﻿ 复数域约束最小二乘拓频
 石油地球物理勘探  2021, Vol. 56 Issue (6): 1244-1253  DOI: 10.13810/j.cnki.issn.1000-7210.2021.06.006 0
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### 引用本文

YANG Peijie. Constrained complex-domain least-squares spectrum blueing. Oil Geophysical Prospecting, 2021, 56(6): 1244-1253. DOI: 10.13810/j.cnki.issn.1000-7210.2021.06.006.

### 文章历史

Constrained complex-domain least-squares spectrum blueing
YANG Peijie
Research Institute of Exploration and Development, Sinopec Shengli Oilfield Company, Dong-ying, Shandong 257015, China
Abstract: The purpose of frequency extension is to improve the dominant frequency of seismic data, broaden the frequency band, and thereby better describe smaller and thinner geological bodies. A forward mathematical model is built for spectrum extension, and the forward problem of spectrum extension is transformed into an inverse problem through the broadband constrained target spectrum (CTS). The spectrum extension operator (SEO) in the frequency domain is acquired through solving the inverse problem by the complex-domain least-squares method. Spectrum extension of seismic data is achieved by applying SEO to the frequency spectrum of the original seismic data and the inverse Fourier transform. This method can effectively improve the dominant frequency, broaden the frequency band, and improve the resolution of seismic data without changing the phase spectrum of the data. The distance between the frequency spectrum after frequency extension and the CTS can be adjusted by setting different spectrum control factors to obtain spectrum extension results with different resolutions. This method has been applied to the fine identification of thin layers and overlap pinch-outs in the Jiyang Depression, and has a broad application prospect.
Keywords: complex-domain least-squares method    constrained target spectrum    spectrum extension operator    spectrum control factor
0 引言

1 数学模型及其解

 $\boldsymbol{E}(\omega) \boldsymbol{D}(\omega)+\boldsymbol{N}(\omega)=\boldsymbol{F}(\omega)$ (1)

 $\boldsymbol{E}(\omega)=\left[\begin{array}{c} e_{\mathrm{R} 1}+\mathrm{i} e_{\mathrm{I} 1} \\ e_{\mathrm{R} 2}+\mathrm{i} e_{\mathrm{I} 2} \\ \vdots \\ e_{\mathrm{R} n}+\mathrm{i} e_{\mathrm{I} n} \end{array}\right]$ (2)
 $\boldsymbol{D}(\omega)=\left[\begin{array}{llll} d_{\mathrm{R} 1}+\mathrm{i} d_{\mathrm{I} 1} & & & \\ & d_{\mathrm{R} 2}+\mathrm{i} d_{\mathrm{I} 2} & & \\ & & \ddots & \\ & & & d_{\mathrm{R} n}+\mathrm{i} d_{\mathrm{I} n} \end{array}\right]$ (3)
 $\boldsymbol{F}(\omega)=\left[\begin{array}{c} f_{\mathrm{R} 1}+\mathrm{i} f_{\mathrm{I} 1} \\ f_{\mathrm{R} 2}+\mathrm{i} f_{\mathrm{I} 2} \\ \vdots \\ f_{\mathrm{R} n}+\mathrm{i} f_{\mathrm{I} n} \end{array}\right]$ (4)

E(ω)定义为下式的最优解

 $\|\boldsymbol{E}(\omega) \boldsymbol{D}(\omega)-\boldsymbol{F}(\omega)\|^{2} \Rightarrow \min$ (5)

 $\left\{\begin{array}{l} \boldsymbol{E}=\boldsymbol{E}_{\mathrm{R}}+\mathrm{i} \boldsymbol{E}_{\mathrm{I}} \\ \boldsymbol{D}=\boldsymbol{D}_{\mathrm{R}}+\mathrm{i} \boldsymbol{D}_{\mathrm{I}} \\ \boldsymbol{F}=\boldsymbol{F}_{\mathrm{R}}+\mathrm{i} \boldsymbol{F}_{\mathrm{I}} \end{array}\right.$ (6)

