} /\beta )}} } & 1 & {{\text{e}}^{{ - i\omega d_{n} (\cos \theta_{{\text{p}}} /\alpha - \cos \theta_{{\text{S}}} /\beta )}} } & {{\text{e}}^{{i\omega d_{n} (\cos \theta_{{\text{p}}} /\alpha - \cos \theta_{{\text{S}}} /\beta )}} } & 1 & {{\text{e}}^{{i\omega d_{n} (\cos \theta_{{\text{p}}} /\alpha + \cos \theta_{{\text{S}}} /\beta )}} } \\ \end{array} } \right].$$

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The partial derivative of E_n with respect to the parameters M_n can be calculated analytically:

$$\frac{{\partial {\mathbf{E}}_{n} }}{{\partial {\mathbf{M}}_{n} }}=\left\{ {\begin{array}{*{20}c} {\frac{{\partial {\mathbf{E}}_{n} }}{\partial \alpha }=i\omega d_{n} {\text{diag}}\left[ {\begin{array}{*{20}c} {A_{\alpha } {\text{e}}^{{ - i\omega d_{n} (\cos \theta_{{\text{p}}} /\alpha + \cos \theta_{{\text{S}}} /\beta )}} } & 0 & {B_{\alpha } {\text{e}}^{{ - i\omega d_{n} (\cos \theta_{{\text{p}}} /\alpha - \cos \theta_{{\text{S}}} /\beta )}} } & { - B_{\alpha } {\text{e}}^{{i\omega d_{n} (\cos \theta_{{\text{p}}} /\alpha - \cos \theta_{{\text{S}}} /\beta )}} } & 0 & { - A_{\alpha } {\text{e}}^{{i\omega d_{n} (\cos \theta_{{\text{p}}} /\alpha + \cos \theta_{{\text{S}}} /\beta )}} } \\ \end{array} } \right],} \\ {\frac{{\partial {\mathbf{E}}_{n} }}{\partial \beta }=i\omega d_{n} {\text{diag}}\left[ {\begin{array}{*{20}c} {A_{\beta } {\text{e}}^{{ - i\omega d_{n} (\cos \theta_{{\text{p}}} /\alpha + \cos \theta_{{\text{S}}} /\beta )}} } & 0 & {{\text{B}}_{\beta } {\text{e}}^{{ - i\omega d_{n} (\cos \theta_{{\text{p}}} /\alpha - \cos \theta_{{\text{S}}} /\beta )}} } & { - {\text{B}}_{\beta } {\text{e}}^{{i\omega d_{n} (\cos \theta_{{\text{p}}} /\alpha - \cos \theta_{{\text{S}}} /\beta )}} } & 0 & { - A_{\beta } {\text{e}}^{{i\omega d_{n} (\cos \theta_{{\text{p}}} /\alpha + \cos \theta_{{\text{S}}} /\beta )}} } \\ \end{array} } \right],} \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{\partial {\mathbf{E}}_{n} }}{\partial \rho } = 0} & , & {\begin{array}{*{20}c} {} & {} \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} \\ \end{array} } & {} \\ \end{array} } & {} \\ \end{array} } & {} \\ \end{array} } & {} \\ \end{array} } & {} \\ \end{array} } & {} \\ \end{array} } & {} \\ \end{array} } \\ \end{array} } & {} & {} & {} \\ \end{array} } & {} & {} & {} \\ \end{array} } & {} & {} & {} \\ \end{array} } & {} & {} & {} \\ \end{array} } & {} & {} & {} \\ \end{array} } & {} & {} & {} \\ \end{array} } \\ \end{array} } \right.$$

