﻿ 基于Zoeppritz方程的叠前和叠后混合多参数非线性地震反演
 石油地球物理勘探  2021, Vol. 56 Issue (1): 164-171  DOI: 10.13810/j.cnki.issn.1000-7210.2021.01.019 0
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ZHANG Lingyuan, ZHANG Hongbing, SHANG Zuoping, YAN Lizhi, REN Quan. Nonlinear multi-parameter hybrid inversion of pre-stack and post-stack seismic data based on Zoeppritz equation. Oil Geophysical Prospecting, 2021, 56(1): 164-171. DOI: 10.13810/j.cnki.issn.1000-7210.2021.01.019.

文章历史

① 河海大学地球科学与工程学院, 江苏南京 211100;
② 河海大学力学与材料学院, 江苏南京 211100

Nonlinear multi-parameter hybrid inversion of pre-stack and post-stack seismic data based on Zoeppritz equation
ZHANG Lingyuan , ZHANG Hongbing , SHANG Zuoping , YAN Lizhi , REN Quan
① College of Earth Science and Engineering, Hohai University, Nanjing, Jiangsu 211100, China;
② College of Mechanics and Materials, Hohai University, Nanjing, Jiangsu 211100, China
Abstract: At present, Shuey or Gray approximate formula and basis pursuit theory are widely applied in inversion of Poisson ratio. The same is true for the velocity ratio which is calculated from P- and S- waves velocity obtained from pre-stack inversion. But it is not avoidable to reduce the accuracy of inversion parameters. We try to apply Zoeppritz equation in inversion for velocity ratio and Poisson ratio. First we developed a hybrid pre-stack and post-stack seismic inversion workflow by the convolution model based on the reflection coefficient of vertical incidence and Zoeppritz equation. Then we constructed a new objective function which includes edge-preserving regularization to reduce the adverse effect of inversion unsuitability. We applied a fast simulated annealing algorithm to achieve global non-linear optimization. To reduce the influence from the magnitude difference of multiple parameters and the poor stability of multi-parameter simultaneous random search, we used an improved Fatti equation in inversion, which introduces two fitting formulas, a density and P-wave velocity formula, and a S-velocity and P-wave velocity formula, to improve the accuracy and stability of inversion. The inverted results of field data indicate that the velocity ratio obtained by direct Zoeppritz equation inversion is better than that obtained by approximate formulas. The inversion results from the new workflow have good consistency with the logging data. In a word, post-stack inversion obtained more accurate P-wave velocity and density, while the accuracy of velocity ration obtained by pre-stack inversion is higher than that obtained by post-stack inversion, and the inversion of 18°~24° angular trace gathers is better than that of 3°~9° and 33°~39° gathers.
Keywords: pre-stack seismic inversion    hybrid inversion    Zoeppritz equation    edge-preserving regularization    velocity ratio    gas zone identification
0 引言

1 叠前和叠后地震混合多参数反演 1.1 叠前和叠后地震响应数学模型

 $\mathit{\boldsymbol{Y}} = \mathit{\boldsymbol{R}} * \mathit{\boldsymbol{W}} + \mathit{\boldsymbol{N}}$ (1)

 ${R_i} = \frac{{{Z_{i + 1}} - {Z_i}}}{{{Z_{i + 1}} + {Z_i}}} = \frac{{{v_{{\rm{P}}i + 1}}{\rho _{{\rm{P}}i + 1}} - {v_{{\rm{P}}i}}{\rho _i}{\rm{ }}}}{{{v_{{\rm{P}}i + 1}}{\rho _{{\rm{P}}i + 1}} + {v_{{\rm{P}}i}}{\rho _i}{\rm{ }}}}$ (2)

 $\mathit{\boldsymbol{Y}}(\theta ) = \mathit{\boldsymbol{R}}(\theta ) * \mathit{\boldsymbol{W}}(\theta ) + \mathit{\boldsymbol{N}}$ (3)

 $\mathit{\boldsymbol{M}}\left[ {\begin{array}{*{20}{c}} {{R_{{\rm{PP}}}}}\\ {{R_{{\rm{PS}}}}}\\ {{T_{{\rm{PP}}}}}\\ {{T_{{\rm{PS}}}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - \sin {\theta _1}}\\ { - \cos {\theta _1}}\\ {\sin 2{\theta _1}}\\ { - \cos 2{\varphi _1}} \end{array}} \right]$ (4)