 $\left[\begin{array}{cc} \boldsymbol{D}_{\mathrm{R}} & -\boldsymbol{D}_{\mathrm{I}} \\ \boldsymbol{D}_{\mathrm{I}} & \boldsymbol{D}_{\mathrm{R}} \end{array}\right]\left[\begin{array}{c} \boldsymbol{E}_{\mathrm{R}} \\ \boldsymbol{E}_{\mathrm{I}} \end{array}\right]=\left[\begin{array}{c} \boldsymbol{F}_{\mathrm{R}} \\ \boldsymbol{F}_{\mathrm{I}} \end{array}\right]$ (7)

 $\begin{gathered} {\left[\begin{array}{cc} \boldsymbol{D}_{\mathrm{R}}^{\mathrm{T}} & \boldsymbol{D}_{\mathrm{I}}^{\mathrm{T}} \\ -\boldsymbol{D}_{\mathrm{I}}^{\mathrm{T}} & \boldsymbol{D}_{\mathrm{R}}^{\mathrm{T}} \end{array}\right]\left[\begin{array}{cc} \boldsymbol{D}_{\mathrm{R}} & -\boldsymbol{D}_{\mathrm{I}} \\ \boldsymbol{D}_{\mathrm{I}} & \boldsymbol{D}_{\mathrm{R}} \end{array}\right]\left[\begin{array}{c} \boldsymbol{E}_{\mathrm{R}} \\ \boldsymbol{E}_{\mathrm{I}} \end{array}\right]} \\ =\left[\begin{array}{cc} \boldsymbol{D}_{\mathrm{R}}^{\mathrm{T}} & \boldsymbol{D}_{\mathrm{I}}^{\mathrm{T}} \\ -\boldsymbol{D}_{\mathrm{I}}^{\mathrm{T}} & \boldsymbol{D}_{\mathrm{R}}^{\mathrm{T}} \end{array}\right]\left[\begin{array}{c} \boldsymbol{F}_{\mathrm{R}} \\ \boldsymbol{F}_{\mathrm{I}} \end{array}\right] \end{gathered}$ (8)

 $\left[\begin{array}{cc} \boldsymbol{P} & \boldsymbol{Q} \\ -\boldsymbol{Q} & \boldsymbol{P} \end{array}\right]\left[\begin{array}{c} \boldsymbol{E}_{\mathrm{R}} \\ \boldsymbol{E}_{\mathrm{I}} \end{array}\right]=\left[\begin{array}{cc} \boldsymbol{D}_{\mathrm{R}}^{\mathrm{T}} & \boldsymbol{D}_{\mathrm{I}}^{\mathrm{T}} \\ -\boldsymbol{D}_{1}^{\mathrm{T}} & \boldsymbol{D}_{\mathrm{R}}^{\mathrm{T}} \end{array}\right]\left[\begin{array}{c} \boldsymbol{F}_{\mathrm{R}} \\ \boldsymbol{F}_{\mathrm{I}} \end{array}\right]$ (9)

 $\left\{\begin{array}{l} \boldsymbol{P}=\boldsymbol{D}_{\mathrm{R}}^{\mathrm{T}} \boldsymbol{D}_{\mathrm{R}}+\boldsymbol{D}_{\mathrm{I}}^{\mathrm{T}} \boldsymbol{D}_{\mathrm{I}} \\ \boldsymbol{Q}=\boldsymbol{D}_{\mathrm{l}}^{\mathrm{T}} \boldsymbol{D}_{\mathrm{R}}-\boldsymbol{D}_{\mathrm{R}}^{\mathrm{T}} \boldsymbol{D}_{\mathrm{I}} \end{array}\right.$ (10)

 $\left[\begin{array}{c} \boldsymbol{E}_{\mathrm{R}} \\ \boldsymbol{E}_{\mathrm{I}} \end{array}\right]=\left[\begin{array}{cc} \boldsymbol{P} & \boldsymbol{Q} \\ -\boldsymbol{Q} & \boldsymbol{P} \end{array}\right]^{-1}\left[\begin{array}{cc} \boldsymbol{D}_{\mathrm{R}}^{\mathrm{T}} & \boldsymbol{D}_{1}^{\mathrm{T}} \\ -\boldsymbol{D}_{1}^{\mathrm{T}} & \boldsymbol{D}_{\mathrm{R}}^{\mathrm{T}} \end{array}\right]\left[\begin{array}{c} \boldsymbol{F}_{\mathrm{R}} \\ \boldsymbol{F}_{\mathrm{I}} \end{array}\right]$ (11)