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where

$$\left\{ {\begin{array}{*{20}l} {{\text{A}}_{\alpha } = \left( {\frac{{\cos \theta_{{\text{P}}} }}{{\alpha^{2} }} + \frac{{\cos \theta_{S} }}{\alpha \beta }} \right),} \hfill \\ {{\text{B}}_{\alpha } = \left( {\frac{{\cos \theta_{{\text{P}}} }}{{\alpha^{2} }} - \frac{{\cos \theta_{S} }}{\alpha \beta }} \right),} \hfill \\ {{\text{A}}_{\beta } = \left( {\frac{{\cos \theta_{{\text{P}}} }}{\alpha \beta } + \frac{{\cos \theta_{S} }}{{\beta^{2} }}} \right),} \hfill \\ {{\text{B}}_{\beta } = \left( {\frac{{\cos \theta_{{\text{P}}} }}{\alpha \beta } - \frac{{\cos \theta_{S} }}{{\beta^{2} }}} \right),} \hfill \\ \end{array} } \right.$$

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in which α and β are the P- and S-wave velocities at the nth layer, respectively.

The partial derivatives of F_n-1 and F_n with respect to the parameters M_n can also be calculated analytically:

$$\frac{{\partial {\mathbf{F}}_{n} }}{{\partial {\mathbf{M}}_{n} }}=\frac{{\partial {\mathbf{T}}_{n}^{ - 1} }}{{\partial {\mathbf{M}}_{n} }}{\mathbf{T}}_{n + 1} ,$$

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$$\frac{{\partial {\mathbf{F}}_{{n{ - }1}} }}{{\partial {\mathbf{M}}_{n} }}={\mathbf{T}}_{n - 1}^{ - 1} \frac{{\partial {\mathbf{T}}_{n} }}{{\partial {\mathbf{M}}_{n} }},$$

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where

$$\frac{{\partial {\mathbf{T}}_{n} }}{{\partial {\mathbf{M}}_{n} }}=\left\{ {\begin{array}{*{20}c} {\frac{{\partial {\mathbf{T}}_{n} }}{\partial \alpha }} \\ {\frac{{\partial {\mathbf{T}}_{n} }}{\partial \beta }} \\ {\frac{{\partial {\mathbf{T}}_{n} }}{\partial \rho }} \\ \end{array} } \right..$$

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The matrices $\frac{{\partial {\mathbf{T}}_{n} }}{\partial \alpha }$, $\frac{{\partial {\mathbf{T}}_{{n{ + }1}} }}{\partial \beta }$, and $\frac{{\partial {\mathbf{T}}_{{n{ + }1}} }}{\partial \rho }$ contain 16 independent elements.