 $\mathit{\boldsymbol{M}} = \left[ {\begin{array}{*{20}{c}} {\sin {\theta _1}}&{\cos {\varphi _1}}&{ - \sin {\theta _2}}&{\cos {\varphi _2}}\\ { - \cos {\theta _1}}&{\sin {\varphi _1}}&{ - \cos {\theta _2}}&{ - \sin {\varphi _2}}\\ {\sin 2{\theta _1}}&{\frac{{{v_{{\rm{P}}1}}}}{{{v_{{\rm{S}}1}}}}\cos 2{\varphi _1}}&{\frac{{{\rho _2}}}{{{\rho _1}}}\frac{{{v_{{\rm{P}}1}}{\kern 1pt} v_{{\rm{S}}2}^2}}{{{v_{{\rm{P}}2}}{\kern 1pt} v_{{\rm{S}}1}^2}}\sin 2{\theta _2}}&{ - \frac{{{\rho _2}}}{{{\rho _1}}}\frac{{{v_{{\rm{P}}1}}{\kern 1pt} {v_{{\rm{S}}2}}}}{{{v_{{\rm{P}}2}}{\kern 1pt} {v_{{\rm{S}}1}}}}\cos 2{\varphi _2}}\\ {\cos 2{\varphi _1}}&{ - \frac{{{v_{{\rm{S}}1}}}}{{{v_{{\rm{P}}1}}}}\sin 2{\varphi _1}}&{ - \frac{{{\rho _2}}}{{{\rho _1}}}\frac{{{v_{{\rm{P}}2}}}}{{{v_{{\rm{P}}1}}}}\cos 2{\varphi _2}}&{ - \frac{{{\rho _2}}}{{{\rho _1}}}\frac{{{v_{{\rm{S}}2}}}}{{{v_{{\rm{P}}1}}}}\sin 2{\varphi _2}} \end{array}} \right]$ (5)

1.2 基于边界保护正则化的反演目标函数

 $\begin{array}{l} J(X) = {J_1}(X) + \lambda \cdot {J_2}(X)\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{1}{{2{\sigma ^2}}}{\left\| {\mathit{\boldsymbol{Y}}(\theta ) - \mathit{\boldsymbol{W}}(\theta ) * \mathit{\boldsymbol{R}}(\theta )} \right\|_2} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \lambda \sum\limits_{{\mathit{\boldsymbol{C}}_k}} \varPhi \left[ {\frac{{{D_{{\mathit{\boldsymbol{C}}_k}}}(X)}}{\mathit{\boldsymbol{\delta }}}} \right] \end{array}$ (6)

 ${J_2} = {J_{{\rm{2P}}}}({v_{\rm{P}}}) + {J_{{\rm{2S}}}}({v_{\rm{S}}}) + {J_{{\rm{2D}}}}(\rho )$ (7)

 $\begin{array}{l} {J_2}(X) = \sum\limits_{{\mathit{\boldsymbol{C}}_k}} \varPhi [{D_{{\mathit{\boldsymbol{C}}_k}}}({v_{\rm{P}}})/{\delta _{\rm{P}}}] + \sum\limits_{{\mathit{\boldsymbol{C}}_k}} \varPhi [{D_{{\mathit{\boldsymbol{C}}_k}}}({v_{\rm{S}}})/{\delta _{\rm{S}}}] + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{{\mathit{\boldsymbol{C}}_k}} \varPhi [{D_{{\mathit{\boldsymbol{C}}_k}}}(\rho )/{\delta _\rho }] \end{array}$ (8)

 $\begin{array}{l} {J_2}(X) = \sum\limits_{{\mathit{\boldsymbol{C}}_k}} \varPhi [{D_{{\mathit{\boldsymbol{C}}_k}}}({v_{\rm{P}}})/{\delta _{\rm{P}}}] + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\nolimits_{{\mathit{\boldsymbol{C}}_k}} \varPhi [{D_{{\mathit{\boldsymbol{C}}_k}}}(\rho )/{\delta _\rho }] \end{array}$ (9)

 ${J_2}(X) = \sum\limits_{{\mathit{\boldsymbol{C}}_k}} \varPhi [{D_{{\mathit{\boldsymbol{C}}_k}}}({v_{\rm{S}}})/{\delta _{\rm{S}}}]$ (10)

 ${J_2}(X) = \sum\limits_{{\mathit{\boldsymbol{C}}_k}} \varPhi [{D_{{\mathit{\boldsymbol{C}}_k}}}(\gamma )/{\delta _\gamma }]$ (11)