 $\|\boldsymbol{E}(\omega) \boldsymbol{D}(\omega)-\boldsymbol{F}(\omega)\|^{2}+\operatorname{Regu} [\boldsymbol{E}(\omega)] \Rightarrow \min$ (12)

 $\begin{gathered} {\left[\begin{array}{c} \boldsymbol{E}_{\mathrm{R}} \\ \boldsymbol{E}_{\mathrm{I}} \end{array}\right]=\left\{\left[\begin{array}{cc} \boldsymbol{P} & \boldsymbol{Q} \\ -\boldsymbol{Q} & \boldsymbol{P} \end{array}\right]+\left[\begin{array}{cc} \mu \boldsymbol{I} & 0 \\ 0 & \mu \boldsymbol{I} \end{array}\right]\right\}^{-1} \times} \\ {\left[\begin{array}{cc} \boldsymbol{D}_{\mathrm{R}}^{\mathrm{T}} & \boldsymbol{D}_{1}^{\mathrm{T}} \\ -\boldsymbol{D}_{1}^{\mathrm{T}} & \boldsymbol{D}_{\mathrm{R}}^{\mathrm{T}} \end{array}\right]\left[\begin{array}{c} \boldsymbol{F}_{\mathrm{R}} \\ \boldsymbol{F}_{\mathrm{I}} \end{array}\right]} \end{gathered}$ (13)

 $\boldsymbol{E}_{\mathrm{O}}(\omega)=\boldsymbol{E}_{\mathrm{R}}(\omega)+\mathrm{i} \boldsymbol{E}_{\mathrm{I}}(\omega)$ (14)

 $\boldsymbol{D}_{\mathrm{Y}}(\omega)=\boldsymbol{E}_{\mathrm{O}}(\omega) \boldsymbol{D}(\omega)$ (15)

 $\boldsymbol{d}_{\mathrm{Y}}=\operatorname{IFFT}\left[\boldsymbol{D}_{\mathrm{Y}}(\omega)\right]$ (16)

2 宽频约束目标谱的设计

 图 1 常用理论目标谱 (a)海宁窗；(b)梯形窗；(c)高斯窗；(d)广义高斯窗

 $G_{\mathrm{g}}=\left\{\begin{array}{c} \exp \left[-\frac{\left(\omega-\omega_{\mathrm{L}}\right)^{2}}{2 \sigma_{\mathrm{L}}^{2}}\right] & \omega \leqslant \omega_{\mathrm{L}} \\ 1 & \omega_{\mathrm{L}}<\omega<\omega_{\mathrm{H}} \\ \exp \left[-\frac{\left(\omega-\omega_{\mathrm{H}}\right)^{2}}{2 \sigma_{\mathrm{H}}^{2}}\right] & \omega \geqslant \omega_{\mathrm{H}} \end{array}\right.$ (17)

 图 2 实际井资料的波阻抗频谱

3 模型验证

 图 3 前积体地质模型

 图 4 前积体模型不同主频子波地震正演结果(左)及其频谱(右) (a)30Hz；(b)50Hz

 图 5 前积体模型不同控频因子μ的拓频后剖面(左)及CDP=180单道频谱(右)的对比 (a)μ=0.01；(b)μ=0.001；(b)μ=0.0001

 图 6 50Hz记录(上)与30Hz记录拓频后(下)的对比

 图 7 时间域拓频算子

 图 8 30Hz合成记录拓频前、后与50Hz合成记录的波形对比

 图 9 30Hz合成记录拓频前、后与50Hz合成记录相关性分析(CDP165)

 图 10 30Hz合成记录拓频前(黑色)、后(红色)相位谱对比

4 实际应用 4.1 薄互层识别

 图 11 CD油田约束目标谱、拓频算子、原始单道及拓频后的频谱对比

 图 12 拓频前(a)、后(b)Line1783地震剖面(左)及其频谱(右)的对比

 图 13 过cb81井地震剖面拓频前(a)、后(b)的对比

 图 14 cb81井拓频前、后井震标定对比
4.2 超覆尖灭点识别

 图 15 GB洼陷原始地震单道频谱、约束目标谱、拓频算子及拓频后频谱对比

 图 16 拓频前(a)、后(b)连井剖面(左)及其频谱(右)的对比

 图 17 拓频结果的带通滤波后地震剖面
5 结束语

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