$$\begin{aligned} \frac{{\partial t_{11} }}{\partial \alpha } & = - \frac{4}{\alpha }t_{11} = \frac{{\partial t_{16} }}{\partial \alpha }, \\ \frac{{\partial t_{12} }}{\partial \alpha } & = - \frac{4}{\alpha }t_{12} , \\ \frac{{\partial t_{13} }}{\partial \alpha } & = - \frac{{4}}{\alpha }t_{13} = - \frac{{\partial t_{14} }}{\partial \alpha }, \\ \frac{{\partial t_{15} }}{\partial \alpha } & = - \frac{{4}}{\alpha }t_{15} , \\ \frac{{\partial t_{21} }}{\partial \alpha } & = - \frac{{3}}{\alpha }t_{21} = - \frac{{\partial t_{23} }}{\partial \alpha } = - \frac{{\partial t_{24} }}{\partial \alpha } = - \frac{{\partial t_{26} }}{\partial \alpha }, \\ \frac{{\partial t_{31} }}{\partial \alpha } & = - \frac{{3}}{\alpha }t_{31} = \frac{{\partial t_{36} }}{\partial \alpha } = \frac{{\partial t_{41} }}{\partial \alpha } = \frac{{\partial t_{46} }}{\partial \alpha }, \\ \frac{{\partial t_{32} }}{\partial \alpha } & = - \frac{3}{\alpha }t_{32} , \\ \frac{{\partial t_{33} }}{\partial \alpha } & = - \frac{{3}}{\alpha }t_{33} = \frac{{\partial t_{43} }}{\partial \alpha } = - \frac{{\partial t_{34} }}{\partial \alpha } = - \frac{{\partial t_{44} }}{\partial \alpha }, \\ \frac{{\partial t_{35} }}{\partial \alpha } & = - \frac{{3}}{\alpha }t_{35} , \\ \frac{{\partial t_{42} }}{\partial \alpha } & = - \frac{3}{\alpha }t_{42} , \\ \frac{{\partial t_{45} }}{\partial \alpha } & = - \frac{{3}}{\alpha }t_{45} , \\ \frac{{\partial t_{{{5}1}} }}{\partial \alpha } & = - \frac{{3}}{\alpha }t_{{{5}1}} = \frac{{\partial t_{{{53}}} }}{\partial \alpha } = \frac{{\partial t_{{{54}}} }}{\partial \alpha } = - \frac{{\partial t_{{{56}}} }}{\partial \alpha }, \\ \frac{{\partial t_{61} }}{\partial \alpha } & = - \frac{2}{\alpha }t_{61} = \frac{{\partial t_{66} }}{\partial \alpha }, \\ \frac{{\partial t_{62} }}{\partial \alpha } & = - \frac{2}{\alpha }t_{62} , \\ \frac{{\partial t_{{6{3}}} }}{\partial \alpha } & = - \frac{2}{\alpha }t_{{6{3}}} = - \frac{{\partial t_{64} }}{\partial \alpha }, \\ \frac{{\partial t_{65} }}{\partial \alpha } & = - \frac{2}{\alpha }t_{65} , \\ \frac{{\partial t_{22} }}{\partial \alpha } & = \frac{{\partial t_{25} }}{\partial \alpha }=\frac{{\partial t_{55} }}{\partial \alpha }=\frac{{\partial t_{52} }}{\partial \alpha }=0. \\ \end{aligned}$$

$$\begin{aligned} \frac{{\partial t_{11} }}{\partial \beta } & = - \frac{4}{\beta }t_{11} = \frac{{\partial t_{16} }}{\partial \beta }, \\ \frac{{\partial t_{12} }}{\partial \beta } & = - \frac{4}{\beta }t_{12} , \\ \frac{{\partial t_{13} }}{\partial \beta } & = - \frac{{4}}{\beta }t_{13} = - \frac{{\partial t_{14} }}{\partial \beta }, \\ \frac{{\partial t_{15} }}{\partial \beta } & = - \frac{{4}}{\beta }t_{15} , \\ \frac{{\partial t_{21} }}{\partial \beta } & = - \frac{{3}}{\beta }t_{21} = - \frac{{\partial t_{23} }}{\partial \beta } = - \frac{{\partial t_{24} }}{\partial \beta } = - \frac{{\partial t_{26} }}{\partial \beta }, \\ \frac{{\partial t_{31} }}{\partial \beta } & = - \frac{{3}}{\beta }t_{31} = \frac{{\partial t_{36} }}{\partial \beta } = \frac{{\partial t_{41} }}{\partial \beta } = \frac{{\partial t_{46} }}{\partial \beta }, \\ \frac{{\partial t_{33} }}{\partial \beta } & = - \frac{{3}}{\beta }t_{33} = \frac{{\partial t_{43} }}{\partial \beta } = - \frac{{\partial t_{34} }}{\partial \beta } = - \frac{{\partial t_{44} }}{\partial \beta }, \\ \frac{{\partial t_{35} }}{\partial \beta } & = - \frac{{3}}{\beta }t_{35} , \\ \frac{{\partial t_{42} }}{\partial \beta } & = - \frac{3}{\beta }t_{42} , \\ \frac{{\partial t_{45} }}{\partial \beta } & = - \frac{{3}}{\beta }t_{45} , \\ \frac{{\partial t_{{{5}1}} }}{\partial \beta } & = - \frac{{3}}{\beta }t_{{{5}1}} = \frac{{\partial t_{{{53}}} }}{\partial \beta } = \frac{{\partial t_{{{54}}} }}{\partial \beta } = - \frac{{\partial t_{{{56}}} }}{\partial \beta }, \\ \frac{{\partial t_{61} }}{\partial \beta } & = - \frac{2}{\beta }t_{61} = \frac{{\partial t_{66} }}{\partial \beta }, \\ \frac{{\partial t_{62} }}{\partial \beta } & = - \frac{2}{\beta }t_{62} , \\ \frac{{\partial t_{{6{3}}} }}{\partial \beta } & = - \frac{2}{\beta }t_{{6{3}}} = - \frac{{\partial t_{64} }}{\partial \beta }, \\ \frac{{\partial t_{65} }}{\partial \beta } & = - \frac{2}{\beta }t_{65} , \\ \frac{{\partial t_{22} }}{\partial \beta } & = \frac{{\partial t_{25} }}{\partial \beta }=\frac{{\partial t_{55} }}{\partial \beta }=\frac{{\partial t_{52} }}{\partial \beta }=0, \\ \end{aligned}$$