1.3 基于模拟退火的反演搜索方法

 $\left\{ {\begin{array}{*{20}{l}} {v_{\rm{P}}^{(m + 1)} = v_{\rm{P}}^{(m)} + {T^{(m)}}{\mathop{\rm sign}\nolimits} \left( {{\zeta _1} - 0.5} \right) \times }\\ {\qquad \left\{ {{{\left[ {1 + \frac{1}{{{T^{(m)}}}}} \right]}^{\left| {2{\xi _1} - 1} \right|}} - 1} \right\}\left( {{v_{{\rm{Pmax}}}} - {v_{{\rm{Pmin}}}}} \right)}\\ {\begin{array}{*{20}{l}} {{\rho ^{(m + 1)}} = {\rho ^{(m)}} + {T^{(m)}}{\mathop{\rm sign}\nolimits} \left( {{\zeta _2} - 0.5} \right) \times }\\ {\qquad \left\{ {{{\left[ {1 + \frac{1}{{{T^{(m)}}}}} \right]}^{\left| {2{\xi _2} - 1} \right|}} - 1} \right\}\left( {{\rho _{\max }} - {\rho _{\min }}} \right)} \end{array}}\\ {\qquad m = 0,1, \cdots } \end{array}} \right.$ (12)
 $\left\{ {\begin{array}{*{20}{l}} {v_{\rm{S}}^{(m + 1)} = v_{\rm{S}}^{(m)} + {T^{(m)}}{\rm{sign}}({\zeta _3} - 0.5) \times }\\ {\qquad \left\{ {{{[1 + 1/{T^{(m)}}]}^{\left| {2{\xi _3} - 1} \right|}} - 1} \right\}({v_{{\rm{S}}\max }} - {v_{{\rm{S}}\min }})}\\ {\qquad m = 0,1, \cdots } \end{array}} \right.$ (13)
 $\begin{array}{l} {\gamma ^{(m + 1)}} = {\gamma ^{(m)}} + {T^{(m)}}{\rm{sign}}({\zeta _4} - 0.5) \times \\ \qquad \left\{ {{{[1 + 1/{T^{(m)}}]}^{\left| {2{\xi _4} - 1} \right|}} - 1} \right\}({\gamma _{\max }} - {\gamma _{\min }})\\ \qquad m = 0,1, \cdots \end{array}$ (14)

2 实测叠前和叠后地震数据混合反演

 图 1 研究区过井二维地震剖面A 井1和井2分别位于CDP 2226和CDP 918。叠后地震数据包含2542个CDP，时间窗口为2200~2600ms，采样率为2ms

 图 2 叠前地震数据角度道集 每个CDP有15个角度道集，角度范围为3°~45°，角度间隔为3°

 图 3 小、中、大角度子波

 图 4 vP(a)、ρ(b)、vS(c)和vP/vS(d)初始模型

(1) 叠后反演vPρ。使用vP(图 4a)、ρ(图 4b)初始模型反演vPρ，在搜索时利用式(12)进行扰动。

(2) 利用叠后反演获得的vPρ开展叠前反演。将vPρ作为已知值代入Zoeppritz方程，分别使用vS初始模型(图 4c)、vP/vS初始模型(图 4d)反演vSvP/vS，在搜索时分别使用式(13)、式(14)进行扰动。

2.1 vPvSρ的混合反演结果

 $\rho = 1.036 + 0.0003825{v_{\rm{P}}}$ (14)

 ${v_{\rm{S}}} = 1775.2 + 0.08818{v_{\rm{P}}}$ (15)

 图 5 叠后地震反演的vP(a)和ρ(b) vP的值域为(3200m/s，4100m/s)，ρ的值域为(2.25g/cm3，2.60g/cm3)。模拟退火算法的初始温度为0.5，终止温度为0.00001，温度衰减系数为0.9。正则化势函数为ΦGM，正则参数λ=0.3，δ=(δP δS δρ δγ)=(150.0, 110.0, 0.15, 0.11)。利用两口井的测井参数约束反演

 图 6 不同角度叠前地震反演vS (a)3°~9°；(b)18°~24°；(c)33°~39° vS的值域为(1600m/s，2300m/s)，模拟退火算法的初始温度为0.5，终止温度为0.00001，温度衰减系数为0.9。正则化势函数为ΦGM，正则参数λ=0.3，δ=(δP δS δρ δγ)=(150.0, 110.0, 0.15, 0.11)。利用两口井的测井参数约束反演

 图 7 18°~24°角度道集混合反演结果与井2(左)、井1(右)的测井数据对比

2.2 vP/vS直接反演结果

 图 8 不同角度vP/vS间接反演结果 (a)18°~24°；(b)33°~39°

 图 9 由不同角度叠前地震数据直接反演的vP/vS (a)18°~24°；(b)33°~39°

 图 10 不同角度反演获得的vP/vS与井2(上)、井1(下)的测井数据对比 (a)18°~24°；(b)33°~39°
3 结论

(1) 直接使用Zoeppritz方程进行反演，可以避免Zoeppritz方程近似式反演引起的误差，对于大角度道集尤其重要。使用Zoeppritz方程直接反演几乎并不影响多参数反演的速度和效果。此外，由于多参数之间的量级差异以及多参数同步随机搜索增大了不确定性，在反演中引入两个附加约束条件，提高了多参数反演结果稳定性。

(2) 叠后和叠前混合多参数反演可以克服叠前地震多参数同步反演中不同参数数量级、参数敏感性、叠前地震资料品质差异引起的问题。由于研究区的实际叠前地震道集品质较差，而叠后地震数据品质较好，故可由叠后反演获得纵波速度和密度，由叠前反演获得横波速度、纵横波速度比、泊松比等。结果显示，叠后反演获得的纵波速度和密度精度较高，叠前反演获得的纵横波速度比的精度高于叠后反演，18°~24°角度道集的反演效果好于3°~9°和33°~39°角度道集。

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