$$\begin{aligned} \frac{{\partial t_{11} }}{\partial \rho } & = - \frac{1}{\rho }t_{11} = \frac{{\partial t_{16} }}{\partial \rho }, \\ \frac{{\partial t_{12} }}{\partial \rho } & = - \frac{1}{\rho }t_{12} , \\ \frac{{\partial t_{13} }}{\partial \rho } & = - \frac{1}{\rho }t_{13} = - \frac{{\partial t_{14} }}{\partial \rho }, \\ \frac{{\partial t_{15} }}{\partial \rho } & = - \frac{1}{\rho }t_{15} , \\ \frac{{\partial t_{21} }}{\partial \rho } & =\frac{{\partial t_{22} }}{\partial \rho } = \frac{{\partial t_{23} }}{\partial \rho }=\frac{{\partial t_{24} }}{\partial \rho }=\frac{{\partial t_{25} }}{\partial \rho }=\frac{{\partial t_{26} }}{\partial \rho } = 0,\;\;\frac{{\partial t_{31} }}{\partial \rho }=\frac{{\partial t_{32} }}{\partial \rho } = \frac{{\partial t_{33} }}{\partial \rho }=\frac{{\partial t_{34} }}{\partial \rho }=\frac{{\partial t_{35} }}{\partial \rho }=\frac{{\partial t_{36} }}{\partial \rho } = 0, \\ \frac{{\partial t_{41} }}{\partial \rho } & =\frac{{\partial t_{42} }}{\partial \rho } = \frac{{\partial t_{43} }}{\partial \rho }=\frac{{\partial t_{44} }}{\partial \rho }=\frac{{\partial t_{45} }}{\partial \rho }=\frac{{\partial t_{46} }}{\partial \rho } = 0,\;\;\frac{{\partial t_{51} }}{\partial \rho }=\frac{{\partial t_{52} }}{\partial \rho } = \frac{{\partial t_{53} }}{\partial \rho }=\frac{{\partial t_{54} }}{\partial \rho }=\frac{{\partial t_{55} }}{\partial \rho }=\frac{{\partial t_{56} }}{\partial \rho } = 0, \\ \frac{{\partial t_{61} }}{\partial \rho } & = \frac{1}{\rho }t_{61} = \frac{{\partial t_{66} }}{\partial \rho },\;\;\frac{{\partial t_{62} }}{\partial \rho } = \frac{1}{\rho }t_{62} ,\;\;\frac{{\partial t_{63} }}{\partial \rho } = \frac{1}{\rho }t_{63} = { - }\frac{{\partial t_{64} }}{\partial \rho },\;\;\frac{{\partial t_{65} }}{\partial \rho } = \frac{1}{\rho }t_{63} . \\ \end{aligned}$$